Subgroups — C₇×#23002 #13200218

Order 1792.

1018 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#461	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#440	C₂	1	2	yes	maximal
3	C₇×#2365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1935	C₂	1	2	yes	maximal, complemented
8	C₇×#1877	C₂	1	2	yes	maximal
9	C₇×#1863	C₂	1	2	yes	maximal, complemented
10	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#102	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
15	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
16	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
17	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
18	C₇×#461	C₄	1	8	yes	complemented
19	C₇×#440	—	2	8	no	—
20	C₂×C₇×#107	C₂²	1	2	yes	—
21	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₇×#103	C₂²	1	2	yes	—
23	C₂×C₇×#103	C₂²	1	2	yes	—
24	C₂×C₇×#103	C₂²	1	2	yes	—
25	C₂×C₇×#102	C₂²	1	1	yes	characteristic
26	C₂×C₇×#102	C₂²	1	1	yes	characteristic
27	C₂×C₇×#97	C₂²	1	2	yes	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#391	C₂²	1	2	yes	—
30	C₇×#389	—	2	1	no	—
31	C₇×#389	C₂²	1	2	yes	—
32	C₇×#387	—	2	2	no	—
33	C₇×#385	—	2	2	no	—
34	C₇×#384	C₂²	1	1	yes	characteristic
35	C₇×#380	—	2	1	no	—
36	C₄×C₇×C₄⋊C₄	—	2	1	no	—
37	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₇×#340	C₂²	1	1	yes	characteristic
40	C₇×#340	C₂²	1	1	yes	characteristic
41	C₇×#340	C₂²	1	2	yes	—
42	C₇×#336	C₂²	1	2	yes	—
43	#23002	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₇×Q₈	C₂³	1	2	yes	—
45	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₇	—	2	1	no	—
47	C₂³×C₄×C₇	—	2	2	no	—
48	C₂×C₇×SD₁₆	—	2	4	no	—
49	C₂×C₇×SD₁₆	—	2	4	no	—
50	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	—
51	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₇×C₈	C₂³	1	2	yes	—
53	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₇×C₄⋊Q₈	—	2	2	no	—
55	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
56	C₄×C₇×Q₈	—	2	4	no	—
57	C₄×C₇×Q₈	—	4	2	no	—
58	C₄×C₇×D₄	—	2	4	no	—
59	C₄×C₇×D₄	—	2	2	no	—
60	C₄×C₇×D₄	—	4	2	no	—
61	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
62	C₂×C₇×C₄⋊C₄	—	2	1	no	—
63	C₂×C₇×C₄⋊C₄	—	2	1	no	—
64	C₂×C₇×C₄⋊C₄	—	2	1	no	—
65	C₂×C₇×C₄⋊C₄	—	2	2	no	—
66	C₂×C₇×C₄⋊C₄	—	2	2	no	—
67	C₂×C₇×C₄⋊C₄	—	2	1	no	—
68	C₂×C₇×C₄⋊C₄	—	4	4	no	—
69	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
70	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
73	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
74	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
76	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
77	C₂×C₄²×C₇	—	2	1	no	—
78	C₂×C₄²×C₇	—	2	2	no	—
79	C₂×C₄²×C₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
81	C₇×#107	—	2	4	no	—
82	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
83	C₇×#103	—	2	4	no	—
84	C₇×#103	—	4	2	no	—
85	C₇×#103	C₂×C₄	1	8	yes	complemented
86	C₇×#102	—	2	2	no	—
87	C₇×#102	D₄	1	4	yes	—
88	C₇×#97	—	2	4	no	—
89	C₄×C₇×C₈	—	2	2	no	—
90	C₇×#95	—	2	2	no	—
91	C₇×#95	—	2	1	no	—
92	C₇×#95	—	2	2	no	—
93	C₇×#95	—	2	2	no	—
94	C₇×#95	D₄	1	2	yes	—
95	C₂×#461	C₂×C₇	1	1	yes	characteristic
96	C₂×#440	C₂×C₇	1	2	yes	—
97	#2365	C₂×C₇	1	1	yes	characteristic, complemented
98	D₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
99	#1956	C₂×C₇	1	1	yes	characteristic, complemented
100	#1947	C₂×C₇	1	1	yes	characteristic, complemented
101	#1935	C₂×C₇	1	2	yes	complemented
102	#1877	C₂×C₇	1	2	yes	—
103	#1863	C₂×C₇	1	2	yes	complemented
104	#1770	C₂×C₇	1	1	yes	characteristic, complemented
105	C₄×#102	C₂×C₇	1	1	yes	characteristic
106	C₂⁴×C₇	—	2	1	no	—
107	C₂×C₇×Q₈	—	2	2	no	—
108	C₂×C₇×Q₈	—	2	2	no	—
109	C₂×C₇×Q₈	—	2	2	no	—
110	C₂×C₇×Q₈	—	2	2	no	—
111	C₂×C₇×Q₈	—	2	2	no	—
112	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
113	C₂×C₇×D₄	—	2	1	no	—
114	C₂×C₇×D₄	—	2	1	no	—
115	C₂×C₇×D₄	—	2	1	no	—
116	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
117	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
118	C₂×C₇×D₄	#41	1	2	yes	—
119	C₂²×C₄×C₇	—	2	2	no	—
120	C₂²×C₄×C₇	—	2	1	no	—
121	C₂²×C₄×C₇	—	2	1	no	—
122	C₂²×C₄×C₇	—	2	4	no	—
123	C₂²×C₄×C₇	—	2	1	no	—
124	C₂²×C₄×C₇	—	2	2	no	—
125	C₂²×C₄×C₇	—	4	4	no	—
126	C₂²×C₄×C₇	—	4	4	no	—
127	C₂²×C₄×C₇	—	4	2	no	—
128	C₂²×C₄×C₇	—	4	2	no	—
129	C₂²×C₄×C₇	—	8	2	no	—
130	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
135	C₇×SD₁₆	—	2	16	no	—
136	C₂×C₇×C₈	—	2	2	no	—
137	C₂×C₇×C₈	—	2	1	no	—
138	C₂×C₇×C₈	—	2	2	no	—
139	C₂×C₇×C₈	—	2	1	no	—
140	C₂×C₇×C₈	—	2	1	no	—
141	C₂×C₇×C₈	—	2	1	no	—
142	C₂×C₇×C₈	—	2	1	no	—
143	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
144	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
145	C₇×C₄⋊C₄	—	2	4	no	—
146	C₇×C₄⋊C₄	—	2	4	no	—
147	C₇×C₄⋊C₄	—	2	2	no	—
148	C₇×C₄⋊C₄	—	2	2	no	—
149	C₇×C₄⋊C₄	—	2	1	no	—
150	C₇×C₄⋊C₄	—	2	1	no	—
151	C₇×C₄⋊C₄	—	2	2	no	—
152	C₇×C₄⋊C₄	—	2	1	no	—
153	C₇×C₄⋊C₄	—	2	2	no	—
154	C₇×C₄⋊C₄	—	2	1	no	—
155	C₇×C₄⋊C₄	—	4	4	no	—
156	C₇×C₄⋊C₄	—	4	4	no	—
157	C₇×C₄⋊C₄	—	4	2	no	—
158	C₇×C₄⋊C₄	—	8	1	no	—
159	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
160	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
161	C₇×C₂²⋊C₄	—	2	4	no	—
162	C₇×C₂²⋊C₄	—	4	4	no	—
163	C₄²×C₇	—	2	2	no	—
164	C₄²×C₇	—	2	2	no	—
165	C₄²×C₇	—	4	4	no	—
166	C₄²×C₇	—	4	2	no	—
167	C₄²×C₇	C₂²×C₄	1	2	yes	—
168	C₄²×C₇	#41	1	2	yes	—
169	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
171	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
172	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
173	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	—	2	1	no	—
175	#461	C₄×C₇	1	8	yes	complemented
176	#440	—	2	8	no	—
177	C₂×#107	C₂²×C₇	1	2	yes	—
178	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
179	C₂×#103	C₂²×C₇	1	2	yes	—
180	C₂×#103	C₂²×C₇	1	2	yes	—
181	C₂×#103	C₂²×C₇	1	2	yes	—
182	C₂×#102	C₂²×C₇	1	1	yes	characteristic
183	C₂×#102	C₂²×C₇	1	1	yes	characteristic
184	C₂×#97	C₂²×C₇	1	2	yes	—
185	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
186	#391	C₂²×C₇	1	2	yes	—
187	#389	—	2	1	no	—
188	#389	C₂²×C₇	1	2	yes	—
189	#387	—	2	2	no	—
190	#385	—	2	2	no	—
191	#384	C₂²×C₇	1	1	yes	characteristic
192	#380	—	2	1	no	—
193	C₄×C₄⋊C₄	—	2	1	no	—
194	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
195	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
196	#340	C₂²×C₇	1	1	yes	characteristic
197	#340	C₂²×C₇	1	2	yes	—
198	#340	C₂²×C₇	1	1	yes	characteristic
199	#336	C₂²×C₇	1	2	yes	—
200	C₂³×C₇	—	2	2	no	—
201	C₂³×C₇	—	2	2	no	—
202	C₂³×C₇	—	2	2	no	—
203	C₂³×C₇	—	4	1	no	—
204	C₂³×C₇	—	4	1	no	—
205	C₂³×C₇	—	4	1	no	—
206	C₂³×C₇	—	4	1	no	—
207	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
208	C₇×Q₈	—	2	4	no	—
209	C₇×Q₈	—	2	4	no	—
210	C₇×Q₈	—	4	2	no	—
211	C₇×Q₈	—	4	2	no	—
212	C₇×D₄	—	2	2	no	—
213	C₇×D₄	—	2	2	no	—
214	C₇×D₄	—	2	2	no	—
215	C₇×D₄	C₄×D₄	1	4	yes	—
216	C₂×C₄×C₇	—	2	1	no	—
217	C₂×C₄×C₇	—	2	1	no	—
218	C₂×C₄×C₇	—	2	2	no	—
219	C₂×C₄×C₇	—	2	4	no	—
220	C₂×C₄×C₇	—	2	2	no	—
221	C₂×C₄×C₇	—	2	2	no	—
222	C₂×C₄×C₇	—	2	2	no	—
223	C₂×C₄×C₇	—	2	2	no	—
224	C₂×C₄×C₇	—	2	2	no	—
225	C₂×C₄×C₇	—	2	2	no	—
226	C₂×C₄×C₇	—	2	2	no	—
227	C₂×C₄×C₇	—	4	2	no	—
228	C₂×C₄×C₇	—	4	2	no	—
229	C₂×C₄×C₇	—	4	1	no	—
230	C₂×C₄×C₇	—	4	1	no	—
231	C₂×C₄×C₇	—	4	1	no	—
232	C₂×C₄×C₇	—	4	1	no	—
233	C₂×C₄×C₇	—	4	8	no	—
234	C₂×C₄×C₇	—	4	2	no	—
235	C₂×C₄×C₇	—	4	1	no	—
236	C₂×C₄×C₇	—	4	1	no	—
237	C₂×C₄×C₇	—	4	2	no	—
238	C₂×C₄×C₇	—	4	2	no	—
239	C₂×C₄×C₇	—	4	1	no	—
240	C₂×C₄×C₇	—	8	8	no	—
241	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
242	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
243	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
244	C₂×C₄×C₇	#137	1	2	yes	—
245	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
251	C₇×C₈	—	2	4	no	—
252	C₇×C₈	—	2	2	no	—
253	C₇×C₈	—	4	1	no	—
254	C₂²×Q₈	C₂³×C₇	1	2	yes	—
255	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
256	C₂³×C₄	—	2	1	no	—
257	C₂³×C₄	—	2	2	no	—
258	C₂×SD₁₆	—	2	4	no	—
259	C₂×SD₁₆	—	2	4	no	—
260	C₂×SD₁₆	C₂×C₄×C₇	1	8	yes	—
261	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₇	1	2	yes	—
263	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
264	C₄⋊Q₈	—	2	2	no	—
265	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	complemented
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	4	2	no	—
268	C₄×D₄	—	2	4	no	—
269	C₄×D₄	—	2	2	no	—
270	C₄×D₄	—	4	2	no	—
271	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	4	4	no	—
279	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
285	C₂×C₂²⋊C₄	—	2	1	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₄²	—	2	2	no	—
288	C₂×C₄²	—	2	1	no	—
289	C₂×C₄²	C₂³×C₇	1	2	yes	—
290	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
291	#107	—	2	4	no	—
292	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
293	#103	—	2	4	no	—
294	#103	—	4	2	no	—
295	#103	C₂×C₄×C₇	1	8	yes	complemented
296	#102	—	2	2	no	—
297	#102	C₇×D₄	1	4	yes	—
298	#97	—	2	4	no	—
299	C₄×C₈	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	1	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	C₇×D₄	1	2	yes	—
305	C₂²×C₇	—	2	4	no	—
306	C₂²×C₇	—	4	2	no	—
307	C₂²×C₇	—	4	2	no	—
308	C₂²×C₇	—	4	2	no	—
309	C₂²×C₇	—	4	2	no	—
310	C₂²×C₇	—	4	2	no	—
311	C₂²×C₇	—	4	2	no	—
312	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
313	C₂²×C₇	D₄²	1	1	yes	characteristic
314	C₂²×C₇	#548	1	1	yes	characteristic
315	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂²×C₇	C₂×#137	1	1	yes	characteristic
317	C₂²×C₇	#577	1	1	yes	characteristic
318	C₂²×C₇	#578	1	1	yes	characteristic
319	C₄×C₇	—	2	2	no	—
320	C₄×C₇	—	4	4	no	—
321	C₄×C₇	—	4	2	no	—
322	C₄×C₇	—	4	2	no	—
323	C₄×C₇	—	4	2	no	—
324	C₄×C₇	—	4	2	no	—
325	C₄×C₇	—	8	1	no	—
326	C₄×C₇	—	8	2	no	—
327	C₄×C₇	#440	1	4	yes	complemented
328	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
329	C₄×C₇	#520	1	1	yes	characteristic
330	C₄×C₇	#547	1	1	yes	characteristic
331	C₄×C₇	#553	1	1	yes	characteristic
332	C₂⁴	—	2	1	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
343	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
344	C₂×D₄	C₇×#41	1	2	yes	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	—	4	4	no	—
353	C₂²×C₄	—	4	2	no	—
354	C₂²×C₄	—	4	2	no	—
355	C₂²×C₄	—	8	2	no	—
356	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
357	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
358	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
359	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
360	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
361	SD₁₆	—	2	16	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
370	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
371	C₄⋊C₄	—	2	4	no	—
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	4	4	no	—
382	C₄⋊C₄	—	4	4	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	8	1	no	—
385	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
386	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
387	C₂²⋊C₄	—	2	4	no	—
388	C₂²⋊C₄	—	4	4	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	4	4	no	—
392	C₄²	—	4	2	no	—
393	C₄²	C₂²×C₄×C₇	1	2	yes	—
394	C₄²	C₇×#41	1	2	yes	—
395	C₂×C₇	—	4	4	no	—
396	C₂×C₇	#2361	1	1	yes	characteristic
397	C₂×C₇	C₂×#440	1	1	yes	characteristic
398	C₂×C₇	#2947	1	1	yes	characteristic
399	C₂×C₇	#3272	1	1	yes	characteristic
400	C₂×C₇	#3276	1	1	yes	characteristic
401	C₂×C₇	#3367	1	1	yes	characteristic
402	C₂×C₇	#3372	1	1	yes	characteristic
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	4	1	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	1	no	—
410	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
411	Q₈	—	2	4	no	—
412	Q₈	—	2	4	no	—
413	Q₈	—	4	2	no	—
414	Q₈	—	4	2	no	—
415	D₄	—	2	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	2	2	no	—
418	D₄	C₄×C₇×D₄	1	4	yes	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	8	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	8	8	no	—
444	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
445	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
446	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
447	C₂×C₄	C₇×#137	1	2	yes	—
448	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
449	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
450	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
451	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
452	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
453	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
454	C₈	—	2	4	no	—
455	C₈	—	2	2	no	—
456	C₈	—	4	1	no	—
457	C₇	#23002	1	1	yes	characteristic, direct product factor, Sylow
458	C₂²	—	2	4	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
466	C₂²	C₇×D₄²	1	1	yes	characteristic
467	C₂²	C₇×#548	1	1	yes	characteristic
468	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
469	C₂²	C₂×C₇×#137	1	1	yes	characteristic
470	C₂²	C₇×#577	1	1	yes	characteristic
471	C₂²	C₇×#578	1	1	yes	characteristic
472	C₄	—	2	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	4	no	—
478	C₄	—	8	2	no	—
479	C₄	—	8	1	no	—
480	C₄	C₇×#440	1	4	yes	complemented
481	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
482	C₄	C₇×#520	1	1	yes	characteristic
483	C₄	C₇×#547	1	1	yes	characteristic
484	C₄	C₇×#553	1	1	yes	characteristic
485	C₂	—	4	4	no	—
486	C₂	C₇×#2361	1	1	yes	characteristic
487	C₂	C₂×C₇×#440	1	1	yes	characteristic
488	C₂	C₇×#2947	1	1	yes	characteristic
489	C₂	C₇×#3272	1	1	yes	characteristic
490	C₂	C₇×#3276	1	1	yes	characteristic
491	C₂	C₇×#3367	1	1	yes	characteristic
492	C₂	C₇×#3372	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	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2365	C₂	semidirect
5. C₇×#1956	C₂	semidirect
6. C₇×#1947	C₂	semidirect
7. C₇×#1935	C₂	semidirect
9. C₇×#1863	C₂	semidirect
10. C₇×#1770	C₂	semidirect
18. C₇×#461	C₄	semidirect
43. #23002	C₇	direct (N × H)
55. C₇×C₄⋊Q₈	C₂×C₄	semidirect
82. C₇×C₄⋊C₈	C₂×C₄	semidirect
85. C₇×#103	C₂×C₄	semidirect
97. #2365	C₂×C₇	semidirect
99. #1956	C₂×C₇	semidirect
100. #1947	C₂×C₇	semidirect
101. #1935	C₂×C₇	semidirect
103. #1863	C₂×C₇	semidirect
104. #1770	C₂×C₇	semidirect
175. #461	C₄×C₇	semidirect
265. C₄⋊Q₈	C₂×C₄×C₇	semidirect
292. C₄⋊C₈	C₂×C₄×C₇	semidirect
295. #103	C₂×C₄×C₇	semidirect
327. C₄×C₇	#440	semidirect
457. C₇	#23002	direct (N × H)
480. C₄	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×#461	2
C₂×C₇×#440	2
C₇×#2365	2
C₇×D₄×C₄⋊C₄	2
C₇×#1956	2
C₇×#1947	2
C₇×#1935	2
C₇×#1877	2
C₇×#1863	2
C₇×#1770	2
C₄×C₇×#102	2
#23002	7

Subgroups — C7×#23002 #13200218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23002 #13200218