Subgroups — C₇×#22855 #13200071

Order 1792.

1170 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#448	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#440	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2361	C₂	1	1	yes	maximal, characteristic
5	C₇×Q₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1973	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1928	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1893	C₂	1	1	yes	maximal, characteristic
10	C₇×#1880	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1877	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1863	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1852	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1763	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1752	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
17	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×#121	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇×D₄	—	2	1	no	—
24	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
25	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₇×#448	C₄	1	8	yes	complemented
27	C₇×#440	—	2	4	no	—
28	C₂×C₇×#107	C₂²	1	1	yes	characteristic
29	C₂×C₇×#107	C₂²	1	1	yes	characteristic
30	C₂×C₇×#103	—	2	4	no	—
31	C₂×C₇×#103	C₂²	1	1	yes	characteristic
32	C₂×C₇×#103	C₂²	1	1	yes	characteristic
33	C₂×C₇×#103	C₂²	1	1	yes	characteristic
34	C₂×C₇×#103	C₂²	1	1	yes	characteristic
35	C₂×C₇×#103	C₂²	1	1	yes	characteristic
36	C₂×C₇×#103	C₂²	1	1	yes	characteristic
37	C₂×C₇×#102	C₂²	1	1	yes	characteristic
38	C₂×C₇×#102	C₂²	1	1	yes	characteristic
39	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₇×#97	C₂²	1	1	yes	characteristic
41	C₂×C₇×#97	C₂²	1	1	yes	characteristic
42	C₇×#391	—	2	1	no	—
43	C₇×#391	—	2	1	no	—
44	C₇×#391	C₂²	1	1	yes	characteristic
45	C₇×#391	C₂²	1	1	yes	characteristic
46	C₇×#390	C₂²	1	1	yes	characteristic
47	C₇×#389	C₂²	1	1	yes	characteristic
48	C₇×#388	—	2	1	no	—
49	C₇×#386	—	2	1	no	—
50	C₇×#385	—	2	1	no	—
51	C₇×#380	—	2	1	no	—
52	C₇×#380	C₂²	1	1	yes	characteristic
53	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
54	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
55	C₄×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
56	C₇×#340	C₂²	1	1	yes	characteristic
57	C₇×#340	C₂²	1	1	yes	characteristic
58	C₇×#340	C₂²	1	1	yes	characteristic
59	C₇×#340	C₂²	1	1	yes	characteristic
60	C₇×#336	C₂²	1	1	yes	characteristic
61	C₇×#336	C₂²	1	1	yes	characteristic
62	#22855	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₇×Q₈	—	2	1	no	—
64	C₂²×C₇×Q₈	—	2	2	no	—
65	C₂²×C₇×Q₈	—	2	2	no	—
66	C₂²×C₇×Q₈	—	2	2	no	—
67	C₂²×C₇×Q₈	—	4	1	no	—
68	C₂²×C₇×Q₈	—	4	1	no	—
69	C₂²×C₇×Q₈	C₂×C₄	1	4	yes	complemented
70	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
71	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
72	C₂²×C₇×D₄	—	2	1	no	—
73	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₇	—	2	1	no	—
75	C₂³×C₄×C₇	—	2	1	no	—
76	C₂³×C₄×C₇	—	2	1	no	—
77	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
78	C₂×C₇×SD₁₆	—	2	2	no	—
79	C₂×C₇×SD₁₆	—	2	2	no	—
80	C₂×C₇×SD₁₆	—	4	1	no	—
81	C₂×C₇×SD₁₆	—	4	1	no	—
82	C₂×C₇×SD₁₆	C₂×C₄	1	4	yes	complemented
83	C₂×C₇×SD₁₆	C₂×C₄	1	4	yes	complemented
84	C₂²×C₇×C₈	—	2	2	no	—
85	C₂²×C₇×C₈	—	2	2	no	—
86	C₂²×C₇×C₈	—	4	1	no	—
87	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
88	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
89	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
90	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
91	C₇×#121	—	2	2	no	—
92	C₇×#121	C₂×C₄	1	4	yes	—
93	C₄×C₇×Q₈	—	2	2	no	—
94	C₄×C₇×Q₈	—	2	2	no	—
95	C₄×C₇×Q₈	—	4	1	no	—
96	C₄×C₇×Q₈	—	4	1	no	—
97	C₄×C₇×D₄	—	2	2	no	—
98	C₄×C₇×D₄	—	4	1	no	—
99	C₄×C₇×D₄	—	4	4	no	—
100	C₂×C₇×C₄⋊C₄	—	2	2	no	—
101	C₂×C₇×C₄⋊C₄	—	2	1	no	—
102	C₂×C₇×C₄⋊C₄	—	2	1	no	—
103	C₂×C₇×C₄⋊C₄	—	2	1	no	—
104	C₂×C₇×C₄⋊C₄	—	2	1	no	—
105	C₂×C₇×C₄⋊C₄	—	4	1	no	—
106	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
107	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
108	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
109	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
110	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
111	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
113	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
114	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
115	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
116	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
117	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₇	—	2	1	no	—
119	C₂×C₄²×C₇	—	2	1	no	—
120	C₂×C₄²×C₇	—	2	1	no	—
121	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
123	C₇×#107	—	2	2	no	—
124	C₇×#107	—	4	1	no	—
125	C₇×#103	—	2	2	no	—
126	C₇×#103	—	2	2	no	—
127	C₇×#103	—	2	2	no	—
128	C₇×#103	C₂×C₄	1	4	yes	complemented
129	C₇×#103	C₂×C₄	1	4	yes	complemented
130	C₇×#103	D₄	1	4	yes	complemented
131	C₇×#102	—	2	2	no	—
132	C₇×#102	—	4	1	no	—
133	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
134	C₇×#97	—	2	2	no	—
135	C₇×#97	—	4	1	no	—
136	C₇×#95	—	2	1	no	—
137	C₇×#95	—	2	1	no	—
138	C₇×#95	—	2	1	no	—
139	C₇×#95	—	2	1	no	—
140	C₇×#95	—	2	1	no	—
141	C₇×#95	—	2	1	no	—
142	C₇×#95	—	2	1	no	—
143	C₇×#95	D₄	1	1	yes	characteristic
144	C₇×#95	D₄	1	1	yes	characteristic
145	C₂×#448	C₂×C₇	1	1	yes	characteristic
146	C₂×#440	C₂×C₇	1	1	yes	characteristic, complemented
147	C₂²×#103	C₂×C₇	1	1	yes	characteristic, complemented
148	#2361	C₂×C₇	1	1	yes	characteristic
149	Q₈×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
150	#1973	C₂×C₇	1	1	yes	characteristic, complemented
151	#1935	C₂×C₇	1	1	yes	characteristic, complemented
152	#1928	C₂×C₇	1	1	yes	characteristic, complemented
153	#1893	C₂×C₇	1	1	yes	characteristic
154	#1880	C₂×C₇	1	1	yes	characteristic, complemented
155	#1877	C₂×C₇	1	1	yes	characteristic, complemented
156	#1863	C₂×C₇	1	1	yes	characteristic, complemented
157	#1852	C₂×C₇	1	1	yes	characteristic, complemented
158	#1763	C₂×C₇	1	1	yes	characteristic, complemented
159	#1752	C₂×C₇	1	1	yes	characteristic, complemented
160	C₂⁴×C₇	—	2	1	no	—
161	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
162	C₂×C₇×Q₈	—	2	4	no	—
163	C₂×C₇×Q₈	—	2	1	no	—
164	C₂×C₇×Q₈	—	2	1	no	—
165	C₂×C₇×Q₈	—	2	1	no	—
166	C₂×C₇×Q₈	—	2	1	no	—
167	C₂×C₇×Q₈	—	2	8	no	—
168	C₂×C₇×Q₈	—	2	1	no	—
169	C₂×C₇×Q₈	—	2	1	no	—
170	C₂×C₇×Q₈	—	2	1	no	—
171	C₂×C₇×Q₈	—	2	1	no	—
172	C₂×C₇×Q₈	—	4	2	no	—
173	C₂×C₇×Q₈	—	4	2	no	—
174	C₂×C₇×Q₈	—	4	4	no	—
175	C₂×C₇×Q₈	—	4	4	no	—
176	C₂×C₇×Q₈	—	4	4	no	—
177	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
178	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
179	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
180	C₂×C₇×Q₈	#41	1	2	yes	—
181	C₂×C₇×D₄	—	2	4	no	—
182	C₂×C₇×D₄	—	2	1	no	—
183	C₂×C₇×D₄	—	2	1	no	—
184	C₂×C₇×D₄	—	2	1	no	—
185	C₂×C₇×D₄	—	2	1	no	—
186	C₂×C₇×D₄	—	2	1	no	—
187	C₂×C₇×D₄	—	4	2	no	—
188	C₂×C₇×D₄	—	4	2	no	—
189	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
190	C₂²×C₄×C₇	—	2	1	no	—
191	C₂²×C₄×C₇	—	2	1	no	—
192	C₂²×C₄×C₇	—	2	1	no	—
193	C₂²×C₄×C₇	—	2	1	no	—
194	C₂²×C₄×C₇	—	2	1	no	—
195	C₂²×C₄×C₇	—	2	4	no	—
196	C₂²×C₄×C₇	—	2	1	no	—
197	C₂²×C₄×C₇	—	2	1	no	—
198	C₂²×C₄×C₇	—	2	1	no	—
199	C₂²×C₄×C₇	—	2	1	no	—
200	C₂²×C₄×C₇	—	2	1	no	—
201	C₂²×C₄×C₇	—	2	1	no	—
202	C₂²×C₄×C₇	—	4	2	no	—
203	C₂²×C₄×C₇	—	4	2	no	—
204	C₂²×C₄×C₇	—	4	2	no	—
205	C₂²×C₄×C₇	—	4	2	no	—
206	C₂²×C₄×C₇	—	4	2	no	—
207	C₂²×C₄×C₇	—	4	2	no	—
208	C₂²×C₄×C₇	—	8	1	no	—
209	C₂²×C₄×C₇	—	8	1	no	—
210	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
211	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
215	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
216	C₂²×C₄×C₇	#41	1	2	yes	—
217	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
218	C₇×SD₁₆	—	2	8	no	—
219	C₇×SD₁₆	—	4	4	no	—
220	C₂×C₇×C₈	—	2	1	no	—
221	C₂×C₇×C₈	—	2	1	no	—
222	C₂×C₇×C₈	—	2	1	no	—
223	C₂×C₇×C₈	—	2	1	no	—
224	C₂×C₇×C₈	—	2	1	no	—
225	C₂×C₇×C₈	—	2	1	no	—
226	C₂×C₇×C₈	—	2	1	no	—
227	C₂×C₇×C₈	—	2	1	no	—
228	C₂×C₇×C₈	—	2	1	no	—
229	C₂×C₇×C₈	—	4	4	no	—
230	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
231	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
232	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
233	C₇×C₄⋊C₄	—	2	1	no	—
234	C₇×C₄⋊C₄	—	2	1	no	—
235	C₇×C₄⋊C₄	—	2	1	no	—
236	C₇×C₄⋊C₄	—	2	1	no	—
237	C₇×C₄⋊C₄	—	2	1	no	—
238	C₇×C₄⋊C₄	—	2	1	no	—
239	C₇×C₄⋊C₄	—	2	1	no	—
240	C₇×C₄⋊C₄	—	2	1	no	—
241	C₇×C₄⋊C₄	—	4	2	no	—
242	C₇×C₄⋊C₄	—	4	2	no	—
243	C₇×C₄⋊C₄	—	4	2	no	—
244	C₇×C₄⋊C₄	—	4	2	no	—
245	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
246	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
247	C₇×C₂²⋊C₄	—	2	2	no	—
248	C₇×C₂²⋊C₄	—	2	2	no	—
249	C₇×C₂²⋊C₄	—	2	4	no	—
250	C₇×C₂²⋊C₄	—	4	2	no	—
251	C₇×C₂²⋊C₄	—	4	2	no	—
252	C₇×C₂²⋊C₄	—	4	2	no	—
253	C₇×C₂²⋊C₄	—	8	1	no	—
254	C₄²×C₇	—	2	1	no	—
255	C₄²×C₇	—	2	1	no	—
256	C₄²×C₇	—	2	1	no	—
257	C₄²×C₇	—	2	1	no	—
258	C₄²×C₇	—	4	2	no	—
259	C₄²×C₇	—	4	2	no	—
260	C₄²×C₇	—	4	2	no	—
261	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
262	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
263	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
264	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
265	C₂×#121	C₂²×C₇	1	1	yes	characteristic
266	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
267	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
268	C₂×C₄×D₄	—	2	1	no	—
269	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
270	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
271	#448	C₄×C₇	1	8	yes	complemented
272	#440	—	2	4	no	—
273	C₂×#107	C₂²×C₇	1	1	yes	characteristic
274	C₂×#107	C₂²×C₇	1	1	yes	characteristic
275	C₂×#103	—	2	4	no	—
276	C₂×#103	C₂²×C₇	1	1	yes	characteristic
277	C₂×#103	C₂²×C₇	1	1	yes	characteristic
278	C₂×#103	C₂²×C₇	1	1	yes	characteristic
279	C₂×#103	C₂²×C₇	1	1	yes	characteristic
280	C₂×#103	C₂²×C₇	1	1	yes	characteristic
281	C₂×#103	C₂²×C₇	1	1	yes	characteristic
282	C₂×#102	C₂²×C₇	1	1	yes	characteristic
283	C₂×#102	C₂²×C₇	1	1	yes	characteristic
284	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
285	C₂×#97	C₂²×C₇	1	1	yes	characteristic
286	C₂×#97	C₂²×C₇	1	1	yes	characteristic
287	#391	—	2	1	no	—
288	#391	—	2	1	no	—
289	#391	C₂²×C₇	1	1	yes	characteristic
290	#391	C₂²×C₇	1	1	yes	characteristic
291	#390	C₂²×C₇	1	1	yes	characteristic
292	#389	C₂²×C₇	1	1	yes	characteristic
293	#388	—	2	1	no	—
294	#386	—	2	1	no	—
295	#385	—	2	1	no	—
296	#380	—	2	1	no	—
297	#380	C₂²×C₇	1	1	yes	characteristic
298	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
299	C₄×C₂²⋊C₄	—	2	1	no	—
300	C₄×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
301	#340	C₂²×C₇	1	1	yes	characteristic
302	#340	C₂²×C₇	1	1	yes	characteristic
303	#340	C₂²×C₇	1	1	yes	characteristic
304	#340	C₂²×C₇	1	1	yes	characteristic
305	#336	C₂²×C₇	1	1	yes	characteristic
306	#336	C₂²×C₇	1	1	yes	characteristic
307	C₂³×C₇	—	2	1	no	—
308	C₂³×C₇	—	2	1	no	—
309	C₂³×C₇	—	2	1	no	—
310	C₂³×C₇	—	2	1	no	—
311	C₂³×C₇	—	2	2	no	—
312	C₂³×C₇	—	4	1	no	—
313	C₂³×C₇	—	4	1	no	—
314	C₂³×C₇	—	4	1	no	—
315	C₂³×C₇	—	4	1	no	—
316	C₂³×C₇	—	4	1	no	—
317	C₂³×C₇	—	4	1	no	—
318	C₂³×C₇	C₄×D₄	1	2	yes	—
319	C₂³×C₇	AGL(1,ℤ/8)	1	2	yes	—
320	C₂³×C₇	#137	1	2	yes	—
321	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
322	C₇×Q₈	—	2	2	no	—
323	C₇×Q₈	—	2	2	no	—
324	C₇×Q₈	—	2	2	no	—
325	C₇×Q₈	—	2	2	no	—
326	C₇×Q₈	—	2	2	no	—
327	C₇×Q₈	—	4	8	no	—
328	C₇×Q₈	—	4	1	no	—
329	C₇×Q₈	—	4	1	no	—
330	C₇×Q₈	C₄×D₄	1	4	yes	complemented
331	C₇×D₄	—	2	2	no	—
332	C₇×D₄	—	2	2	no	—
333	C₇×D₄	—	4	8	no	—
334	C₇×D₄	—	4	1	no	—
335	C₇×D₄	—	4	1	no	—
336	C₂×C₄×C₇	—	2	2	no	—
337	C₂×C₄×C₇	—	2	2	no	—
338	C₂×C₄×C₇	—	2	1	no	—
339	C₂×C₄×C₇	—	2	2	no	—
340	C₂×C₄×C₇	—	2	1	no	—
341	C₂×C₄×C₇	—	2	2	no	—
342	C₂×C₄×C₇	—	2	2	no	—
343	C₂×C₄×C₇	—	2	2	no	—
344	C₂×C₄×C₇	—	2	2	no	—
345	C₂×C₄×C₇	—	2	2	no	—
346	C₂×C₄×C₇	—	2	2	no	—
347	C₂×C₄×C₇	—	2	1	no	—
348	C₂×C₄×C₇	—	2	1	no	—
349	C₂×C₄×C₇	—	2	2	no	—
350	C₂×C₄×C₇	—	2	1	no	—
351	C₂×C₄×C₇	—	2	1	no	—
352	C₂×C₄×C₇	—	2	1	no	—
353	C₂×C₄×C₇	—	2	1	no	—
354	C₂×C₄×C₇	—	2	1	no	—
355	C₂×C₄×C₇	—	2	1	no	—
356	C₂×C₄×C₇	—	2	1	no	—
357	C₂×C₄×C₇	—	2	1	no	—
358	C₂×C₄×C₇	—	2	1	no	—
359	C₂×C₄×C₇	—	2	1	no	—
360	C₂×C₄×C₇	—	4	8	no	—
361	C₂×C₄×C₇	—	4	1	no	—
362	C₂×C₄×C₇	—	4	1	no	—
363	C₂×C₄×C₇	—	4	1	no	—
364	C₂×C₄×C₇	—	4	1	no	—
365	C₂×C₄×C₇	—	4	1	no	—
366	C₂×C₄×C₇	—	4	1	no	—
367	C₂×C₄×C₇	—	4	1	no	—
368	C₂×C₄×C₇	—	4	1	no	—
369	C₂×C₄×C₇	—	4	1	no	—
370	C₂×C₄×C₇	—	4	1	no	—
371	C₂×C₄×C₇	—	4	1	no	—
372	C₂×C₄×C₇	—	4	1	no	—
373	C₂×C₄×C₇	—	4	1	no	—
374	C₂×C₄×C₇	—	4	1	no	—
375	C₂×C₄×C₇	—	4	1	no	—
376	C₂×C₄×C₇	—	8	4	no	—
377	C₂×C₄×C₇	—	8	4	no	—
378	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
379	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
380	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
381	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
382	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
383	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
384	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
385	C₇×C₈	—	2	2	no	—
386	C₇×C₈	—	2	2	no	—
387	C₇×C₈	—	2	2	no	—
388	C₇×C₈	—	4	1	no	—
389	C₂²×Q₈	—	2	1	no	—
390	C₂²×Q₈	—	2	2	no	—
391	C₂²×Q₈	—	2	2	no	—
392	C₂²×Q₈	—	2	2	no	—
393	C₂²×Q₈	—	4	1	no	—
394	C₂²×Q₈	—	4	1	no	—
395	C₂²×Q₈	C₂×C₄×C₇	1	4	yes	complemented
396	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
397	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
398	C₂²×D₄	—	2	1	no	—
399	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
400	C₂³×C₄	—	2	1	no	—
401	C₂³×C₄	—	2	1	no	—
402	C₂³×C₄	—	2	1	no	—
403	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
404	C₂×SD₁₆	—	2	2	no	—
405	C₂×SD₁₆	—	2	2	no	—
406	C₂×SD₁₆	—	4	1	no	—
407	C₂×SD₁₆	—	4	1	no	—
408	C₂×SD₁₆	C₂×C₄×C₇	1	4	yes	complemented
409	C₂×SD₁₆	C₂×C₄×C₇	1	4	yes	complemented
410	C₂²×C₈	—	2	2	no	—
411	C₂²×C₈	—	2	2	no	—
412	C₂²×C₈	—	4	1	no	—
413	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
414	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
415	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
416	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
417	#121	—	2	2	no	—
418	#121	C₂×C₄×C₇	1	4	yes	—
419	C₄×Q₈	—	2	2	no	—
420	C₄×Q₈	—	2	2	no	—
421	C₄×Q₈	—	4	1	no	—
422	C₄×Q₈	—	4	1	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	—	4	1	no	—
425	C₄×D₄	—	4	4	no	—
426	C₂×C₄⋊C₄	—	2	2	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	—	4	1	no	—
432	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
443	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
444	C₂×C₄²	—	2	1	no	—
445	C₂×C₄²	—	2	1	no	—
446	C₂×C₄²	—	2	1	no	—
447	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
449	#107	—	2	2	no	—
450	#107	—	4	1	no	—
451	#103	—	2	2	no	—
452	#103	—	2	2	no	—
453	#103	—	2	2	no	—
454	#103	C₂×C₄×C₇	1	4	yes	complemented
455	#103	C₂×C₄×C₇	1	4	yes	complemented
456	#103	C₇×D₄	1	4	yes	complemented
457	#102	—	2	2	no	—
458	#102	—	4	1	no	—
459	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
460	#97	—	2	2	no	—
461	#97	—	4	1	no	—
462	#95	—	2	1	no	—
463	#95	—	2	1	no	—
464	#95	—	2	1	no	—
465	#95	—	2	1	no	—
466	#95	—	2	1	no	—
467	#95	—	2	1	no	—
468	#95	—	2	1	no	—
469	#95	C₇×D₄	1	1	yes	characteristic
470	#95	C₇×D₄	1	1	yes	characteristic
471	C₂²×C₇	—	2	2	no	—
472	C₂²×C₇	—	2	2	no	—
473	C₂²×C₇	—	2	2	no	—
474	C₂²×C₇	—	2	2	no	—
475	C₂²×C₇	—	2	2	no	—
476	C₂²×C₇	—	2	2	no	—
477	C₂²×C₇	—	4	2	no	—
478	C₂²×C₇	—	4	2	no	—
479	C₂²×C₇	—	4	2	no	—
480	C₂²×C₇	—	8	1	no	—
481	C₂²×C₇	—	8	1	no	—
482	C₂²×C₇	—	8	1	no	—
483	C₂²×C₇	—	8	1	no	—
484	C₂²×C₇	#440	1	4	yes	complemented
485	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
486	C₂²×C₇	D₄²	1	1	yes	characteristic
487	C₂²×C₇	#546	1	1	yes	characteristic
488	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
489	C₂²×C₇	C₂×#137	1	1	yes	characteristic
490	C₂²×C₇	#577	1	1	yes	characteristic
491	C₂²×C₇	#578	1	1	yes	characteristic
492	C₄×C₇	—	2	2	no	—
493	C₄×C₇	—	2	4	no	—
494	C₄×C₇	—	4	2	no	—
495	C₄×C₇	—	4	2	no	—
496	C₄×C₇	—	4	1	no	—
497	C₄×C₇	—	4	2	no	—
498	C₄×C₇	—	4	1	no	—
499	C₄×C₇	—	4	1	no	—
500	C₄×C₇	—	4	1	no	—
501	C₄×C₇	—	8	1	no	—
502	C₄×C₇	—	8	1	no	—
503	C₄×C₇	—	8	1	no	—
504	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
505	C₄×C₇	#518	1	1	yes	characteristic
506	C₄×C₇	#546	1	1	yes	characteristic
507	C₄×C₇	#551	1	1	yes	characteristic
508	C₂⁴	—	2	1	no	—
509	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
510	C₂×Q₈	—	2	4	no	—
511	C₂×Q₈	—	2	1	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	2	1	no	—
515	C₂×Q₈	—	2	8	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	2	1	no	—
518	C₂×Q₈	—	2	1	no	—
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	—	4	2	no	—
521	C₂×Q₈	—	4	2	no	—
522	C₂×Q₈	—	4	4	no	—
523	C₂×Q₈	—	4	4	no	—
524	C₂×Q₈	—	4	4	no	—
525	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
526	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
527	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
528	C₂×Q₈	C₇×#41	1	2	yes	—
529	C₂×D₄	—	2	4	no	—
530	C₂×D₄	—	2	1	no	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	1	no	—
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	4	2	no	—
536	C₂×D₄	—	4	2	no	—
537	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	4	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	2	1	no	—
549	C₂²×C₄	—	2	1	no	—
550	C₂²×C₄	—	4	2	no	—
551	C₂²×C₄	—	4	2	no	—
552	C₂²×C₄	—	4	2	no	—
553	C₂²×C₄	—	4	2	no	—
554	C₂²×C₄	—	4	2	no	—
555	C₂²×C₄	—	4	2	no	—
556	C₂²×C₄	—	8	1	no	—
557	C₂²×C₄	—	8	1	no	—
558	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
559	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
560	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
561	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
562	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
563	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
564	C₂²×C₄	C₇×#41	1	2	yes	—
565	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
566	SD₁₆	—	2	8	no	—
567	SD₁₆	—	4	4	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	2	1	no	—
570	C₂×C₈	—	2	1	no	—
571	C₂×C₈	—	2	1	no	—
572	C₂×C₈	—	2	1	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	4	4	no	—
578	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
579	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
580	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	4	2	no	—
590	C₄⋊C₄	—	4	2	no	—
591	C₄⋊C₄	—	4	2	no	—
592	C₄⋊C₄	—	4	2	no	—
593	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
594	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
595	C₂²⋊C₄	—	2	2	no	—
596	C₂²⋊C₄	—	2	2	no	—
597	C₂²⋊C₄	—	2	4	no	—
598	C₂²⋊C₄	—	4	2	no	—
599	C₂²⋊C₄	—	4	2	no	—
600	C₂²⋊C₄	—	4	2	no	—
601	C₂²⋊C₄	—	8	1	no	—
602	C₄²	—	2	1	no	—
603	C₄²	—	2	1	no	—
604	C₄²	—	2	1	no	—
605	C₄²	—	2	1	no	—
606	C₄²	—	4	2	no	—
607	C₄²	—	4	2	no	—
608	C₄²	—	4	2	no	—
609	C₂×C₇	—	2	4	no	—
610	C₂×C₇	—	8	2	no	—
611	C₂×C₇	#2358	1	1	yes	characteristic
612	C₂×C₇	C₂×#440	1	1	yes	characteristic
613	C₂×C₇	#2947	1	1	yes	characteristic
614	C₂×C₇	#3265	1	1	yes	characteristic
615	C₂×C₇	#3270	1	1	yes	characteristic
616	C₂×C₇	#3308	1	1	yes	characteristic
617	C₂×C₇	#3320	1	1	yes	characteristic
618	C₂³	—	2	1	no	—
619	C₂³	—	2	1	no	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	1	no	—
622	C₂³	—	2	2	no	—
623	C₂³	—	4	1	no	—
624	C₂³	—	4	1	no	—
625	C₂³	—	4	1	no	—
626	C₂³	—	4	1	no	—
627	C₂³	—	4	1	no	—
628	C₂³	—	4	1	no	—
629	C₂³	C₄×C₇×D₄	1	2	yes	—
630	C₂³	C₇×AGL(1,ℤ/8)	1	2	yes	—
631	C₂³	C₇×#137	1	2	yes	—
632	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
633	Q₈	—	2	2	no	—
634	Q₈	—	2	2	no	—
635	Q₈	—	2	2	no	—
636	Q₈	—	2	2	no	—
637	Q₈	—	2	2	no	—
638	Q₈	—	4	8	no	—
639	Q₈	—	4	1	no	—
640	Q₈	—	4	1	no	—
641	Q₈	C₄×C₇×D₄	1	4	yes	complemented
642	D₄	—	2	2	no	—
643	D₄	—	2	2	no	—
644	D₄	—	4	8	no	—
645	D₄	—	4	1	no	—
646	D₄	—	4	1	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	1	no	—
668	C₂×C₄	—	2	1	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	1	no	—
671	C₂×C₄	—	4	8	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	4	1	no	—
683	C₂×C₄	—	4	1	no	—
684	C₂×C₄	—	4	1	no	—
685	C₂×C₄	—	4	1	no	—
686	C₂×C₄	—	4	1	no	—
687	C₂×C₄	—	8	4	no	—
688	C₂×C₄	—	8	4	no	—
689	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
690	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
691	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
692	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
693	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
694	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
695	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
696	C₈	—	2	2	no	—
697	C₈	—	2	2	no	—
698	C₈	—	2	2	no	—
699	C₈	—	4	1	no	—
700	C₇	#22855	1	1	yes	characteristic, direct product factor, Sylow
701	C₂²	—	2	2	no	—
702	C₂²	—	2	2	no	—
703	C₂²	—	2	2	no	—
704	C₂²	—	2	2	no	—
705	C₂²	—	2	2	no	—
706	C₂²	—	2	2	no	—
707	C₂²	—	4	2	no	—
708	C₂²	—	4	2	no	—
709	C₂²	—	4	2	no	—
710	C₂²	—	8	1	no	—
711	C₂²	—	8	1	no	—
712	C₂²	—	8	1	no	—
713	C₂²	—	8	1	no	—
714	C₂²	C₇×#440	1	4	yes	complemented
715	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
716	C₂²	C₇×D₄²	1	1	yes	characteristic
717	C₂²	C₇×#546	1	1	yes	characteristic
718	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
719	C₂²	C₂×C₇×#137	1	1	yes	characteristic
720	C₂²	C₇×#577	1	1	yes	characteristic
721	C₂²	C₇×#578	1	1	yes	characteristic
722	C₄	—	2	2	no	—
723	C₄	—	2	4	no	—
724	C₄	—	4	2	no	—
725	C₄	—	4	1	no	—
726	C₄	—	4	2	no	—
727	C₄	—	4	1	no	—
728	C₄	—	4	1	no	—
729	C₄	—	4	1	no	—
730	C₄	—	4	2	no	—
731	C₄	—	8	1	no	—
732	C₄	—	8	1	no	—
733	C₄	—	8	1	no	—
734	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
735	C₄	C₇×#518	1	1	yes	characteristic
736	C₄	C₇×#546	1	1	yes	characteristic
737	C₄	C₇×#551	1	1	yes	characteristic
738	C₂	—	2	4	no	—
739	C₂	—	8	2	no	—
740	C₂	C₇×#2358	1	1	yes	characteristic
741	C₂	C₂×C₇×#440	1	1	yes	characteristic
742	C₂	C₇×#2947	1	1	yes	characteristic
743	C₂	C₇×#3265	1	1	yes	characteristic
744	C₂	C₇×#3270	1	1	yes	characteristic
745	C₂	C₇×#3308	1	1	yes	characteristic
746	C₂	C₇×#3320	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₂²⋊C₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₇×#440	C₂	semidirect
3. C₂²×C₇×#103	C₂	semidirect
5. C₇×Q₈×C₂²⋊C₄	C₂	semidirect
6. C₇×#1973	C₂	semidirect
7. C₇×#1935	C₂	semidirect
8. C₇×#1928	C₂	semidirect
10. C₇×#1880	C₂	semidirect
11. C₇×#1877	C₂	semidirect
12. C₇×#1863	C₂	semidirect
13. C₇×#1852	C₂	semidirect
14. C₇×#1763	C₂	semidirect
15. C₇×#1752	C₂	semidirect
26. C₇×#448	C₄	semidirect
62. #22855	C₇	direct (N × H)
69. C₂²×C₇×Q₈	C₂×C₄	semidirect
82. C₂×C₇×SD₁₆	C₂×C₄	semidirect
83. C₂×C₇×SD₁₆	C₂×C₄	semidirect
128. C₇×#103	C₂×C₄	semidirect
129. C₇×#103	C₂×C₄	semidirect
130. C₇×#103	D₄	semidirect
133. C₇×C₂²⋊C₈	C₂×C₄	semidirect
146. C₂×#440	C₂×C₇	semidirect
147. C₂²×#103	C₂×C₇	semidirect
149. Q₈×C₂²⋊C₄	C₂×C₇	semidirect
150. #1973	C₂×C₇	semidirect
151. #1935	C₂×C₇	semidirect
152. #1928	C₂×C₇	semidirect
154. #1880	C₂×C₇	semidirect
155. #1877	C₂×C₇	semidirect
156. #1863	C₂×C₇	semidirect
157. #1852	C₂×C₇	semidirect
158. #1763	C₂×C₇	semidirect
159. #1752	C₂×C₇	semidirect
271. #448	C₄×C₇	semidirect
330. C₇×Q₈	C₄×D₄	semidirect
395. C₂²×Q₈	C₂×C₄×C₇	semidirect
408. C₂×SD₁₆	C₂×C₄×C₇	semidirect
409. C₂×SD₁₆	C₂×C₄×C₇	semidirect
454. #103	C₂×C₄×C₇	semidirect
455. #103	C₂×C₄×C₇	semidirect
456. #103	C₇×D₄	semidirect
459. C₂²⋊C₈	C₂×C₄×C₇	semidirect
484. C₂²×C₇	#440	semidirect
641. Q₈	C₄×C₇×D₄	semidirect
700. C₇	#22855	direct (N × H)
714. C₂²	C₇×#440	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#448	2
C₂×C₇×#440	2
C₂²×C₇×#103	2
C₇×#2361	2
C₇×Q₈×C₂²⋊C₄	2
C₇×#1973	2
C₇×#1935	2
C₇×#1928	2
C₇×#1893	2
C₇×#1880	2
C₇×#1877	2
C₇×#1863	2
C₇×#1852	2
C₇×#1763	2
C₇×#1752	2
#22855	7

Subgroups — C7×#22855 #13200071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22855 #13200071