Subgroups — #11725494

Order 896.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#407	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
3	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
4	C₂×C₇×M₁₆	C₄	1	2	yes	—
5	C₂²×C₇⋊C₈	—	2	2	no	—
6	C₇×C₂²⋊C₈	—	2	2	no	—
7	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
8	C₂²⋊(C₇⋊C₈)	C₄	1	4	yes	—
9	#1298	—	7	1	no	maximal, Sylow
10	C₂⁴×C₇	Q₈	1	1	yes	characteristic
11	C₂²×C₄×C₇	—	4	2	no	—
12	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₇	D₄	1	2	yes	—
15	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
16	C₇×M₁₆	—	4	2	no	—
17	C₂×C₇×C₈	—	2	2	no	—
18	C₂×C₇⋊C₈	—	2	4	no	—
19	C₂×C₇⋊C₈	—	4	4	no	—
20	#407	D₇	1	1	yes	characteristic
21	C₂×C₂²⋊C₈	—	7	2	no	—
22	C₂³×C₇	—	2	4	no	—
23	C₂³×C₇	—	4	1	no	—
24	C₂³×C₇	C₄⋊C₄	1	2	yes	—
25	C₂³×C₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₇	—	2	4	no	—
27	C₂×C₄×C₇	—	2	2	no	—
28	C₂×C₄×C₇	—	4	2	no	—
29	C₂×C₄×C₇	—	4	1	no	—
30	C₂×C₄×C₇	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₇×C₈	—	4	2	no	—
34	C₇⋊C₈	—	4	4	no	—
35	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
36	C₂×M₁₆	Dic₇	1	2	yes	—
37	C₂²×C₈	—	14	2	no	—
38	C₂²⋊C₈	—	2	2	no	—
39	C₂²⋊C₈	—	7	4	no	—
40	C₂²⋊C₈	—	14	2	no	—
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	—	2	2	no	—
43	C₂²×C₇	—	2	2	no	—
44	C₂²×C₇	—	2	1	no	—
45	C₂²×C₇	—	4	4	no	—
46	C₂²×C₇	#95	1	1	yes	characteristic
47	C₂²×C₇	#108	1	4	yes	—
48	C₄×C₇	—	2	2	no	—
49	C₄×C₇	—	2	1	no	—
50	C₄×C₇	#99	1	2	yes	—
51	C₂⁴	Dic₁₄	1	1	yes	characteristic
52	C₂²×C₄	—	4	2	no	—
53	C₂²×C₄	C₄×D₇	1	2	yes	—
54	C₂²×C₄	D₂₈	1	1	yes	characteristic
55	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
56	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
57	M₁₆	—	4	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	14	4	no	—
60	C₂×C₈	—	28	4	no	—
61	C₂×C₇	—	2	4	no	—
62	C₂×C₇	—	4	1	no	—
63	C₂×C₇	#341	1	2	yes	—
64	C₂×C₇	#342	1	1	yes	characteristic
65	C₂³	—	2	4	no	—
66	C₂³	—	4	1	no	—
67	C₂³	Dic₇⋊C₄	1	2	yes	—
68	C₂³	C₄⋊Dic₇	1	1	yes	characteristic
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₄×Dic₇	1	1	yes	characteristic
74	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
75	C₂×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
76	C₈	—	4	2	no	—
77	C₈	—	28	4	no	—
78	C₇	#1298	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	4	4	no	—
84	C₂²	#6396	1	4	yes	—
85	C₂²	C₇⋊#95	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	#6408	1	2	yes	—
89	C₂	—	2	4	no	—
90	C₂	—	4	1	no	—
91	C₂	#89321	1	1	yes	characteristic
92	C₂	#89353	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₄²
Frattini	C₂³×C₄	C₂×D₇
Fitting	C₇×#407	C₂
socle	C₂²×C₇	#95
prime radical	C₂⁴×C₇	Q₈
cosocle	C₂³×C₄×C₇	C₂²
hypercenter	#407	D₇
hypocenter	C₇	#1298
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
78. C₇	#1298	semidirect

Maximal subgroups

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

subgroup	index
C₇×#407	2
C₂×C₂²⋊(C₇⋊C₈)	2
#1298	7