Subgroups — C₇×#60944 #13238160

Order 1792.

1846 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3444	C₂	1	2	yes	maximal, complemented
2	C₇×#3250	C₂	1	4	yes	maximal, complemented
3	C₇×#3231	C₂	1	2	yes	maximal, complemented
4	C₇×#3228	C₂	1	2	yes	maximal, complemented
5	C₇×#3210	C₂	1	2	yes	maximal, complemented
6	C₇×#3207	C₂	1	2	yes	maximal, complemented
7	C₇×#3191	C₂	1	4	yes	maximal, complemented
8	C₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3167	C₂	1	2	yes	maximal, complemented
11	C₇×#3102	C₂	1	8	yes	maximal, complemented
12	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#533	C₂²	1	2	yes	—
14	C₇×#533	C₂²	1	2	yes	—
15	C₇×#533	C₂²	1	2	yes	—
16	C₇×#532	C₂²	1	2	yes	—
17	C₇×#532	C₂²	1	2	yes	—
18	C₇×#532	C₂²	1	2	yes	—
19	C₇×#529	—	2	2	no	—
20	C₇×#529	—	2	2	no	—
21	C₇×#529	—	2	2	no	—
22	C₇×#529	—	2	2	no	—
23	C₇×#529	C₂²	1	4	yes	—
24	C₇×#529	C₂²	1	4	yes	—
25	C₇×#525	—	2	2	no	—
26	C₇×#525	—	2	2	no	—
27	C₇×#525	C₂²	1	4	yes	—
28	C₄×C₇×#41	—	2	2	no	—
29	C₄×C₇×#41	—	2	2	no	—
30	C₄×C₇×#41	C₂²	1	4	yes	—
31	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
32	C₇×#500	C₂²	1	1	yes	characteristic, complemented
33	C₇×#500	C₂²	1	1	yes	characteristic, complemented
34	C₇×#498	C₂²	1	2	yes	complemented
35	C₇×#494	C₂²	1	1	yes	characteristic, complemented
36	C₇×#493	C₂²	1	1	yes	characteristic, complemented
37	C₇×#492	C₂²	1	2	yes	complemented
38	C₇×#487	C₂²	1	2	yes	complemented
39	C₇×#487	C₂²	1	2	yes	complemented
40	C₇×#486	C₂²	1	2	yes	complemented
41	C₇×#486	C₂²	1	2	yes	complemented
42	C₇×#485	C₂²	1	8	yes	—
43	C₇×#482	C₂²	1	8	yes	—
44	C₇×#479	C₂²	1	8	yes	complemented
45	C₇×#478	C₂²	1	8	yes	complemented
46	C₇×#467	C₂²	1	4	yes	—
47	C₇×#466	C₂²	1	4	yes	—
48	C₇×#465	C₂²	1	8	yes	—
49	C₇×#464	C₂²	1	8	yes	complemented
50	C₇×#463	C₂²	1	8	yes	complemented
51	C₇×#452	C₂²	1	8	yes	—
52	C₇×#449	C₂²	1	8	yes	—
53	C₄×C₇×Q₁₆	C₂²	1	4	yes	complemented
54	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
55	C₄×C₇×D₈	C₂²	1	4	yes	complemented
56	C₇×#431	C₂²	1	2	yes	—
57	C₇×#431	C₂²	1	2	yes	—
58	C₇×#431	C₂²	1	4	yes	—
59	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
60	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
61	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
62	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
63	#60944	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₇×#41	—	2	2	no	—
65	C₂×C₇×#41	—	2	2	no	—
66	C₂×C₇×#41	C₂³	1	4	yes	—
67	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
68	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
69	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
70	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
71	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
72	C₂×C₇×D₈	C₂³	1	4	yes	—
73	C₂²×C₇×C₈	C₂³	1	2	yes	—
74	C₂²×C₇×C₈	C₂³	1	2	yes	—
75	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
77	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
78	C₇×#127	C₂³	1	2	yes	—
79	C₇×#126	—	2	4	no	—
80	C₇×#126	—	2	4	no	—
81	C₇×#125	—	2	2	no	—
82	C₇×#125	—	2	2	no	—
83	C₇×#125	C₂³	1	4	yes	—
84	C₇×#124	—	2	2	no	—
85	C₇×#124	—	2	2	no	—
86	C₇×#124	C₂³	1	4	yes	—
87	C₇×#123	—	2	4	no	—
88	C₇×#123	—	2	4	no	—
89	C₇×#123	—	2	4	no	—
90	C₇×C₂²⋊Q₈	—	2	2	no	—
91	C₇×C₂²⋊Q₈	—	2	2	no	—
92	C₇×C₂²⋊Q₈	—	2	2	no	—
93	C₇×C₂²⋊Q₈	—	2	2	no	—
94	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
95	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
96	C₇×#121	—	2	2	no	—
97	C₇×#121	—	2	2	no	—
98	C₇×#121	—	2	2	no	—
99	C₇×#121	—	2	2	no	—
100	C₇×#121	C₂³	1	4	yes	—
101	C₇×#121	C₂³	1	4	yes	—
102	C₇×#120	—	2	4	no	—
103	C₄×C₇×Q₈	—	2	2	no	—
104	C₄×C₇×Q₈	—	2	2	no	—
105	C₄×C₇×Q₈	—	2	2	no	—
106	C₄×C₇×Q₈	—	2	2	no	—
107	C₄×C₇×Q₈	C₂³	1	4	yes	—
108	C₄×C₇×Q₈	C₂³	1	4	yes	—
109	C₄×C₇×D₄	—	2	4	no	—
110	C₄×C₇×D₄	—	2	4	no	—
111	C₄×C₇×D₄	—	2	4	no	—
112	C₄×C₇×D₄	—	2	4	no	—
113	C₄×C₇×D₄	—	2	4	no	—
114	C₄×C₇×D₄	—	2	4	no	—
115	C₄×C₇×D₄	—	2	2	no	—
116	C₄×C₇×D₄	—	2	2	no	—
117	C₄×C₇×D₄	—	2	2	no	—
118	C₄×C₇×D₄	—	2	2	no	—
119	C₄×C₇×D₄	C₂³	1	4	yes	—
120	C₄×C₇×D₄	C₂³	1	4	yes	—
121	C₇×#117	—	2	2	no	—
122	C₇×#117	—	2	2	no	—
123	C₇×#117	—	2	4	no	—
124	C₇×#117	—	2	4	no	—
125	C₇×#117	—	2	4	no	—
126	C₇×#117	C₂³	1	4	yes	—
127	C₂×C₄²×C₇	—	2	4	no	—
128	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
129	C₂×C₄²×C₇	D₄	1	2	yes	—
130	C₂×C₄²×C₇	D₄	1	2	yes	—
131	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
132	C₂×C₄²×C₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₇	C₂³	1	2	yes	—
136	C₇×#107	C₂³	1	4	yes	—
137	C₇×#107	C₂³	1	4	yes	—
138	C₇×#106	C₂³	1	8	yes	—
139	C₇×C₄⋊C₈	C₂³	1	4	yes	—
140	C₇×C₄⋊C₈	C₂³	1	4	yes	—
141	C₇×#103	C₂³	1	8	yes	—
142	C₇×#103	C₂³	1	8	yes	—
143	C₇×#102	C₂³	1	8	yes	—
144	C₇×#102	C₂³	1	8	yes	—
145	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
146	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
147	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
148	C₄×C₇×C₈	C₂³	1	2	yes	—
149	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
150	C₄×C₇×C₈	C₂³	1	2	yes	—
151	C₄×C₇×C₈	C₂³	1	2	yes	—
152	#3444	C₂×C₇	1	2	yes	complemented
153	#3250	C₂×C₇	1	4	yes	complemented
154	#3231	C₂×C₇	1	2	yes	complemented
155	#3228	C₂×C₇	1	2	yes	complemented
156	#3210	C₂×C₇	1	2	yes	complemented
157	#3207	C₂×C₇	1	2	yes	complemented
158	#3191	C₂×C₇	1	4	yes	complemented
159	#3170	C₂×C₇	1	1	yes	characteristic, complemented
160	#3169	C₂×C₇	1	1	yes	characteristic, complemented
161	#3167	C₂×C₇	1	2	yes	complemented
162	#3102	C₂×C₇	1	8	yes	complemented
163	C₂×#431	C₂×C₇	1	1	yes	characteristic, complemented
164	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
165	C₇×#41	—	4	8	no	—
166	C₇×#41	—	4	8	no	—
167	C₇×#41	—	4	8	no	—
168	C₂×C₇×Q₈	—	2	2	no	—
169	C₂×C₇×Q₈	—	2	2	no	—
170	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
171	C₂×C₇×D₄	—	2	4	no	—
172	C₂×C₇×D₄	—	2	4	no	—
173	C₂×C₇×D₄	—	2	4	no	—
174	C₂×C₇×D₄	—	2	2	no	—
175	C₂×C₇×D₄	—	2	2	no	—
176	C₂×C₇×D₄	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₇	—	2	4	no	—
178	C₂²×C₄×C₇	—	2	4	no	—
179	C₂²×C₄×C₇	—	2	4	no	—
180	C₂²×C₄×C₇	—	2	4	no	—
181	C₂²×C₄×C₇	—	4	2	no	—
182	C₂²×C₄×C₇	—	4	2	no	—
183	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
190	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
192	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
195	C₇×Q₁₆	—	4	4	no	—
196	C₇×SD₁₆	—	4	8	no	—
197	C₇×D₈	—	4	4	no	—
198	C₂×C₇×C₈	—	4	2	no	—
199	C₂×C₇×C₈	—	4	2	no	—
200	C₂×C₇×C₈	C₂⁴	1	2	yes	—
201	C₂×C₇×C₈	C₂⁴	1	4	yes	—
202	C₂×C₇×C₈	C₂⁴	1	2	yes	—
203	C₇×C₄⋊C₄	—	2	2	no	—
204	C₇×C₄⋊C₄	—	2	2	no	—
205	C₇×C₄⋊C₄	—	2	2	no	—
206	C₇×C₄⋊C₄	—	2	2	no	—
207	C₇×C₄⋊C₄	—	2	4	no	—
208	C₇×C₄⋊C₄	—	2	4	no	—
209	C₇×C₄⋊C₄	—	2	4	no	—
210	C₇×C₄⋊C₄	—	2	4	no	—
211	C₇×C₄⋊C₄	—	2	4	no	—
212	C₇×C₄⋊C₄	—	2	4	no	—
213	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
214	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
215	C₇×C₂²⋊C₄	—	2	4	no	—
216	C₇×C₂²⋊C₄	—	2	4	no	—
217	C₇×C₂²⋊C₄	—	2	4	no	—
218	C₇×C₂²⋊C₄	—	2	4	no	—
219	C₇×C₂²⋊C₄	—	2	4	no	—
220	C₇×C₂²⋊C₄	—	2	4	no	—
221	C₇×C₂²⋊C₄	—	2	4	no	—
222	C₇×C₂²⋊C₄	—	2	4	no	—
223	C₇×C₂²⋊C₄	—	2	4	no	—
224	C₄²×C₇	—	2	4	no	—
225	C₄²×C₇	—	2	4	no	—
226	C₄²×C₇	—	2	4	no	—
227	C₄²×C₇	C₂×D₄	1	2	yes	—
228	C₄²×C₇	C₂×D₄	1	2	yes	—
229	C₄²×C₇	C₂×D₄	1	2	yes	—
230	C₄²×C₇	C₂×D₄	1	2	yes	—
231	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
232	C₄²×C₇	C₂×D₄	1	2	yes	—
233	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₇	C₂⁴	1	1	yes	characteristic
235	C₄²×C₇	C₂⁴	1	2	yes	—
236	C₄²×C₇	C₂⁴	1	1	yes	characteristic
237	#533	C₂²×C₇	1	2	yes	—
238	#533	C₂²×C₇	1	2	yes	—
239	#533	C₂²×C₇	1	2	yes	—
240	#532	C₂²×C₇	1	2	yes	—
241	#532	C₂²×C₇	1	2	yes	—
242	#532	C₂²×C₇	1	2	yes	—
243	#529	—	2	2	no	—
244	#529	—	2	2	no	—
245	#529	—	2	2	no	—
246	#529	—	2	2	no	—
247	#529	C₂²×C₇	1	4	yes	—
248	#529	C₂²×C₇	1	4	yes	—
249	#525	—	2	2	no	—
250	#525	—	2	2	no	—
251	#525	C₂²×C₇	1	4	yes	—
252	C₄×#41	—	2	2	no	—
253	C₄×#41	—	2	2	no	—
254	C₄×#41	C₂²×C₇	1	4	yes	—
255	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
256	#500	C₂²×C₇	1	1	yes	characteristic, complemented
257	#500	C₂²×C₇	1	1	yes	characteristic, complemented
258	#498	C₂²×C₇	1	2	yes	complemented
259	#494	C₂²×C₇	1	1	yes	characteristic, complemented
260	#493	C₂²×C₇	1	1	yes	characteristic, complemented
261	#492	C₂²×C₇	1	2	yes	complemented
262	#487	C₂²×C₇	1	2	yes	complemented
263	#487	C₂²×C₇	1	2	yes	complemented
264	#486	C₂²×C₇	1	2	yes	complemented
265	#486	C₂²×C₇	1	2	yes	complemented
266	#485	C₂²×C₇	1	8	yes	—
267	#482	C₂²×C₇	1	8	yes	—
268	#479	C₂²×C₇	1	8	yes	complemented
269	#478	C₂²×C₇	1	8	yes	complemented
270	#467	C₂²×C₇	1	4	yes	—
271	#466	C₂²×C₇	1	4	yes	—
272	#465	C₂²×C₇	1	8	yes	—
273	#464	C₂²×C₇	1	8	yes	complemented
274	#463	C₂²×C₇	1	8	yes	complemented
275	#452	C₂²×C₇	1	8	yes	—
276	#449	C₂²×C₇	1	8	yes	—
277	C₄×Q₁₆	C₂²×C₇	1	4	yes	complemented
278	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
279	C₄×D₈	C₂²×C₇	1	4	yes	complemented
280	#431	C₂²×C₇	1	4	yes	—
281	#431	C₂²×C₇	1	2	yes	—
282	#431	C₂²×C₇	1	2	yes	—
283	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
284	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
285	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
286	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
287	C₂³×C₇	—	2	2	no	—
288	C₂³×C₇	—	2	4	no	—
289	C₂³×C₇	—	4	1	no	—
290	C₂³×C₇	—	4	1	no	—
291	C₂³×C₇	C₂²×D₄	1	2	yes	—
292	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
293	C₇×Q₈	—	2	8	no	—
294	C₇×Q₈	—	4	4	no	—
295	C₇×Q₈	—	4	4	no	—
296	C₇×D₄	—	2	8	no	—
297	C₇×D₄	—	4	8	no	—
298	C₇×D₄	—	4	8	no	—
299	C₇×D₄	—	4	8	no	—
300	C₇×D₄	—	4	4	no	—
301	C₇×D₄	—	4	4	no	—
302	C₂×C₄×C₇	—	2	2	no	—
303	C₂×C₄×C₇	—	2	4	no	—
304	C₂×C₄×C₇	—	2	4	no	—
305	C₂×C₄×C₇	—	2	2	no	—
306	C₂×C₄×C₇	—	2	4	no	—
307	C₂×C₄×C₇	—	2	2	no	—
308	C₂×C₄×C₇	—	2	4	no	—
309	C₂×C₄×C₇	—	2	4	no	—
310	C₂×C₄×C₇	—	2	4	no	—
311	C₂×C₄×C₇	—	2	4	no	—
312	C₂×C₄×C₇	—	4	1	no	—
313	C₂×C₄×C₇	—	4	2	no	—
314	C₂×C₄×C₇	—	4	1	no	—
315	C₂×C₄×C₇	—	4	2	no	—
316	C₂×C₄×C₇	—	4	8	no	—
317	C₂×C₄×C₇	—	4	8	no	—
318	C₂×C₄×C₇	—	4	8	no	—
319	C₂×C₄×C₇	#135	1	4	yes	—
320	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
321	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
322	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
324	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
325	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
326	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
327	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
328	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
329	C₇×C₈	—	2	2	no	—
330	C₇×C₈	—	2	4	no	—
331	C₇×C₈	—	2	2	no	—
332	C₂×#41	—	2	2	no	—
333	C₂×#41	—	2	2	no	—
334	C₂×#41	C₂³×C₇	1	4	yes	—
335	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
336	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
338	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
339	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
340	C₂×D₈	C₂³×C₇	1	4	yes	—
341	C₂²×C₈	C₂³×C₇	1	2	yes	—
342	C₂²×C₈	C₂³×C₇	1	2	yes	—
343	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
344	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
345	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
346	#127	C₂³×C₇	1	2	yes	—
347	#126	—	2	4	no	—
348	#126	—	2	4	no	—
349	#125	—	2	2	no	—
350	#125	—	2	2	no	—
351	#125	C₂³×C₇	1	4	yes	—
352	#124	—	2	2	no	—
353	#124	—	2	2	no	—
354	#124	C₂³×C₇	1	4	yes	—
355	#123	—	2	4	no	—
356	#123	—	2	4	no	—
357	#123	—	2	4	no	—
358	C₂²⋊Q₈	—	2	2	no	—
359	C₂²⋊Q₈	—	2	2	no	—
360	C₂²⋊Q₈	—	2	2	no	—
361	C₂²⋊Q₈	—	2	2	no	—
362	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
363	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
364	#121	—	2	2	no	—
365	#121	—	2	2	no	—
366	#121	—	2	2	no	—
367	#121	—	2	2	no	—
368	#121	C₂³×C₇	1	4	yes	—
369	#121	C₂³×C₇	1	4	yes	—
370	#120	—	2	4	no	—
371	C₄×Q₈	—	2	2	no	—
372	C₄×Q₈	—	2	2	no	—
373	C₄×Q₈	—	2	2	no	—
374	C₄×Q₈	—	2	2	no	—
375	C₄×Q₈	C₂³×C₇	1	4	yes	—
376	C₄×Q₈	C₂³×C₇	1	4	yes	—
377	C₄×D₄	—	2	4	no	—
378	C₄×D₄	—	2	4	no	—
379	C₄×D₄	—	2	4	no	—
380	C₄×D₄	—	2	4	no	—
381	C₄×D₄	—	2	4	no	—
382	C₄×D₄	—	2	4	no	—
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	2	2	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	C₂³×C₇	1	4	yes	—
388	C₄×D₄	C₂³×C₇	1	4	yes	—
389	#117	—	2	2	no	—
390	#117	—	2	2	no	—
391	#117	—	2	4	no	—
392	#117	—	2	4	no	—
393	#117	—	2	4	no	—
394	#117	C₂³×C₇	1	4	yes	—
395	C₂×C₄²	—	2	4	no	—
396	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
397	C₂×C₄²	C₇×D₄	1	2	yes	—
398	C₂×C₄²	C₇×D₄	1	2	yes	—
399	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₇	1	2	yes	—
401	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₇	1	2	yes	—
404	#107	C₂³×C₇	1	4	yes	—
405	#107	C₂³×C₇	1	4	yes	—
406	#106	C₂³×C₇	1	8	yes	—
407	C₄⋊C₈	C₂³×C₇	1	4	yes	—
408	C₄⋊C₈	C₂³×C₇	1	4	yes	—
409	#103	C₂³×C₇	1	8	yes	—
410	#103	C₂³×C₇	1	8	yes	—
411	#102	C₂³×C₇	1	8	yes	—
412	#102	C₂³×C₇	1	8	yes	—
413	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
414	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
415	C₄×C₈	C₂³×C₇	1	2	yes	—
416	C₄×C₈	C₂³×C₇	1	2	yes	—
417	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
418	C₄×C₈	C₂³×C₇	1	2	yes	—
419	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
420	C₂²×C₇	—	2	1	no	—
421	C₂²×C₇	—	2	1	no	—
422	C₂²×C₇	—	2	2	no	—
423	C₂²×C₇	—	2	2	no	—
424	C₂²×C₇	—	2	2	no	—
425	C₂²×C₇	—	4	4	no	—
426	C₂²×C₇	—	4	4	no	—
427	C₂²×C₇	—	4	4	no	—
428	C₂²×C₇	—	4	4	no	—
429	C₂²×C₇	C₂×#135	1	2	yes	—
430	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
431	C₄×C₇	—	2	2	no	—
432	C₄×C₇	—	2	1	no	—
433	C₄×C₇	—	2	2	no	—
434	C₄×C₇	—	2	2	no	—
435	C₄×C₇	—	2	1	no	—
436	C₄×C₇	—	4	4	no	—
437	C₄×C₇	—	4	4	no	—
438	C₄×C₇	—	4	4	no	—
439	C₄×C₇	C₂×#135	1	2	yes	—
440	C₄×C₇	C₂×#135	1	2	yes	—
441	C₄×C₇	#575	1	2	yes	—
442	C₄×C₇	#585	1	2	yes	—
443	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
444	#41	—	4	8	no	—
445	#41	—	4	8	no	—
446	#41	—	4	8	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	C₂⁴×C₇	1	4	yes	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	4	2	no	—
461	C₂²×C₄	—	4	2	no	—
462	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
463	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
464	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
465	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
466	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
467	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
468	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
469	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
470	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
471	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
472	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
474	Q₁₆	—	4	4	no	—
475	SD₁₆	—	4	8	no	—
476	D₈	—	4	4	no	—
477	C₂×C₈	—	4	2	no	—
478	C₂×C₈	—	4	2	no	—
479	C₂×C₈	C₂⁴×C₇	1	2	yes	—
480	C₂×C₈	C₂⁴×C₇	1	4	yes	—
481	C₂×C₈	C₂⁴×C₇	1	2	yes	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
493	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₄²	—	2	4	no	—
504	C₄²	—	2	4	no	—
505	C₄²	—	2	4	no	—
506	C₄²	C₂×C₇×D₄	1	2	yes	—
507	C₄²	C₂×C₇×D₄	1	2	yes	—
508	C₄²	C₂×C₇×D₄	1	2	yes	—
509	C₄²	C₂×C₇×D₄	1	2	yes	—
510	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
511	C₄²	C₂×C₇×D₄	1	2	yes	—
512	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
513	C₄²	C₂⁴×C₇	1	1	yes	characteristic
514	C₄²	C₂⁴×C₇	1	1	yes	characteristic
515	C₄²	C₂⁴×C₇	1	2	yes	—
516	C₂×C₇	—	2	2	no	—
517	C₂×C₇	—	4	2	no	—
518	C₂×C₇	—	8	4	no	—
519	C₂×C₇	#3449	1	1	yes	characteristic
520	C₂×C₇	C₂²×#135	1	1	yes	characteristic
521	C₂×C₇	C₂×#575	1	1	yes	characteristic
522	C₂³	—	2	2	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	4	1	no	—
525	C₂³	—	4	1	no	—
526	C₂³	C₂²×C₇×D₄	1	2	yes	—
527	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
528	Q₈	—	2	8	no	—
529	Q₈	—	4	4	no	—
530	Q₈	—	4	4	no	—
531	D₄	—	2	8	no	—
532	D₄	—	4	8	no	—
533	D₄	—	4	8	no	—
534	D₄	—	4	8	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	4	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	8	no	—
552	C₂×C₄	—	4	8	no	—
553	C₂×C₄	—	4	8	no	—
554	C₂×C₄	C₇×#135	1	4	yes	—
555	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
556	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
557	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
558	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
559	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
560	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
561	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
562	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
563	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
564	C₈	—	2	2	no	—
565	C₈	—	2	4	no	—
566	C₈	—	2	2	no	—
567	C₇	#60944	1	1	yes	characteristic, direct product factor, Sylow
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	4	no	—
575	C₂²	—	4	4	no	—
576	C₂²	—	4	4	no	—
577	C₂²	C₂×C₇×#135	1	2	yes	—
578	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	2	no	—
583	C₄	—	2	1	no	—
584	C₄	—	4	4	no	—
585	C₄	—	4	4	no	—
586	C₄	—	4	4	no	—
587	C₄	C₂×C₇×#135	1	2	yes	—
588	C₄	C₂×C₇×#135	1	2	yes	—
589	C₄	C₇×#575	1	2	yes	—
590	C₄	C₇×#585	1	2	yes	—
591	C₂	—	2	2	no	—
592	C₂	—	4	2	no	—
593	C₂	—	8	4	no	—
594	C₂	C₇×#3449	1	1	yes	characteristic
595	C₂	C₂²×C₇×#135	1	1	yes	characteristic
596	C₂	C₂×C₇×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3444	C₂	semidirect
2. C₇×#3250	C₂	semidirect
3. C₇×#3231	C₂	semidirect
4. C₇×#3228	C₂	semidirect
5. C₇×#3210	C₂	semidirect
6. C₇×#3207	C₂	semidirect
7. C₇×#3191	C₂	semidirect
8. C₇×#3170	C₂	semidirect
9. C₇×#3169	C₂	semidirect
10. C₇×#3167	C₂	semidirect
11. C₇×#3102	C₂	semidirect
12. C₂×C₇×#431	C₂	semidirect
32. C₇×#500	C₂²	semidirect
33. C₇×#500	C₂²	semidirect
34. C₇×#498	C₂²	semidirect
35. C₇×#494	C₂²	semidirect
36. C₇×#493	C₂²	semidirect
37. C₇×#492	C₂²	semidirect
38. C₇×#487	C₂²	semidirect
39. C₇×#487	C₂²	semidirect
40. C₇×#486	C₂²	semidirect
41. C₇×#486	C₂²	semidirect
44. C₇×#479	C₂²	semidirect
45. C₇×#478	C₂²	semidirect
49. C₇×#464	C₂²	semidirect
50. C₇×#463	C₂²	semidirect
53. C₄×C₇×Q₁₆	C₂²	semidirect
54. C₄×C₇×SD₁₆	C₂²	semidirect
55. C₄×C₇×D₈	C₂²	semidirect
63. #60944	C₇	direct (N × H)
152. #3444	C₂×C₇	semidirect
153. #3250	C₂×C₇	semidirect
154. #3231	C₂×C₇	semidirect
155. #3228	C₂×C₇	semidirect
156. #3210	C₂×C₇	semidirect
157. #3207	C₂×C₇	semidirect
158. #3191	C₂×C₇	semidirect
159. #3170	C₂×C₇	semidirect
160. #3169	C₂×C₇	semidirect
161. #3167	C₂×C₇	semidirect
162. #3102	C₂×C₇	semidirect
163. C₂×#431	C₂×C₇	semidirect
256. #500	C₂²×C₇	semidirect
257. #500	C₂²×C₇	semidirect
258. #498	C₂²×C₇	semidirect
259. #494	C₂²×C₇	semidirect
260. #493	C₂²×C₇	semidirect
261. #492	C₂²×C₇	semidirect
262. #487	C₂²×C₇	semidirect
263. #487	C₂²×C₇	semidirect
264. #486	C₂²×C₇	semidirect
265. #486	C₂²×C₇	semidirect
268. #479	C₂²×C₇	semidirect
269. #478	C₂²×C₇	semidirect
273. #464	C₂²×C₇	semidirect
274. #463	C₂²×C₇	semidirect
277. C₄×Q₁₆	C₂²×C₇	semidirect
278. C₄×SD₁₆	C₂²×C₇	semidirect
279. C₄×D₈	C₂²×C₇	semidirect
567. C₇	#60944	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×#3444	2
C₇×#3250	2
C₇×#3231	2
C₇×#3228	2
C₇×#3210	2
C₇×#3207	2
C₇×#3191	2
C₇×#3170	2
C₇×#3169	2
C₇×#3167	2
C₇×#3102	2
C₂×C₇×#431	2
#60944	7

Subgroups — C7×#60944 #13238160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60944 #13238160