Subgroups — #90015

Order 448.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#6498	C₂	1	3	yes	maximal, complemented
3	#6408	C₂	1	3	yes	maximal, complemented
4	C₇×#41	—	2	3	no	—
5	C₂×C₇×D₄	—	2	3	no	—
6	C₂×C₇×D₄	C₂²	1	3	yes	—
7	C₂²×Dic₇	C₂²	1	1	yes	characteristic, complemented
8	C₂²⋊Dic₇	—	2	3	no	—
9	C₂²⋊Dic₇	C₂²	1	3	yes	complemented
10	Dic₇⋊C₄	—	2	3	no	—
11	#458	—	7	1	no	maximal, Sylow
12	C₂³×C₇	—	2	1	no	—
13	C₂³×C₇	D₄	1	3	yes	—
14	C₂³×C₇	C₂³	1	1	yes	characteristic
15	C₇×Q₈	—	2	1	no	—
16	C₇×D₄	—	2	3	no	—
17	C₇×D₄	—	2	6	no	—
18	C₂×C₄×C₇	—	2	3	no	—
19	C₂×C₄×C₇	D₄	1	3	yes	—
20	C₂×Dic₇	—	2	3	no	—
21	C₂×Dic₇	—	2	3	no	—
22	H(5,𝔽₂)	D₇	1	1	yes	characteristic
23	#123	—	7	3	no	—
24	#99	—	7	3	no	—
25	C₂²×C₇	—	2	3	no	—
26	C₂²×C₇	—	4	1	no	—
27	C₂²×C₇	—	4	3	no	—
28	C₂²×C₇	—	8	1	no	—
29	C₂²×C₇	C₂×D₄	1	3	yes	—
30	C₄×C₇	—	2	3	no	—
31	Dic₇	—	4	3	no	—
32	Dic₇	—	4	1	no	—
33	#41	—	2	3	no	—
34	C₂×D₄	—	2	3	no	—
35	C₂×D₄	C₂×D₇	1	3	yes	—
36	C₂²×C₄	—	7	1	no	—
37	C₄⋊C₄	—	14	3	no	—
38	C₂²⋊C₄	—	7	3	no	—
39	C₂²⋊C₄	—	14	3	no	—
40	C₂×C₇	—	2	3	no	—
41	C₂×C₇	—	4	3	no	—
42	C₂×C₇	#120	1	1	yes	characteristic
43	C₂³	—	2	1	no	—
44	C₂³	C₂²⋊D₇	1	3	yes	—
45	C₂³	C₂²×D₇	1	1	yes	characteristic
46	Q₈	—	2	1	no	—
47	D₄	—	2	3	no	—
48	D₄	—	2	6	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	14	3	no	—
51	C₂×C₄	—	14	3	no	—
52	C₂×C₄	C₂²⋊D₇	1	3	yes	—
53	C₇	#458	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	2	3	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	4	3	no	—
57	C₂²	—	8	1	no	—
58	C₂²	C₂×C₂²⋊D₇	1	3	yes	—
59	C₄	—	2	3	no	—
60	C₄	—	28	3	no	—
61	C₄	—	28	1	no	—
62	C₂	—	2	3	no	—
63	C₂	—	4	3	no	—
64	C₂	#6516	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6516
commutator, cosocle	C₂³×C₇	C₂³
Frattini	C₂³	C₂²×D₇
Fitting, prime radical	C₇×H(5,𝔽₂)	C₂
socle	C₂×C₇	#120
hypercenter	H(5,𝔽₂)	D₇
hypocenter	C₇	#458
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. #6498	C₂	semidirect
3. #6408	C₂	semidirect
7. C₂²×Dic₇	C₂²	semidirect
9. C₂²⋊Dic₇	C₂²	semidirect
53. C₇	#458	semidirect

Maximal subgroups

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

subgroup	index
C₇×H(5,𝔽₂)	2
#6498	2
#6408	2
#458	7