Subgroups — C₇×#22816 #13200032

Order 1792.

850 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	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2357	C₂	1	2	yes	maximal, complemented
3	C₇×#1972	C₂	1	2	yes	maximal
4	C₇×#1925	C₂	1	4	yes	maximal, complemented
5	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1804	C₂	1	2	yes	maximal, complemented
7	C₇×#1776	C₂	1	2	yes	maximal, complemented
8	C₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
11	C₄×C₇×D₈	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×#102	C₂²	1	4	yes	—
17	C₂×C₇×#102	C₂²	1	2	yes	—
18	C₂×C₇×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#390	C₂²	1	4	yes	—
21	C₇×#388	—	2	4	no	—
22	C₇×#382	C₂²	1	2	yes	—
23	C₇×#381	—	2	2	no	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	#22816	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	C₂³	1	2	yes	—
31	C₂³×C₄×C₇	—	2	2	no	—
32	C₂×C₇×D₈	—	2	2	no	—
33	C₂×C₇×D₈	—	2	2	no	—
34	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₇×D₄	—	2	4	no	—
39	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
40	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
46	C₂×C₄²×C₇	—	2	4	no	—
47	C₂×C₄²×C₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₇×#107	—	4	2	no	—
50	C₇×#107	C₂×C₄	1	4	yes	—
51	C₇×#106	—	2	4	no	—
52	C₇×C₄⋊C₈	—	2	2	no	—
53	C₇×#102	—	4	4	no	—
54	C₇×#102	C₂×C₄	1	8	yes	—
55	C₇×#97	—	2	4	no	—
56	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
57	C₇×#95	—	2	2	no	—
58	C₇×#95	—	2	4	no	—
59	C₇×#95	—	2	2	no	—
60	C₇×#95	D₄	1	2	yes	—
61	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
62	#2357	C₂×C₇	1	2	yes	complemented
63	#1972	C₂×C₇	1	2	yes	—
64	#1925	C₂×C₇	1	4	yes	complemented
65	#1846	C₂×C₇	1	1	yes	characteristic, complemented
66	#1804	C₂×C₇	1	2	yes	complemented
67	#1776	C₂×C₇	1	2	yes	complemented
68	#1770	C₂×C₇	1	1	yes	characteristic, complemented
69	C₂⁴×C₇	—	2	2	no	—
70	C₂×C₇×D₄	—	2	2	no	—
71	C₂×C₇×D₄	—	2	2	no	—
72	C₂×C₇×D₄	—	2	2	no	—
73	C₂×C₇×D₄	—	2	2	no	—
74	C₂×C₇×D₄	—	2	2	no	—
75	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₇	—	2	2	no	—
77	C₂²×C₄×C₇	—	2	8	no	—
78	C₂²×C₄×C₇	—	2	4	no	—
79	C₂²×C₄×C₇	—	8	4	no	—
80	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
84	C₇×D₈	—	2	8	no	—
85	C₂×C₇×C₈	—	2	1	no	—
86	C₂×C₇×C₈	—	2	1	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₇×C₈	#41	1	2	yes	—
93	C₂×C₇×C₈	#41	1	2	yes	—
94	C₇×C₄⋊C₄	—	2	2	no	—
95	C₇×C₄⋊C₄	—	2	4	no	—
96	C₇×C₄⋊C₄	—	2	4	no	—
97	C₇×C₄⋊C₄	—	4	2	no	—
98	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
99	C₇×C₂²⋊C₄	—	2	8	no	—
100	C₇×C₂²⋊C₄	—	4	8	no	—
101	C₇×C₂²⋊C₄	—	8	4	no	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	—	2	2	no	—
104	C₄²×C₇	—	8	4	no	—
105	C₄²×C₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₇	#41	1	2	yes	—
107	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
109	C₄×D₈	C₄×C₇	1	8	yes	complemented
110	C₂×#107	C₂²×C₇	1	2	yes	—
111	C₂×#107	C₂²×C₇	1	1	yes	characteristic
112	C₂×#106	C₂²×C₇	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
114	C₂×#102	C₂²×C₇	1	4	yes	—
115	C₂×#102	C₂²×C₇	1	2	yes	—
116	C₂×#97	C₂²×C₇	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
118	#390	C₂²×C₇	1	4	yes	—
119	#388	—	2	4	no	—
120	#382	C₂²×C₇	1	2	yes	—
121	#381	—	2	2	no	—
122	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
123	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
124	C₄×C₂²⋊C₄	—	2	2	no	—
125	#340	C₂²×C₇	1	4	yes	—
126	#336	C₂²×C₇	1	2	yes	—
127	C₂³×C₇	—	2	4	no	—
128	C₂³×C₇	—	4	2	no	—
129	C₂³×C₇	—	4	2	no	—
130	C₂³×C₇	—	4	4	no	—
131	C₂³×C₇	—	4	4	no	—
132	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
133	C₇×D₄	—	2	4	no	—
134	C₇×D₄	—	2	4	no	—
135	C₇×D₄	—	4	2	no	—
136	C₇×D₄	—	4	2	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	2	4	no	—
140	C₂×C₄×C₇	—	2	4	no	—
141	C₂×C₄×C₇	—	2	4	no	—
142	C₂×C₄×C₇	—	2	4	no	—
143	C₂×C₄×C₇	—	4	4	no	—
144	C₂×C₄×C₇	—	4	4	no	—
145	C₂×C₄×C₇	—	4	4	no	—
146	C₂×C₄×C₇	—	4	4	no	—
147	C₂×C₄×C₇	—	8	8	no	—
148	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
156	C₇×C₈	—	2	2	no	—
157	C₇×C₈	—	4	2	no	—
158	C₇×C₈	#117	1	4	yes	—
159	C₂²×D₄	C₂³×C₇	1	2	yes	—
160	C₂³×C₄	—	2	2	no	—
161	C₂×D₈	—	2	2	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
164	C₂²×C₈	C₂³×C₇	1	2	yes	—
165	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
167	C₄×D₄	—	2	4	no	—
168	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
169	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
170	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
171	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₂²⋊C₄	—	2	4	no	—
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	C₂³×C₇	1	2	yes	—
177	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
178	#107	—	4	2	no	—
179	#107	C₂×C₄×C₇	1	4	yes	—
180	#106	—	2	4	no	—
181	C₄⋊C₈	—	2	2	no	—
182	#102	—	4	4	no	—
183	#102	C₂×C₄×C₇	1	8	yes	—
184	#97	—	2	4	no	—
185	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
186	#95	—	2	2	no	—
187	#95	—	2	4	no	—
188	#95	—	2	2	no	—
189	#95	C₇×D₄	1	2	yes	—
190	C₂²×C₇	—	4	4	no	—
191	C₂²×C₇	—	4	4	no	—
192	C₂²×C₇	—	4	4	no	—
193	C₂²×C₇	—	8	4	no	—
194	C₂²×C₇	—	8	4	no	—
195	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×C₇	#547	1	1	yes	characteristic
197	C₂²×C₇	#550	1	1	yes	characteristic
198	C₂²×C₇	#575	1	2	yes	—
199	C₂²×C₇	#576	1	2	yes	—
200	C₄×C₇	—	2	2	no	—
201	C₄×C₇	—	2	2	no	—
202	C₄×C₇	—	4	4	no	—
203	C₄×C₇	—	4	4	no	—
204	C₄×C₇	—	8	4	no	—
205	C₄×C₇	C₂×#117	1	1	yes	characteristic
206	C₄×C₇	#520	1	1	yes	characteristic
207	C₄×C₇	#555	1	1	yes	characteristic
208	C₄×C₇	#556	1	1	yes	characteristic
209	C₂⁴	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	8	4	no	—
220	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
221	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
224	D₈	—	2	8	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
232	C₂×C₈	C₇×#41	1	2	yes	—
233	C₂×C₈	C₇×#41	1	2	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
239	C₂²⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	4	8	no	—
241	C₂²⋊C₄	—	8	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	8	4	no	—
245	C₄²	C₂²×C₄×C₇	1	2	yes	—
246	C₄²	C₇×#41	1	2	yes	—
247	C₂×C₇	—	8	4	no	—
248	C₂×C₇	#2354	1	1	yes	characteristic
249	C₂×C₇	#2943	1	1	yes	characteristic
250	C₂×C₇	#2948	1	1	yes	characteristic
251	C₂×C₇	#3355	1	2	yes	—
252	C₂×C₇	#3403	1	2	yes	—
253	C₂³	—	2	4	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
259	D₄	—	2	4	no	—
260	D₄	—	2	4	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	8	8	no	—
274	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
275	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
276	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
278	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
279	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
280	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	C₇×#117	1	4	yes	—
285	C₇	#22816	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	4	no	—
291	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
292	C₂²	C₇×#547	1	1	yes	characteristic
293	C₂²	C₇×#550	1	1	yes	characteristic
294	C₂²	C₇×#575	1	2	yes	—
295	C₂²	C₇×#576	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	4	4	no	—
300	C₄	—	8	4	no	—
301	C₄	C₂×C₇×#117	1	1	yes	characteristic
302	C₄	C₇×#520	1	1	yes	characteristic
303	C₄	C₇×#555	1	1	yes	characteristic
304	C₄	C₇×#556	1	1	yes	characteristic
305	C₂	—	8	4	no	—
306	C₂	C₇×#2354	1	1	yes	characteristic
307	C₂	C₇×#2943	1	1	yes	characteristic
308	C₂	C₇×#2948	1	1	yes	characteristic
309	C₂	C₇×#3355	1	2	yes	—
310	C₂	C₇×#3403	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2357	C₂	semidirect
4. C₇×#1925	C₂	semidirect
5. C₇×#1846	C₂	semidirect
6. C₇×#1804	C₂	semidirect
7. C₇×#1776	C₂	semidirect
8. C₇×#1770	C₂	semidirect
11. C₄×C₇×D₈	C₄	semidirect
29. #22816	C₇	direct (N × H)
62. #2357	C₂×C₇	semidirect
64. #1925	C₂×C₇	semidirect
65. #1846	C₂×C₇	semidirect
66. #1804	C₂×C₇	semidirect
67. #1776	C₂×C₇	semidirect
68. #1770	C₂×C₇	semidirect
109. C₄×D₈	C₄×C₇	semidirect
285. C₇	#22816	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×D₈	2
C₇×#2357	2
C₇×#1972	2
C₇×#1925	2
C₇×#1846	2
C₇×#1804	2
C₇×#1776	2
C₇×#1770	2
#22816	7

Subgroups — C7×#22816 #13200032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22816 #13200032