Subgroups — #89323

Order 448.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
3	C₂²×C₄×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×C₄⋊C₄	—	2	2	no	—
7	C₂×C₇⋊C₈	—	2	2	no	—
8	#324	—	7	1	no	maximal, Sylow
9	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₇	—	2	1	no	—
11	C₂×C₄×C₇	—	2	1	no	—
12	C₂×C₄×C₇	—	2	1	no	—
13	C₂×C₄×C₇	—	4	1	no	—
14	C₂×C₄×C₇	—	4	1	no	—
15	C₂×C₄×C₇	C₈	1	2	yes	complemented
16	C₂×C₄×C₇	D₄	1	2	yes	—
17	C₇⋊C₈	—	4	2	no	—
18	C₂×C₄⋊C₄	D₇	1	1	yes	characteristic
19	C₂²⋊C₈	—	7	2	no	—
20	C₂²×C₇	—	2	1	no	—
21	C₂²×C₇	—	2	1	no	—
22	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
24	C₂²×C₇	M₁₆	1	1	yes	characteristic
25	C₄×C₇	—	2	2	no	—
26	C₄×C₇	—	2	2	no	—
27	C₄×C₇	—	4	1	no	—
28	C₂²×C₄	Dic₇	1	1	yes	characteristic
29	C₂²×C₄	Dic₇	1	1	yes	characteristic
30	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
31	C₂×C₈	—	14	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₂×C₇	—	2	1	no	—
34	C₂×C₇	—	2	1	no	—
35	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₂×C₇	#99	1	1	yes	characteristic
37	C₂×C₇	#101	1	1	yes	characteristic
38	C₂³	C₂×Dic₇	1	1	yes	characteristic
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	C₇⋊C₈	1	2	yes	complemented
45	C₂×C₄	C₂²⋊D₇	1	2	yes	—
46	C₈	—	28	2	no	—
47	C₇	#324	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₂×C₇⋊C₈	1	1	yes	characteristic
51	C₂²	#1136	1	1	yes	characteristic
52	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	2	2	no	—
55	C₄	—	4	1	no	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
59	C₂	#6408	1	1	yes	characteristic
60	C₂	#6410	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂×C₈
Frattini	C₂²×C₄	C₂×D₇
Fitting	C₂×C₇×C₄⋊C₄	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂²×C₄×C₇	C₂²
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	#324
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
15. C₂×C₄×C₇	C₈	semidirect
44. C₂×C₄	C₇⋊C₈	semidirect
47. C₇	#324	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊C₄	2
C₂²⋊(C₇⋊C₈)	2
#324	7