Subgroups — C₇×#60720 #13237936

Order 1792.

1846 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3510	C₂	1	1	yes	maximal, characteristic
2	C₇×#3509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3181	C₂	1	4	yes	maximal, complemented
4	C₇×#3178	C₂	1	8	yes	maximal, complemented
5	C₇×#3172	C₂	1	4	yes	maximal, complemented
6	C₇×#3162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#497	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3059	C₂	1	2	yes	maximal, complemented
11	C₇×#3057	C₂	1	2	yes	maximal
12	C₇×#2950	C₂	1	2	yes	maximal, complemented
13	C₇×#2949	C₂	1	2	yes	maximal, complemented
14	C₇×#2875	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
16	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₇×#131	C₂²	1	2	yes	—
18	C₇×#559	—	2	2	no	—
19	C₇×#557	—	2	2	no	—
20	C₇×#556	—	2	2	no	—
21	C₇×#556	—	2	2	no	—
22	C₇×#552	—	2	4	no	—
23	C₇×#551	—	2	4	no	—
24	C₇×#542	C₂²	1	4	yes	—
25	C₇×#541	C₂²	1	4	yes	complemented
26	C₇×#536	C₂²	1	2	yes	—
27	C₇×#536	C₂²	1	1	yes	characteristic
28	C₇×#535	C₂²	1	1	yes	characteristic
29	C₇×#533	C₂²	1	1	yes	characteristic
30	C₇×#533	C₂²	1	1	yes	characteristic
31	C₇×#533	C₂²	1	4	yes	complemented
32	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
33	C₇×#529	C₂²	1	4	yes	—
34	C₂×C₇×#124	C₂²	1	1	yes	characteristic
35	C₇×#519	C₂²	1	2	yes	—
36	C₇×#519	C₂²	1	1	yes	characteristic
37	C₇×#518	C₂²	1	1	yes	characteristic
38	C₄×C₇×#41	C₂²	1	1	yes	characteristic
39	C₇×#501	C₂²	1	2	yes	complemented
40	C₇×#501	C₂²	1	4	yes	complemented
41	C₇×#501	C₂²	1	2	yes	complemented
42	C₇×#497	C₂²	1	2	yes	—
43	C₇×#497	C₂²	1	4	yes	—
44	C₇×#497	C₂²	1	2	yes	—
45	C₇×#491	C₂²	1	4	yes	complemented
46	C₇×#489	C₂²	1	4	yes	—
47	C₇×#488	C₂²	1	4	yes	—
48	C₇×#487	C₂²	1	4	yes	complemented
49	C₇×#470	C₂²	1	8	yes	complemented
50	C₇×#468	C₂²	1	8	yes	—
51	C₇×#467	C₂²	1	8	yes	complemented
52	C₇×#465	C₂²	1	8	yes	—
53	C₇×#441	C₂²	1	8	yes	complemented
54	C₇×#440	C₂²	1	8	yes	—
55	C₄×C₇×Q₁₆	C₂²	1	8	yes	complemented
56	C₄×C₇×SD₁₆	C₂²	1	8	yes	—
57	C₇×#427	C₂²	1	2	yes	—
58	C₇×#427	C₂²	1	2	yes	—
59	C₇×#419	C₂²	1	2	yes	—
60	C₇×#419	C₂²	1	2	yes	—
61	C₇×#419	C₂²	1	2	yes	—
62	C₇×#418	C₂²	1	2	yes	—
63	C₇×#405	C₂²	1	2	yes	—
64	C₇×#405	C₂²	1	4	yes	complemented
65	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
66	C₄×C₇×M₁₆	C₂²	1	2	yes	complemented
67	C₂×C₇×#97	C₂²	1	1	yes	characteristic
68	C₂×C₇×#97	C₂²	1	2	yes	complemented
69	#60720	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₇×#41	C₂³	1	1	yes	characteristic
71	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
72	C₂²×C₇×Q₈	C₂³	1	2	yes	—
73	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
74	C₂×C₇×Q₁₆	—	2	8	no	—
75	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
76	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
77	C₂×C₇×SD₁₆	—	2	8	no	—
78	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
79	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
80	C₇×#131	—	2	8	no	—
81	C₂×C₇×M₁₆	C₂³	1	4	yes	—
82	C₂×C₇×M₁₆	C₂³	1	2	yes	—
83	C₂²×C₇×C₈	C₂³	1	4	yes	—
84	C₂²×C₇×C₈	C₂³	1	2	yes	—
85	C₇×C₄⋊Q₈	—	2	1	no	—
86	C₇×C₄⋊Q₈	—	2	2	no	—
87	C₇×C₄⋊Q₈	—	2	2	no	—
88	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
89	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
90	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
91	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
92	C₇×#127	—	2	1	no	—
93	C₇×#126	—	2	4	no	—
94	C₇×#126	—	2	8	no	—
95	C₇×#126	—	2	4	no	—
96	C₇×#125	—	2	8	no	—
97	C₇×#125	—	2	4	no	—
98	C₇×#125	C₂³	1	2	yes	—
99	C₇×#125	C₂³	1	2	yes	—
100	C₇×#124	—	2	1	no	—
101	C₇×#124	—	2	2	no	—
102	C₇×#124	—	2	1	no	—
103	C₇×#124	—	2	4	no	—
104	C₇×#124	—	2	2	no	—
105	C₇×#124	C₂³	1	2	yes	—
106	C₇×#124	C₂³	1	2	yes	—
107	C₇×#123	—	2	2	no	—
108	C₇×#123	—	2	4	no	—
109	C₇×#123	—	2	4	no	—
110	C₇×#123	—	2	4	no	—
111	C₇×C₂²⋊Q₈	—	2	2	no	—
112	C₇×C₂²⋊Q₈	—	2	2	no	—
113	C₇×C₂²⋊Q₈	—	2	4	no	—
114	C₇×C₂²⋊Q₈	—	2	4	no	—
115	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
116	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
117	C₇×#121	C₂³	1	4	yes	—
118	C₇×#120	—	2	2	no	—
119	C₄×C₇×Q₈	—	2	4	no	—
120	C₄×C₇×Q₈	—	2	2	no	—
121	C₄×C₇×Q₈	—	2	2	no	—
122	C₄×C₇×Q₈	—	2	2	no	—
123	C₄×C₇×Q₈	—	2	4	no	—
124	C₄×C₇×Q₈	D₄	1	2	yes	—
125	C₄×C₇×Q₈	C₂³	1	4	yes	—
126	C₄×C₇×Q₈	C₂³	1	8	yes	—
127	C₄×C₇×D₄	—	2	4	no	—
128	C₄×C₇×D₄	—	2	2	no	—
129	C₄×C₇×D₄	D₄	1	4	yes	—
130	C₄×C₇×D₄	D₄	1	2	yes	—
131	C₄×C₇×D₄	C₂³	1	4	yes	—
132	C₇×#117	—	2	4	no	—
133	C₇×#117	—	2	2	no	—
134	C₇×#117	—	2	4	no	—
135	C₇×#117	—	2	4	no	—
136	C₇×#117	C₂³	1	1	yes	characteristic
137	C₇×#117	C₂³	1	1	yes	characteristic
138	C₇×#117	C₂³	1	2	yes	—
139	C₇×#117	C₂³	1	1	yes	characteristic
140	C₇×#117	C₂³	1	2	yes	—
141	C₂×C₇×C₄⋊C₄	—	2	2	no	—
142	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
143	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
144	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
145	C₂×C₄²×C₇	C₂³	1	2	yes	—
146	C₇×#107	C₂³	1	4	yes	—
147	C₇×#107	C₂³	1	4	yes	—
148	C₇×#106	C₂³	1	4	yes	—
149	C₇×#106	C₂³	1	4	yes	—
150	C₇×#103	C₂³	1	8	yes	—
151	C₇×#103	C₂³	1	8	yes	—
152	C₇×#103	C₂³	1	8	yes	—
153	C₇×#102	C₂³	1	8	yes	—
154	C₇×#97	C₂³	1	2	yes	—
155	C₇×#97	C₂³	1	1	yes	characteristic
156	C₇×#97	C₂³	1	1	yes	characteristic
157	C₇×#97	C₂³	1	2	yes	—
158	C₇×#97	C₂³	1	4	yes	—
159	C₄×C₇×C₈	C₂³	1	2	yes	—
160	C₄×C₇×C₈	C₂³	1	4	yes	—
161	#3510	C₂×C₇	1	1	yes	characteristic
162	#3509	C₂×C₇	1	1	yes	characteristic, complemented
163	#3181	C₂×C₇	1	4	yes	complemented
164	#3178	C₂×C₇	1	8	yes	complemented
165	#3172	C₂×C₇	1	4	yes	complemented
166	#3162	C₂×C₇	1	1	yes	characteristic, complemented
167	C₂×#497	C₂×C₇	1	1	yes	characteristic, complemented
168	#3140	C₂×C₇	1	1	yes	characteristic, complemented
169	#3139	C₂×C₇	1	1	yes	characteristic, complemented
170	#3059	C₂×C₇	1	2	yes	complemented
171	#3057	C₂×C₇	1	2	yes	—
172	#2950	C₂×C₇	1	2	yes	complemented
173	#2949	C₂×C₇	1	2	yes	complemented
174	#2875	C₂×C₇	1	1	yes	characteristic, complemented
175	C₂⁴×C₇	—	2	1	no	—
176	C₇×#41	—	2	4	no	—
177	C₂×C₇×Q₈	—	2	2	no	—
178	C₂×C₇×Q₈	—	2	1	no	—
179	C₂×C₇×Q₈	—	2	2	no	—
180	C₂×C₇×Q₈	—	2	2	no	—
181	C₂×C₇×Q₈	—	2	4	no	—
182	C₂×C₇×Q₈	—	2	4	no	—
183	C₂×C₇×Q₈	—	2	4	no	—
184	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
185	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
186	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
187	C₂×C₇×D₄	—	2	4	no	—
188	C₂×C₇×D₄	—	2	1	no	—
189	C₂×C₇×D₄	—	2	4	no	—
190	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
191	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
192	C₂×C₇×D₄	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₇	—	2	1	no	—
194	C₂²×C₄×C₇	—	2	2	no	—
195	C₂²×C₄×C₇	—	2	4	no	—
196	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
198	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
200	C₇×Q₁₆	—	2	8	no	—
201	C₇×Q₁₆	—	2	8	no	—
202	C₇×SD₁₆	—	2	8	no	—
203	C₇×SD₁₆	—	2	8	no	—
204	C₇×M₁₆	—	2	4	no	—
205	C₇×M₁₆	—	2	8	no	—
206	C₂×C₇×C₈	—	2	8	no	—
207	C₂×C₇×C₈	#41	1	8	yes	—
208	C₂×C₇×C₈	C₂⁴	1	4	yes	—
209	C₂×C₇×C₈	C₂⁴	1	2	yes	—
210	C₂×C₇×C₈	C₂⁴	1	2	yes	—
211	C₇×C₄⋊C₄	—	2	4	no	—
212	C₇×C₄⋊C₄	—	2	4	no	—
213	C₇×C₄⋊C₄	—	2	2	no	—
214	C₇×C₄⋊C₄	—	2	2	no	—
215	C₇×C₄⋊C₄	—	2	4	no	—
216	C₇×C₄⋊C₄	—	2	8	no	—
217	C₇×C₄⋊C₄	—	2	4	no	—
218	C₇×C₄⋊C₄	—	2	2	no	—
219	C₇×C₄⋊C₄	—	2	2	no	—
220	C₇×C₄⋊C₄	—	2	4	no	—
221	C₇×C₄⋊C₄	—	2	8	no	—
222	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
223	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
224	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
225	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
226	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
227	C₇×C₂²⋊C₄	—	2	1	no	—
228	C₇×C₂²⋊C₄	—	2	1	no	—
229	C₇×C₂²⋊C₄	—	2	2	no	—
230	C₇×C₂²⋊C₄	—	2	8	no	—
231	C₇×C₂²⋊C₄	—	2	4	no	—
232	C₇×C₂²⋊C₄	—	2	4	no	—
233	C₇×C₂²⋊C₄	—	2	4	no	—
234	C₇×C₂²⋊C₄	—	2	4	no	—
235	C₇×C₂²⋊C₄	—	2	4	no	—
236	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
237	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
238	C₄²×C₇	—	2	4	no	—
239	C₄²×C₇	—	2	8	no	—
240	C₄²×C₇	—	2	1	no	—
241	C₄²×C₇	—	2	1	no	—
242	C₄²×C₇	—	2	2	no	—
243	C₄²×C₇	C₂×D₄	1	4	yes	—
244	C₄²×C₇	C₂×D₄	1	2	yes	—
245	C₄²×C₇	C₂⁴	1	2	yes	—
246	C₄²×C₇	C₂⁴	1	1	yes	characteristic
247	C₄²×C₇	C₂⁴	1	1	yes	characteristic
248	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
249	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
250	C₂×#131	C₂²×C₇	1	2	yes	—
251	#559	—	2	2	no	—
252	#557	—	2	2	no	—
253	#556	—	2	2	no	—
254	#556	—	2	2	no	—
255	#552	—	2	4	no	—
256	#551	—	2	4	no	—
257	#542	C₂²×C₇	1	4	yes	—
258	#541	C₂²×C₇	1	4	yes	complemented
259	#536	C₂²×C₇	1	2	yes	—
260	#536	C₂²×C₇	1	1	yes	characteristic
261	#535	C₂²×C₇	1	1	yes	characteristic
262	#533	C₂²×C₇	1	1	yes	characteristic
263	#533	C₂²×C₇	1	1	yes	characteristic
264	#533	C₂²×C₇	1	4	yes	complemented
265	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
266	#529	C₂²×C₇	1	4	yes	—
267	C₂×#124	C₂²×C₇	1	1	yes	characteristic
268	#519	C₂²×C₇	1	2	yes	—
269	#519	C₂²×C₇	1	1	yes	characteristic
270	#518	C₂²×C₇	1	1	yes	characteristic
271	C₄×#41	C₂²×C₇	1	1	yes	characteristic
272	#501	C₂²×C₇	1	2	yes	complemented
273	#501	C₂²×C₇	1	4	yes	complemented
274	#501	C₂²×C₇	1	2	yes	complemented
275	#497	C₂²×C₇	1	2	yes	—
276	#497	C₂²×C₇	1	4	yes	—
277	#497	C₂²×C₇	1	2	yes	—
278	#491	C₂²×C₇	1	4	yes	complemented
279	#489	C₂²×C₇	1	4	yes	—
280	#488	C₂²×C₇	1	4	yes	—
281	#487	C₂²×C₇	1	4	yes	complemented
282	#470	C₂²×C₇	1	8	yes	complemented
283	#468	C₂²×C₇	1	8	yes	—
284	#467	C₂²×C₇	1	8	yes	complemented
285	#465	C₂²×C₇	1	8	yes	—
286	#441	C₂²×C₇	1	8	yes	complemented
287	#440	C₂²×C₇	1	8	yes	—
288	C₄×Q₁₆	C₂²×C₇	1	8	yes	complemented
289	C₄×SD₁₆	C₂²×C₇	1	8	yes	—
290	#427	C₂²×C₇	1	2	yes	—
291	#427	C₂²×C₇	1	2	yes	—
292	#419	C₂²×C₇	1	2	yes	—
293	#419	C₂²×C₇	1	2	yes	—
294	#419	C₂²×C₇	1	2	yes	—
295	#418	C₂²×C₇	1	2	yes	—
296	#405	C₂²×C₇	1	2	yes	—
297	#405	C₂²×C₇	1	4	yes	complemented
298	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
299	C₄×M₁₆	C₂²×C₇	1	2	yes	complemented
300	C₂×#97	C₂²×C₇	1	1	yes	characteristic
301	C₂×#97	C₂²×C₇	1	2	yes	complemented
302	C₂³×C₇	—	2	2	no	—
303	C₂³×C₇	—	4	2	no	—
304	C₂³×C₇	—	8	2	no	—
305	C₂³×C₇	C₂²×D₄	1	2	yes	—
306	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
307	C₇×Q₈	—	2	2	no	—
308	C₇×Q₈	—	2	4	no	—
309	C₇×Q₈	—	2	4	no	—
310	C₇×Q₈	—	2	4	no	—
311	C₇×Q₈	—	4	4	no	—
312	C₇×Q₈	—	4	2	no	—
313	C₇×Q₈	—	4	4	no	—
314	C₇×Q₈	—	8	2	no	—
315	C₇×D₄	—	2	4	no	—
316	C₇×D₄	—	2	2	no	—
317	C₇×D₄	—	2	4	no	—
318	C₇×D₄	—	4	2	no	—
319	C₇×D₄	—	8	4	no	—
320	C₂×C₄×C₇	—	2	4	no	—
321	C₂×C₄×C₇	—	2	4	no	—
322	C₂×C₄×C₇	—	2	2	no	—
323	C₂×C₄×C₇	—	2	2	no	—
324	C₂×C₄×C₇	—	2	4	no	—
325	C₂×C₄×C₇	—	2	2	no	—
326	C₂×C₄×C₇	—	4	4	no	—
327	C₂×C₄×C₇	—	4	2	no	—
328	C₂×C₄×C₇	—	4	2	no	—
329	C₂×C₄×C₇	—	4	4	no	—
330	C₂×C₄×C₇	—	8	4	no	—
331	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
332	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
333	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
334	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
335	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×C₇	C₂×#41	1	4	yes	—
337	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
338	C₇×C₈	—	2	4	no	—
339	C₇×C₈	C₂×#41	1	4	yes	—
340	C₇×C₈	C₂×#41	1	4	yes	—
341	C₂×#41	C₂³×C₇	1	1	yes	characteristic
342	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
343	C₂²×Q₈	C₂³×C₇	1	2	yes	—
344	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
345	C₂×Q₁₆	—	2	8	no	—
346	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
347	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
348	C₂×SD₁₆	—	2	8	no	—
349	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
350	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
351	#131	—	2	8	no	—
352	C₂×M₁₆	C₂³×C₇	1	2	yes	—
353	C₂×M₁₆	C₂³×C₇	1	4	yes	—
354	C₂²×C₈	C₂³×C₇	1	4	yes	—
355	C₂²×C₈	C₂³×C₇	1	2	yes	—
356	C₄⋊Q₈	—	2	1	no	—
357	C₄⋊Q₈	—	2	2	no	—
358	C₄⋊Q₈	—	2	2	no	—
359	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
360	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
361	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
362	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
363	#127	—	2	1	no	—
364	#126	—	2	4	no	—
365	#126	—	2	8	no	—
366	#126	—	2	4	no	—
367	#125	—	2	8	no	—
368	#125	—	2	4	no	—
369	#125	C₂³×C₇	1	2	yes	—
370	#125	C₂³×C₇	1	2	yes	—
371	#124	—	2	1	no	—
372	#124	—	2	2	no	—
373	#124	—	2	1	no	—
374	#124	—	2	4	no	—
375	#124	—	2	2	no	—
376	#124	C₂³×C₇	1	2	yes	—
377	#124	C₂³×C₇	1	2	yes	—
378	#123	—	2	2	no	—
379	#123	—	2	4	no	—
380	#123	—	2	4	no	—
381	#123	—	2	4	no	—
382	C₂²⋊Q₈	—	2	2	no	—
383	C₂²⋊Q₈	—	2	2	no	—
384	C₂²⋊Q₈	—	2	4	no	—
385	C₂²⋊Q₈	—	2	4	no	—
386	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
387	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
388	#121	C₂³×C₇	1	4	yes	—
389	#120	—	2	2	no	—
390	C₄×Q₈	—	2	4	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	—	2	2	no	—
393	C₄×Q₈	—	2	2	no	—
394	C₄×Q₈	—	2	4	no	—
395	C₄×Q₈	C₇×D₄	1	2	yes	—
396	C₄×Q₈	C₂³×C₇	1	4	yes	—
397	C₄×Q₈	C₂³×C₇	1	8	yes	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	C₇×D₄	1	4	yes	—
401	C₄×D₄	C₇×D₄	1	2	yes	—
402	C₄×D₄	C₂³×C₇	1	4	yes	—
403	#117	—	2	4	no	—
404	#117	—	2	2	no	—
405	#117	—	2	4	no	—
406	#117	—	2	4	no	—
407	#117	C₂³×C₇	1	1	yes	characteristic
408	#117	C₂³×C₇	1	1	yes	characteristic
409	#117	C₂³×C₇	1	2	yes	—
410	#117	C₂³×C₇	1	1	yes	characteristic
411	#117	C₂³×C₇	1	2	yes	—
412	C₂×C₄⋊C₄	—	2	2	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	—	2	2	no	—
415	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₇	1	2	yes	—
417	#107	C₂³×C₇	1	4	yes	—
418	#107	C₂³×C₇	1	4	yes	—
419	#106	C₂³×C₇	1	4	yes	—
420	#106	C₂³×C₇	1	4	yes	—
421	#103	C₂³×C₇	1	8	yes	—
422	#103	C₂³×C₇	1	8	yes	—
423	#103	C₂³×C₇	1	8	yes	—
424	#102	C₂³×C₇	1	8	yes	—
425	#97	C₂³×C₇	1	1	yes	characteristic
426	#97	C₂³×C₇	1	1	yes	characteristic
427	#97	C₂³×C₇	1	2	yes	—
428	#97	C₂³×C₇	1	2	yes	—
429	#97	C₂³×C₇	1	4	yes	—
430	C₄×C₈	C₂³×C₇	1	2	yes	—
431	C₄×C₈	C₂³×C₇	1	4	yes	—
432	C₂²×C₇	—	2	2	no	—
433	C₂²×C₇	—	2	4	no	—
434	C₂²×C₇	—	2	2	no	—
435	C₂²×C₇	—	4	4	no	—
436	C₂²×C₇	—	4	2	no	—
437	C₂²×C₇	—	16	2	no	—
438	C₂²×C₇	#532	1	2	yes	—
439	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
440	C₄×C₇	—	2	2	no	—
441	C₄×C₇	—	2	4	no	—
442	C₄×C₇	—	2	2	no	—
443	C₄×C₇	—	2	2	no	—
444	C₄×C₇	—	4	2	no	—
445	C₄×C₇	—	4	2	no	—
446	C₄×C₇	—	4	4	no	—
447	C₄×C₇	—	4	2	no	—
448	C₄×C₇	—	4	4	no	—
449	C₄×C₇	#532	1	2	yes	—
450	C₄×C₇	C₂²×#41	1	2	yes	—
451	C₂⁴	—	2	1	no	—
452	#41	—	2	4	no	—
453	C₂×Q₈	—	2	2	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×Q₈	—	2	4	no	—
460	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
461	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
462	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
467	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
468	C₂×D₄	C₂⁴×C₇	1	2	yes	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
473	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
476	Q₁₆	—	2	8	no	—
477	Q₁₆	—	2	8	no	—
478	SD₁₆	—	2	8	no	—
479	SD₁₆	—	2	8	no	—
480	M₁₆	—	2	4	no	—
481	M₁₆	—	2	8	no	—
482	C₂×C₈	—	2	8	no	—
483	C₂×C₈	C₇×#41	1	8	yes	—
484	C₂×C₈	C₂⁴×C₇	1	4	yes	—
485	C₂×C₈	C₂⁴×C₇	1	2	yes	—
486	C₂×C₈	C₂⁴×C₇	1	2	yes	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	8	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	8	no	—
498	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
499	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
500	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
501	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
502	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	2	1	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	8	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
513	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
514	C₄²	—	2	4	no	—
515	C₄²	—	2	8	no	—
516	C₄²	—	2	1	no	—
517	C₄²	—	2	1	no	—
518	C₄²	—	2	2	no	—
519	C₄²	C₂×C₇×D₄	1	4	yes	—
520	C₄²	C₂×C₇×D₄	1	2	yes	—
521	C₄²	C₂⁴×C₇	1	2	yes	—
522	C₄²	C₂⁴×C₇	1	1	yes	characteristic
523	C₄²	C₂⁴×C₇	1	1	yes	characteristic
524	C₂×C₇	—	2	2	no	—
525	C₂×C₇	—	4	2	no	—
526	C₂×C₇	—	8	2	no	—
527	C₂×C₇	C₂×#532	1	1	yes	characteristic
528	C₂×C₇	#3586	1	2	yes	—
529	C₂³	—	2	2	no	—
530	C₂³	—	4	2	no	—
531	C₂³	—	8	2	no	—
532	C₂³	C₂²×C₇×D₄	1	2	yes	—
533	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
534	Q₈	—	2	2	no	—
535	Q₈	—	2	4	no	—
536	Q₈	—	2	4	no	—
537	Q₈	—	2	4	no	—
538	Q₈	—	4	4	no	—
539	Q₈	—	4	2	no	—
540	Q₈	—	4	4	no	—
541	Q₈	—	8	2	no	—
542	D₄	—	2	4	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	2	no	—
545	D₄	—	4	2	no	—
546	D₄	—	8	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	8	4	no	—
558	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
559	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
560	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
561	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
562	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
563	C₂×C₄	C₂×C₇×#41	1	4	yes	—
564	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
565	C₈	—	2	4	no	—
566	C₈	C₂×C₇×#41	1	4	yes	—
567	C₈	C₂×C₇×#41	1	4	yes	—
568	C₇	#60720	1	1	yes	characteristic, direct product factor, Sylow
569	C₂²	—	2	2	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	16	2	no	—
575	C₂²	C₇×#532	1	2	yes	—
576	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
577	C₄	—	2	2	no	—
578	C₄	—	2	4	no	—
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	4	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	4	no	—
586	C₄	C₇×#532	1	2	yes	—
587	C₄	C₂²×C₇×#41	1	2	yes	—
588	C₂	—	2	2	no	—
589	C₂	—	4	2	no	—
590	C₂	—	8	2	no	—
591	C₂	C₂×C₇×#532	1	1	yes	characteristic
592	C₂	C₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#535	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3509	C₂	semidirect
3. C₇×#3181	C₂	semidirect
4. C₇×#3178	C₂	semidirect
5. C₇×#3172	C₂	semidirect
6. C₇×#3162	C₂	semidirect
7. C₂×C₇×#497	C₂	semidirect
8. C₇×#3140	C₂	semidirect
9. C₇×#3139	C₂	semidirect
10. C₇×#3059	C₂	semidirect
12. C₇×#2950	C₂	semidirect
13. C₇×#2949	C₂	semidirect
14. C₇×#2875	C₂	semidirect
25. C₇×#541	C₂²	semidirect
31. C₇×#533	C₂²	semidirect
39. C₇×#501	C₂²	semidirect
40. C₇×#501	C₂²	semidirect
41. C₇×#501	C₂²	semidirect
45. C₇×#491	C₂²	semidirect
48. C₇×#487	C₂²	semidirect
49. C₇×#470	C₂²	semidirect
51. C₇×#467	C₂²	semidirect
53. C₇×#441	C₂²	semidirect
55. C₄×C₇×Q₁₆	C₂²	semidirect
64. C₇×#405	C₂²	semidirect
66. C₄×C₇×M₁₆	C₂²	semidirect
68. C₂×C₇×#97	C₂²	semidirect
69. #60720	C₇	direct (N × H)
162. #3509	C₂×C₇	semidirect
163. #3181	C₂×C₇	semidirect
164. #3178	C₂×C₇	semidirect
165. #3172	C₂×C₇	semidirect
166. #3162	C₂×C₇	semidirect
167. C₂×#497	C₂×C₇	semidirect
168. #3140	C₂×C₇	semidirect
169. #3139	C₂×C₇	semidirect
170. #3059	C₂×C₇	semidirect
172. #2950	C₂×C₇	semidirect
173. #2949	C₂×C₇	semidirect
174. #2875	C₂×C₇	semidirect
258. #541	C₂²×C₇	semidirect
264. #533	C₂²×C₇	semidirect
272. #501	C₂²×C₇	semidirect
273. #501	C₂²×C₇	semidirect
274. #501	C₂²×C₇	semidirect
278. #491	C₂²×C₇	semidirect
281. #487	C₂²×C₇	semidirect
282. #470	C₂²×C₇	semidirect
284. #467	C₂²×C₇	semidirect
286. #441	C₂²×C₇	semidirect
288. C₄×Q₁₆	C₂²×C₇	semidirect
297. #405	C₂²×C₇	semidirect
299. C₄×M₁₆	C₂²×C₇	semidirect
301. C₂×#97	C₂²×C₇	semidirect
568. C₇	#60720	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×#3510	2
C₇×#3509	2
C₇×#3181	2
C₇×#3178	2
C₇×#3172	2
C₇×#3162	2
C₂×C₇×#497	2
C₇×#3140	2
C₇×#3139	2
C₇×#3059	2
C₇×#3057	2
C₇×#2950	2
C₇×#2949	2
C₇×#2875	2
#60720	7

Subgroups — C7×#60720 #13237936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60720 #13237936