Subgroups — C₇×#19629 #13196845

Order 1792.

1050 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3205	C₂	1	2	yes	maximal, complemented
2	C₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3151	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1721	C₂	1	4	yes	maximal, complemented
6	C₇×#1712	C₂	1	2	yes	maximal, complemented
7	C₇×#1706	C₂	1	2	yes	maximal, complemented
8	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×AGL(1,ℤ/8)	—	2	4	no	—
10	C₂²×C₇×D₈	—	2	2	no	—
11	C₇×#532	C₂²	1	2	yes	—
12	C₂×C₇×#127	C₂²	1	2	yes	—
13	C₇×#500	C₂²	1	1	yes	characteristic, complemented
14	C₇×#496	—	2	4	no	—
15	C₇×#493	—	2	2	no	—
16	C₇×#493	C₂²	1	2	yes	—
17	C₇×#493	C₂²	1	2	yes	—
18	C₇×#493	C₂²	1	1	yes	characteristic, complemented
19	C₇×#493	C₂²	1	4	yes	complemented
20	C₇×#493	C₂²	1	4	yes	complemented
21	C₇×#492	C₂²	1	4	yes	complemented
22	C₇×#486	C₂²	1	2	yes	complemented
23	C₇×#486	C₂²	1	4	yes	complemented
24	C₇×#466	—	2	2	no	—
25	C₇×#459	—	2	4	no	—
26	C₇×#447	—	2	4	no	—
27	C₄×C₇×D₈	—	2	2	no	—
28	C₇×#431	C₂²	1	1	yes	characteristic
29	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#322	C₂²	1	2	yes	complemented
32	C₇×C₈²	C₂²	1	2	yes	complemented
33	#19629	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	—	2	2	no	—
35	C₂²×C₇×D₄	—	2	2	no	—
36	C₂²×C₇×D₄	—	2	2	no	—
37	C₂²×C₇×D₄	—	2	2	no	—
38	C₇×AGL(1,ℤ/8)	—	4	16	no	—
39	C₂×C₇×SD₁₆	—	2	8	no	—
40	C₂×C₇×D₈	—	2	2	no	—
41	C₂×C₇×D₈	—	2	4	no	—
42	C₂×C₇×D₈	—	2	4	no	—
43	C₂×C₇×D₈	—	2	8	no	—
44	C₂×C₇×D₈	—	2	8	no	—
45	C₂×C₇×D₈	—	4	8	no	—
46	C₂×C₇×M₁₆	D₄	1	2	yes	—
47	C₂²×C₇×C₈	D₄	1	2	yes	—
48	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₇×#127	—	2	2	no	—
50	C₇×#127	—	2	2	no	—
51	C₇×#127	—	2	2	no	—
52	C₇×#127	C₂³	1	4	yes	—
53	C₇×#127	C₂³	1	2	yes	—
54	C₇×#124	—	2	2	no	—
55	C₇×#121	—	2	2	no	—
56	C₇×#121	—	2	2	no	—
57	C₄×C₇×D₄	—	2	2	no	—
58	C₄×C₇×D₄	—	2	2	no	—
59	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
60	C₇×#107	—	2	2	no	—
61	C₇×C₄⋊C₈	D₄	1	2	yes	—
62	C₇×#102	—	2	4	no	—
63	C₇×#102	—	2	8	no	—
64	C₇×C₂²⋊C₈	D₄	1	2	yes	—
65	C₇×#97	D₄	1	2	yes	—
66	C₄×C₇×C₈	D₄	1	2	yes	—
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	2	yes	—
69	C₄×C₇×C₈	C₂³	1	2	yes	—
70	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
71	#3205	C₂×C₇	1	2	yes	complemented
72	#3169	C₂×C₇	1	1	yes	characteristic, complemented
73	#3151	C₂×C₇	1	2	yes	complemented
74	C₂×#493	C₂×C₇	1	1	yes	characteristic, complemented
75	#1721	C₂×C₇	1	4	yes	complemented
76	#1712	C₂×C₇	1	2	yes	complemented
77	#1706	C₂×C₇	1	2	yes	complemented
78	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
79	C₂⁴×C₇	—	4	2	no	—
80	C₇×#41	—	4	8	no	—
81	C₂×C₇×Q₈	—	2	2	no	—
82	C₂×C₇×D₄	—	2	2	no	—
83	C₂×C₇×D₄	—	2	4	no	—
84	C₂×C₇×D₄	—	2	2	no	—
85	C₂×C₇×D₄	—	2	2	no	—
86	C₂×C₇×D₄	—	2	4	no	—
87	C₂×C₇×D₄	—	2	4	no	—
88	C₂×C₇×D₄	—	4	8	no	—
89	C₂×C₇×D₄	—	4	2	no	—
90	C₂×C₇×D₄	—	4	2	no	—
91	C₂×C₇×D₄	—	4	2	no	—
92	C₂×C₇×D₄	—	4	2	no	—
93	C₂×C₇×D₄	—	4	8	no	—
94	C₂×C₇×D₄	—	8	4	no	—
95	C₂²×C₄×C₇	—	4	2	no	—
96	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
99	C₇×SD₁₆	—	4	16	no	—
100	C₇×D₈	—	4	8	no	—
101	C₇×D₈	—	4	16	no	—
102	C₇×D₈	—	4	4	no	—
103	C₇×D₈	—	4	8	no	—
104	C₇×D₈	—	8	8	no	—
105	C₇×M₁₆	D₈	1	8	yes	complemented
106	C₂×C₇×C₈	—	2	4	no	—
107	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
109	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
110	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
111	C₇×C₄⋊C₄	—	2	2	no	—
112	C₇×C₄⋊C₄	—	2	2	no	—
113	C₇×C₂²⋊C₄	—	4	2	no	—
114	C₇×C₂²⋊C₄	—	4	2	no	—
115	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₇	C₂⁴	1	1	yes	characteristic
119	C₂×AGL(1,ℤ/8)	—	2	4	no	—
120	C₂²×D₈	—	2	2	no	—
121	#532	C₂²×C₇	1	2	yes	—
122	C₂×#127	C₂²×C₇	1	2	yes	—
123	#500	C₂²×C₇	1	1	yes	characteristic, complemented
124	#496	—	2	4	no	—
125	#493	—	2	2	no	—
126	#493	C₂²×C₇	1	2	yes	—
127	#493	C₂²×C₇	1	2	yes	—
128	#493	C₂²×C₇	1	4	yes	complemented
129	#493	C₂²×C₇	1	1	yes	characteristic, complemented
130	#493	C₂²×C₇	1	4	yes	complemented
131	#492	C₂²×C₇	1	4	yes	complemented
132	#486	C₂²×C₇	1	2	yes	complemented
133	#486	C₂²×C₇	1	4	yes	complemented
134	#466	—	2	2	no	—
135	#459	—	2	4	no	—
136	#447	—	2	4	no	—
137	C₄×D₈	—	2	2	no	—
138	#431	C₂²×C₇	1	1	yes	characteristic
139	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
141	#322	C₂²×C₇	1	2	yes	complemented
142	C₈²	C₂²×C₇	1	2	yes	complemented
143	C₂³×C₇	—	4	2	no	—
144	C₂³×C₇	—	4	4	no	—
145	C₂³×C₇	—	8	4	no	—
146	C₂³×C₇	—	8	4	no	—
147	C₂³×C₇	—	8	4	no	—
148	C₂³×C₇	#127	1	1	yes	characteristic
149	C₇×Q₈	—	4	4	no	—
150	C₇×D₄	—	4	4	no	—
151	C₇×D₄	—	4	8	no	—
152	C₇×D₄	—	4	8	no	—
153	C₇×D₄	—	4	4	no	—
154	C₇×D₄	—	4	4	no	—
155	C₇×D₄	—	4	8	no	—
156	C₇×D₄	—	8	4	no	—
157	C₇×D₄	—	8	4	no	—
158	C₇×D₄	—	8	4	no	—
159	C₇×D₄	—	8	4	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	2	no	—
162	C₂×C₄×C₇	—	4	2	no	—
163	C₂×C₄×C₇	—	8	4	no	—
164	C₂×C₄×C₇	#127	1	1	yes	characteristic
165	C₂×C₄×C₇	#127	1	1	yes	characteristic
166	C₂×C₄×C₇	#127	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
168	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
171	C₇×C₈	—	2	2	no	—
172	C₇×C₈	—	2	4	no	—
173	C₇×C₈	C₂×D₈	1	8	yes	complemented
174	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	complemented
175	C₂×#41	—	2	2	no	—
176	C₂²×D₄	—	2	2	no	—
177	C₂²×D₄	—	2	2	no	—
178	C₂²×D₄	—	2	2	no	—
179	AGL(1,ℤ/8)	—	4	16	no	—
180	C₂×SD₁₆	—	2	8	no	—
181	C₂×D₈	—	2	4	no	—
182	C₂×D₈	—	2	4	no	—
183	C₂×D₈	—	2	8	no	—
184	C₂×D₈	—	2	2	no	—
185	C₂×D₈	—	2	8	no	—
186	C₂×D₈	—	4	8	no	—
187	C₂×M₁₆	C₇×D₄	1	2	yes	—
188	C₂²×C₈	C₇×D₄	1	2	yes	—
189	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
190	#127	—	2	2	no	—
191	#127	—	2	2	no	—
192	#127	—	2	2	no	—
193	#127	C₂³×C₇	1	4	yes	—
194	#127	C₂³×C₇	1	2	yes	—
195	#124	—	2	2	no	—
196	#121	—	2	2	no	—
197	#121	—	2	2	no	—
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
201	#107	—	2	2	no	—
202	C₄⋊C₈	C₇×D₄	1	2	yes	—
203	#102	—	2	4	no	—
204	#102	—	2	8	no	—
205	C₂²⋊C₈	C₇×D₄	1	2	yes	—
206	#97	C₇×D₄	1	2	yes	—
207	C₄×C₈	C₇×D₄	1	2	yes	—
208	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₇	1	2	yes	—
210	C₄×C₈	C₂³×C₇	1	2	yes	—
211	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
212	C₂²×C₇	—	2	1	no	—
213	C₂²×C₇	—	2	1	no	—
214	C₂²×C₇	—	8	2	no	—
215	C₂²×C₇	—	8	4	no	—
216	C₂²×C₇	—	8	4	no	—
217	C₂²×C₇	—	8	2	no	—
218	C₂²×C₇	—	8	4	no	—
219	C₂²×C₇	—	8	2	no	—
220	C₂²×C₇	—	16	8	no	—
221	C₂²×C₇	#493	1	2	yes	—
222	C₂²×C₇	C₂×#127	1	1	yes	characteristic
223	C₄×C₇	—	2	1	no	—
224	C₄×C₇	—	2	1	no	—
225	C₄×C₇	—	8	2	no	—
226	C₄×C₇	#493	1	2	yes	—
227	C₄×C₇	C₂²×D₈	1	2	yes	—
228	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
229	C₄×C₇	#576	1	2	yes	—
230	C₂⁴	—	4	2	no	—
231	#41	—	4	8	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	4	8	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	8	no	—
245	C₂×D₄	—	8	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
250	SD₁₆	—	4	16	no	—
251	D₈	—	4	8	no	—
252	D₈	—	4	16	no	—
253	D₈	—	4	4	no	—
254	D₈	—	4	8	no	—
255	D₈	—	8	8	no	—
256	M₁₆	C₇×D₈	1	8	yes	complemented
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
259	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
260	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
261	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
267	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
268	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
269	C₄²	C₂⁴×C₇	1	1	yes	characteristic
270	C₂×C₇	—	2	2	no	—
271	C₂×C₇	—	16	2	no	—
272	C₂×C₇	—	16	4	no	—
273	C₂×C₇	C₂×#493	1	1	yes	characteristic
274	C₂×C₇	#3151	1	1	yes	characteristic
275	C₂×C₇	#3155	1	1	yes	characteristic
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	—	8	4	no	—
280	C₂³	—	8	4	no	—
281	C₂³	C₇×#127	1	1	yes	characteristic
282	Q₈	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	8	no	—
285	D₄	—	4	8	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	8	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	4	no	—
291	D₄	—	8	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	C₇×#127	1	1	yes	characteristic
298	C₂×C₄	C₇×#127	1	1	yes	characteristic
299	C₂×C₄	C₇×#127	1	1	yes	characteristic
300	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
301	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	4	no	—
306	C₈	C₂×C₇×D₈	1	8	yes	complemented
307	C₈	C₇×AGL(1,ℤ/8)	1	4	yes	complemented
308	C₇	#19629	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	4	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	16	8	no	—
318	C₂²	C₇×#493	1	2	yes	—
319	C₂²	C₂×C₇×#127	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	C₇×#493	1	2	yes	—
324	C₄	C₂²×C₇×D₈	1	2	yes	—
325	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
326	C₄	C₇×#576	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	16	4	no	—
329	C₂	—	16	2	no	—
330	C₂	C₂×C₇×#493	1	1	yes	characteristic
331	C₂	C₇×#3151	1	1	yes	characteristic
332	C₂	C₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3205	C₂	semidirect
2. C₇×#3169	C₂	semidirect
3. C₇×#3151	C₂	semidirect
4. C₂×C₇×#493	C₂	semidirect
5. C₇×#1721	C₂	semidirect
6. C₇×#1712	C₂	semidirect
7. C₇×#1706	C₂	semidirect
8. C₇×C₈×M₁₆	C₂	semidirect
13. C₇×#500	C₂²	semidirect
18. C₇×#493	C₂²	semidirect
19. C₇×#493	C₂²	semidirect
20. C₇×#493	C₂²	semidirect
21. C₇×#492	C₂²	semidirect
22. C₇×#486	C₂²	semidirect
23. C₇×#486	C₂²	semidirect
31. C₇×#322	C₂²	semidirect
32. C₇×C₈²	C₂²	semidirect
33. #19629	C₇	direct (N × H)
71. #3205	C₂×C₇	semidirect
72. #3169	C₂×C₇	semidirect
73. #3151	C₂×C₇	semidirect
74. C₂×#493	C₂×C₇	semidirect
75. #1721	C₂×C₇	semidirect
76. #1712	C₂×C₇	semidirect
77. #1706	C₂×C₇	semidirect
78. C₈×M₁₆	C₂×C₇	semidirect
105. C₇×M₁₆	D₈	semidirect
123. #500	C₂²×C₇	semidirect
128. #493	C₂²×C₇	semidirect
129. #493	C₂²×C₇	semidirect
130. #493	C₂²×C₇	semidirect
131. #492	C₂²×C₇	semidirect
132. #486	C₂²×C₇	semidirect
133. #486	C₂²×C₇	semidirect
141. #322	C₂²×C₇	semidirect
142. C₈²	C₂²×C₇	semidirect
173. C₇×C₈	C₂×D₈	semidirect
174. C₇×C₈	AGL(1,ℤ/8)	semidirect
256. M₁₆	C₇×D₈	semidirect
306. C₈	C₂×C₇×D₈	semidirect
307. C₈	C₇×AGL(1,ℤ/8)	semidirect
308. C₇	#19629	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×#3205	2
C₇×#3169	2
C₇×#3151	2
C₂×C₇×#493	2
C₇×#1721	2
C₇×#1712	2
C₇×#1706	2
C₇×C₈×M₁₆	2
#19629	7

Subgroups — C7×#19629 #13196845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19629 #13196845