Subgroups — C₇×#60626 #13237842

Order 1792.

2134 conjugacy classes of subgroups.

The subgroup lattice has 342 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, complemented
2	C₇×#3498	C₂	1	1	yes	maximal, characteristic
3	C₇×#3142	C₂	1	3	yes	maximal, complemented
4	C₇×#3138	C₂	1	3	yes	maximal, complemented
5	C₇×#3136	C₂	1	12	yes	maximal, complemented
6	C₇×#3135	C₂	1	3	yes	maximal, complemented
7	C₂×C₇×#490	C₂	1	3	yes	maximal, complemented
8	C₇×#2897	C₂	1	4	yes	maximal, complemented
9	C₇×#2875	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
11	C₂×C₇×#131	C₂²	1	2	yes	—
12	C₇×#557	—	2	2	no	—
13	C₇×#554	—	2	6	no	—
14	C₇×#550	—	2	2	no	—
15	C₇×D₄²	—	2	6	no	—
16	C₇×#544	C₂²	1	6	yes	complemented
17	C₇×#535	C₂²	1	6	yes	—
18	C₇×#533	C₂²	1	3	yes	complemented
19	C₇×#532	C₂²	1	3	yes	—
20	C₂×C₇×#127	C₂²	1	3	yes	—
21	C₂×C₇×#125	C₂²	1	3	yes	complemented
22	C₇×#520	C₂²	1	2	yes	—
23	C₄×C₇×#41	C₂²	1	1	yes	characteristic
24	C₇×#491	C₂²	1	1	yes	characteristic, complemented
25	C₇×#491	C₂²	1	3	yes	complemented
26	C₇×#490	C₂²	1	3	yes	complemented
27	C₇×#490	C₂²	1	12	yes	complemented
28	C₇×#490	C₂²	1	6	yes	complemented
29	C₇×#490	C₂²	1	3	yes	complemented
30	C₇×#489	C₂²	1	12	yes	complemented
31	C₇×#488	C₂²	1	12	yes	complemented
32	C₇×#486	C₂²	1	12	yes	complemented
33	C₇×#418	C₂²	1	12	yes	complemented
34	C₇×#417	C₂²	1	12	yes	complemented
35	C₇×#416	C₂²	1	12	yes	complemented
36	C₂×C₇×#102	C₂²	1	12	yes	complemented
37	C₇×#405	C₂²	1	6	yes	complemented
38	C₄×C₇×M₁₆	C₂²	1	3	yes	complemented
39	C₂×C₇×#97	C₂²	1	3	yes	complemented
40	#60626	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₇×H(5,𝔽₂)	—	2	16	no	—
42	C₂×C₇×#41	—	2	2	no	—
43	C₂×C₇×#41	C₂³	1	6	yes	—
44	C₂×C₇×#41	C₂³	1	1	yes	characteristic
45	C₂²×C₇×D₄	—	2	6	no	—
46	C₂²×C₇×D₄	—	2	6	no	—
47	C₂²×C₇×D₄	C₂³	1	6	yes	—
48	C₇×#131	—	2	8	no	—
49	C₂×C₇×M₁₆	C₂³	1	6	yes	—
50	C₂²×C₇×C₈	C₂³	1	6	yes	—
51	C₇×C₄⋊Q₈	—	2	6	no	—
52	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
53	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
54	C₇×#127	—	2	6	no	—
55	C₇×#127	C₂³	1	6	yes	—
56	C₇×#127	C₂³	1	3	yes	—
57	C₇×#125	—	2	6	no	—
58	C₇×#125	—	2	6	no	—
59	C₇×#125	C₂³	1	6	yes	—
60	C₇×#125	C₂³	1	3	yes	—
61	C₇×#125	C₂³	1	1	yes	characteristic
62	C₇×#124	C₂³	1	6	yes	—
63	C₇×C₂²⋊Q₈	—	2	6	no	—
64	C₇×C₂²⋊Q₈	—	2	12	no	—
65	C₇×#121	—	2	6	no	—
66	C₇×#121	—	2	12	no	—
67	C₇×#120	—	2	12	no	—
68	C₄×C₇×Q₈	—	2	2	no	—
69	C₄×C₇×Q₈	—	2	6	no	—
70	C₄×C₇×Q₈	D₄	1	2	yes	—
71	C₄×C₇×D₄	—	2	6	no	—
72	C₄×C₇×D₄	—	2	6	no	—
73	C₄×C₇×D₄	D₄	1	6	yes	—
74	C₇×#117	C₂³	1	6	yes	—
75	C₇×#117	C₂³	1	3	yes	—
76	C₂×C₇×C₄⋊C₄	—	2	6	no	—
77	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
78	C₂×C₄²×C₇	C₂³	1	3	yes	—
79	C₇×#103	C₂³	1	12	yes	—
80	C₇×#103	C₂³	1	4	yes	—
81	C₇×#102	C₂³	1	12	yes	—
82	C₇×#102	C₂³	1	24	yes	—
83	C₇×#102	C₂³	1	12	yes	—
84	C₇×#97	C₂³	1	6	yes	—
85	C₇×#97	C₂³	1	1	yes	characteristic
86	C₇×#97	C₂³	1	3	yes	—
87	C₄×C₇×C₈	C₂³	1	6	yes	—
88	#3504	C₂×C₇	1	1	yes	characteristic, complemented
89	#3498	C₂×C₇	1	1	yes	characteristic
90	#3142	C₂×C₇	1	3	yes	complemented
91	#3138	C₂×C₇	1	3	yes	complemented
92	#3136	C₂×C₇	1	12	yes	complemented
93	#3135	C₂×C₇	1	3	yes	complemented
94	C₂×#490	C₂×C₇	1	3	yes	complemented
95	#2897	C₂×C₇	1	4	yes	complemented
96	#2875	C₂×C₇	1	1	yes	characteristic, complemented
97	C₂⁴×C₇	—	2	6	no	—
98	C₇×#41	—	2	24	no	—
99	C₇×#41	—	4	8	no	—
100	C₇×#41	#41	1	8	yes	—
101	C₂×C₇×Q₈	—	2	6	no	—
102	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
104	C₂×C₇×D₄	—	2	6	no	—
105	C₂×C₇×D₄	—	2	6	no	—
106	C₂×C₇×D₄	—	2	6	no	—
107	C₂×C₇×D₄	—	2	12	no	—
108	C₂×C₇×D₄	—	2	24	no	—
109	C₂×C₇×D₄	—	4	24	no	—
110	C₂×C₇×D₄	—	8	12	no	—
111	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
112	C₂×C₇×D₄	C₂⁴	1	6	yes	—
113	C₂²×C₄×C₇	—	2	6	no	—
114	C₂²×C₄×C₇	—	2	6	no	—
115	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
116	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
117	C₇×M₁₆	—	2	12	no	—
118	C₂×C₇×C₈	—	2	12	no	—
119	C₂×C₇×C₈	C₂⁴	1	6	yes	—
120	C₂×C₇×C₈	C₂⁴	1	2	yes	—
121	C₇×C₄⋊C₄	—	2	6	no	—
122	C₇×C₄⋊C₄	—	2	12	no	—
123	C₇×C₄⋊C₄	—	2	6	no	—
124	C₇×C₄⋊C₄	—	2	2	no	—
125	C₇×C₄⋊C₄	—	2	2	no	—
126	C₇×C₄⋊C₄	—	2	6	no	—
127	C₇×C₄⋊C₄	C₂×D₄	1	6	yes	—
128	C₇×C₄⋊C₄	C₂⁴	1	6	yes	—
129	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
130	C₇×C₂²⋊C₄	—	2	6	no	—
131	C₇×C₂²⋊C₄	—	2	12	no	—
132	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
133	C₄²×C₇	—	2	6	no	—
134	C₄²×C₇	C₂×D₄	1	6	yes	—
135	C₄²×C₇	C₂⁴	1	3	yes	—
136	C₄²×C₇	C₂⁴	1	1	yes	characteristic
137	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
138	C₂×#131	C₂²×C₇	1	2	yes	—
139	#557	—	2	2	no	—
140	#554	—	2	6	no	—
141	#550	—	2	2	no	—
142	D₄²	—	2	6	no	—
143	#544	C₂²×C₇	1	6	yes	complemented
144	#535	C₂²×C₇	1	6	yes	—
145	#533	C₂²×C₇	1	3	yes	complemented
146	#532	C₂²×C₇	1	3	yes	—
147	C₂×#127	C₂²×C₇	1	3	yes	—
148	C₂×#125	C₂²×C₇	1	3	yes	complemented
149	#520	C₂²×C₇	1	2	yes	—
150	C₄×#41	C₂²×C₇	1	1	yes	characteristic
151	#491	C₂²×C₇	1	1	yes	characteristic, complemented
152	#491	C₂²×C₇	1	3	yes	complemented
153	#490	C₂²×C₇	1	3	yes	complemented
154	#490	C₂²×C₇	1	12	yes	complemented
155	#490	C₂²×C₇	1	6	yes	complemented
156	#490	C₂²×C₇	1	3	yes	complemented
157	#489	C₂²×C₇	1	12	yes	complemented
158	#488	C₂²×C₇	1	12	yes	complemented
159	#486	C₂²×C₇	1	12	yes	complemented
160	#418	C₂²×C₇	1	12	yes	complemented
161	#417	C₂²×C₇	1	12	yes	complemented
162	#416	C₂²×C₇	1	12	yes	complemented
163	C₂×#102	C₂²×C₇	1	12	yes	complemented
164	#405	C₂²×C₇	1	6	yes	complemented
165	C₄×M₁₆	C₂²×C₇	1	3	yes	complemented
166	C₂×#97	C₂²×C₇	1	3	yes	complemented
167	C₂³×C₇	—	2	6	no	—
168	C₂³×C₇	—	4	12	no	—
169	C₂³×C₇	—	8	12	no	—
170	C₂³×C₇	C₂²×D₄	1	3	yes	—
171	C₇×Q₈	—	2	4	no	—
172	C₇×Q₈	—	8	6	no	—
173	C₇×Q₈	#124	1	4	yes	—
174	C₇×D₄	—	2	12	no	—
175	C₇×D₄	—	4	12	no	—
176	C₇×D₄	—	4	12	no	—
177	C₇×D₄	—	4	12	no	—
178	C₇×D₄	—	8	12	no	—
179	C₇×D₄	#124	1	12	yes	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	—	2	6	no	—
182	C₂×C₄×C₇	—	2	2	no	—
183	C₂×C₄×C₇	—	4	6	no	—
184	C₂×C₄×C₇	—	4	12	no	—
185	C₂×C₄×C₇	—	8	6	no	—
186	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
188	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
189	C₂×C₄×C₇	C₂×#41	1	12	yes	—
190	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
191	C₇×C₈	—	2	2	no	—
192	C₇×C₈	—	2	6	no	—
193	H(5,𝔽₂)	—	2	16	no	—
194	C₂×#41	—	2	2	no	—
195	C₂×#41	C₂³×C₇	1	6	yes	—
196	C₂×#41	C₂³×C₇	1	1	yes	characteristic
197	C₂²×D₄	—	2	6	no	—
198	C₂²×D₄	—	2	6	no	—
199	C₂²×D₄	C₂³×C₇	1	6	yes	—
200	#131	—	2	8	no	—
201	C₂×M₁₆	C₂³×C₇	1	6	yes	—
202	C₂²×C₈	C₂³×C₇	1	6	yes	—
203	C₄⋊Q₈	—	2	6	no	—
204	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
205	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
206	#127	—	2	6	no	—
207	#127	C₂³×C₇	1	3	yes	—
208	#127	C₂³×C₇	1	6	yes	—
209	#125	—	2	6	no	—
210	#125	—	2	6	no	—
211	#125	C₂³×C₇	1	3	yes	—
212	#125	C₂³×C₇	1	6	yes	—
213	#125	C₂³×C₇	1	1	yes	characteristic
214	#124	C₂³×C₇	1	6	yes	—
215	C₂²⋊Q₈	—	2	6	no	—
216	C₂²⋊Q₈	—	2	12	no	—
217	#121	—	2	6	no	—
218	#121	—	2	12	no	—
219	#120	—	2	12	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	2	6	no	—
222	C₄×Q₈	C₇×D₄	1	2	yes	—
223	C₄×D₄	—	2	6	no	—
224	C₄×D₄	—	2	6	no	—
225	C₄×D₄	C₇×D₄	1	6	yes	—
226	#117	C₂³×C₇	1	6	yes	—
227	#117	C₂³×C₇	1	3	yes	—
228	C₂×C₄⋊C₄	—	2	6	no	—
229	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
230	C₂×C₄²	C₂³×C₇	1	3	yes	—
231	#103	C₂³×C₇	1	4	yes	—
232	#103	C₂³×C₇	1	12	yes	—
233	#102	C₂³×C₇	1	12	yes	—
234	#102	C₂³×C₇	1	24	yes	—
235	#102	C₂³×C₇	1	12	yes	—
236	#97	C₂³×C₇	1	1	yes	characteristic
237	#97	C₂³×C₇	1	3	yes	—
238	#97	C₂³×C₇	1	6	yes	—
239	C₄×C₈	C₂³×C₇	1	6	yes	—
240	C₂²×C₇	—	2	6	no	—
241	C₂²×C₇	—	4	12	no	—
242	C₂²×C₇	—	4	6	no	—
243	C₂²×C₇	—	16	12	no	—
244	C₂²×C₇	C₂×#124	1	6	yes	—
245	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₇	—	2	2	no	—
247	C₄×C₇	—	2	6	no	—
248	C₄×C₇	—	4	6	no	—
249	C₄×C₇	—	4	2	no	—
250	C₄×C₇	—	4	2	no	—
251	C₄×C₇	C₂×#124	1	6	yes	—
252	C₄×C₇	C₂²×#41	1	2	yes	—
253	C₂⁴	—	2	6	no	—
254	#41	—	2	24	no	—
255	#41	—	4	8	no	—
256	#41	C₇×#41	1	8	yes	—
257	C₂×Q₈	—	2	6	no	—
258	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
259	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
260	C₂×D₄	—	2	6	no	—
261	C₂×D₄	—	2	6	no	—
262	C₂×D₄	—	2	6	no	—
263	C₂×D₄	—	2	12	no	—
264	C₂×D₄	—	2	24	no	—
265	C₂×D₄	—	4	24	no	—
266	C₂×D₄	—	8	12	no	—
267	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
268	C₂×D₄	C₂⁴×C₇	1	6	yes	—
269	C₂²×C₄	—	2	6	no	—
270	C₂²×C₄	—	2	6	no	—
271	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
272	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
273	M₁₆	—	2	12	no	—
274	C₂×C₈	—	2	12	no	—
275	C₂×C₈	C₂⁴×C₇	1	6	yes	—
276	C₂×C₈	C₂⁴×C₇	1	2	yes	—
277	C₄⋊C₄	—	2	6	no	—
278	C₄⋊C₄	—	2	12	no	—
279	C₄⋊C₄	—	2	6	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	6	no	—
283	C₄⋊C₄	C₂×C₇×D₄	1	6	yes	—
284	C₄⋊C₄	C₂⁴×C₇	1	6	yes	—
285	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
286	C₂²⋊C₄	—	2	6	no	—
287	C₂²⋊C₄	—	2	12	no	—
288	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
289	C₄²	—	2	6	no	—
290	C₄²	C₂×C₇×D₄	1	6	yes	—
291	C₄²	C₂⁴×C₇	1	3	yes	—
292	C₄²	C₂⁴×C₇	1	1	yes	characteristic
293	C₂×C₇	—	2	6	no	—
294	C₂×C₇	—	8	6	no	—
295	C₂×C₇	C₂²×#124	1	1	yes	characteristic
296	C₂×C₇	#3585	1	2	yes	—
297	C₂³	—	2	6	no	—
298	C₂³	—	4	12	no	—
299	C₂³	—	8	12	no	—
300	C₂³	C₂²×C₇×D₄	1	3	yes	—
301	Q₈	—	2	4	no	—
302	Q₈	—	8	6	no	—
303	Q₈	C₇×#124	1	4	yes	—
304	D₄	—	2	12	no	—
305	D₄	—	4	12	no	—
306	D₄	—	4	12	no	—
307	D₄	—	4	12	no	—
308	D₄	—	8	12	no	—
309	D₄	C₇×#124	1	12	yes	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	6	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	6	no	—
314	C₂×C₄	—	4	12	no	—
315	C₂×C₄	—	8	6	no	—
316	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
317	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
318	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
319	C₂×C₄	C₂×C₇×#41	1	12	yes	—
320	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	2	6	no	—
323	C₇	#60626	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²	—	2	6	no	—
325	C₂²	—	4	12	no	—
326	C₂²	—	4	6	no	—
327	C₂²	—	16	12	no	—
328	C₂²	C₂×C₇×#124	1	6	yes	—
329	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	6	no	—
332	C₄	—	4	6	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	2	no	—
335	C₄	C₂×C₇×#124	1	6	yes	—
336	C₄	C₂²×C₇×#41	1	2	yes	—
337	C₂	—	2	6	no	—
338	C₂	—	8	6	no	—
339	C₂	C₂²×C₇×#124	1	1	yes	characteristic
340	C₂	C₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3498	C₂
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×#3504	C₂	semidirect
3. C₇×#3142	C₂	semidirect
4. C₇×#3138	C₂	semidirect
5. C₇×#3136	C₂	semidirect
6. C₇×#3135	C₂	semidirect
7. C₂×C₇×#490	C₂	semidirect
8. C₇×#2897	C₂	semidirect
9. C₇×#2875	C₂	semidirect
16. C₇×#544	C₂²	semidirect
18. C₇×#533	C₂²	semidirect
21. C₂×C₇×#125	C₂²	semidirect
24. C₇×#491	C₂²	semidirect
25. C₇×#491	C₂²	semidirect
26. C₇×#490	C₂²	semidirect
27. C₇×#490	C₂²	semidirect
28. C₇×#490	C₂²	semidirect
29. C₇×#490	C₂²	semidirect
30. C₇×#489	C₂²	semidirect
31. C₇×#488	C₂²	semidirect
32. C₇×#486	C₂²	semidirect
33. C₇×#418	C₂²	semidirect
34. C₇×#417	C₂²	semidirect
35. C₇×#416	C₂²	semidirect
36. C₂×C₇×#102	C₂²	semidirect
37. C₇×#405	C₂²	semidirect
38. C₄×C₇×M₁₆	C₂²	semidirect
39. C₂×C₇×#97	C₂²	semidirect
40. #60626	C₇	direct (N × H)
88. #3504	C₂×C₇	semidirect
90. #3142	C₂×C₇	semidirect
91. #3138	C₂×C₇	semidirect
92. #3136	C₂×C₇	semidirect
93. #3135	C₂×C₇	semidirect
94. C₂×#490	C₂×C₇	semidirect
95. #2897	C₂×C₇	semidirect
96. #2875	C₂×C₇	semidirect
143. #544	C₂²×C₇	semidirect
145. #533	C₂²×C₇	semidirect
148. C₂×#125	C₂²×C₇	semidirect
151. #491	C₂²×C₇	semidirect
152. #491	C₂²×C₇	semidirect
153. #490	C₂²×C₇	semidirect
154. #490	C₂²×C₇	semidirect
155. #490	C₂²×C₇	semidirect
156. #490	C₂²×C₇	semidirect
157. #489	C₂²×C₇	semidirect
158. #488	C₂²×C₇	semidirect
159. #486	C₂²×C₇	semidirect
160. #418	C₂²×C₇	semidirect
161. #417	C₂²×C₇	semidirect
162. #416	C₂²×C₇	semidirect
163. C₂×#102	C₂²×C₇	semidirect
164. #405	C₂²×C₇	semidirect
165. C₄×M₁₆	C₂²×C₇	semidirect
166. C₂×#97	C₂²×C₇	semidirect
323. C₇	#60626	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×#3498	2
C₇×#3142	2
C₇×#3138	2
C₇×#3136	2
C₇×#3135	2
C₂×C₇×#490	2
C₇×#2897	2
C₇×#2875	2
#60626	7

Subgroups — C7×#60626 #13237842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60626 #13237842