Subgroups — C₇×#12792 #13190008

Order 1792.

440 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2117	C₂	1	2	yes	maximal
5	C₇×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#131	C₄	1	1	yes	characteristic
7	C₂×C₇×#131	C₂²	1	1	yes	characteristic
8	C₄×C₇×#41	C₄	1	1	yes	characteristic
9	C₂×C₇×M₃₂	C₂²	1	2	yes	—
10	C₂×C₇×M₃₂	C₂²	1	2	yes	—
11	C₇×#405	—	2	1	no	—
12	C₇×#405	—	2	1	no	—
13	C₇×#405	C₄	1	1	yes	characteristic
14	C₇×#405	C₂²	1	1	yes	characteristic
15	C₄×C₇×M₁₆	—	2	1	no	—
16	C₄×C₇×M₁₆	C₄	1	1	yes	characteristic
17	C₂×C₄×C₇×C₈	—	2	1	no	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×#364	—	2	1	no	—
20	C₇×#364	—	2	1	no	—
21	C₇×#364	—	2	1	no	—
22	C₇×#364	—	2	1	no	—
23	C₇×#350	—	2	4	no	—
24	#12792	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
26	C₇×#131	—	2	2	no	—
27	C₇×#131	—	2	1	no	—
28	C₇×#131	—	2	1	no	—
29	C₇×#131	C₈	1	4	yes	—
30	C₇×#131	D₄	1	2	yes	—
31	C₇×#131	Q₈	1	2	yes	—
32	C₂×C₇×M₁₆	—	2	1	no	—
33	C₂×C₇×M₁₆	—	2	1	no	—
34	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	—	2	1	no	—
37	C₂²×C₇×C₈	—	2	1	no	—
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₇×Q₈	—	2	1	no	—
41	C₄×C₇×D₄	—	2	1	no	—
42	C₄×C₇×D₄	—	2	2	no	—
43	C₇×#117	—	2	1	no	—
44	C₇×#117	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇	—	2	1	no	—
46	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
47	C₇×M₃₂	—	2	2	no	—
48	C₇×M₃₂	—	2	2	no	—
49	C₇×M₃₂	—	2	2	no	—
50	C₇×M₃₂	—	2	2	no	—
51	C₂×C₇×C₁₆	—	2	2	no	—
52	C₂×C₇×C₁₆	—	2	2	no	—
53	C₇×#97	—	2	1	no	—
54	C₇×#97	—	2	1	no	—
55	C₇×#97	—	2	1	no	—
56	C₄×C₇×C₈	—	2	1	no	—
57	C₄×C₇×C₈	—	2	1	no	—
58	C₄×C₇×C₈	—	2	1	no	—
59	C₄×C₇×C₈	D₄	1	1	yes	characteristic
60	C₄×C₇×C₈	D₄	1	1	yes	characteristic
61	C₄×C₇×C₈	D₄	1	1	yes	characteristic
62	C₄×C₇×C₈	D₄	1	1	yes	characteristic
63	C₄×#131	C₂×C₇	1	1	yes	characteristic
64	C₂×#364	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂×#364	C₂×C₇	1	1	yes	characteristic, complemented
66	#2117	C₂×C₇	1	2	yes	—
67	#1378	C₂×C₇	1	2	yes	complemented
68	C₇×#41	—	2	1	no	—
69	C₇×#41	—	2	1	no	—
70	C₇×#41	C₄⋊C₄	1	2	yes	—
71	C₇×#41	C₂×C₈	1	2	yes	—
72	C₂×C₇×Q₈	M₁₆	1	1	yes	characteristic
73	C₂×C₇×D₄	—	2	1	no	—
74	C₂×C₇×D₄	M₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₇	—	2	1	no	—
76	C₂²×C₄×C₇	—	2	2	no	—
77	C₂²×C₄×C₇	—	2	1	no	—
78	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
79	C₇×M₁₆	—	2	1	no	—
80	C₇×M₁₆	—	2	1	no	—
81	C₇×M₁₆	—	2	4	no	—
82	C₇×M₁₆	—	2	2	no	—
83	C₇×M₁₆	—	2	2	no	—
84	C₇×M₁₆	C₄⋊C₄	1	2	yes	—
85	C₇×M₁₆	C₂×C₈	1	2	yes	—
86	C₂×C₇×C₈	—	2	1	no	—
87	C₂×C₇×C₈	—	2	1	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
94	C₂×C₇×C₈	M₁₆	1	2	yes	—
95	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₇×C₈	#41	1	1	yes	characteristic
100	C₂×C₇×C₈	#41	1	1	yes	characteristic
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	1	no	—
103	C₇×C₂²⋊C₄	—	2	2	no	—
104	C₇×C₂²⋊C₄	M₁₆	1	2	yes	—
105	C₄²×C₇	—	2	2	no	—
106	C₄²×C₇	—	2	1	no	—
107	C₄²×C₇	—	2	1	no	—
108	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
109	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
110	C₇×C₁₆	—	4	2	no	—
111	C₇×C₁₆	—	4	2	no	—
112	C₂×#131	C₄×C₇	1	1	yes	characteristic
113	C₂×#131	C₂²×C₇	1	1	yes	characteristic
114	C₄×#41	C₄×C₇	1	1	yes	characteristic
115	C₂×M₃₂	C₂²×C₇	1	2	yes	—
116	C₂×M₃₂	C₂²×C₇	1	2	yes	—
117	#405	—	2	1	no	—
118	#405	—	2	1	no	—
119	#405	C₄×C₇	1	1	yes	characteristic
120	#405	C₂²×C₇	1	1	yes	characteristic
121	C₄×M₁₆	—	2	1	no	—
122	C₄×M₁₆	C₄×C₇	1	1	yes	characteristic
123	C₂×C₄×C₈	—	2	1	no	—
124	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
125	#364	—	2	1	no	—
126	#364	—	2	1	no	—
127	#364	—	2	1	no	—
128	#364	—	2	1	no	—
129	#350	—	2	4	no	—
130	C₂³×C₇	—	2	1	no	—
131	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
132	C₇×Q₈	—	2	1	no	—
133	C₇×Q₈	C₄⋊C₈	1	2	yes	—
134	C₇×D₄	—	2	2	no	—
135	C₇×D₄	—	2	2	no	—
136	C₇×D₄	—	2	1	no	—
137	C₇×D₄	C₄⋊C₈	1	2	yes	—
138	C₂×C₄×C₇	—	2	1	no	—
139	C₂×C₄×C₇	—	2	1	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	1	no	—
144	C₂×C₄×C₇	—	4	2	no	—
145	C₂×C₄×C₇	—	4	4	no	—
146	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₇	#117	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
152	C₇×C₈	—	2	2	no	—
153	C₇×C₈	—	2	2	no	—
154	C₇×C₈	C₂²⋊C₈	1	2	yes	—
155	C₇×C₈	C₂²⋊C₈	1	2	yes	—
156	C₇×C₈	#121	1	1	yes	characteristic
157	C₇×C₈	#121	1	1	yes	characteristic
158	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
159	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×#41	C₂×C₄×C₇	1	1	yes	characteristic
161	#131	—	2	2	no	—
162	#131	—	2	1	no	—
163	#131	—	2	1	no	—
164	#131	C₇×C₈	1	4	yes	—
165	#131	C₇×D₄	1	2	yes	—
166	#131	C₇×Q₈	1	2	yes	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
170	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
175	C₄×Q₈	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	2	2	no	—
178	#117	—	2	1	no	—
179	#117	C₂×C₄×C₇	1	1	yes	characteristic
180	C₂×C₄²	—	2	1	no	—
181	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
182	M₃₂	—	2	2	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	2	2	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	#97	—	2	1	no	—
189	#97	—	2	1	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₄×C₈	C₇×D₄	1	1	yes	characteristic
195	C₄×C₈	C₇×D₄	1	1	yes	characteristic
196	C₄×C₈	C₇×D₄	1	1	yes	characteristic
197	C₄×C₈	C₇×D₄	1	1	yes	characteristic
198	C₂²×C₇	—	2	2	no	—
199	C₂²×C₇	—	4	1	no	—
200	C₂²×C₇	—	4	2	no	—
201	C₂²×C₇	#407	1	1	yes	characteristic
202	C₂²×C₇	#423	1	1	yes	characteristic
203	C₂²×C₇	#431	1	1	yes	characteristic
204	C₄×C₇	—	2	2	no	—
205	C₄×C₇	—	2	2	no	—
206	C₄×C₇	—	4	2	no	—
207	C₄×C₇	—	4	1	no	—
208	C₄×C₇	#380	1	1	yes	characteristic
209	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
210	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
211	C₄×C₇	#432	1	1	yes	characteristic
212	#41	—	2	1	no	—
213	#41	—	2	1	no	—
214	#41	C₇×C₄⋊C₄	1	2	yes	—
215	#41	C₂×C₇×C₈	1	2	yes	—
216	C₂×Q₈	C₇×M₁₆	1	1	yes	characteristic
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	C₇×M₁₆	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
223	M₁₆	—	2	1	no	—
224	M₁₆	—	2	1	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	C₇×C₄⋊C₄	1	2	yes	—
229	M₁₆	C₂×C₇×C₈	1	2	yes	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
238	C₂×C₈	C₇×M₁₆	1	2	yes	—
239	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
240	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
241	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
243	C₂×C₈	C₇×#41	1	1	yes	characteristic
244	C₂×C₈	C₇×#41	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	C₇×M₁₆	1	2	yes	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
253	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
254	C₁₆	—	4	2	no	—
255	C₁₆	—	4	2	no	—
256	C₂×C₇	—	2	1	no	—
257	C₂×C₇	—	2	1	no	—
258	C₂×C₇	—	2	1	no	—
259	C₂×C₇	—	4	2	no	—
260	C₂×C₇	#1797	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
263	Q₈	—	2	1	no	—
264	Q₈	C₇×C₄⋊C₈	1	2	yes	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	2	1	no	—
268	D₄	C₇×C₄⋊C₈	1	2	yes	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₇×#117	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
282	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	C₇×C₂²⋊C₈	1	2	yes	—
286	C₈	C₇×C₂²⋊C₈	1	2	yes	—
287	C₈	C₇×#121	1	1	yes	characteristic
288	C₈	C₇×#121	1	1	yes	characteristic
289	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
290	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
291	C₇	#12792	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₇×#407	1	1	yes	characteristic
296	C₂²	C₇×#423	1	1	yes	characteristic
297	C₂²	C₇×#431	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	C₇×#380	1	1	yes	characteristic
303	C₄	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
304	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
305	C₄	C₇×#432	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	C₇×#1797	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	#121
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1797
prime radical	C₂×C₇×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#364	C₂	semidirect
3. C₂×C₇×#364	C₂	semidirect
5. C₇×#1378	C₂	semidirect
24. #12792	C₇	direct (N × H)
64. C₂×#364	C₂×C₇	semidirect
65. C₂×#364	C₂×C₇	semidirect
67. #1378	C₂×C₇	semidirect
291. C₇	#12792	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×#131	2
C₂×C₇×#364	2
C₇×#2117	2
C₇×#1378	2
#12792	7

Subgroups — C7×#12792 #13190008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12792 #13190008