Subgroups — C₇×#10805 #13188021

Order 1792.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#344	C₂	1	2	yes	maximal
3	C₂×C₇×#339	C₂	1	2	yes	maximal
4	C₂×C₇×#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₇×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₄×C₇×M₁₆	—	2	2	no	—
12	C₄×C₇×M₁₆	—	2	2	no	—
13	C₂×C₇×#97	C₄	1	2	yes	complemented
14	C₂×C₄×C₇×C₈	C₄	1	2	yes	complemented
15	C₇×#344	C₄	1	8	yes	complemented
16	C₇×#339	—	2	4	no	—
17	C₇×#338	—	2	2	no	—
18	C₇×H(3,ℤ/4)	—	2	2	no	—
19	#10805	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₇	—	2	1	no	—
21	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×C₇×M₁₆	—	2	1	no	—
23	C₂×C₇×M₁₆	—	2	1	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	—	2	2	no	—
27	C₂×C₇×M₁₆	—	2	2	no	—
28	C₂×C₇×M₁₆	—	2	2	no	—
29	C₂×C₇×M₁₆	—	2	2	no	—
30	C₂×C₇×M₁₆	—	2	2	no	—
31	C₂×C₇×M₁₆	—	2	2	no	—
32	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₇×M₁₆	D₄	1	2	yes	—
35	C₂×C₇×M₁₆	D₄	1	2	yes	—
36	C₂²×C₇×C₈	—	2	2	no	—
37	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
39	C₇×#117	—	2	2	no	—
40	C₇×#117	—	2	2	no	—
41	C₇×#117	—	2	2	no	—
42	C₇×#117	C₂×C₄	1	4	yes	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₄²×C₇	—	2	1	no	—
46	C₂×C₄²×C₇	—	2	1	no	—
47	C₂×C₄²×C₇	—	2	1	no	—
48	C₂×C₄²×C₇	—	2	2	no	—
49	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇	D₄	1	2	yes	—
52	C₂×C₄²×C₇	Q₈	1	2	yes	—
53	C₇×#97	—	2	4	no	—
54	C₄×C₇×C₈	—	2	4	no	—
55	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
56	C₂×#344	C₂×C₇	1	2	yes	—
57	C₂×#339	C₂×C₇	1	2	yes	—
58	C₂×#338	C₂×C₇	1	1	yes	characteristic
59	C₂×H(3,ℤ/4)	C₂×C₇	1	1	yes	characteristic
60	C₂⁴×C₇	#41	1	1	yes	characteristic
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	—	2	1	no	—
63	C₂²×C₄×C₇	—	2	2	no	—
64	C₂²×C₄×C₇	—	2	2	no	—
65	C₂²×C₄×C₇	—	4	4	no	—
66	C₂²×C₄×C₇	—	4	2	no	—
67	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₇	#41	1	1	yes	characteristic
74	C₂²×C₄×C₇	#41	1	2	yes	—
75	C₂²×C₄×C₇	#41	1	2	yes	—
76	C₂²×C₄×C₇	#41	1	2	yes	—
77	C₇×M₁₆	—	2	4	no	—
78	C₇×M₁₆	—	2	4	no	—
79	C₇×M₁₆	—	2	4	no	—
80	C₇×M₁₆	—	2	4	no	—
81	C₇×M₁₆	—	2	4	no	—
82	C₇×M₁₆	—	2	4	no	—
83	C₇×M₁₆	—	4	2	no	—
84	C₇×M₁₆	—	4	2	no	—
85	C₂×C₇×C₈	—	2	2	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	4	no	—
90	C₂×C₇×C₈	—	4	2	no	—
91	C₂×C₇×C₈	—	4	2	no	—
92	C₂×C₇×C₈	C₄²	1	4	yes	—
93	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₄⋊C₄	—	4	2	no	—
96	C₇×C₂²⋊C₄	—	2	4	no	—
97	C₇×C₂²⋊C₄	—	4	2	no	—
98	C₄²×C₇	—	2	1	no	—
99	C₄²×C₇	—	2	1	no	—
100	C₄²×C₇	—	2	2	no	—
101	C₄²×C₇	—	2	2	no	—
102	C₄²×C₇	—	2	4	no	—
103	C₄²×C₇	—	4	2	no	—
104	C₄²×C₇	C₄⋊C₄	1	2	yes	—
105	C₄²×C₇	C₄⋊C₄	1	2	yes	—
106	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
107	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
108	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
109	C₂×#117	C₂²×C₇	1	2	yes	—
110	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
111	C₄×M₁₆	—	2	2	no	—
112	C₄×M₁₆	—	2	2	no	—
113	C₂×#97	C₄×C₇	1	2	yes	complemented
114	C₂×C₄×C₈	C₄×C₇	1	2	yes	complemented
115	#344	C₄×C₇	1	8	yes	complemented
116	#339	—	2	4	no	—
117	#338	—	2	2	no	—
118	H(3,ℤ/4)	—	2	2	no	—
119	C₂³×C₇	—	4	2	no	—
120	C₂³×C₇	#117	1	1	yes	characteristic
121	C₂³×C₇	#117	1	2	yes	—
122	C₂³×C₇	#121	1	1	yes	characteristic
123	C₂³×C₇	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂³×C₇	#126	1	2	yes	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	4	4	no	—
129	C₂×C₄×C₇	—	4	4	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	—	8	4	no	—
133	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇	#117	1	1	yes	characteristic
137	C₂×C₄×C₇	#117	1	1	yes	characteristic
138	C₂×C₄×C₇	#117	1	2	yes	—
139	C₂×C₄×C₇	#117	1	1	yes	characteristic
140	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
144	C₂×C₄×C₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄×C₇	#123	1	2	yes	—
147	C₂×C₄×C₇	#123	1	2	yes	—
148	C₂×C₄×C₇	#124	1	2	yes	—
149	C₂×C₄×C₇	#125	1	2	yes	—
150	C₂×C₄×C₇	#126	1	2	yes	—
151	C₇×C₈	—	2	4	no	—
152	C₇×C₈	—	2	4	no	—
153	C₇×C₈	—	4	4	no	—
154	C₂³×C₄	—	2	1	no	—
155	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
156	C₂×M₁₆	—	2	1	no	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
167	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
168	C₂×M₁₆	C₇×D₄	1	2	yes	—
169	C₂×M₁₆	C₇×D₄	1	2	yes	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
172	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
173	#117	—	2	2	no	—
174	#117	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	C₂×C₄×C₇	1	4	yes	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₂²⋊C₄	—	2	2	no	—
179	C₂×C₄²	—	2	1	no	—
180	C₂×C₄²	—	2	1	no	—
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
185	C₂×C₄²	C₇×D₄	1	2	yes	—
186	C₂×C₄²	C₇×Q₈	1	2	yes	—
187	#97	—	2	4	no	—
188	C₄×C₈	—	2	4	no	—
189	C₂²×C₇	—	2	2	no	—
190	C₂²×C₇	—	2	4	no	—
191	C₂²×C₇	—	4	1	no	—
192	C₂²×C₇	—	4	1	no	—
193	C₂²×C₇	—	4	2	no	—
194	C₂²×C₇	#376	1	1	yes	characteristic
195	C₂²×C₇	#380	1	1	yes	characteristic
196	C₂²×C₇	#383	1	1	yes	characteristic
197	C₂²×C₇	#387	1	2	yes	—
198	C₂²×C₇	#388	1	2	yes	—
199	C₄×C₇	—	2	4	no	—
200	C₄×C₇	—	4	2	no	—
201	C₄×C₇	—	8	4	no	—
202	C₄×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
203	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
204	C₄×C₇	#381	1	1	yes	characteristic
205	C₄×C₇	#382	1	1	yes	characteristic
206	C₄×C₇	#387	1	2	yes	—
207	C₄×C₇	#388	1	2	yes	—
208	C₂⁴	C₇×#41	1	1	yes	characteristic
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
217	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
221	C₂²×C₄	C₇×#41	1	1	yes	characteristic
222	C₂²×C₄	C₇×#41	1	2	yes	—
223	C₂²×C₄	C₇×#41	1	2	yes	—
224	C₂²×C₄	C₇×#41	1	2	yes	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	4	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	C₄²×C₇	1	4	yes	—
241	C₂×C₈	C₇×C₂²⋊C₄	1	4	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	4	2	no	—
252	C₄²	C₇×C₄⋊C₄	1	2	yes	—
253	C₄²	C₇×C₄⋊C₄	1	2	yes	—
254	C₂×C₇	—	2	1	no	—
255	C₂×C₇	—	2	2	no	—
256	C₂×C₇	—	2	1	no	—
257	C₂×C₇	—	2	2	no	—
258	C₂×C₇	#1434	1	1	yes	characteristic
259	C₂×C₇	#1841	1	2	yes	—
260	C₂³	—	4	2	no	—
261	C₂³	C₇×#117	1	1	yes	characteristic
262	C₂³	C₇×#117	1	2	yes	—
263	C₂³	C₇×#121	1	1	yes	characteristic
264	C₂³	C₇×C₂²⋊Q₈	1	1	yes	characteristic
265	C₂³	C₇×#126	1	2	yes	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₇×#117	1	1	yes	characteristic
278	C₂×C₄	C₇×#117	1	1	yes	characteristic
279	C₂×C₄	C₇×#117	1	1	yes	characteristic
280	C₂×C₄	C₇×#117	1	2	yes	—
281	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
282	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
283	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
284	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
285	C₂×C₄	C₇×#121	1	1	yes	characteristic
286	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
287	C₂×C₄	C₇×#123	1	2	yes	—
288	C₂×C₄	C₇×#123	1	2	yes	—
289	C₂×C₄	C₇×#124	1	2	yes	—
290	C₂×C₄	C₇×#125	1	2	yes	—
291	C₂×C₄	C₇×#126	1	2	yes	—
292	C₈	—	2	4	no	—
293	C₈	—	2	4	no	—
294	C₈	—	4	4	no	—
295	C₇	#10805	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₇×#376	1	1	yes	characteristic
302	C₂²	C₇×#380	1	1	yes	characteristic
303	C₂²	C₇×#383	1	1	yes	characteristic
304	C₂²	C₇×#387	1	2	yes	—
305	C₂²	C₇×#388	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	4	no	—
309	C₄	C₄×C₇×C₂²⋊C₄	1	1	yes	characteristic
310	C₄	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
311	C₄	C₇×#381	1	1	yes	characteristic
312	C₄	C₇×#382	1	1	yes	characteristic
313	C₄	C₇×#387	1	2	yes	—
314	C₄	C₇×#388	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	C₇×#1434	1	1	yes	characteristic
320	C₂	C₇×#1841	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	#117
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle	C₂²×C₇	#383
prime radical	C₂⁴×C₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₇×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₇×#97	C₄	semidirect
14. C₂×C₄×C₇×C₈	C₄	semidirect
15. C₇×#344	C₄	semidirect
19. #10805	C₇	direct (N × H)
113. C₂×#97	C₄×C₇	semidirect
114. C₂×C₄×C₈	C₄×C₇	semidirect
115. #344	C₄×C₇	semidirect
295. C₇	#10805	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×M₁₆	2
C₂×C₇×#344	2
C₂×C₇×#339	2
C₂×C₇×#338	2
C₂×C₇×H(3,ℤ/4)	2
#10805	7

Subgroups — C7×#10805 #13188021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10805 #13188021