Subgroups — #13269167

Order 1792.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1770	C₂	1	1	yes	maximal, characteristic
2	C₂×#89261	C₂	1	2	yes	maximal
3	#11727497	C₂	1	1	yes	maximal, characteristic
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic
5	#11725469	C₂	1	2	yes	maximal
6	C₂×C₇×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₇×#97	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂×#6386	C₂²	1	1	yes	characteristic
11	C₂×#6386	C₂²	1	2	yes	—
12	C₂×#6386	C₂²	1	1	yes	characteristic
13	C₇×#336	C₄	1	2	yes	—
14	#89261	—	2	4	no	—
15	C₄⋊(C₇⋊C₁₆)	C₄	1	2	yes	complemented
16	C₄⋊(C₇⋊C₁₆)	C₄	1	2	yes	complemented
17	#89256	C₄	1	2	yes	complemented
18	C₄×C₇⋊C₁₆	C₄	1	2	yes	complemented
19	#12666	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₄	—	2	1	no	—
24	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
26	C₇×C₄⋊C₈	—	2	1	no	—
27	C₇×C₄⋊C₈	—	2	1	no	—
28	C₇×#97	—	2	2	no	—
29	C₇×#97	—	2	2	no	—
30	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₇×#95	—	2	1	no	—
35	#6386	—	2	2	no	—
36	#6386	—	2	1	no	—
37	#6386	—	2	1	no	—
38	#6386	—	2	2	no	—
39	#6386	D₄	1	2	yes	—
40	#6386	Q₈	1	2	yes	—
41	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
42	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
43	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
44	#2150	—	7	1	no	—
45	C₂×#347	—	7	2	no	—
46	C₄×M₃₂	—	7	1	no	—
47	#1770	D₇	1	1	yes	characteristic
48	#1375	—	7	2	no	—
49	C₂²×C₄×C₇	—	2	2	no	—
50	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₇×C₈	—	2	1	no	—
54	C₂×C₇×C₈	—	2	1	no	—
55	C₂×C₇×C₈	—	2	2	no	—
56	C₂×C₇×C₈	—	2	2	no	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
59	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₇×C₈	#41	1	1	yes	characteristic
65	C₂×C₇×C₈	#41	1	1	yes	characteristic
66	C₇×C₄⋊C₄	—	4	2	no	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	C₄²	1	1	yes	characteristic
70	C₄²×C₇	C₄²	1	1	yes	characteristic
71	C₄²×C₇	M₁₆	1	1	yes	characteristic
72	C₄²×C₇	M₁₆	1	1	yes	characteristic
73	C₇⋊C₁₆	—	2	2	no	—
74	C₇⋊C₁₆	—	2	4	no	—
75	C₇⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
76	C₂×M₃₂	—	7	1	no	—
77	C₂×M₃₂	—	7	1	no	—
78	C₂×M₃₂	—	7	2	no	—
79	C₂×C₄⋊C₈	Dic₇	1	1	yes	characteristic
80	C₂×#97	C₂×D₇	1	2	yes	—
81	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
82	C₄×C₄⋊C₄	Dic₇	1	1	yes	characteristic
83	C₄⋊C₁₆	—	7	2	no	—
84	C₄⋊C₁₆	—	7	2	no	—
85	#347	—	14	4	no	—
86	#346	—	7	2	no	—
87	C₄×C₁₆	—	7	2	no	—
88	#336	Dic₇	1	2	yes	—
89	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	4	1	no	—
92	C₂×C₄×C₇	—	4	2	no	—
93	C₂×C₄×C₇	—	4	2	no	—
94	C₂×C₄×C₇	—	4	2	no	—
95	C₂×C₄×C₇	#97	1	1	yes	characteristic
96	C₂×C₄×C₇	#97	1	1	yes	characteristic
97	C₂×C₄×C₇	#97	1	1	yes	characteristic
98	C₂×C₄×C₇	#97	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇	#131	1	1	yes	characteristic
104	C₂×C₄×C₇	#131	1	1	yes	characteristic
105	C₇×C₈	—	2	1	no	—
106	C₇×C₈	—	2	1	no	—
107	C₇×C₈	—	4	2	no	—
108	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₇×C₈	#117	1	1	yes	characteristic
110	C₇×C₈	C₄×D₄	1	1	yes	characteristic
111	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₇	1	2	yes	—
113	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄²	C₂×Dic₇	1	2	yes	—
117	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
118	M₃₂	—	7	2	no	—
119	M₃₂	—	7	2	no	—
120	M₃₂	—	14	2	no	—
121	M₃₂	—	14	1	no	—
122	M₃₂	—	14	1	no	—
123	M₃₂	—	14	2	no	—
124	C₂×C₁₆	—	7	2	no	—
125	C₂×C₁₆	—	7	2	no	—
126	C₂×C₁₆	—	7	4	no	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	—	2	1	no	—
129	#97	—	2	2	no	—
130	#97	—	2	2	no	—
131	C₄×C₈	C₄×D₇	1	1	yes	characteristic
132	C₄×C₈	C₄×D₇	1	1	yes	characteristic
133	C₄×C₈	C₄×D₇	1	1	yes	characteristic
134	C₄×C₈	C₄×D₇	1	1	yes	characteristic
135	#95	—	2	1	no	—
136	C₂²×C₇	—	2	1	no	—
137	C₂²×C₇	—	2	1	no	—
138	C₂²×C₇	C₂×#97	1	1	yes	characteristic
139	C₂²×C₇	#436	1	1	yes	characteristic
140	C₂²×C₇	#446	1	1	yes	characteristic
141	C₄×C₇	—	2	1	no	—
142	C₄×C₇	—	2	1	no	—
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	—	8	2	no	—
145	C₄×C₇	#347	1	2	yes	complemented
146	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₇	#405	1	1	yes	characteristic
148	C₄×C₇	#436	1	1	yes	characteristic
149	C₄×C₇	#446	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	#1136	1	1	yes	characteristic
152	C₂²×C₄	#1136	1	1	yes	characteristic
153	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
160	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
162	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
163	C₂×C₈	D₄×D₇	1	1	yes	characteristic
164	C₂×C₈	#1159	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
166	C₂×C₈	#1161	1	1	yes	characteristic
167	C₄⋊C₄	—	4	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	#1136	1	1	yes	characteristic
171	C₄²	#1136	1	1	yes	characteristic
172	C₄²	C₄×Dic₇	1	1	yes	characteristic
173	C₄²	C₄×Dic₇	1	1	yes	characteristic
174	C₁₆	—	7	4	no	—
175	C₁₆	—	14	2	no	—
176	C₁₆	—	14	4	no	—
177	C₂×C₇	—	2	1	no	—
178	C₂×C₇	—	2	1	no	—
179	C₂×C₇	#1770	1	1	yes	characteristic
180	C₂×C₇	C₂×#347	1	1	yes	characteristic
181	C₂×C₇	#2110	1	1	yes	characteristic
182	C₂³	C₂×#1136	1	1	yes	characteristic
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	#6377	1	1	yes	characteristic
189	C₂×C₄	#6377	1	1	yes	characteristic
190	C₂×C₄	#6377	1	1	yes	characteristic
191	C₂×C₄	#6377	1	1	yes	characteristic
192	C₂×C₄	C₂×#1136	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
194	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
195	C₂×C₄	Q₈×Dic₇	1	1	yes	characteristic
196	C₂×C₄	#6511	1	1	yes	characteristic
197	C₂×C₄	#6511	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	4	2	no	—
201	C₈	#6450	1	1	yes	characteristic
202	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
203	C₈	#6455	1	1	yes	characteristic
204	C₈	#6456	1	1	yes	characteristic
205	C₇	#12666	1	1	yes	characteristic, complemented, Sylow
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#6377	1	1	yes	characteristic
209	C₂²	#89784	1	1	yes	characteristic
210	C₂²	#89796	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	#89261	1	2	yes	complemented
216	C₄	C₄⋊C₄×Dic₇	1	1	yes	characteristic
217	C₄	#89769	1	1	yes	characteristic
218	C₄	#89784	1	1	yes	characteristic
219	C₄	#89796	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#11727231	1	1	yes	characteristic
223	C₂	C₂×#89261	1	1	yes	characteristic
224	C₂	#11727638	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1770	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×C₇	C₂×M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1770	D₇
hypocenter	C₇	#12666
LCS	G ▷ C₂×C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
15. C₄⋊(C₇⋊C₁₆)	C₄	semidirect
16. C₄⋊(C₇⋊C₁₆)	C₄	semidirect
17. #89256	C₄	semidirect
18. C₄×C₇⋊C₁₆	C₄	semidirect
75. C₇⋊C₁₆	C₄⋊C₄	semidirect
145. C₄×C₇	#347	semidirect
205. C₇	#12666	semidirect
215. C₄	#89261	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1770	2
C₂×#89261	2
#11727497	2
C₄×#6386	2
#11725469	2
#12666	7