Subgroups — C₇×#60718 #13237934

Order 1792.

1942 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	C₇×#3511	C₂	1	2	yes	maximal, complemented
2	C₇×#3182	C₂	1	4	yes	maximal, complemented
3	C₇×#3175	C₂	1	8	yes	maximal, complemented
4	C₇×#3170	C₂	1	2	yes	maximal, complemented
5	C₇×#3169	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3141	C₂	1	2	yes	maximal, complemented
9	C₇×#3061	C₂	1	4	yes	maximal, complemented
10	C₇×#2951	C₂	1	4	yes	maximal, complemented
11	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	4	yes	—
13	C₂×C₇×#131	C₂²	1	2	yes	—
14	C₇×#556	—	2	2	no	—
15	C₇×#555	—	2	2	no	—
16	C₇×#554	—	2	2	no	—
17	C₇×#553	—	2	4	no	—
18	C₇×#548	—	2	4	no	—
19	C₇×#547	—	2	2	no	—
20	C₇×#542	C₂²	1	8	yes	—
21	C₇×#537	C₂²	1	4	yes	—
22	C₇×#533	C₂²	1	4	yes	—
23	C₇×#532	C₂²	1	2	yes	—
24	C₇×#532	C₂²	1	4	yes	—
25	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₇×#520	C₂²	1	4	yes	—
27	C₄×C₇×#41	C₂²	1	1	yes	characteristic
28	C₇×#500	C₂²	1	2	yes	—
29	C₇×#500	C₂²	1	4	yes	—
30	C₇×#500	C₂²	1	2	yes	—
31	C₇×#495	C₂²	1	4	yes	—
32	C₇×#494	C₂²	1	1	yes	characteristic
33	C₇×#493	C₂²	1	1	yes	characteristic
34	C₇×#492	C₂²	1	2	yes	—
35	C₇×#489	C₂²	1	8	yes	—
36	C₇×#487	C₂²	1	4	yes	—
37	C₇×#486	C₂²	1	4	yes	—
38	C₇×#468	C₂²	1	16	yes	—
39	C₇×#467	C₂²	1	8	yes	—
40	C₇×#466	C₂²	1	8	yes	—
41	C₇×#440	C₂²	1	16	yes	—
42	C₄×C₇×Q₁₆	C₂²	1	8	yes	—
43	C₄×C₇×D₈	C₂²	1	8	yes	—
44	C₂×C₇×#107	C₂²	1	4	yes	—
45	C₇×#417	C₂²	1	8	yes	—
46	C₇×#405	C₂²	1	4	yes	—
47	C₇×#405	C₂²	1	2	yes	—
48	C₄×C₇×M₁₆	C₂²	1	2	yes	—
49	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
50	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
51	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
52	#60718	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₇×#41	—	2	4	no	—
54	C₂×C₇×#41	C₂³	1	4	yes	—
55	C₂×C₇×#41	C₂³	1	1	yes	characteristic
56	C₂²×C₇×Q₈	—	2	2	no	—
57	C₇×#135	—	2	16	no	—
58	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
59	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
60	C₂×C₇×D₈	C₂³	1	4	yes	—
61	C₇×#131	—	2	8	no	—
62	C₂×C₇×M₁₆	C₂³	1	4	yes	—
63	C₂×C₇×M₁₆	C₂³	1	2	yes	—
64	C₂²×C₇×C₈	C₂³	1	4	yes	—
65	C₂²×C₇×C₈	C₂³	1	2	yes	—
66	C₇×C₄⋊Q₈	—	2	2	no	—
67	C₇×C₄⋊Q₈	—	2	2	no	—
68	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
70	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
71	C₇×#127	C₂³	1	2	yes	—
72	C₇×#126	—	2	8	no	—
73	C₇×#125	—	2	4	no	—
74	C₇×#124	—	2	8	no	—
75	C₇×#124	—	2	4	no	—
76	C₇×#124	C₂³	1	4	yes	—
77	C₇×#123	—	2	8	no	—
78	C₇×C₂²⋊Q₈	—	2	4	no	—
79	C₇×C₂²⋊Q₈	—	2	4	no	—
80	C₇×C₂²⋊Q₈	—	2	8	no	—
81	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
82	C₇×#121	—	2	4	no	—
83	C₇×#121	—	2	8	no	—
84	C₇×#121	—	2	2	no	—
85	C₇×#121	—	2	2	no	—
86	C₇×#121	—	2	4	no	—
87	C₇×#121	C₂³	1	8	yes	—
88	C₄×C₇×Q₈	—	2	4	no	—
89	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
90	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
91	C₄×C₇×Q₈	C₂³	1	8	yes	—
92	C₄×C₇×D₄	—	2	8	no	—
93	C₄×C₇×D₄	—	2	2	no	—
94	C₄×C₇×D₄	—	2	4	no	—
95	C₄×C₇×D₄	—	2	2	no	—
96	C₄×C₇×D₄	D₄	1	1	yes	characteristic
97	C₄×C₇×D₄	D₄	1	2	yes	—
98	C₄×C₇×D₄	D₄	1	2	yes	—
99	C₄×C₇×D₄	D₄	1	1	yes	characteristic
100	C₄×C₇×D₄	C₂³	1	8	yes	—
101	C₇×#117	—	2	4	no	—
102	C₇×#117	—	2	4	no	—
103	C₇×#117	C₂³	1	1	yes	characteristic
104	C₇×#117	C₂³	1	2	yes	—
105	C₂×C₇×C₄⋊C₄	—	2	2	no	—
106	C₂×C₇×C₄⋊C₄	—	2	4	no	—
107	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
108	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₇	C₂³	1	2	yes	—
110	C₇×#107	C₂³	1	8	yes	—
111	C₇×#107	C₂³	1	8	yes	—
112	C₇×#103	C₂³	1	16	yes	—
113	C₇×#102	C₂³	1	16	yes	—
114	C₇×#97	C₂³	1	2	yes	—
115	C₇×#97	C₂³	1	4	yes	—
116	C₄×C₇×C₈	C₂³	1	2	yes	—
117	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
118	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
119	C₄×C₇×C₈	C₂³	1	2	yes	—
120	C₄×C₇×C₈	C₂³	1	4	yes	—
121	#3511	C₂×C₇	1	2	yes	complemented
122	#3182	C₂×C₇	1	4	yes	complemented
123	#3175	C₂×C₇	1	8	yes	complemented
124	#3170	C₂×C₇	1	2	yes	complemented
125	#3169	C₂×C₇	1	2	yes	complemented
126	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
127	#3147	C₂×C₇	1	1	yes	characteristic, complemented
128	#3141	C₂×C₇	1	2	yes	complemented
129	#3061	C₂×C₇	1	4	yes	complemented
130	#2951	C₂×C₇	1	4	yes	complemented
131	C₄×#131	C₂×C₇	1	1	yes	characteristic, complemented
132	C₇×#41	—	2	4	no	—
133	C₇×#41	—	2	16	no	—
134	C₇×#41	—	8	8	no	—
135	C₂×C₇×Q₈	—	2	4	no	—
136	C₂×C₇×Q₈	—	2	4	no	—
137	C₂×C₇×Q₈	—	8	4	no	—
138	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
139	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
140	C₂×C₇×D₄	—	2	4	no	—
141	C₂×C₇×D₄	—	2	8	no	—
142	C₂×C₇×D₄	—	2	2	no	—
143	C₂×C₇×D₄	—	2	2	no	—
144	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
145	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
146	C₂×C₇×D₄	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₇	—	2	4	no	—
148	C₂²×C₄×C₇	—	2	4	no	—
149	C₂²×C₄×C₇	—	2	8	no	—
150	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
156	C₇×Q₁₆	—	2	8	no	—
157	C₇×SD₁₆	—	2	16	no	—
158	C₇×D₈	—	2	8	no	—
159	C₇×M₁₆	—	2	4	no	—
160	C₇×M₁₆	—	2	8	no	—
161	C₂×C₇×C₈	—	2	8	no	—
162	C₂×C₇×C₈	#41	1	8	yes	complemented
163	C₂×C₇×C₈	C₂⁴	1	4	yes	—
164	C₂×C₇×C₈	C₂⁴	1	2	yes	—
165	C₂×C₇×C₈	C₂⁴	1	2	yes	—
166	C₇×C₄⋊C₄	—	2	4	no	—
167	C₇×C₄⋊C₄	—	2	4	no	—
168	C₇×C₄⋊C₄	—	2	8	no	—
169	C₇×C₄⋊C₄	—	2	4	no	—
170	C₇×C₄⋊C₄	—	2	2	no	—
171	C₇×C₄⋊C₄	—	2	2	no	—
172	C₇×C₄⋊C₄	—	2	8	no	—
173	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
174	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
175	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
176	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
177	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
178	C₇×C₂²⋊C₄	—	2	8	no	—
179	C₇×C₂²⋊C₄	—	2	8	no	—
180	C₇×C₂²⋊C₄	—	2	8	no	—
181	C₇×C₂²⋊C₄	—	2	4	no	—
182	C₇×C₂²⋊C₄	—	2	4	no	—
183	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
184	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
185	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
186	C₄²×C₇	—	2	8	no	—
187	C₄²×C₇	C₂×D₄	1	2	yes	—
188	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₇	C₂×D₄	1	2	yes	—
191	C₄²×C₇	C₂⁴	1	1	yes	characteristic
192	C₄²×C₇	C₂⁴	1	2	yes	—
193	C₄²×C₇	C₂⁴	1	1	yes	characteristic
194	C₂×#135	C₂²×C₇	1	4	yes	—
195	C₂×#131	C₂²×C₇	1	2	yes	—
196	#556	—	2	2	no	—
197	#555	—	2	2	no	—
198	#554	—	2	2	no	—
199	#553	—	2	4	no	—
200	#548	—	2	4	no	—
201	#547	—	2	2	no	—
202	#542	C₂²×C₇	1	8	yes	—
203	#537	C₂²×C₇	1	4	yes	—
204	#533	C₂²×C₇	1	4	yes	—
205	#532	C₂²×C₇	1	2	yes	—
206	#532	C₂²×C₇	1	4	yes	—
207	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
208	#520	C₂²×C₇	1	4	yes	—
209	C₄×#41	C₂²×C₇	1	1	yes	characteristic
210	#500	C₂²×C₇	1	2	yes	—
211	#500	C₂²×C₇	1	4	yes	—
212	#500	C₂²×C₇	1	2	yes	—
213	#495	C₂²×C₇	1	4	yes	—
214	#494	C₂²×C₇	1	1	yes	characteristic
215	#493	C₂²×C₇	1	1	yes	characteristic
216	#492	C₂²×C₇	1	2	yes	—
217	#489	C₂²×C₇	1	8	yes	—
218	#487	C₂²×C₇	1	4	yes	—
219	#486	C₂²×C₇	1	4	yes	—
220	#468	C₂²×C₇	1	16	yes	—
221	#467	C₂²×C₇	1	8	yes	—
222	#466	C₂²×C₇	1	8	yes	—
223	#440	C₂²×C₇	1	16	yes	—
224	C₄×Q₁₆	C₂²×C₇	1	8	yes	—
225	C₄×D₈	C₂²×C₇	1	8	yes	—
226	C₂×#107	C₂²×C₇	1	4	yes	—
227	#417	C₂²×C₇	1	8	yes	—
228	#405	C₂²×C₇	1	4	yes	—
229	#405	C₂²×C₇	1	2	yes	—
230	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
231	C₄×M₁₆	C₂²×C₇	1	2	yes	—
232	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
233	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
234	C₂³×C₇	—	2	4	no	—
235	C₂³×C₇	C₂²×D₄	1	2	yes	—
236	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
237	C₇×Q₈	—	2	2	no	—
238	C₇×Q₈	—	2	8	no	—
239	C₇×Q₈	—	4	8	no	—
240	C₇×Q₈	—	8	4	no	—
241	C₇×D₄	—	2	4	no	—
242	C₇×D₄	—	2	2	no	—
243	C₇×D₄	—	2	8	no	—
244	C₇×D₄	—	4	8	no	—
245	C₇×D₄	—	4	4	no	—
246	C₇×D₄	—	4	4	no	—
247	C₇×D₄	—	8	8	no	—
248	C₂×C₄×C₇	—	2	2	no	—
249	C₂×C₄×C₇	—	2	4	no	—
250	C₂×C₄×C₇	—	2	2	no	—
251	C₂×C₄×C₇	—	2	4	no	—
252	C₂×C₄×C₇	—	2	4	no	—
253	C₂×C₄×C₇	—	2	4	no	—
254	C₂×C₄×C₇	—	2	8	no	—
255	C₂×C₄×C₇	—	4	8	no	—
256	C₂×C₄×C₇	—	8	8	no	—
257	C₂×C₄×C₇	—	8	4	no	—
258	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₇	C₂×#41	1	4	yes	—
267	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
268	C₇×C₈	—	2	4	no	—
269	C₇×C₈	C₂×#41	1	4	yes	—
270	C₇×C₈	C₂×#41	1	4	yes	—
271	C₂×#41	—	2	4	no	—
272	C₂×#41	C₂³×C₇	1	4	yes	—
273	C₂×#41	C₂³×C₇	1	1	yes	characteristic
274	C₂²×Q₈	—	2	2	no	—
275	#135	—	2	16	no	—
276	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
277	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
278	C₂×D₈	C₂³×C₇	1	4	yes	—
279	#131	—	2	8	no	—
280	C₂×M₁₆	C₂³×C₇	1	2	yes	—
281	C₂×M₁₆	C₂³×C₇	1	4	yes	—
282	C₂²×C₈	C₂³×C₇	1	4	yes	—
283	C₂²×C₈	C₂³×C₇	1	2	yes	—
284	C₄⋊Q₈	—	2	2	no	—
285	C₄⋊Q₈	—	2	2	no	—
286	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
287	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
288	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
289	#127	C₂³×C₇	1	2	yes	—
290	#126	—	2	8	no	—
291	#125	—	2	4	no	—
292	#124	—	2	8	no	—
293	#124	—	2	4	no	—
294	#124	C₂³×C₇	1	4	yes	—
295	#123	—	2	8	no	—
296	C₂²⋊Q₈	—	2	4	no	—
297	C₂²⋊Q₈	—	2	4	no	—
298	C₂²⋊Q₈	—	2	8	no	—
299	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
300	#121	—	2	4	no	—
301	#121	—	2	8	no	—
302	#121	—	2	2	no	—
303	#121	—	2	2	no	—
304	#121	—	2	4	no	—
305	#121	C₂³×C₇	1	8	yes	—
306	C₄×Q₈	—	2	4	no	—
307	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
308	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
309	C₄×Q₈	C₂³×C₇	1	8	yes	—
310	C₄×D₄	—	2	8	no	—
311	C₄×D₄	—	2	2	no	—
312	C₄×D₄	—	2	4	no	—
313	C₄×D₄	—	2	2	no	—
314	C₄×D₄	C₇×D₄	1	1	yes	characteristic
315	C₄×D₄	C₇×D₄	1	2	yes	—
316	C₄×D₄	C₇×D₄	1	2	yes	—
317	C₄×D₄	C₇×D₄	1	1	yes	characteristic
318	C₄×D₄	C₂³×C₇	1	8	yes	—
319	#117	—	2	4	no	—
320	#117	—	2	4	no	—
321	#117	C₂³×C₇	1	1	yes	characteristic
322	#117	C₂³×C₇	1	2	yes	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	4	no	—
325	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
326	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₇	1	2	yes	—
328	#107	C₂³×C₇	1	8	yes	—
329	#107	C₂³×C₇	1	8	yes	—
330	#103	C₂³×C₇	1	16	yes	—
331	#102	C₂³×C₇	1	16	yes	—
332	#97	C₂³×C₇	1	4	yes	—
333	#97	C₂³×C₇	1	2	yes	—
334	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
335	C₄×C₈	C₂³×C₇	1	2	yes	—
336	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
337	C₄×C₈	C₂³×C₇	1	4	yes	—
338	C₄×C₈	C₂³×C₇	1	2	yes	—
339	C₂²×C₇	—	2	1	no	—
340	C₂²×C₇	—	2	1	no	—
341	C₂²×C₇	—	2	2	no	—
342	C₂²×C₇	—	2	2	no	—
343	C₂²×C₇	—	2	2	no	—
344	C₂²×C₇	—	4	4	no	—
345	C₂²×C₇	—	4	4	no	—
346	C₂²×C₇	—	4	4	no	—
347	C₂²×C₇	#532	1	2	yes	—
348	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
349	C₄×C₇	—	2	2	no	—
350	C₄×C₇	—	2	2	no	—
351	C₄×C₇	—	2	1	no	—
352	C₄×C₇	—	2	2	no	—
353	C₄×C₇	—	2	1	no	—
354	C₄×C₇	—	4	4	no	—
355	C₄×C₇	—	4	8	no	—
356	C₄×C₇	#532	1	2	yes	—
357	C₄×C₇	#576	1	2	yes	—
358	C₄×C₇	#578	1	2	yes	—
359	C₄×C₇	C₂²×#41	1	2	yes	—
360	#41	—	2	4	no	—
361	#41	—	2	16	no	—
362	#41	—	8	8	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	8	4	no	—
366	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
367	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
373	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
374	C₂×D₄	C₂⁴×C₇	1	4	yes	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	8	no	—
378	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
379	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
380	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
381	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
382	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
383	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
384	Q₁₆	—	2	8	no	—
385	SD₁₆	—	2	16	no	—
386	D₈	—	2	8	no	—
387	M₁₆	—	2	4	no	—
388	M₁₆	—	2	8	no	—
389	C₂×C₈	—	2	8	no	—
390	C₂×C₈	C₇×#41	1	8	yes	complemented
391	C₂×C₈	C₂⁴×C₇	1	4	yes	—
392	C₂×C₈	C₂⁴×C₇	1	2	yes	—
393	C₂×C₈	C₂⁴×C₇	1	2	yes	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	8	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
402	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
403	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
404	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
405	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₂²⋊C₄	—	2	8	no	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₂²⋊C₄	—	2	4	no	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
412	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
413	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
414	C₄²	—	2	8	no	—
415	C₄²	C₂×C₇×D₄	1	2	yes	—
416	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
417	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
418	C₄²	C₂×C₇×D₄	1	2	yes	—
419	C₄²	C₂⁴×C₇	1	1	yes	characteristic
420	C₄²	C₂⁴×C₇	1	2	yes	—
421	C₄²	C₂⁴×C₇	1	1	yes	characteristic
422	C₂×C₇	—	2	2	no	—
423	C₂×C₇	—	4	2	no	—
424	C₂×C₇	—	8	4	no	—
425	C₂×C₇	C₂×#532	1	1	yes	characteristic
426	C₂×C₇	C₂×#576	1	1	yes	characteristic
427	C₂×C₇	C₂×#578	1	1	yes	characteristic
428	C₂³	—	2	4	no	—
429	C₂³	C₂²×C₇×D₄	1	2	yes	—
430	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
431	Q₈	—	2	2	no	—
432	Q₈	—	2	8	no	—
433	Q₈	—	4	8	no	—
434	Q₈	—	8	4	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	8	no	—
437	D₄	—	2	2	no	—
438	D₄	—	4	8	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	4	no	—
441	D₄	—	8	8	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	4	8	no	—
450	C₂×C₄	—	8	8	no	—
451	C₂×C₄	—	8	4	no	—
452	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
453	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
454	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
455	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
456	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
457	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
458	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
459	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
460	C₂×C₄	C₂×C₇×#41	1	4	yes	—
461	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
462	C₈	—	2	4	no	—
463	C₈	C₂×C₇×#41	1	4	yes	—
464	C₈	C₂×C₇×#41	1	4	yes	—
465	C₇	#60718	1	1	yes	characteristic, direct product factor, Sylow
466	C₂²	—	2	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	C₇×#532	1	2	yes	—
475	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
476	C₄	—	2	2	no	—
477	C₄	—	2	1	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	2	no	—
481	C₄	—	4	8	no	—
482	C₄	—	4	4	no	—
483	C₄	C₇×#532	1	2	yes	—
484	C₄	C₇×#576	1	2	yes	—
485	C₄	C₇×#578	1	2	yes	—
486	C₄	C₂²×C₇×#41	1	2	yes	—
487	C₂	—	2	2	no	—
488	C₂	—	4	2	no	—
489	C₂	—	8	4	no	—
490	C₂	C₂×C₇×#532	1	1	yes	characteristic
491	C₂	C₂×C₇×#576	1	1	yes	characteristic
492	C₂	C₂×C₇×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
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
1. C₇×#3511	C₂	semidirect
2. C₇×#3182	C₂	semidirect
3. C₇×#3175	C₂	semidirect
4. C₇×#3170	C₂	semidirect
5. C₇×#3169	C₂	semidirect
6. C₂×C₇×#500	C₂	semidirect
7. C₇×#3147	C₂	semidirect
8. C₇×#3141	C₂	semidirect
9. C₇×#3061	C₂	semidirect
10. C₇×#2951	C₂	semidirect
11. C₄×C₇×#131	C₂	semidirect
52. #60718	C₇	direct (N × H)
121. #3511	C₂×C₇	semidirect
122. #3182	C₂×C₇	semidirect
123. #3175	C₂×C₇	semidirect
124. #3170	C₂×C₇	semidirect
125. #3169	C₂×C₇	semidirect
126. C₂×#500	C₂×C₇	semidirect
127. #3147	C₂×C₇	semidirect
128. #3141	C₂×C₇	semidirect
129. #3061	C₂×C₇	semidirect
130. #2951	C₂×C₇	semidirect
131. C₄×#131	C₂×C₇	semidirect
162. C₂×C₇×C₈	#41	semidirect
390. C₂×C₈	C₇×#41	semidirect
465. C₇	#60718	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×#3511	2
C₇×#3182	2
C₇×#3175	2
C₇×#3170	2
C₇×#3169	2
C₂×C₇×#500	2
C₇×#3147	2
C₇×#3141	2
C₇×#3061	2
C₇×#2951	2
C₄×C₇×#131	2
#60718	7

Subgroups — C7×#60718 #13237934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60718 #13237934