Subgroups — #89309

Order 448.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1131	C₂	1	1	yes	maximal, characteristic
2	C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#6386	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₇	C₄	1	1	yes	characteristic
5	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₇⋊C₈	C₄	1	1	yes	characteristic
7	#1131	—	2	1	no	—
8	#1131	—	2	1	no	—
9	C₇×C₁₆	—	2	1	no	—
10	C₇⋊C₁₆	—	2	1	no	—
11	#349	—	7	1	no	maximal, Sylow
12	C₂²×D₇	C₈	1	1	yes	characteristic
13	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₇	C₈	1	1	yes	characteristic
15	C₄×D₇	—	2	1	no	—
16	C₄×D₇	—	2	1	no	—
17	C₇×C₈	D₄	1	1	yes	characteristic
18	C₇×C₈	D₄	1	1	yes	characteristic
19	C₇⋊C₈	—	2	1	no	—
20	C₂×M₁₆	—	7	1	no	—
21	M₃₂	—	7	1	no	—
22	M₃₂	D₇	1	1	yes	characteristic, complemented
23	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
24	C₂×D₇	—	2	1	no	—
25	C₂×D₇	—	4	1	no	—
26	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×C₇	M₁₆	1	1	yes	characteristic
28	Dic₇	—	2	1	no	—
29	C₂²×C₄	—	7	1	no	—
30	M₁₆	—	14	1	no	—
31	M₁₆	—	14	1	no	—
32	C₂×C₈	—	7	1	no	—
33	C₂×C₈	C₂×D₇	1	1	yes	characteristic
34	C₁₆	—	2	1	no	—
35	C₁₆	—	14	1	no	—
36	C₂×C₇	—	2	1	no	—
37	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
38	D₇	—	4	1	no	—
39	C₂³	—	7	1	no	—
40	C₂×C₄	—	7	1	no	—
41	C₂×C₄	—	14	1	no	—
42	C₂×C₄	—	14	1	no	—
43	C₂×C₄	C₄×D₇	1	1	yes	characteristic
44	C₈	—	14	1	no	—
45	C₈	D₂₈	1	1	yes	characteristic
46	C₈	C₂²⋊D₇	1	1	yes	characteristic
47	C₇	#349	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	14	1	no	—
49	C₂²	—	28	1	no	—
50	C₂²	C₈×D₇	1	1	yes	characteristic
51	C₄	—	14	1	no	—
52	C₄	#1131	1	1	yes	characteristic
53	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	28	1	no	—
56	C₂	(C₂×D₇)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₇)⋊C₄
commutator	C₂²×C₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×M₃₂	C₂
socle	C₂×C₇	C₂²⋊C₈
prime radical	C₂²×D₇	C₈
cosocle	C₂×C₇×C₈	C₂²
hypercenter	M₃₂	D₇
hypocenter	C₇	#349
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₇×M₃₂	C₂	semidirect
3. #6386	C₂	semidirect
22. M₃₂	D₇	semidirect
47. C₇	#349	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1131	2
C₇×M₃₂	2
#6386	2
#349	7