Subgroups — #89320

Order 448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₇	C₄	1	1	yes	characteristic
4	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₇×C₂²⋊C₄	—	2	2	no	—
7	C₂×C₇⋊C₈	—	2	2	no	—
8	#323	—	7	1	no	maximal, Sylow
9	C₂³×C₇	—	2	1	no	—
10	C₂³×C₇	—	2	1	no	—
11	C₂³×C₇	—	4	1	no	—
12	C₂³×C₇	—	4	1	no	—
13	C₂³×C₇	C₈	1	2	yes	complemented
14	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇	—	2	1	no	—
16	C₂×C₄×C₇	—	4	1	no	—
17	C₂×C₄×C₇	—	4	1	no	—
18	C₂×C₄×C₇	D₄	1	2	yes	—
19	C₇⋊C₈	—	4	2	no	—
20	C₂×C₂²⋊C₄	D₇	1	1	yes	characteristic
21	C₂²⋊C₈	—	7	2	no	—
22	C₂²×C₇	—	2	1	no	—
23	C₂²×C₇	—	2	1	no	—
24	C₂²×C₇	—	2	2	no	—
25	C₂²×C₇	—	4	2	no	—
26	C₂²×C₇	—	4	2	no	—
27	C₂²×C₇	—	4	1	no	—
28	C₂²×C₇	—	4	1	no	—
29	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₇	M₁₆	1	1	yes	characteristic
32	C₄×C₇	—	2	2	no	—
33	C₄×C₇	—	4	1	no	—
34	C₂⁴	Dic₇	1	1	yes	characteristic
35	C₂²×C₄	Dic₇	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
37	C₂×C₈	—	14	2	no	—
38	C₂²⋊C₄	—	2	2	no	—
39	C₂×C₇	—	2	1	no	—
40	C₂×C₇	—	2	1	no	—
41	C₂×C₇	—	4	2	no	—
42	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₇	#99	1	1	yes	characteristic
44	C₂×C₇	#100	1	1	yes	characteristic
45	C₂³	—	2	1	no	—
46	C₂³	—	2	1	no	—
47	C₂³	—	4	1	no	—
48	C₂³	—	4	1	no	—
49	C₂³	C₇⋊C₈	1	2	yes	complemented
50	C₂³	C₂×Dic₇	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	C₂²⋊D₇	1	2	yes	—
55	C₈	—	28	2	no	—
56	C₇	#323	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	1	no	—
64	C₂²	C₂×C₇⋊C₈	1	1	yes	characteristic
65	C₂²	#1136	1	1	yes	characteristic
66	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	4	1	no	—
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	4	2	no	—
72	C₂	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
73	C₂	#6407	1	1	yes	characteristic
74	C₂	#6408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂×C₈
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₂×C₇×C₂²⋊C₄	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂⁴×C₇	C₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂×C₂²⋊C₄	D₇
hypocenter	C₇	#323
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₇⋊C₈)	C₂	semidirect
13. C₂³×C₇	C₈	semidirect
49. C₂³	C₇⋊C₈	semidirect
56. C₇	#323	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₂²⋊C₄	2
C₂²⋊(C₇⋊C₈)	2
#323	7