Subgroups — #89334

Order 448.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#123	C₂	1	1	yes	maximal, characteristic
2	#6408	C₂	1	1	yes	maximal, characteristic
3	#6407	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₇	C₄	1	1	yes	characteristic, complemented
6	C₇×C₄⋊C₄	—	2	1	no	—
7	C₇×C₂²⋊C₄	—	2	1	no	—
8	C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic, complemented
9	C₂²⋊Dic₇	—	2	1	no	—
10	#1136	—	2	1	no	—
11	#352	—	7	1	no	maximal, Sylow
12	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₇	D₄	1	1	yes	characteristic
14	C₇×D₄	—	4	1	no	—
15	C₂×C₄×C₇	—	2	1	no	—
16	C₂×C₄×C₇	—	2	1	no	—
17	C₂×C₄×C₇	—	4	1	no	—
18	C₂×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂×Dic₇	—	4	1	no	—
20	C₇⋊C₈	—	4	1	no	—
21	#123	D₇	1	1	yes	characteristic
22	#100	—	7	1	no	—
23	#99	—	7	1	no	—
24	C₂²×C₇	—	2	1	no	—
25	C₂²×C₇	—	2	1	no	—
26	C₂²×C₇	—	4	1	no	—
27	C₂²×C₇	—	4	1	no	—
28	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₄×C₇	—	2	1	no	—
30	C₄×C₇	—	4	1	no	—
31	C₄×C₇	—	4	1	no	—
32	Dic₇	—	8	1	no	—
33	C₂×D₄	C₂×D₇	1	1	yes	characteristic
34	C₂²×C₄	Dic₇	1	1	yes	characteristic, complemented
35	M₁₆	—	14	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₂²⋊C₄	—	2	1	no	—
38	C₂²⋊C₄	—	14	1	no	—
39	C₂²⋊C₄	Dic₇	1	1	yes	characteristic, complemented
40	C₂×C₇	—	2	1	no	—
41	C₂×C₇	—	4	1	no	—
42	C₂×C₇	—	4	1	no	—
43	C₂×C₇	#99	1	1	yes	characteristic
44	C₂³	C₂×Dic₇	1	1	yes	characteristic
45	C₂³	C₂²⋊D₇	1	1	yes	characteristic
46	D₄	—	4	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	28	1	no	—
51	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
52	C₈	—	28	1	no	—
53	C₇	#352	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	4	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	56	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	4	1	no	—
66	C₂	#6408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6408
commutator	C₂³×C₇	C₂×C₄
Frattini	C₂×D₄	C₂×D₇
Fitting	C₇×#123	C₂
socle	C₂×C₇	#99
prime radical, cosocle	C₂×C₇×D₄	C₂²
hypercenter	#123	D₇
hypocenter	C₇	#352
LCS	G ▷ C₂³×C₇ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #123

Complements

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

N	H ≅ G/N	type
5. C₂²×C₄×C₇	C₄	semidirect
8. C₇×C₂²⋊C₄	C₄	semidirect
34. C₂²×C₄	Dic₇	semidirect
39. C₂²⋊C₄	Dic₇	semidirect
53. C₇	#352	semidirect

Maximal subgroups

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

subgroup	index
C₇×#123	2
#6408	2
#6407	2
#352	7