Subgroups — #89273

Order 448.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6507	C₂	1	1	yes	maximal, characteristic
2	C₇×#101	C₂	1	1	yes	maximal, characteristic
3	#6410	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
5	C₂×Dic₁₄	C₄	1	1	yes	characteristic
6	C₇×M₁₆	—	2	1	no	—
7	Dic₇⋊C₄	—	2	1	no	—
8	Dic₇⋊C₄	—	2	1	no	—
9	C₄×Dic₇	C₄	1	1	yes	characteristic
10	#1136	—	2	1	no	—
11	#356	—	7	1	no	maximal, Sylow
12	C₇×Q₈	—	4	1	no	—
13	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×Dic₇	—	2	1	no	—
17	C₂×Dic₇	—	2	1	no	—
18	Dic₁₄	—	4	1	no	—
19	C₇×C₈	—	4	1	no	—
20	C₇⋊C₈	—	4	1	no	—
21	C₄⋊Q₈	—	7	1	no	—
22	#101	—	7	1	no	—
23	#101	D₇	1	1	yes	characteristic
24	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₇	—	2	1	no	—
26	C₄×C₇	—	2	1	no	—
27	C₄×C₇	—	2	1	no	—
28	Dic₇	—	2	2	no	—
29	Dic₇	—	4	1	no	—
30	C₂×Q₈	—	7	1	no	—
31	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
32	M₁₆	—	2	1	no	—
33	M₁₆	—	14	1	no	—
34	C₄⋊C₄	—	14	1	no	—
35	C₄⋊C₄	—	14	1	no	—
36	C₄²	—	7	1	no	—
37	C₂×C₇	—	2	1	no	—
38	C₂×C₇	#99	1	1	yes	characteristic
39	Q₈	—	4	1	no	—
40	Q₈	—	28	1	no	—
41	C₂×C₄	—	14	1	no	—
42	C₂×C₄	—	14	1	no	—
43	C₂×C₄	C₄×D₇	1	1	yes	characteristic
44	C₂×C₄	D₂₈	1	1	yes	characteristic
45	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
46	C₈	—	4	1	no	—
47	C₈	—	28	1	no	—
48	C₇	#356	1	1	yes	characteristic, complemented, Sylow
49	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	14	2	no	—
54	C₄	—	28	1	no	—
55	C₂	—	2	1	no	—
56	C₂	#6380	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6380
commutator	C₂×C₄×C₇	C₂×C₄
Frattini	C₂×Q₈	C₂×D₇
Fitting	C₇×#101	C₂
socle	C₂×C₇	#99
prime radical	C₂²×C₇	C₂²⋊C₄
cosocle	C₂×C₇×Q₈	C₂²
hypercenter	#101	D₇
hypocenter	C₇	#356
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #101

Complements

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

N	H ≅ G/N	type
48. C₇	#356	semidirect

Maximal subgroups

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

subgroup	index
#6507	2
C₇×#101	2
#6410	2
#356	7