Subgroups — #90016

Order 448.

198 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, complemented
2	#6500	C₂	1	3	yes	maximal, complemented
3	#6408	C₂	1	3	yes	maximal, complemented
4	C₂³×D₇	C₂²	1	1	yes	characteristic, complemented
5	C₇×#41	—	2	3	no	—
6	C₂×C₇×D₄	—	2	3	no	—
7	C₂×C₇×D₄	C₂²	1	3	yes	—
8	C₂×C₂²⋊D₇	—	2	3	no	—
9	C₂²⋊Dic₇	C₂²	1	3	yes	complemented
10	(C₂×D₇)⋊C₄	—	2	3	no	—
11	B(4,𝔽₂)	—	7	1	no	maximal, Sylow
12	C₂³×C₇	—	2	1	no	—
13	C₂³×C₇	D₄	1	3	yes	complemented
14	C₂³×C₇	C₂³	1	1	yes	characteristic
15	C₂²×D₇	—	2	3	no	—
16	C₂²×D₇	—	4	2	no	—
17	C₇×Q₈	—	2	1	no	—
18	C₇×D₄	—	2	3	no	—
19	C₇×D₄	—	2	6	no	—
20	C₂×C₄×C₇	—	2	3	no	—
21	C₂×C₄×C₇	D₄	1	3	yes	complemented
22	C₂²⋊D₇	—	4	6	no	—
23	C₂×Dic₇	—	2	3	no	—
24	H(5,𝔽₂)	D₇	1	1	yes	characteristic, complemented
25	#120	—	7	3	no	—
26	#99	—	7	3	no	—
27	C₂²×C₇	—	2	3	no	—
28	C₂²×C₇	—	4	1	no	—
29	C₂²×C₇	—	4	3	no	—
30	C₂²×C₇	—	8	1	no	—
31	C₂²×C₇	C₂×D₄	1	3	yes	—
32	C₂×D₇	—	2	6	no	—
33	C₂×D₇	—	4	3	no	—
34	C₂×D₇	—	4	1	no	—
35	C₄×C₇	—	2	3	no	—
36	Dic₇	—	4	3	no	—
37	C₂⁴	—	7	1	no	—
38	#41	—	2	3	no	—
39	C₂×D₄	—	2	3	no	—
40	C₂×D₄	—	14	3	no	—
41	C₂×D₄	C₂×D₇	1	3	yes	—
42	C₂²⋊C₄	—	7	3	no	—
43	C₂²⋊C₄	—	14	3	no	—
44	C₂×C₇	—	2	3	no	—
45	C₂×C₇	—	4	3	no	—
46	C₂×C₇	#120	1	1	yes	characteristic
47	D₇	—	4	2	no	—
48	C₂³	—	2	1	no	—
49	C₂³	—	14	3	no	—
50	C₂³	—	28	2	no	—
51	C₂³	C₂²⋊D₇	1	3	yes	complemented
52	C₂³	C₂²×D₇	1	1	yes	characteristic
53	Q₈	—	2	1	no	—
54	D₄	—	2	3	no	—
55	D₄	—	2	6	no	—
56	D₄	—	28	6	no	—
57	C₂×C₄	—	2	3	no	—
58	C₂×C₄	—	14	3	no	—
59	C₂×C₄	C₂²⋊D₇	1	3	yes	complemented
60	C₇	B(4,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	2	3	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	3	no	—
64	C₂²	—	8	1	no	—
65	C₂²	—	14	6	no	—
66	C₂²	—	28	3	no	—
67	C₂²	—	28	1	no	—
68	C₂²	C₂×C₂²⋊D₇	1	3	yes	—
69	C₄	—	2	3	no	—
70	C₄	—	28	3	no	—
71	C₂	—	2	3	no	—
72	C₂	—	4	3	no	—
73	C₂	—	28	2	no	—
74	C₂	#6516	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6516
commutator, cosocle	C₂³×C₇	C₂³
Frattini	C₂³	C₂²×D₇
Fitting	C₇×H(5,𝔽₂)	C₂
socle	C₂×C₇	#120
hypercenter	H(5,𝔽₂)	D₇
hypocenter	C₇	B(4,𝔽₂)
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
1. C₇×H(5,𝔽₂)	C₂	semidirect
2. #6500	C₂	semidirect
3. #6408	C₂	semidirect
4. C₂³×D₇	C₂²	semidirect
9. C₂²⋊Dic₇	C₂²	semidirect
13. C₂³×C₇	D₄	semidirect
21. C₂×C₄×C₇	D₄	semidirect
24. H(5,𝔽₂)	D₇	semidirect
51. C₂³	C₂²⋊D₇	semidirect
59. C₂×C₄	C₂²⋊D₇	semidirect
60. C₇	B(4,𝔽₂)	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
#6500	2
#6408	2
B(4,𝔽₂)	7