Subgroups — #11739747

Order 896.

456 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	#89979	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#89783	C₂	1	8	yes	maximal, complemented
5	#89698	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×D₄	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₄×C₇×D₄	C₄	1	4	yes	—
10	C₄×C₇×D₄	C₂²	1	4	yes	—
11	C₂×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
13	C₂×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
15	C₂×#1136	C₂²	1	4	yes	complemented
16	C₂²×C₇⋊C₈	C₂²	1	4	yes	complemented
17	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	—
18	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
19	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
20	C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
21	#6377	C₂²	1	4	yes	complemented
22	#2973	—	7	1	no	maximal, Sylow
23	C₂⁴×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂⁴×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇×D₄	—	2	4	no	—
26	C₂×C₇×D₄	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₇	—	4	2	no	—
28	C₂²×C₄×C₇	—	4	2	no	—
29	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
40	C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₄²×C₇	C₂×C₄	1	2	yes	—
42	C₄²×C₇	C₂³	1	2	yes	—
43	#1136	—	4	4	no	—
44	C₂×C₇⋊C₈	—	4	4	no	—
45	C₂×C₇⋊C₈	C₂³	1	8	yes	—
46	C₂×C₄×D₄	D₇	1	1	yes	characteristic
47	#435	—	7	8	no	—
48	#432	—	7	2	no	—
49	#407	—	7	2	no	—
50	C₂×C₂²⋊C₈	—	7	2	no	—
51	C₂³×C₇	—	2	2	no	—
52	C₂³×C₇	—	4	1	no	—
53	C₂³×C₇	—	4	1	no	—
54	C₂³×C₇	—	4	1	no	—
55	C₂³×C₇	—	4	1	no	—
56	C₂³×C₇	M₁₆	1	4	yes	complemented
57	C₂³×C₇	C₂²×C₄	1	2	yes	—
58	C₂³×C₇	C₂²×C₄	1	2	yes	—
59	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₇×D₄	—	2	8	no	—
61	C₂×C₄×C₇	—	2	2	no	—
62	C₂×C₄×C₇	—	2	4	no	—
63	C₂×C₄×C₇	—	2	4	no	—
64	C₂×C₄×C₇	—	2	2	no	—
65	C₂×C₄×C₇	—	4	2	no	—
66	C₂×C₄×C₇	—	4	2	no	—
67	C₂×C₄×C₇	—	4	1	no	—
68	C₂×C₄×C₇	—	4	1	no	—
69	C₂×C₄×C₇	—	4	1	no	—
70	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
76	C₇⋊C₈	—	2	8	no	—
77	C₂²×D₄	Dic₇	1	1	yes	characteristic
78	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
79	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
80	C₂×M₁₆	—	7	4	no	—
81	C₂²×C₈	—	7	4	no	—
82	C₄×D₄	Dic₇	1	4	yes	—
83	C₄×D₄	C₂×D₇	1	4	yes	—
84	C₂×C₄⋊C₄	Dic₇	1	1	yes	characteristic
85	C₂×C₂²⋊C₄	Dic₇	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	Dic₇	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
88	C₄⋊C₈	—	7	4	no	—
89	C₂²⋊C₈	—	7	4	no	—
90	C₂²⋊C₈	—	7	4	no	—
91	C₂²⋊C₈	—	7	4	no	—
92	#97	—	7	4	no	—
93	C₂²×C₇	—	2	1	no	—
94	C₂²×C₇	—	2	2	no	—
95	C₂²×C₇	—	2	1	no	—
96	C₂²×C₇	—	2	2	no	—
97	C₂²×C₇	—	2	2	no	—
98	C₂²×C₇	—	2	2	no	—
99	C₂²×C₇	—	2	2	no	—
100	C₂²×C₇	—	4	4	no	—
101	C₂²×C₇	—	4	4	no	—
102	C₂²×C₇	C₂×M₁₆	1	2	yes	—
103	C₂²×C₇	C₂×M₁₆	1	4	yes	—
104	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
105	C₄×C₇	—	2	1	no	—
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	—	2	2	no	—
108	C₄×C₇	—	2	2	no	—
109	C₄×C₇	—	2	2	no	—
110	C₄×C₇	H(5,𝔽₂)	1	2	yes	—
111	C₂⁴	C₂×Dic₇	1	1	yes	characteristic
112	C₂⁴	C₂×Dic₇	1	1	yes	characteristic
113	C₂×D₄	—	2	4	no	—
114	C₂×D₄	C₂×Dic₇	1	4	yes	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
118	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
119	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
120	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
121	C₂²×C₄	C₂×Dic₇	1	2	yes	—
122	C₂²×C₄	C₂×Dic₇	1	2	yes	—
123	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₇	1	2	yes	—
125	C₂²×C₄	C₂²×D₇	1	2	yes	—
126	M₁₆	—	28	4	no	—
127	C₂×C₈	—	7	8	no	—
128	C₂×C₈	—	28	4	no	—
129	C₄⋊C₄	C₂×Dic₇	1	4	yes	—
130	C₂²⋊C₄	C₂×Dic₇	1	4	yes	—
131	C₂²⋊C₄	C₂×Dic₇	1	4	yes	—
132	C₄²	C₂×Dic₇	1	2	yes	—
133	C₄²	C₂²×D₇	1	2	yes	—
134	C₂×C₇	—	2	2	no	—
135	C₂×C₇	—	2	4	no	—
136	C₂×C₇	—	4	2	no	—
137	C₂×C₇	#518	1	1	yes	characteristic
138	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
139	C₂×C₇	#568	1	1	yes	characteristic
140	C₂³	—	2	2	no	—
141	C₂³	—	4	1	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	—	4	1	no	—
145	C₂³	#1136	1	4	yes	complemented
146	C₂³	C₂²×Dic₇	1	2	yes	—
147	C₂³	C₂²×Dic₇	1	2	yes	—
148	C₂³	C₂²×Dic₇	1	1	yes	characteristic
149	D₄	—	2	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₇	1	2	yes	—
161	C₂×C₄	C₂²×Dic₇	1	2	yes	—
162	C₂×C₄	C₂²×Dic₇	1	2	yes	—
163	C₂×C₄	C₂²×Dic₇	1	2	yes	—
164	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
165	C₈	—	14	8	no	—
166	C₇	#2973	1	1	yes	characteristic, complemented, Sylow
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂×#1136	1	2	yes	—
177	C₂²	C₂×#1136	1	4	yes	—
178	C₂²	C₂³×Dic₇	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	#6548	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	2	4	no	—
187	C₂	—	4	2	no	—
188	C₂	C₂²×#1136	1	1	yes	characteristic
189	C₂	#90489	1	1	yes	characteristic
190	C₂	#90510	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₂×C₄×C₇×D₄	C₂
prime radical	C₂²×C₇×D₄	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₇
hypocenter	C₇	#2973
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #89979	C₂	semidirect
3. C₂×C₂²⋊(C₇⋊C₈)	C₂	semidirect
4. #89783	C₂	semidirect
5. #89698	C₂	semidirect
15. C₂×#1136	C₂²	semidirect
16. C₂²×C₇⋊C₈	C₂²	semidirect
18. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
19. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
20. C₄⋊(C₇⋊C₈)	C₂²	semidirect
21. #6377	C₂²	semidirect
56. C₂³×C₇	M₁₆	semidirect
145. C₂³	#1136	semidirect
166. C₇	#2973	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×D₄	2
#89979	2
C₂×C₂²⋊(C₇⋊C₈)	2
#89783	2
#89698	2
#2973	7