Subgroups — #89340

Order 448.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#127	C₂	1	1	yes	maximal, characteristic
2	#6408	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×D₄	—	2	2	no	—
4	C₂×C₇×D₄	C₄	1	1	yes	characteristic, complemented
5	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
6	C₄²×C₇	C₄	1	1	yes	characteristic, complemented
7	C₂²⋊Dic₇	—	2	2	no	—
8	#353	—	7	1	no	maximal, Sylow
9	C₂³×C₇	—	2	1	no	—
10	C₂³×C₇	D₄	1	2	yes	—
11	C₇×D₄	—	4	1	no	—
12	C₇×D₄	—	4	1	no	—
13	C₇×D₄	—	4	4	no	—
14	C₂×C₄×C₇	—	2	1	no	—
15	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₇	—	4	2	no	—
17	#127	D₇	1	1	yes	characteristic
18	#99	—	7	2	no	—
19	C₂²×C₇	—	2	2	no	—
20	C₂²×C₇	—	4	2	no	—
21	C₂²×C₇	—	4	2	no	—
22	C₂²×C₇	—	4	1	no	—
23	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₄×C₇	—	2	2	no	—
25	C₄×C₇	—	2	1	no	—
26	Dic₇	—	8	2	no	—
27	C₂×D₄	—	2	2	no	—
28	C₂×D₄	Dic₇	1	1	yes	characteristic, complemented
29	C₂×D₄	C₂×D₇	1	1	yes	characteristic
30	C₂²⋊C₄	—	14	2	no	—
31	C₄²	Dic₇	1	1	yes	characteristic, complemented
32	C₂×C₇	—	2	1	no	—
33	C₂×C₇	—	4	2	no	—
34	C₂×C₇	—	8	1	no	—
35	C₂×C₇	#99	1	1	yes	characteristic
36	C₂³	—	2	1	no	—
37	C₂³	C₂²⋊D₇	1	2	yes	—
38	D₄	—	4	1	no	—
39	D₄	—	4	1	no	—
40	D₄	—	4	4	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	28	2	no	—
43	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
44	C₇	#353	1	1	yes	characteristic, complemented, Sylow
45	C₂²	—	2	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	2	no	—
48	C₂²	—	4	1	no	—
49	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	—	56	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	4	2	no	—
55	C₂	—	8	1	no	—
56	C₂	#6408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6408
commutator	C₂×C₄×C₇	C₂×C₄
Frattini	C₂×D₄	C₂×D₇
Fitting, prime radical	C₇×#127	C₂
socle	C₂×C₇	#99
cosocle	C₂×C₇×D₄	C₂²
hypercenter	#127	D₇
hypocenter	C₇	#353
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #127

Complements

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

N	H ≅ G/N	type
2. #6408	C₂	semidirect
4. C₂×C₇×D₄	C₄	semidirect
6. C₄²×C₇	C₄	semidirect
28. C₂×D₄	Dic₇	semidirect
31. C₄²	Dic₇	semidirect
44. C₇	#353	semidirect

Maximal subgroups

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

subgroup	index
C₇×#127	2
#6408	2
#353	7