Subgroups — C₇×#22744 #13199960

Order 1792.

714 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#446	C₂	1	1	yes	maximal, characteristic
2	C₇×#2355	C₂	1	2	yes	maximal
3	C₇×#1949	C₂	1	2	yes	maximal
4	C₇×#1949	C₂	1	2	yes	maximal
5	C₇×#1947	C₂	1	2	yes	maximal
6	C₇×#1836	C₂	1	2	yes	maximal
7	C₇×#1835	C₂	1	1	yes	maximal, characteristic
8	C₇×#1776	C₂	1	2	yes	maximal
9	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₇×#446	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×#391	—	2	2	no	—
20	C₇×#387	—	2	2	no	—
21	C₇×#387	—	2	2	no	—
22	C₇×#384	C₂²	1	2	yes	—
23	C₇×#382	C₂²	1	2	yes	—
24	C₇×#382	C₂²	1	2	yes	—
25	C₄×C₇×C₄⋊C₄	—	2	2	no	—
26	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
29	C₇×#340	C₂²	1	1	yes	characteristic
30	C₇×#340	C₂²	1	1	yes	characteristic
31	C₇×#340	C₂²	1	2	yes	—
32	C₇×#340	C₂²	1	1	yes	characteristic
33	C₇×#340	C₂²	1	1	yes	characteristic
34	C₇×#340	C₂²	1	2	yes	—
35	C₇×#340	C₂²	1	2	yes	—
36	#22744	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂³	1	2	yes	—
41	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
42	C₄×C₇×Q₈	Q₈	1	4	yes	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	—	2	2	no	—
52	C₂×C₄²×C₇	—	2	2	no	—
53	C₂×C₄²×C₇	—	2	2	no	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇	C₂³	1	2	yes	—
56	C₇×#107	—	2	4	no	—
57	C₇×#107	—	2	4	no	—
58	C₇×#106	—	2	4	no	—
59	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
60	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
61	C₇×#97	C₂×C₄	1	8	yes	—
62	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
63	C₇×#95	—	2	2	no	—
64	C₇×#95	—	2	2	no	—
65	C₇×#95	—	2	2	no	—
66	C₇×#95	—	2	2	no	—
67	C₇×#95	—	2	2	no	—
68	C₇×#95	—	2	2	no	—
69	C₇×#95	—	2	2	no	—
70	C₂×#446	C₂×C₇	1	1	yes	characteristic
71	#2355	C₂×C₇	1	2	yes	—
72	#1949	C₂×C₇	1	2	yes	—
73	#1949	C₂×C₇	1	2	yes	—
74	#1947	C₂×C₇	1	2	yes	—
75	#1836	C₂×C₇	1	2	yes	—
76	#1835	C₂×C₇	1	1	yes	characteristic
77	#1776	C₂×C₇	1	2	yes	—
78	C₄×#107	C₂×C₇	1	1	yes	characteristic
79	C₂×C₇×Q₈	—	4	2	no	—
80	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
81	C₂²×C₄×C₇	—	2	2	no	—
82	C₂²×C₄×C₇	—	2	2	no	—
83	C₂²×C₄×C₇	—	2	2	no	—
84	C₂²×C₄×C₇	—	2	2	no	—
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	2	2	no	—
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	4	4	no	—
105	C₇×C₄⋊C₄	—	8	2	no	—
106	C₇×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
107	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
108	C₇×C₄⋊C₄	#41	1	4	yes	—
109	C₄²×C₇	—	4	4	no	—
110	C₄²×C₇	—	8	2	no	—
111	C₄²×C₇	—	8	2	no	—
112	C₄²×C₇	C₂²×C₄	1	2	yes	—
113	C₄²×C₇	C₂²×C₄	1	4	yes	—
114	C₄²×C₇	C₂×Q₈	1	2	yes	—
115	C₄²×C₇	C₂×Q₈	1	4	yes	—
116	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
117	#446	C₄×C₇	1	8	yes	complemented
118	C₂×#107	C₂²×C₇	1	2	yes	—
119	C₂×#107	C₂²×C₇	1	2	yes	—
120	C₂×#106	C₂²×C₇	1	2	yes	—
121	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
123	C₂×#97	C₂²×C₇	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
125	#391	—	2	2	no	—
126	#387	—	2	2	no	—
127	#387	—	2	2	no	—
128	#384	C₂²×C₇	1	2	yes	—
129	#382	C₂²×C₇	1	2	yes	—
130	#382	C₂²×C₇	1	2	yes	—
131	C₄×C₄⋊C₄	—	2	2	no	—
132	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
133	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
134	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
135	#340	C₂²×C₇	1	2	yes	—
136	#340	C₂²×C₇	1	1	yes	characteristic
137	#340	C₂²×C₇	1	1	yes	characteristic
138	#340	C₂²×C₇	1	1	yes	characteristic
139	#340	C₂²×C₇	1	2	yes	—
140	#340	C₂²×C₇	1	2	yes	—
141	#340	C₂²×C₇	1	1	yes	characteristic
142	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
143	C₇×Q₈	—	4	4	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	1	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	1	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	2	4	no	—
152	C₂×C₄×C₇	—	4	2	no	—
153	C₂×C₄×C₇	—	4	2	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	—	4	2	no	—
159	C₂×C₄×C₇	—	4	2	no	—
160	C₂×C₄×C₇	—	4	2	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	—	4	2	no	—
163	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
165	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
167	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂×#41	1	2	yes	—
170	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
171	C₇×C₈	—	2	4	no	—
172	C₇×C₈	—	2	2	no	—
173	C₇×C₈	—	2	2	no	—
174	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₇	1	2	yes	—
178	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
179	C₄×Q₈	C₇×Q₈	1	4	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
183	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
187	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₇	1	2	yes	—
193	#107	—	2	4	no	—
194	#107	—	2	4	no	—
195	#106	—	2	4	no	—
196	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
197	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
198	#97	C₂×C₄×C₇	1	8	yes	—
199	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
208	C₂²×C₇	#548	1	2	yes	—
209	C₂²×C₇	#575	1	2	yes	—
210	C₂²×C₇	#576	1	1	yes	characteristic
211	C₂²×C₇	#578	1	1	yes	characteristic
212	C₄×C₇	—	2	4	no	—
213	C₄×C₇	—	2	2	no	—
214	C₄×C₇	—	2	2	no	—
215	C₄×C₇	—	2	4	no	—
216	C₄×C₇	—	4	4	no	—
217	C₄×C₇	—	8	2	no	—
218	C₄×C₇	—	8	2	no	—
219	C₄×C₇	—	8	2	no	—
220	C₄×C₇	C₄×#41	1	1	yes	characteristic
221	C₄×C₇	#519	1	1	yes	characteristic
222	C₄×C₇	#557	1	2	yes	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
232	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	8	2	no	—
250	C₄⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
251	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
252	C₄⋊C₄	C₇×#41	1	4	yes	—
253	C₄²	—	4	4	no	—
254	C₄²	—	8	2	no	—
255	C₄²	—	8	2	no	—
256	C₄²	C₂²×C₄×C₇	1	2	yes	—
257	C₄²	C₂²×C₄×C₇	1	4	yes	—
258	C₄²	C₂×C₇×Q₈	1	2	yes	—
259	C₄²	C₂×C₇×Q₈	1	4	yes	—
260	C₂×C₇	#2351	1	1	yes	characteristic
261	C₂×C₇	#2911	1	1	yes	characteristic
262	C₂×C₇	#2913	1	1	yes	characteristic
263	C₂×C₇	#3370	1	2	yes	—
264	C₂×C₇	#3377	1	2	yes	—
265	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
287	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
288	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
289	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
290	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
292	C₂×C₄	C₂×C₇×#41	1	2	yes	—
293	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
294	C₈	—	2	4	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₇	#22744	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
299	C₂²	C₇×#548	1	2	yes	—
300	C₂²	C₇×#575	1	2	yes	—
301	C₂²	C₇×#576	1	1	yes	characteristic
302	C₂²	C₇×#578	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₄×C₇×#41	1	1	yes	characteristic
312	C₄	C₇×#519	1	1	yes	characteristic
313	C₄	C₇×#557	1	2	yes	—
314	C₂	C₇×#2351	1	1	yes	characteristic
315	C₂	C₇×#2911	1	1	yes	characteristic
316	C₂	C₇×#2913	1	1	yes	characteristic
317	C₂	C₇×#3370	1	2	yes	—
318	C₂	C₇×#3377	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₇×#446	C₄	semidirect
36. #22744	C₇	direct (N × H)
117. #446	C₄×C₇	semidirect
297. C₇	#22744	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×C₇×#446	2
C₇×#2355	2
C₇×#1949	2
C₇×#1947	2
C₇×#1836	2
C₇×#1835	2
C₇×#1776	2
C₄×C₇×#107	2
#22744	7

Subgroups — C7×#22744 #13199960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22744 #13199960