Subgroups — C₇×#22055 #13199271

Order 1792.

1024 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#449	C₂	1	1	yes	maximal, characteristic
2	C₇×#2324	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2311	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1973	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1936	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1933	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1928	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1925	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1879	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1877	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1874	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1852	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1788	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1750	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
17	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×#121	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₇×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₇×#449	C₄	1	8	yes	complemented
24	C₂×C₇×#107	C₂²	1	1	yes	characteristic
25	C₂×C₇×#107	C₂²	1	1	yes	characteristic
26	C₂×C₇×#103	C₂²	1	1	yes	characteristic
27	C₂×C₇×#103	C₂²	1	1	yes	characteristic
28	C₂×C₇×#103	C₂²	1	1	yes	characteristic
29	C₂×C₇×#102	C₂²	1	1	yes	characteristic
30	C₂×C₇×#102	C₂²	1	1	yes	characteristic
31	C₂×C₇×#102	C₂²	1	1	yes	characteristic
32	C₂×C₇×#102	C₂²	1	1	yes	characteristic
33	C₂×C₇×#102	C₂²	1	1	yes	characteristic
34	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₇×#97	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
38	C₇×#391	C₂²	1	1	yes	characteristic
39	C₇×#391	C₂²	1	1	yes	characteristic
40	C₇×#390	—	2	1	no	—
41	C₇×#390	C₂²	1	1	yes	characteristic
42	C₇×#390	C₂²	1	1	yes	characteristic
43	C₇×#390	C₂²	1	1	yes	characteristic
44	C₇×#389	C₂²	1	1	yes	characteristic
45	C₇×#388	—	2	1	no	—
46	C₇×#386	—	2	1	no	—
47	C₇×#386	—	2	1	no	—
48	C₇×#386	—	2	1	no	—
49	C₇×#385	—	2	1	no	—
50	C₇×#380	—	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	C₇×#340	C₂²	1	1	yes	characteristic
55	C₇×#340	C₂²	1	1	yes	characteristic
56	C₇×#340	C₂²	1	1	yes	characteristic
57	C₇×#340	C₂²	1	1	yes	characteristic
58	C₇×#336	C₂²	1	1	yes	characteristic
59	C₇×#336	C₂²	1	1	yes	characteristic
60	#22055	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₇×#41	—	2	2	no	—
62	C₂×C₇×#41	—	4	1	no	—
63	C₂×C₇×#41	—	4	1	no	—
64	C₂×C₇×#41	—	4	1	no	—
65	C₂×C₇×#41	—	4	1	no	—
66	C₂×C₇×#41	C₂×C₄	1	4	yes	complemented
67	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
68	C₂²×C₇×D₄	—	2	1	no	—
69	C₂²×C₇×D₄	—	2	1	no	—
70	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
71	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₇	—	2	1	no	—
73	C₂³×C₄×C₇	—	2	1	no	—
74	C₂³×C₄×C₇	—	2	1	no	—
75	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
76	C₂×C₇×SD₁₆	—	4	1	no	—
77	C₂×C₇×SD₁₆	—	4	1	no	—
78	C₂×C₇×SD₁₆	C₂×C₄	1	4	yes	complemented
79	C₂×C₇×D₈	—	4	1	no	—
80	C₂×C₇×D₈	—	4	1	no	—
81	C₂×C₇×D₈	C₂×C₄	1	4	yes	complemented
82	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
83	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
84	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
85	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₇×#121	—	2	2	no	—
87	C₇×#121	C₂×C₄	1	4	yes	—
88	C₄×C₇×D₄	—	4	1	no	—
89	C₄×C₇×D₄	—	4	1	no	—
90	C₇×#117	—	4	1	no	—
91	C₇×#117	—	4	1	no	—
92	C₂×C₇×C₄⋊C₄	—	2	1	no	—
93	C₂×C₇×C₄⋊C₄	—	2	1	no	—
94	C₂×C₇×C₄⋊C₄	—	4	1	no	—
95	C₂×C₇×C₄⋊C₄	—	4	1	no	—
96	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
97	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
98	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
99	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
100	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
101	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
102	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
103	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
104	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
105	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
106	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
107	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
108	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
109	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
110	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
113	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₇	—	2	1	no	—
115	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
117	C₇×#107	—	4	1	no	—
118	C₇×#107	—	4	1	no	—
119	C₇×#103	—	4	1	no	—
120	C₇×#103	—	4	1	no	—
121	C₇×#103	C₂×C₄	1	4	yes	complemented
122	C₇×#102	—	4	1	no	—
123	C₇×#102	—	4	1	no	—
124	C₇×#102	—	4	1	no	—
125	C₇×#102	—	4	1	no	—
126	C₇×#102	C₂×C₄	1	4	yes	complemented
127	C₇×C₂²⋊C₈	—	2	2	no	—
128	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
129	C₇×#97	—	4	1	no	—
130	C₄×C₇×C₈	—	4	1	no	—
131	C₇×#95	—	2	1	no	—
132	C₇×#95	—	2	1	no	—
133	C₇×#95	—	2	1	no	—
134	C₇×#95	—	2	1	no	—
135	C₇×#95	—	2	1	no	—
136	C₇×#95	—	2	1	no	—
137	C₇×#95	—	2	1	no	—
138	C₂×#449	C₂×C₇	1	1	yes	characteristic
139	#2324	C₂×C₇	1	1	yes	characteristic, complemented
140	#2311	C₂×C₇	1	1	yes	characteristic, complemented
141	#1973	C₂×C₇	1	1	yes	characteristic, complemented
142	#1970	C₂×C₇	1	1	yes	characteristic, complemented
143	#1936	C₂×C₇	1	1	yes	characteristic, complemented
144	#1933	C₂×C₇	1	1	yes	characteristic, complemented
145	#1928	C₂×C₇	1	1	yes	characteristic, complemented
146	#1925	C₂×C₇	1	1	yes	characteristic, complemented
147	#1879	C₂×C₇	1	1	yes	characteristic, complemented
148	#1877	C₂×C₇	1	1	yes	characteristic, complemented
149	#1874	C₂×C₇	1	1	yes	characteristic, complemented
150	#1852	C₂×C₇	1	1	yes	characteristic, complemented
151	#1788	C₂×C₇	1	1	yes	characteristic, complemented
152	#1750	C₂×C₇	1	1	yes	characteristic, complemented
153	C₂⁴×C₇	—	2	1	no	—
154	C₂⁴×C₇	—	2	1	no	—
155	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
156	C₇×#41	—	4	4	no	—
157	C₇×#41	—	4	4	no	—
158	C₇×#41	—	4	4	no	—
159	C₇×#41	—	4	4	no	—
160	C₂×C₇×Q₈	—	2	1	no	—
161	C₂×C₇×Q₈	—	2	1	no	—
162	C₂×C₇×Q₈	—	2	1	no	—
163	C₂×C₇×Q₈	—	2	1	no	—
164	C₂×C₇×Q₈	—	2	1	no	—
165	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
166	C₂×C₇×D₄	—	2	4	no	—
167	C₂×C₇×D₄	—	2	4	no	—
168	C₂×C₇×D₄	—	2	1	no	—
169	C₂×C₇×D₄	—	2	1	no	—
170	C₂×C₇×D₄	—	2	1	no	—
171	C₂×C₇×D₄	—	2	1	no	—
172	C₂×C₇×D₄	—	2	1	no	—
173	C₂×C₇×D₄	—	2	1	no	—
174	C₂×C₇×D₄	—	2	1	no	—
175	C₂×C₇×D₄	—	2	1	no	—
176	C₂×C₇×D₄	—	2	1	no	—
177	C₂×C₇×D₄	—	2	1	no	—
178	C₂×C₇×D₄	—	4	2	no	—
179	C₂×C₇×D₄	—	4	2	no	—
180	C₂×C₇×D₄	—	8	1	no	—
181	C₂×C₇×D₄	—	8	1	no	—
182	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
183	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
184	C₂²×C₄×C₇	—	2	1	no	—
185	C₂²×C₄×C₇	—	2	1	no	—
186	C₂²×C₄×C₇	—	2	1	no	—
187	C₂²×C₄×C₇	—	2	1	no	—
188	C₂²×C₄×C₇	—	2	1	no	—
189	C₂²×C₄×C₇	—	2	1	no	—
190	C₂²×C₄×C₇	—	2	4	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	—	4	2	no	—
196	C₂²×C₄×C₇	—	4	2	no	—
197	C₂²×C₄×C₇	—	4	2	no	—
198	C₂²×C₄×C₇	—	4	2	no	—
199	C₂²×C₄×C₇	—	8	1	no	—
200	C₂²×C₄×C₇	—	8	1	no	—
201	C₂²×C₄×C₇	—	8	1	no	—
202	C₂²×C₄×C₇	—	8	1	no	—
203	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
204	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₇	#41	1	2	yes	—
210	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
211	C₇×SD₁₆	—	4	4	no	—
212	C₇×D₈	—	4	4	no	—
213	C₂×C₇×C₈	—	2	1	no	—
214	C₂×C₇×C₈	—	2	1	no	—
215	C₂×C₇×C₈	—	2	1	no	—
216	C₂×C₇×C₈	—	2	1	no	—
217	C₂×C₇×C₈	—	2	1	no	—
218	C₂×C₇×C₈	—	2	1	no	—
219	C₂×C₇×C₈	—	2	1	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	C₂²×C₄	1	2	yes	—
224	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
225	C₇×C₄⋊C₄	—	2	1	no	—
226	C₇×C₄⋊C₄	—	2	1	no	—
227	C₇×C₄⋊C₄	—	2	1	no	—
228	C₇×C₄⋊C₄	—	2	1	no	—
229	C₇×C₄⋊C₄	—	2	1	no	—
230	C₇×C₄⋊C₄	—	2	1	no	—
231	C₇×C₄⋊C₄	—	2	1	no	—
232	C₇×C₄⋊C₄	—	2	1	no	—
233	C₇×C₄⋊C₄	—	2	1	no	—
234	C₇×C₄⋊C₄	—	4	1	no	—
235	C₇×C₄⋊C₄	—	4	1	no	—
236	C₇×C₄⋊C₄	—	4	2	no	—
237	C₇×C₄⋊C₄	—	8	1	no	—
238	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
239	C₇×C₂²⋊C₄	—	2	4	no	—
240	C₇×C₂²⋊C₄	—	4	1	no	—
241	C₇×C₂²⋊C₄	—	4	1	no	—
242	C₇×C₂²⋊C₄	—	4	2	no	—
243	C₇×C₂²⋊C₄	—	8	1	no	—
244	C₇×C₂²⋊C₄	—	8	1	no	—
245	C₇×C₂²⋊C₄	—	8	1	no	—
246	C₇×C₂²⋊C₄	—	8	1	no	—
247	C₇×C₂²⋊C₄	—	8	1	no	—
248	C₇×C₂²⋊C₄	—	8	1	no	—
249	C₇×C₂²⋊C₄	—	8	1	no	—
250	C₄²×C₇	—	2	1	no	—
251	C₄²×C₇	—	2	1	no	—
252	C₄²×C₇	—	2	1	no	—
253	C₄²×C₇	—	2	1	no	—
254	C₄²×C₇	—	8	1	no	—
255	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
256	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
257	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
258	C₂×#121	C₂²×C₇	1	1	yes	characteristic
259	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
260	C₂×#117	C₂²×C₇	1	1	yes	characteristic
261	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
262	#449	C₄×C₇	1	8	yes	complemented
263	C₂×#107	C₂²×C₇	1	1	yes	characteristic
264	C₂×#107	C₂²×C₇	1	1	yes	characteristic
265	C₂×#103	C₂²×C₇	1	1	yes	characteristic
266	C₂×#103	C₂²×C₇	1	1	yes	characteristic
267	C₂×#103	C₂²×C₇	1	1	yes	characteristic
268	C₂×#102	C₂²×C₇	1	1	yes	characteristic
269	C₂×#102	C₂²×C₇	1	1	yes	characteristic
270	C₂×#102	C₂²×C₇	1	1	yes	characteristic
271	C₂×#102	C₂²×C₇	1	1	yes	characteristic
272	C₂×#102	C₂²×C₇	1	1	yes	characteristic
273	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
274	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
275	C₂×#97	C₂²×C₇	1	1	yes	characteristic
276	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
277	#391	C₂²×C₇	1	1	yes	characteristic
278	#391	C₂²×C₇	1	1	yes	characteristic
279	#390	—	2	1	no	—
280	#390	C₂²×C₇	1	1	yes	characteristic
281	#390	C₂²×C₇	1	1	yes	characteristic
282	#390	C₂²×C₇	1	1	yes	characteristic
283	#389	C₂²×C₇	1	1	yes	characteristic
284	#388	—	2	1	no	—
285	#386	—	2	1	no	—
286	#386	—	2	1	no	—
287	#386	—	2	1	no	—
288	#385	—	2	1	no	—
289	#380	—	2	1	no	—
290	#380	—	2	1	no	—
291	#380	C₂²×C₇	1	1	yes	characteristic
292	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
293	#340	C₂²×C₇	1	1	yes	characteristic
294	#340	C₂²×C₇	1	1	yes	characteristic
295	#340	C₂²×C₇	1	1	yes	characteristic
296	#340	C₂²×C₇	1	1	yes	characteristic
297	#336	C₂²×C₇	1	1	yes	characteristic
298	#336	C₂²×C₇	1	1	yes	characteristic
299	C₂³×C₇	—	2	1	no	—
300	C₂³×C₇	—	2	1	no	—
301	C₂³×C₇	—	2	1	no	—
302	C₂³×C₇	—	2	1	no	—
303	C₂³×C₇	—	2	1	no	—
304	C₂³×C₇	—	2	1	no	—
305	C₂³×C₇	—	2	2	no	—
306	C₂³×C₇	—	2	2	no	—
307	C₂³×C₇	—	4	1	no	—
308	C₂³×C₇	—	4	1	no	—
309	C₂³×C₇	—	4	1	no	—
310	C₂³×C₇	—	4	1	no	—
311	C₂³×C₇	—	4	1	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	—	4	1	no	—
319	C₂³×C₇	C₄×D₄	1	2	yes	—
320	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
321	C₇×Q₈	—	2	2	no	—
322	C₇×Q₈	—	2	2	no	—
323	C₇×Q₈	—	4	1	no	—
324	C₇×Q₈	—	4	1	no	—
325	C₇×D₄	—	2	2	no	—
326	C₇×D₄	—	2	2	no	—
327	C₇×D₄	—	2	2	no	—
328	C₇×D₄	—	2	2	no	—
329	C₇×D₄	—	4	8	no	—
330	C₇×D₄	—	4	1	no	—
331	C₇×D₄	—	4	1	no	—
332	C₇×D₄	—	4	1	no	—
333	C₇×D₄	—	4	1	no	—
334	C₇×D₄	—	8	4	no	—
335	C₂×C₄×C₇	—	2	1	no	—
336	C₂×C₄×C₇	—	2	1	no	—
337	C₂×C₄×C₇	—	2	2	no	—
338	C₂×C₄×C₇	—	2	2	no	—
339	C₂×C₄×C₇	—	2	2	no	—
340	C₂×C₄×C₇	—	2	2	no	—
341	C₂×C₄×C₇	—	2	2	no	—
342	C₂×C₄×C₇	—	2	2	no	—
343	C₂×C₄×C₇	—	2	1	no	—
344	C₂×C₄×C₇	—	2	1	no	—
345	C₂×C₄×C₇	—	2	1	no	—
346	C₂×C₄×C₇	—	2	1	no	—
347	C₂×C₄×C₇	—	2	1	no	—
348	C₂×C₄×C₇	—	2	1	no	—
349	C₂×C₄×C₇	—	2	1	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	—	4	8	no	—
356	C₂×C₄×C₇	—	4	1	no	—
357	C₂×C₄×C₇	—	4	1	no	—
358	C₂×C₄×C₇	—	4	1	no	—
359	C₂×C₄×C₇	—	4	1	no	—
360	C₂×C₄×C₇	—	4	1	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	—	8	4	no	—
373	C₂×C₄×C₇	—	8	4	no	—
374	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
375	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
376	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
377	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
378	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
379	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
380	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
381	C₇×C₈	—	2	2	no	—
382	C₇×C₈	—	2	2	no	—
383	C₇×C₈	—	4	1	no	—
384	C₇×C₈	—	4	1	no	—
385	C₂×#41	—	2	2	no	—
386	C₂×#41	—	4	1	no	—
387	C₂×#41	—	4	1	no	—
388	C₂×#41	—	4	1	no	—
389	C₂×#41	—	4	1	no	—
390	C₂×#41	C₂×C₄×C₇	1	4	yes	complemented
391	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
392	C₂²×D₄	—	2	1	no	—
393	C₂²×D₄	—	2	1	no	—
394	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
395	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
396	C₂³×C₄	—	2	1	no	—
397	C₂³×C₄	—	2	1	no	—
398	C₂³×C₄	—	2	1	no	—
399	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
400	C₂×SD₁₆	—	4	1	no	—
401	C₂×SD₁₆	—	4	1	no	—
402	C₂×SD₁₆	C₂×C₄×C₇	1	4	yes	complemented
403	C₂×D₈	—	4	1	no	—
404	C₂×D₈	—	4	1	no	—
405	C₂×D₈	C₂×C₄×C₇	1	4	yes	complemented
406	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
407	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
408	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
409	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
410	#121	—	2	2	no	—
411	#121	C₂×C₄×C₇	1	4	yes	—
412	C₄×D₄	—	4	1	no	—
413	C₄×D₄	—	4	1	no	—
414	#117	—	4	1	no	—
415	#117	—	4	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	4	1	no	—
419	C₂×C₄⋊C₄	—	4	1	no	—
420	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₂²⋊C₄	—	2	1	no	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₂²⋊C₄	—	2	1	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
438	C₂×C₄²	—	2	1	no	—
439	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
441	#107	—	4	1	no	—
442	#107	—	4	1	no	—
443	#103	—	4	1	no	—
444	#103	—	4	1	no	—
445	#103	C₂×C₄×C₇	1	4	yes	complemented
446	#102	—	4	1	no	—
447	#102	—	4	1	no	—
448	#102	—	4	1	no	—
449	#102	—	4	1	no	—
450	#102	C₂×C₄×C₇	1	4	yes	complemented
451	C₂²⋊C₈	—	2	2	no	—
452	C₂²⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
453	#97	—	4	1	no	—
454	C₄×C₈	—	4	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	#95	—	2	1	no	—
458	#95	—	2	1	no	—
459	#95	—	2	1	no	—
460	#95	—	2	1	no	—
461	#95	—	2	1	no	—
462	C₂²×C₇	—	2	2	no	—
463	C₂²×C₇	—	2	2	no	—
464	C₂²×C₇	—	2	2	no	—
465	C₂²×C₇	—	4	1	no	—
466	C₂²×C₇	—	4	1	no	—
467	C₂²×C₇	—	4	1	no	—
468	C₂²×C₇	—	4	1	no	—
469	C₂²×C₇	—	4	2	no	—
470	C₂²×C₇	—	4	2	no	—
471	C₂²×C₇	—	4	2	no	—
472	C₂²×C₇	—	4	2	no	—
473	C₂²×C₇	—	4	2	no	—
474	C₂²×C₇	—	4	2	no	—
475	C₂²×C₇	—	8	1	no	—
476	C₂²×C₇	—	8	1	no	—
477	C₂²×C₇	—	8	1	no	—
478	C₂²×C₇	—	8	1	no	—
479	C₂²×C₇	—	8	1	no	—
480	C₂²×C₇	—	8	1	no	—
481	C₂²×C₇	—	8	1	no	—
482	C₂²×C₇	—	8	1	no	—
483	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
484	C₂²×C₇	#534	1	1	yes	characteristic
485	C₂²×C₇	#535	1	1	yes	characteristic
486	C₂²×C₇	#576	1	1	yes	characteristic
487	C₂²×C₇	#576	1	1	yes	characteristic
488	C₂²×C₇	#577	1	1	yes	characteristic
489	C₂²×C₇	#578	1	1	yes	characteristic
490	C₄×C₇	—	2	2	no	—
491	C₄×C₇	—	4	2	no	—
492	C₄×C₇	—	4	1	no	—
493	C₄×C₇	—	4	1	no	—
494	C₄×C₇	—	4	2	no	—
495	C₄×C₇	—	4	1	no	—
496	C₄×C₇	—	4	1	no	—
497	C₄×C₇	—	8	1	no	—
498	C₄×C₇	—	8	1	no	—
499	C₄×C₇	—	8	1	no	—
500	C₄×C₇	—	8	1	no	—
501	C₄×C₇	#518	1	1	yes	characteristic
502	C₄×C₇	#518	1	1	yes	characteristic
503	C₄×C₇	#538	1	1	yes	characteristic
504	C₄×C₇	#538	1	1	yes	characteristic
505	C₂⁴	—	2	1	no	—
506	C₂⁴	—	2	1	no	—
507	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
508	#41	—	4	4	no	—
509	#41	—	4	4	no	—
510	#41	—	4	4	no	—
511	#41	—	4	4	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	2	1	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
518	C₂×D₄	—	2	4	no	—
519	C₂×D₄	—	2	4	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	—	2	1	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	—	2	1	no	—
527	C₂×D₄	—	2	1	no	—
528	C₂×D₄	—	2	1	no	—
529	C₂×D₄	—	2	1	no	—
530	C₂×D₄	—	4	2	no	—
531	C₂×D₄	—	4	2	no	—
532	C₂×D₄	—	8	1	no	—
533	C₂×D₄	—	8	1	no	—
534	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
535	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
536	C₂²×C₄	—	2	1	no	—
537	C₂²×C₄	—	2	1	no	—
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	4	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	4	2	no	—
548	C₂²×C₄	—	4	2	no	—
549	C₂²×C₄	—	4	2	no	—
550	C₂²×C₄	—	4	2	no	—
551	C₂²×C₄	—	8	1	no	—
552	C₂²×C₄	—	8	1	no	—
553	C₂²×C₄	—	8	1	no	—
554	C₂²×C₄	—	8	1	no	—
555	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
556	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
557	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
558	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
559	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
560	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
561	C₂²×C₄	C₇×#41	1	2	yes	—
562	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
563	SD₁₆	—	4	4	no	—
564	D₈	—	4	4	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	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	C₂²×C₄×C₇	1	2	yes	—
576	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
587	C₄⋊C₄	—	4	1	no	—
588	C₄⋊C₄	—	4	2	no	—
589	C₄⋊C₄	—	8	1	no	—
590	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
591	C₂²⋊C₄	—	2	4	no	—
592	C₂²⋊C₄	—	4	1	no	—
593	C₂²⋊C₄	—	4	1	no	—
594	C₂²⋊C₄	—	4	2	no	—
595	C₂²⋊C₄	—	8	1	no	—
596	C₂²⋊C₄	—	8	1	no	—
597	C₂²⋊C₄	—	8	1	no	—
598	C₂²⋊C₄	—	8	1	no	—
599	C₂²⋊C₄	—	8	1	no	—
600	C₂²⋊C₄	—	8	1	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	—	8	1	no	—
607	C₂×C₇	—	4	2	no	—
608	C₂×C₇	—	8	2	no	—
609	C₂×C₇	—	8	2	no	—
610	C₂×C₇	#2321	1	1	yes	characteristic
611	C₂×C₇	#2946	1	1	yes	characteristic
612	C₂×C₇	#2951	1	1	yes	characteristic
613	C₂×C₇	#3198	1	1	yes	characteristic
614	C₂×C₇	#3203	1	1	yes	characteristic
615	C₂×C₇	#3220	1	1	yes	characteristic
616	C₂×C₇	#3225	1	1	yes	characteristic
617	C₂³	—	2	1	no	—
618	C₂³	—	2	1	no	—
619	C₂³	—	2	1	no	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	1	no	—
622	C₂³	—	2	1	no	—
623	C₂³	—	2	2	no	—
624	C₂³	—	2	2	no	—
625	C₂³	—	4	1	no	—
626	C₂³	—	4	1	no	—
627	C₂³	—	4	1	no	—
628	C₂³	—	4	1	no	—
629	C₂³	—	4	1	no	—
630	C₂³	—	4	1	no	—
631	C₂³	—	4	1	no	—
632	C₂³	—	4	1	no	—
633	C₂³	—	4	1	no	—
634	C₂³	—	4	1	no	—
635	C₂³	—	4	1	no	—
636	C₂³	—	4	1	no	—
637	C₂³	C₄×C₇×D₄	1	2	yes	—
638	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
639	Q₈	—	2	2	no	—
640	Q₈	—	2	2	no	—
641	Q₈	—	4	1	no	—
642	Q₈	—	4	1	no	—
643	D₄	—	2	2	no	—
644	D₄	—	2	2	no	—
645	D₄	—	2	2	no	—
646	D₄	—	2	2	no	—
647	D₄	—	4	8	no	—
648	D₄	—	4	1	no	—
649	D₄	—	4	1	no	—
650	D₄	—	4	1	no	—
651	D₄	—	4	1	no	—
652	D₄	—	8	4	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	2	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	—	2	1	no	—
672	C₂×C₄	—	2	1	no	—
673	C₂×C₄	—	4	8	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	—	4	1	no	—
688	C₂×C₄	—	4	1	no	—
689	C₂×C₄	—	4	1	no	—
690	C₂×C₄	—	8	4	no	—
691	C₂×C₄	—	8	4	no	—
692	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
693	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
694	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
695	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
696	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
697	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
698	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
699	C₈	—	2	2	no	—
700	C₈	—	2	2	no	—
701	C₈	—	4	1	no	—
702	C₈	—	4	1	no	—
703	C₇	#22055	1	1	yes	characteristic, direct product factor, Sylow
704	C₂²	—	2	2	no	—
705	C₂²	—	2	2	no	—
706	C₂²	—	2	2	no	—
707	C₂²	—	4	1	no	—
708	C₂²	—	4	1	no	—
709	C₂²	—	4	1	no	—
710	C₂²	—	4	1	no	—
711	C₂²	—	4	2	no	—
712	C₂²	—	4	2	no	—
713	C₂²	—	4	2	no	—
714	C₂²	—	4	2	no	—
715	C₂²	—	4	2	no	—
716	C₂²	—	4	2	no	—
717	C₂²	—	8	1	no	—
718	C₂²	—	8	1	no	—
719	C₂²	—	8	1	no	—
720	C₂²	—	8	1	no	—
721	C₂²	—	8	1	no	—
722	C₂²	—	8	1	no	—
723	C₂²	—	8	1	no	—
724	C₂²	—	8	1	no	—
725	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
726	C₂²	C₇×#534	1	1	yes	characteristic
727	C₂²	C₇×#535	1	1	yes	characteristic
728	C₂²	C₇×#576	1	1	yes	characteristic
729	C₂²	C₇×#576	1	1	yes	characteristic
730	C₂²	C₇×#577	1	1	yes	characteristic
731	C₂²	C₇×#578	1	1	yes	characteristic
732	C₄	—	2	2	no	—
733	C₄	—	4	1	no	—
734	C₄	—	4	1	no	—
735	C₄	—	4	2	no	—
736	C₄	—	4	1	no	—
737	C₄	—	4	1	no	—
738	C₄	—	4	2	no	—
739	C₄	—	8	1	no	—
740	C₄	—	8	1	no	—
741	C₄	—	8	1	no	—
742	C₄	—	8	1	no	—
743	C₄	C₇×#518	1	1	yes	characteristic
744	C₄	C₇×#518	1	1	yes	characteristic
745	C₄	C₇×#538	1	1	yes	characteristic
746	C₄	C₇×#538	1	1	yes	characteristic
747	C₂	—	4	2	no	—
748	C₂	—	8	2	no	—
749	C₂	—	8	2	no	—
750	C₂	C₇×#2321	1	1	yes	characteristic
751	C₂	C₇×#2946	1	1	yes	characteristic
752	C₂	C₇×#2951	1	1	yes	characteristic
753	C₂	C₇×#3198	1	1	yes	characteristic
754	C₂	C₇×#3203	1	1	yes	characteristic
755	C₂	C₇×#3220	1	1	yes	characteristic
756	C₂	C₇×#3225	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×#449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2324	C₂	semidirect
3. C₇×#2311	C₂	semidirect
4. C₇×#1973	C₂	semidirect
5. C₇×#1970	C₂	semidirect
6. C₇×#1936	C₂	semidirect
7. C₇×#1933	C₂	semidirect
8. C₇×#1928	C₂	semidirect
9. C₇×#1925	C₂	semidirect
10. C₇×#1879	C₂	semidirect
11. C₇×#1877	C₂	semidirect
12. C₇×#1874	C₂	semidirect
13. C₇×#1852	C₂	semidirect
14. C₇×#1788	C₂	semidirect
15. C₇×#1750	C₂	semidirect
23. C₇×#449	C₄	semidirect
60. #22055	C₇	direct (N × H)
66. C₂×C₇×#41	C₂×C₄	semidirect
78. C₂×C₇×SD₁₆	C₂×C₄	semidirect
81. C₂×C₇×D₈	C₂×C₄	semidirect
121. C₇×#103	C₂×C₄	semidirect
126. C₇×#102	C₂×C₄	semidirect
128. C₇×C₂²⋊C₈	C₂×C₄	semidirect
139. #2324	C₂×C₇	semidirect
140. #2311	C₂×C₇	semidirect
141. #1973	C₂×C₇	semidirect
142. #1970	C₂×C₇	semidirect
143. #1936	C₂×C₇	semidirect
144. #1933	C₂×C₇	semidirect
145. #1928	C₂×C₇	semidirect
146. #1925	C₂×C₇	semidirect
147. #1879	C₂×C₇	semidirect
148. #1877	C₂×C₇	semidirect
149. #1874	C₂×C₇	semidirect
150. #1852	C₂×C₇	semidirect
151. #1788	C₂×C₇	semidirect
152. #1750	C₂×C₇	semidirect
262. #449	C₄×C₇	semidirect
390. C₂×#41	C₂×C₄×C₇	semidirect
402. C₂×SD₁₆	C₂×C₄×C₇	semidirect
405. C₂×D₈	C₂×C₄×C₇	semidirect
445. #103	C₂×C₄×C₇	semidirect
450. #102	C₂×C₄×C₇	semidirect
452. C₂²⋊C₈	C₂×C₄×C₇	semidirect
703. C₇	#22055	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#449	2
C₇×#2324	2
C₇×#2311	2
C₇×#1973	2
C₇×#1970	2
C₇×#1936	2
C₇×#1933	2
C₇×#1928	2
C₇×#1925	2
C₇×#1879	2
C₇×#1877	2
C₇×#1874	2
C₇×#1852	2
C₇×#1788	2
C₇×#1750	2
#22055	7

Subgroups — C7×#22055 #13199271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22055 #13199271