Subgroups — C₇×#24325 #13201541

Order 1792.

960 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3504	C₂	1	1	yes	maximal, characteristic
2	C₇×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×#137	C₂	1	1	yes	maximal, characteristic
4	C₇×#2565	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2061	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2060	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2028	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2028	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1998	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1986	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1882	C₂	1	1	yes	maximal, characteristic
12	C₇×#1882	C₂	1	1	yes	maximal, characteristic
13	C₇×#1860	C₂	1	1	yes	maximal, characteristic
14	C₇×#1859	C₂	1	1	yes	maximal, characteristic
15	C₇×#1806	C₂	1	1	yes	maximal, characteristic
16	C₂×C₇×#137	C₂²	1	1	yes	characteristic
17	C₇×#557	—	2	1	no	—
18	C₇×#557	—	2	1	no	—
19	C₇×#554	—	2	1	no	—
20	C₇×#554	—	2	1	no	—
21	C₇×#554	—	2	2	no	—
22	C₇×#554	—	2	2	no	—
23	C₇×#544	—	2	1	no	—
24	C₇×#544	—	2	1	no	—
25	C₇×#544	C₂²	1	1	yes	characteristic
26	C₇×#544	C₂²	1	1	yes	characteristic
27	C₇×#533	—	2	1	no	—
28	C₇×#533	C₂²	1	1	yes	characteristic
29	C₂×C₇×#125	—	2	1	no	—
30	C₂×C₇×#125	C₂²	1	1	yes	characteristic
31	C₇×#520	C₂²	1	1	yes	characteristic
32	C₇×#520	C₂²	1	1	yes	characteristic
33	C₄×C₇×#41	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
35	C₇×#479	—	2	1	no	—
36	C₇×#479	—	2	1	no	—
37	C₇×#477	—	2	1	no	—
38	C₇×#475	—	2	1	no	—
39	C₇×#441	—	2	1	no	—
40	C₇×#440	—	2	1	no	—
41	C₄×C₇×Q₁₆	—	2	1	no	—
42	C₄×C₇×SD₁₆	—	2	1	no	—
43	C₇×#428	C₂²	1	1	yes	characteristic
44	C₇×#428	C₂²	1	1	yes	characteristic
45	C₇×#428	C₂²	1	1	yes	characteristic
46	C₇×#423	C₂²	1	1	yes	characteristic
47	C₂×C₇×#104	C₂²	1	1	yes	characteristic
48	C₂×C₇×#104	C₂²	1	1	yes	characteristic
49	C₇×#419	C₂²	1	1	yes	characteristic
50	C₇×#419	C₂²	1	1	yes	characteristic
51	C₇×#419	C₂²	1	1	yes	characteristic
52	C₇×#417	C₂²	1	1	yes	characteristic
53	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
54	C₇×#398	—	2	2	no	—
55	C₇×#391	C₂²	1	1	yes	characteristic
56	C₇×#391	C₂²	1	1	yes	characteristic
57	C₇×#387	—	2	2	no	—
58	C₇×#384	C₂²	1	1	yes	characteristic
59	C₇×#382	C₂²	1	1	yes	characteristic
60	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
61	C₇×#343	C₂²	1	1	yes	characteristic
62	C₇×#343	C₂²	1	1	yes	characteristic
63	C₇×#343	C₂²	1	1	yes	characteristic
64	C₇×#343	C₂²	1	1	yes	characteristic
65	C₇×#339	C₂²	1	1	yes	characteristic
66	C₇×#339	C₂²	1	1	yes	characteristic
67	C₇×#339	C₂²	1	1	yes	characteristic
68	C₇×#339	C₂²	1	1	yes	characteristic
69	C₇×#339	C₂²	1	1	yes	characteristic
70	C₇×#339	C₂²	1	1	yes	characteristic
71	#24325	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇×#41	C₂³	1	1	yes	characteristic
73	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
74	C₇×#137	—	2	2	no	—
75	C₇×#137	Q₈	1	4	yes	complemented
76	C₂×C₇×Q₁₆	—	2	1	no	—
77	C₂×C₇×SD₁₆	—	2	1	no	—
78	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
82	C₇×C₄⋊Q₈	—	2	2	no	—
83	C₇×C₄⋊Q₈	—	2	2	no	—
84	C₇×C₄⋊Q₈	—	2	1	no	—
85	C₇×C₄⋊Q₈	—	2	1	no	—
86	C₇×C₄⋊Q₈	—	2	1	no	—
87	C₇×C₄⋊Q₈	—	2	1	no	—
88	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
89	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
90	C₇×#125	—	2	2	no	—
91	C₇×#125	—	2	2	no	—
92	C₇×#125	—	2	2	no	—
93	C₇×#125	—	2	2	no	—
94	C₇×#125	—	2	1	no	—
95	C₇×#125	—	2	1	no	—
96	C₇×#125	—	2	1	no	—
97	C₇×#125	—	2	1	no	—
98	C₇×#125	—	2	1	no	—
99	C₇×#125	—	2	1	no	—
100	C₇×#125	—	2	1	no	—
101	C₇×#125	—	2	1	no	—
102	C₇×#125	D₄	1	1	yes	characteristic
103	C₇×#125	D₄	1	1	yes	characteristic
104	C₇×C₂²⋊Q₈	—	2	2	no	—
105	C₇×C₂²⋊Q₈	—	2	2	no	—
106	C₇×C₂²⋊Q₈	—	2	2	no	—
107	C₇×C₂²⋊Q₈	—	2	2	no	—
108	C₇×C₂²⋊Q₈	—	2	2	no	—
109	C₇×C₂²⋊Q₈	—	2	2	no	—
110	C₇×C₂²⋊Q₈	—	2	1	no	—
111	C₇×C₂²⋊Q₈	—	2	1	no	—
112	C₇×C₂²⋊Q₈	—	2	2	no	—
113	C₇×C₂²⋊Q₈	—	2	2	no	—
114	C₄×C₇×Q₈	—	2	2	no	—
115	C₄×C₇×Q₈	—	2	2	no	—
116	C₄×C₇×Q₈	—	2	1	no	—
117	C₄×C₇×Q₈	—	2	1	no	—
118	C₄×C₇×Q₈	—	2	1	no	—
119	C₄×C₇×Q₈	—	2	1	no	—
120	C₄×C₇×Q₈	—	2	1	no	—
121	C₄×C₇×Q₈	—	2	1	no	—
122	C₄×C₇×Q₈	—	2	1	no	—
123	C₄×C₇×Q₈	—	2	1	no	—
124	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
125	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
126	C₄×C₇×D₄	—	2	1	no	—
127	C₄×C₇×D₄	—	2	1	no	—
128	C₄×C₇×D₄	—	2	1	no	—
129	C₄×C₇×D₄	—	2	1	no	—
130	C₄×C₇×D₄	—	2	2	no	—
131	C₄×C₇×D₄	—	2	2	no	—
132	C₄×C₇×D₄	D₄	1	1	yes	characteristic
133	C₄×C₇×D₄	D₄	1	1	yes	characteristic
134	C₇×#117	—	2	1	no	—
135	C₇×#117	—	2	1	no	—
136	C₇×#117	—	2	1	no	—
137	C₇×#117	C₂³	1	1	yes	characteristic
138	C₇×#117	C₂³	1	1	yes	characteristic
139	C₇×#117	C₂³	1	1	yes	characteristic
140	C₂×C₇×C₄⋊C₄	—	2	2	no	—
141	C₂×C₇×C₄⋊C₄	—	2	2	no	—
142	C₂×C₇×C₄⋊C₄	—	2	1	no	—
143	C₂×C₇×C₄⋊C₄	—	2	1	no	—
144	C₂×C₇×C₄⋊C₄	—	2	1	no	—
145	C₂×C₇×C₄⋊C₄	—	2	1	no	—
146	C₂×C₇×C₄⋊C₄	—	2	2	no	—
147	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
148	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
149	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
150	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
151	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₇	—	2	1	no	—
153	C₂×C₄²×C₇	—	2	1	no	—
154	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
155	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
156	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
157	C₇×#107	—	2	1	no	—
158	C₇×#107	—	2	1	no	—
159	C₇×#107	—	2	1	no	—
160	C₇×#106	—	2	1	no	—
161	C₇×#106	—	2	1	no	—
162	C₇×#106	—	2	1	no	—
163	C₇×C₄⋊C₈	—	2	1	no	—
164	C₇×C₄⋊C₈	—	2	1	no	—
165	C₇×#104	—	2	2	no	—
166	C₇×#104	—	2	2	no	—
167	C₇×#103	—	2	1	no	—
168	C₇×#103	—	2	1	no	—
169	C₇×#103	—	2	1	no	—
170	C₇×#103	—	2	1	no	—
171	C₇×#103	—	2	1	no	—
172	C₇×#103	—	2	1	no	—
173	C₇×#103	—	2	1	no	—
174	C₇×#102	—	2	1	no	—
175	C₇×#97	—	2	1	no	—
176	C₄×C₇×C₈	—	2	1	no	—
177	C₇×#95	—	2	1	no	—
178	C₇×#95	—	2	2	no	—
179	C₇×#95	—	2	2	no	—
180	C₇×#95	—	2	2	no	—
181	#3504	C₂×C₇	1	1	yes	characteristic
182	#3078	C₂×C₇	1	1	yes	characteristic, complemented
183	C₄×#137	C₂×C₇	1	1	yes	characteristic
184	#2565	C₂×C₇	1	1	yes	characteristic, complemented
185	#2061	C₂×C₇	1	1	yes	characteristic, complemented
186	#2060	C₂×C₇	1	1	yes	characteristic, complemented
187	#2028	C₂×C₇	1	1	yes	characteristic, complemented
188	#2028	C₂×C₇	1	1	yes	characteristic, complemented
189	#1998	C₂×C₇	1	1	yes	characteristic, complemented
190	#1986	C₂×C₇	1	1	yes	characteristic, complemented
191	#1882	C₂×C₇	1	1	yes	characteristic
192	#1882	C₂×C₇	1	1	yes	characteristic
193	#1860	C₂×C₇	1	1	yes	characteristic
194	#1859	C₂×C₇	1	1	yes	characteristic
195	#1806	C₂×C₇	1	1	yes	characteristic
196	C₇×#41	—	2	1	no	—
197	C₇×#41	—	2	1	no	—
198	C₇×#41	C₂×Q₈	1	2	yes	—
199	C₇×#41	#41	1	2	yes	—
200	C₂×C₇×Q₈	—	2	2	no	—
201	C₂×C₇×Q₈	—	2	2	no	—
202	C₂×C₇×Q₈	—	2	1	no	—
203	C₂×C₇×Q₈	—	2	1	no	—
204	C₂×C₇×Q₈	—	2	1	no	—
205	C₂×C₇×Q₈	—	2	1	no	—
206	C₂×C₇×Q₈	—	2	1	no	—
207	C₂×C₇×Q₈	—	2	1	no	—
208	C₂×C₇×Q₈	—	2	1	no	—
209	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
210	C₂×C₇×Q₈	C₂×Q₈	1	2	yes	—
211	C₂×C₇×D₄	—	2	1	no	—
212	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₇	—	2	1	no	—
214	C₂²×C₄×C₇	—	2	1	no	—
215	C₂²×C₄×C₇	—	2	1	no	—
216	C₂²×C₄×C₇	—	2	2	no	—
217	C₂²×C₄×C₇	—	2	2	no	—
218	C₂²×C₄×C₇	—	2	1	no	—
219	C₂²×C₄×C₇	—	2	1	no	—
220	C₂²×C₄×C₇	—	2	1	no	—
221	C₂²×C₄×C₇	—	2	1	no	—
222	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
223	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
224	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
225	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
226	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
227	C₇×Q₁₆	—	2	4	no	—
228	C₇×SD₁₆	—	2	4	no	—
229	C₇×M₁₆	—	2	1	no	—
230	C₇×M₁₆	—	2	1	no	—
231	C₇×M₁₆	—	2	1	no	—
232	C₇×M₁₆	—	2	1	no	—
233	C₇×M₁₆	—	2	1	no	—
234	C₇×M₁₆	—	2	1	no	—
235	C₇×M₁₆	—	2	1	no	—
236	C₇×M₁₆	C₂×Q₈	1	2	yes	—
237	C₂×C₇×C₈	—	2	1	no	—
238	C₂×C₇×C₈	—	2	1	no	—
239	C₂×C₇×C₈	—	2	1	no	—
240	C₂×C₇×C₈	—	2	1	no	—
241	C₇×C₄⋊C₄	—	2	2	no	—
242	C₇×C₄⋊C₄	—	2	2	no	—
243	C₇×C₄⋊C₄	—	2	2	no	—
244	C₇×C₄⋊C₄	—	2	2	no	—
245	C₇×C₄⋊C₄	—	2	2	no	—
246	C₇×C₄⋊C₄	—	2	2	no	—
247	C₇×C₄⋊C₄	—	2	2	no	—
248	C₇×C₄⋊C₄	—	2	2	no	—
249	C₇×C₄⋊C₄	—	2	1	no	—
250	C₇×C₄⋊C₄	—	2	1	no	—
251	C₇×C₄⋊C₄	—	2	2	no	—
252	C₇×C₄⋊C₄	—	2	2	no	—
253	C₇×C₄⋊C₄	—	2	2	no	—
254	C₇×C₄⋊C₄	—	2	2	no	—
255	C₇×C₄⋊C₄	—	2	1	no	—
256	C₇×C₄⋊C₄	—	2	1	no	—
257	C₇×C₄⋊C₄	—	2	1	no	—
258	C₇×C₄⋊C₄	—	2	1	no	—
259	C₇×C₄⋊C₄	—	2	1	no	—
260	C₇×C₄⋊C₄	—	2	1	no	—
261	C₇×C₄⋊C₄	—	2	1	no	—
262	C₇×C₄⋊C₄	—	2	1	no	—
263	C₇×C₄⋊C₄	—	2	1	no	—
264	C₇×C₄⋊C₄	—	2	1	no	—
265	C₇×C₄⋊C₄	—	2	1	no	—
266	C₇×C₄⋊C₄	—	2	1	no	—
267	C₇×C₄⋊C₄	—	4	2	no	—
268	C₇×C₄⋊C₄	—	4	4	no	—
269	C₇×C₄⋊C₄	—	4	1	no	—
270	C₇×C₄⋊C₄	—	4	1	no	—
271	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
272	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
273	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
274	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
275	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
276	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
277	C₇×C₂²⋊C₄	—	2	2	no	—
278	C₇×C₂²⋊C₄	—	2	2	no	—
279	C₇×C₂²⋊C₄	—	2	1	no	—
280	C₇×C₂²⋊C₄	—	2	1	no	—
281	C₇×C₂²⋊C₄	—	2	1	no	—
282	C₇×C₂²⋊C₄	—	2	1	no	—
283	C₇×C₂²⋊C₄	—	2	1	no	—
284	C₄²×C₇	—	2	2	no	—
285	C₄²×C₇	—	2	2	no	—
286	C₄²×C₇	—	2	1	no	—
287	C₄²×C₇	—	2	1	no	—
288	C₄²×C₇	—	2	1	no	—
289	C₄²×C₇	—	2	1	no	—
290	C₄²×C₇	—	2	1	no	—
291	C₄²×C₇	—	2	1	no	—
292	C₄²×C₇	—	2	1	no	—
293	C₄²×C₇	—	2	1	no	—
294	C₄²×C₇	—	2	1	no	—
295	C₄²×C₇	—	4	1	no	—
296	C₄²×C₇	—	4	1	no	—
297	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
298	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
299	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
300	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
301	C₂×#137	C₂²×C₇	1	1	yes	characteristic
302	#557	—	2	1	no	—
303	#557	—	2	1	no	—
304	#554	—	2	1	no	—
305	#554	—	2	1	no	—
306	#554	—	2	2	no	—
307	#554	—	2	2	no	—
308	#544	—	2	1	no	—
309	#544	—	2	1	no	—
310	#544	C₂²×C₇	1	1	yes	characteristic
311	#544	C₂²×C₇	1	1	yes	characteristic
312	#533	—	2	1	no	—
313	#533	C₂²×C₇	1	1	yes	characteristic
314	C₂×#125	—	2	1	no	—
315	C₂×#125	C₂²×C₇	1	1	yes	characteristic
316	#520	C₂²×C₇	1	1	yes	characteristic
317	#520	C₂²×C₇	1	1	yes	characteristic
318	C₄×#41	C₂²×C₇	1	1	yes	characteristic
319	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
320	#479	—	2	1	no	—
321	#479	—	2	1	no	—
322	#477	—	2	1	no	—
323	#475	—	2	1	no	—
324	#441	—	2	1	no	—
325	#440	—	2	1	no	—
326	C₄×Q₁₆	—	2	1	no	—
327	C₄×SD₁₆	—	2	1	no	—
328	#428	C₂²×C₇	1	1	yes	characteristic
329	#428	C₂²×C₇	1	1	yes	characteristic
330	#428	C₂²×C₇	1	1	yes	characteristic
331	#423	C₂²×C₇	1	1	yes	characteristic
332	C₂×#104	C₂²×C₇	1	1	yes	characteristic
333	C₂×#104	C₂²×C₇	1	1	yes	characteristic
334	#419	C₂²×C₇	1	1	yes	characteristic
335	#419	C₂²×C₇	1	1	yes	characteristic
336	#419	C₂²×C₇	1	1	yes	characteristic
337	#417	C₂²×C₇	1	1	yes	characteristic
338	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
339	#398	—	2	2	no	—
340	#391	C₂²×C₇	1	1	yes	characteristic
341	#391	C₂²×C₇	1	1	yes	characteristic
342	#387	—	2	2	no	—
343	#384	C₂²×C₇	1	1	yes	characteristic
344	#382	C₂²×C₇	1	1	yes	characteristic
345	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
346	#343	C₂²×C₇	1	1	yes	characteristic
347	#343	C₂²×C₇	1	1	yes	characteristic
348	#343	C₂²×C₇	1	1	yes	characteristic
349	#343	C₂²×C₇	1	1	yes	characteristic
350	#339	C₂²×C₇	1	1	yes	characteristic
351	#339	C₂²×C₇	1	1	yes	characteristic
352	#339	C₂²×C₇	1	1	yes	characteristic
353	#339	C₂²×C₇	1	1	yes	characteristic
354	#339	C₂²×C₇	1	1	yes	characteristic
355	#339	C₂²×C₇	1	1	yes	characteristic
356	C₂³×C₇	—	2	1	no	—
357	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
358	C₇×Q₈	—	2	1	no	—
359	C₇×Q₈	—	2	2	no	—
360	C₇×Q₈	—	2	2	no	—
361	C₇×Q₈	—	4	1	no	—
362	C₇×Q₈	—	4	1	no	—
363	C₇×Q₈	—	8	1	no	—
364	C₇×Q₈	—	8	1	no	—
365	C₇×Q₈	—	8	2	no	—
366	C₇×Q₈	—	8	2	no	—
367	C₇×Q₈	C₂²⋊Q₈	1	2	yes	—
368	C₇×D₄	—	2	2	no	—
369	C₇×D₄	—	2	2	no	—
370	C₇×D₄	—	2	1	no	—
371	C₇×D₄	C₂²⋊Q₈	1	2	yes	—
372	C₂×C₄×C₇	—	2	1	no	—
373	C₂×C₄×C₇	—	2	2	no	—
374	C₂×C₄×C₇	—	2	2	no	—
375	C₂×C₄×C₇	—	2	2	no	—
376	C₂×C₄×C₇	—	2	1	no	—
377	C₂×C₄×C₇	—	2	1	no	—
378	C₂×C₄×C₇	—	2	1	no	—
379	C₂×C₄×C₇	—	2	1	no	—
380	C₂×C₄×C₇	—	2	2	no	—
381	C₂×C₄×C₇	—	2	2	no	—
382	C₂×C₄×C₇	—	2	1	no	—
383	C₂×C₄×C₇	—	2	1	no	—
384	C₂×C₄×C₇	—	2	2	no	—
385	C₂×C₄×C₇	—	4	1	no	—
386	C₂×C₄×C₇	—	4	1	no	—
387	C₂×C₄×C₇	—	4	1	no	—
388	C₂×C₄×C₇	—	4	1	no	—
389	C₂×C₄×C₇	—	4	1	no	—
390	C₂×C₄×C₇	—	4	1	no	—
391	C₂×C₄×C₇	—	4	2	no	—
392	C₂×C₄×C₇	—	4	2	no	—
393	C₂×C₄×C₇	—	4	1	no	—
394	C₂×C₄×C₇	—	4	2	no	—
395	C₂×C₄×C₇	—	4	1	no	—
396	C₂×C₄×C₇	—	4	1	no	—
397	C₂×C₄×C₇	—	4	1	no	—
398	C₂×C₄×C₇	—	8	1	no	—
399	C₂×C₄×C₇	—	8	1	no	—
400	C₂×C₄×C₇	—	8	2	no	—
401	C₂×C₄×C₇	—	8	2	no	—
402	C₂×C₄×C₇	#120	1	1	yes	characteristic
403	C₂×C₄×C₇	#120	1	1	yes	characteristic
404	C₂×C₄×C₇	#120	1	1	yes	characteristic
405	C₂×C₄×C₇	#120	1	1	yes	characteristic
406	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
407	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
408	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
409	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
410	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
411	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
412	C₇×C₈	—	2	2	no	—
413	C₇×C₈	—	4	1	no	—
414	C₇×C₈	—	4	1	no	—
415	C₇×C₈	—	4	1	no	—
416	C₂×#41	C₂³×C₇	1	1	yes	characteristic
417	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
418	#137	—	2	2	no	—
419	#137	C₇×Q₈	1	4	yes	complemented
420	C₂×Q₁₆	—	2	1	no	—
421	C₂×SD₁₆	—	2	1	no	—
422	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
423	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
424	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
425	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
426	C₄⋊Q₈	—	2	2	no	—
427	C₄⋊Q₈	—	2	2	no	—
428	C₄⋊Q₈	—	2	1	no	—
429	C₄⋊Q₈	—	2	1	no	—
430	C₄⋊Q₈	—	2	1	no	—
431	C₄⋊Q₈	—	2	1	no	—
432	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
433	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
434	#125	—	2	2	no	—
435	#125	—	2	2	no	—
436	#125	—	2	2	no	—
437	#125	—	2	2	no	—
438	#125	—	2	1	no	—
439	#125	—	2	1	no	—
440	#125	—	2	1	no	—
441	#125	—	2	1	no	—
442	#125	—	2	1	no	—
443	#125	—	2	1	no	—
444	#125	—	2	1	no	—
445	#125	—	2	1	no	—
446	#125	C₇×D₄	1	1	yes	characteristic
447	#125	C₇×D₄	1	1	yes	characteristic
448	C₂²⋊Q₈	—	2	2	no	—
449	C₂²⋊Q₈	—	2	2	no	—
450	C₂²⋊Q₈	—	2	2	no	—
451	C₂²⋊Q₈	—	2	2	no	—
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	—	2	2	no	—
454	C₂²⋊Q₈	—	2	1	no	—
455	C₂²⋊Q₈	—	2	1	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₄×Q₈	—	2	2	no	—
459	C₄×Q₈	—	2	2	no	—
460	C₄×Q₈	—	2	1	no	—
461	C₄×Q₈	—	2	1	no	—
462	C₄×Q₈	—	2	1	no	—
463	C₄×Q₈	—	2	1	no	—
464	C₄×Q₈	—	2	1	no	—
465	C₄×Q₈	—	2	1	no	—
466	C₄×Q₈	—	2	1	no	—
467	C₄×Q₈	—	2	1	no	—
468	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
469	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
470	C₄×D₄	—	2	1	no	—
471	C₄×D₄	—	2	1	no	—
472	C₄×D₄	—	2	1	no	—
473	C₄×D₄	—	2	1	no	—
474	C₄×D₄	—	2	2	no	—
475	C₄×D₄	—	2	2	no	—
476	C₄×D₄	C₇×D₄	1	1	yes	characteristic
477	C₄×D₄	C₇×D₄	1	1	yes	characteristic
478	#117	—	2	1	no	—
479	#117	—	2	1	no	—
480	#117	—	2	1	no	—
481	#117	C₂³×C₇	1	1	yes	characteristic
482	#117	C₂³×C₇	1	1	yes	characteristic
483	#117	C₂³×C₇	1	1	yes	characteristic
484	C₂×C₄⋊C₄	—	2	2	no	—
485	C₂×C₄⋊C₄	—	2	2	no	—
486	C₂×C₄⋊C₄	—	2	1	no	—
487	C₂×C₄⋊C₄	—	2	1	no	—
488	C₂×C₄⋊C₄	—	2	1	no	—
489	C₂×C₄⋊C₄	—	2	1	no	—
490	C₂×C₄⋊C₄	—	2	2	no	—
491	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
492	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
493	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
494	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
495	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
496	C₂×C₄²	—	2	1	no	—
497	C₂×C₄²	—	2	1	no	—
498	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
499	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
500	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
501	#107	—	2	1	no	—
502	#107	—	2	1	no	—
503	#107	—	2	1	no	—
504	#106	—	2	1	no	—
505	#106	—	2	1	no	—
506	#106	—	2	1	no	—
507	C₄⋊C₈	—	2	1	no	—
508	C₄⋊C₈	—	2	1	no	—
509	#104	—	2	2	no	—
510	#104	—	2	2	no	—
511	#103	—	2	1	no	—
512	#103	—	2	1	no	—
513	#103	—	2	1	no	—
514	#103	—	2	1	no	—
515	#103	—	2	1	no	—
516	#103	—	2	1	no	—
517	#103	—	2	1	no	—
518	#102	—	2	1	no	—
519	#97	—	2	1	no	—
520	C₄×C₈	—	2	1	no	—
521	#95	—	2	1	no	—
522	#95	—	2	2	no	—
523	#95	—	2	2	no	—
524	#95	—	2	2	no	—
525	C₂²×C₇	—	2	1	no	—
526	C₂²×C₇	—	2	1	no	—
527	C₂²×C₇	—	2	2	no	—
528	C₂²×C₇	—	4	1	no	—
529	C₂²×C₇	—	4	1	no	—
530	C₂²×C₇	C₂×#120	1	1	yes	characteristic
531	C₂²×C₇	#548	1	1	yes	characteristic
532	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
533	C₄×C₇	—	2	2	no	—
534	C₄×C₇	—	2	1	no	—
535	C₄×C₇	—	2	1	no	—
536	C₄×C₇	—	2	1	no	—
537	C₄×C₇	—	2	1	no	—
538	C₄×C₇	—	4	2	no	—
539	C₄×C₇	—	4	2	no	—
540	C₄×C₇	—	4	2	no	—
541	C₄×C₇	—	4	1	no	—
542	C₄×C₇	—	4	1	no	—
543	C₄×C₇	—	4	1	no	—
544	C₄×C₇	—	4	2	no	—
545	C₄×C₇	—	4	1	no	—
546	C₄×C₇	—	4	1	no	—
547	C₄×C₇	—	4	1	no	—
548	C₄×C₇	—	8	1	no	—
549	C₄×C₇	—	8	1	no	—
550	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
551	C₄×C₇	#536	1	1	yes	characteristic
552	C₄×C₇	#548	1	1	yes	characteristic
553	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
554	#41	—	2	1	no	—
555	#41	—	2	1	no	—
556	#41	C₂×C₇×Q₈	1	2	yes	—
557	#41	C₇×#41	1	2	yes	—
558	C₂×Q₈	—	2	2	no	—
559	C₂×Q₈	—	2	2	no	—
560	C₂×Q₈	—	2	1	no	—
561	C₂×Q₈	—	2	1	no	—
562	C₂×Q₈	—	2	1	no	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	—	2	1	no	—
565	C₂×Q₈	—	2	1	no	—
566	C₂×Q₈	—	2	1	no	—
567	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
568	C₂×Q₈	C₂×C₇×Q₈	1	2	yes	—
569	C₂×D₄	—	2	1	no	—
570	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
571	C₂²×C₄	—	2	1	no	—
572	C₂²×C₄	—	2	1	no	—
573	C₂²×C₄	—	2	1	no	—
574	C₂²×C₄	—	2	2	no	—
575	C₂²×C₄	—	2	2	no	—
576	C₂²×C₄	—	2	1	no	—
577	C₂²×C₄	—	2	1	no	—
578	C₂²×C₄	—	2	1	no	—
579	C₂²×C₄	—	2	1	no	—
580	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
581	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
582	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
583	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
584	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
585	Q₁₆	—	2	4	no	—
586	SD₁₆	—	2	4	no	—
587	M₁₆	—	2	1	no	—
588	M₁₆	—	2	1	no	—
589	M₁₆	—	2	1	no	—
590	M₁₆	—	2	1	no	—
591	M₁₆	—	2	1	no	—
592	M₁₆	—	2	1	no	—
593	M₁₆	—	2	1	no	—
594	M₁₆	C₂×C₇×Q₈	1	2	yes	—
595	C₂×C₈	—	2	1	no	—
596	C₂×C₈	—	2	1	no	—
597	C₂×C₈	—	2	1	no	—
598	C₂×C₈	—	2	1	no	—
599	C₄⋊C₄	—	2	2	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	2	no	—
606	C₄⋊C₄	—	2	2	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	2	no	—
610	C₄⋊C₄	—	2	2	no	—
611	C₄⋊C₄	—	2	2	no	—
612	C₄⋊C₄	—	2	2	no	—
613	C₄⋊C₄	—	2	1	no	—
614	C₄⋊C₄	—	2	1	no	—
615	C₄⋊C₄	—	2	1	no	—
616	C₄⋊C₄	—	2	1	no	—
617	C₄⋊C₄	—	2	1	no	—
618	C₄⋊C₄	—	2	1	no	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	1	no	—
623	C₄⋊C₄	—	2	1	no	—
624	C₄⋊C₄	—	2	1	no	—
625	C₄⋊C₄	—	4	2	no	—
626	C₄⋊C₄	—	4	4	no	—
627	C₄⋊C₄	—	4	1	no	—
628	C₄⋊C₄	—	4	1	no	—
629	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
630	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
631	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
632	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
633	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
634	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
635	C₂²⋊C₄	—	2	2	no	—
636	C₂²⋊C₄	—	2	2	no	—
637	C₂²⋊C₄	—	2	1	no	—
638	C₂²⋊C₄	—	2	1	no	—
639	C₂²⋊C₄	—	2	1	no	—
640	C₂²⋊C₄	—	2	1	no	—
641	C₂²⋊C₄	—	2	1	no	—
642	C₄²	—	2	2	no	—
643	C₄²	—	2	2	no	—
644	C₄²	—	2	1	no	—
645	C₄²	—	2	1	no	—
646	C₄²	—	2	1	no	—
647	C₄²	—	2	1	no	—
648	C₄²	—	2	1	no	—
649	C₄²	—	2	1	no	—
650	C₄²	—	2	1	no	—
651	C₄²	—	2	1	no	—
652	C₄²	—	2	1	no	—
653	C₄²	—	4	1	no	—
654	C₄²	—	4	1	no	—
655	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
656	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
657	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
658	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
659	C₂×C₇	—	2	1	no	—
660	C₂×C₇	—	2	1	no	—
661	C₂×C₇	—	4	2	no	—
662	C₂×C₇	#2409	1	1	yes	characteristic
663	C₂×C₇	#3018	1	1	yes	characteristic
664	C₂×C₇	#3022	1	1	yes	characteristic
665	C₂³	—	2	1	no	—
666	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
667	Q₈	—	2	1	no	—
668	Q₈	—	2	2	no	—
669	Q₈	—	2	2	no	—
670	Q₈	—	4	1	no	—
671	Q₈	—	4	1	no	—
672	Q₈	—	8	1	no	—
673	Q₈	—	8	1	no	—
674	Q₈	—	8	2	no	—
675	Q₈	—	8	2	no	—
676	Q₈	C₇×C₂²⋊Q₈	1	2	yes	—
677	D₄	—	2	2	no	—
678	D₄	—	2	2	no	—
679	D₄	—	2	1	no	—
680	D₄	C₇×C₂²⋊Q₈	1	2	yes	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	2	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	2	1	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	2	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	4	1	no	—
695	C₂×C₄	—	4	1	no	—
696	C₂×C₄	—	4	1	no	—
697	C₂×C₄	—	4	1	no	—
698	C₂×C₄	—	4	1	no	—
699	C₂×C₄	—	4	1	no	—
700	C₂×C₄	—	4	2	no	—
701	C₂×C₄	—	4	2	no	—
702	C₂×C₄	—	4	1	no	—
703	C₂×C₄	—	4	2	no	—
704	C₂×C₄	—	4	1	no	—
705	C₂×C₄	—	4	1	no	—
706	C₂×C₄	—	4	1	no	—
707	C₂×C₄	—	8	1	no	—
708	C₂×C₄	—	8	1	no	—
709	C₂×C₄	—	8	2	no	—
710	C₂×C₄	—	8	2	no	—
711	C₂×C₄	C₇×#120	1	1	yes	characteristic
712	C₂×C₄	C₇×#120	1	1	yes	characteristic
713	C₂×C₄	C₇×#120	1	1	yes	characteristic
714	C₂×C₄	C₇×#120	1	1	yes	characteristic
715	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
716	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
717	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
718	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
719	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
720	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
721	C₈	—	2	2	no	—
722	C₈	—	4	1	no	—
723	C₈	—	4	1	no	—
724	C₈	—	4	1	no	—
725	C₇	#24325	1	1	yes	characteristic, direct product factor, Sylow
726	C₂²	—	2	1	no	—
727	C₂²	—	2	1	no	—
728	C₂²	—	2	2	no	—
729	C₂²	—	4	1	no	—
730	C₂²	—	4	1	no	—
731	C₂²	C₂×C₇×#120	1	1	yes	characteristic
732	C₂²	C₇×#548	1	1	yes	characteristic
733	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
734	C₄	—	2	2	no	—
735	C₄	—	2	1	no	—
736	C₄	—	2	1	no	—
737	C₄	—	2	1	no	—
738	C₄	—	2	1	no	—
739	C₄	—	4	2	no	—
740	C₄	—	4	2	no	—
741	C₄	—	4	2	no	—
742	C₄	—	4	1	no	—
743	C₄	—	4	1	no	—
744	C₄	—	4	1	no	—
745	C₄	—	4	2	no	—
746	C₄	—	4	1	no	—
747	C₄	—	4	1	no	—
748	C₄	—	4	1	no	—
749	C₄	—	8	1	no	—
750	C₄	—	8	1	no	—
751	C₄	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
752	C₄	C₇×#536	1	1	yes	characteristic
753	C₄	C₇×#548	1	1	yes	characteristic
754	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
755	C₂	—	2	1	no	—
756	C₂	—	2	1	no	—
757	C₂	—	4	2	no	—
758	C₂	C₇×#2409	1	1	yes	characteristic
759	C₂	C₇×#3018	1	1	yes	characteristic
760	C₂	C₇×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3078	C₂	semidirect
4. C₇×#2565	C₂	semidirect
5. C₇×#2061	C₂	semidirect
6. C₇×#2060	C₂	semidirect
7. C₇×#2028	C₂	semidirect
8. C₇×#2028	C₂	semidirect
9. C₇×#1998	C₂	semidirect
10. C₇×#1986	C₂	semidirect
71. #24325	C₇	direct (N × H)
75. C₇×#137	Q₈	semidirect
182. #3078	C₂×C₇	semidirect
184. #2565	C₂×C₇	semidirect
185. #2061	C₂×C₇	semidirect
186. #2060	C₂×C₇	semidirect
187. #2028	C₂×C₇	semidirect
188. #2028	C₂×C₇	semidirect
189. #1998	C₂×C₇	semidirect
190. #1986	C₂×C₇	semidirect
419. #137	C₇×Q₈	semidirect
725. C₇	#24325	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3504	2
C₇×#3078	2
C₄×C₇×#137	2
C₇×#2565	2
C₇×#2061	2
C₇×#2060	2
C₇×#2028	2
C₇×#1998	2
C₇×#1986	2
C₇×#1882	2
C₇×#1860	2
C₇×#1859	2
C₇×#1806	2
#24325	7

Subgroups — C7×#24325 #13201541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24325 #13201541