Subgroups — #89357

Order 448.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	#6392	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×D₄	—	2	1	no	—
5	C₇×D₈	—	2	1	no	—
6	C₇×D₈	C₄	1	2	yes	complemented
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₇	—	2	1	no	—
9	C₇⋊C₁₆	—	2	1	no	—
10	#357	—	7	1	no	maximal, Sylow
11	C₂³×C₇	—	4	1	no	—
12	C₇×D₄	—	2	2	no	—
13	C₇×D₄	—	4	1	no	—
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×Dic₇	—	4	1	no	—
16	C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₇×C₈	D₄	1	1	yes	characteristic
18	C₂×D₈	D₇	1	1	yes	characteristic
19	C₂×C₁₆	—	7	1	no	—
20	#107	—	7	1	no	—
21	C₂²×C₇	—	4	2	no	—
22	C₂²×C₇	—	8	1	no	—
23	C₂²×C₇	—	8	1	no	—
24	C₂²×C₇	D₈	1	1	yes	characteristic
25	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₇	SD₁₆	1	1	yes	characteristic
27	Dic₇	—	8	1	no	—
28	C₂×D₄	—	2	1	no	—
29	D₈	—	2	1	no	—
30	D₈	Dic₇	1	2	yes	complemented
31	C₂×C₈	C₂×D₇	1	1	yes	characteristic
32	C₄⋊C₄	—	14	1	no	—
33	C₁₆	—	14	1	no	—
34	C₂×C₇	—	8	2	no	—
35	C₂×C₇	#102	1	1	yes	characteristic
36	C₂×C₇	D₁₆	1	1	yes	characteristic
37	C₂×C₇	SD₃₂	1	1	yes	characteristic
38	C₂³	—	4	1	no	—
39	D₄	—	2	2	no	—
40	D₄	—	4	1	no	—
41	C₂×C₄	—	28	1	no	—
42	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
43	C₈	C₂×Dic₇	1	1	yes	characteristic
44	C₈	C₂²⋊D₇	1	1	yes	characteristic
45	C₇	#357	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	4	2	no	—
47	C₂²	—	8	1	no	—
48	C₂²	—	8	1	no	—
49	C₂²	#1141	1	1	yes	characteristic
50	C₄	—	56	1	no	—
51	C₄	#1142	1	1	yes	characteristic
52	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
53	C₂	—	8	2	no	—
54	C₂	#6400	1	1	yes	characteristic
55	C₂	#6401	1	1	yes	characteristic
56	C₂	#6406	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#1141
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting, prime radical	C₂×C₇×D₈	C₂
socle	C₂²×C₇	D₈
cosocle	C₂×C₇×C₈	C₂²
hypercenter	C₂×D₈	D₇
hypocenter	C₇	#357
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. #6392	C₂	semidirect
3. C₂×C₇⋊C₁₆	C₂	semidirect
6. C₇×D₈	C₄	semidirect
30. D₈	Dic₇	semidirect
45. C₇	#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₈	2
#6392	2
C₂×C₇⋊C₁₆	2
#357	7