Subgroups — #89356

Order 448.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#131	C₂	1	1	yes	maximal, characteristic
2	#6386	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#41	C₄	1	1	yes	characteristic
5	C₇×M₁₆	—	2	1	no	—
6	C₇×M₁₆	C₄	1	1	yes	characteristic
7	C₂×C₇×C₈	—	2	1	no	—
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇⋊C₁₆	—	2	1	no	—
10	C₇⋊C₁₆	—	2	1	no	—
11	#350	—	7	1	no	maximal, Sylow
12	C₇×Q₈	C₈	1	1	yes	characteristic
13	C₇×D₄	—	2	1	no	—
14	C₇×D₄	C₈	1	1	yes	characteristic
15	C₂×C₄×C₇	—	2	1	no	—
16	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₇×C₈	—	2	1	no	—
18	C₇×C₈	D₄	1	1	yes	characteristic
19	C₇×C₈	D₄	1	1	yes	characteristic
20	#131	D₇	1	1	yes	characteristic
21	M₃₂	—	7	1	no	—
22	C₂×C₁₆	—	7	1	no	—
23	C₂²×C₇	—	2	1	no	—
24	C₂²×C₇	M₁₆	1	1	yes	characteristic
25	C₄×C₇	—	2	1	no	—
26	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×C₇	C₂×C₈	1	1	yes	characteristic
28	#41	Dic₇	1	1	yes	characteristic
29	M₁₆	—	2	1	no	—
30	M₁₆	Dic₇	1	1	yes	characteristic
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	C₂×D₇	1	1	yes	characteristic
33	C₁₆	—	14	1	no	—
34	C₁₆	—	14	1	no	—
35	C₂×C₇	—	2	1	no	—
36	C₂×C₇	—	4	1	no	—
37	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
38	Q₈	C₇⋊C₈	1	1	yes	characteristic
39	D₄	—	2	1	no	—
40	D₄	C₇⋊C₈	1	1	yes	characteristic
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	C₂²⋊D₇	1	1	yes	characteristic
45	C₈	C₂²⋊D₇	1	1	yes	characteristic
46	C₇	#350	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	2	1	no	—
48	C₂²	#1136	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
51	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×#131	C₂
socle	C₂×C₇	C₂²⋊C₈
prime radical	C₇×#41	C₄
cosocle	C₂×C₇×C₈	C₂²
hypercenter	#131	D₇
hypocenter	C₇	#350
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. #6386	C₂	semidirect
3. C₂×C₇⋊C₁₆	C₂	semidirect
46. C₇	#350	semidirect

Maximal subgroups

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

subgroup	index
C₇×#131	2
#6386	2
C₂×C₇⋊C₁₆	2
#350	7