Subgroups — #90391

Order 448.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#6459	C₂	1	4	yes	maximal, complemented
3	#6458	C₂	1	4	yes	maximal, complemented
4	#6457	C₂	1	4	yes	maximal, complemented
5	#6438	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₈	C₂²	1	2	yes	—
7	C₂×D₂₈	C₂²	1	4	yes	—
8	C₂×C₄×D₇	C₂²	1	4	yes	—
9	C₂×Dic₁₄	C₂²	1	2	yes	—
10	C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	C₂²	1	4	yes	—
12	C₄²×C₇	C₂²	1	1	yes	characteristic
13	(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
15	(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
16	Dic₇⋊C₄	C₂²	1	4	yes	—
17	#562	—	7	1	no	maximal, Sylow
18	C₂²×D₇	C₂³	1	4	yes	—
19	C₂×C₄×C₇	C₂³	1	4	yes	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂³	1	2	yes	—
22	C₂×Dic₇	C₂³	1	4	yes	—
23	D₂₈	—	4	2	no	—
24	D₂₈	—	4	4	no	—
25	C₄×D₇	—	4	4	no	—
26	Dic₁₄	—	4	2	no	—
27	#125	D₇	1	1	yes	characteristic, complemented
28	#124	—	7	2	no	—
29	#123	—	7	4	no	—
30	C₂²⋊Q₈	—	7	4	no	—
31	#121	—	7	4	no	—
32	C₂²×C₇	C₂⁴	1	1	yes	characteristic
33	C₂×D₇	—	2	4	no	—
34	C₂×D₇	—	2	4	no	—
35	C₂×D₇	—	2	4	no	—
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	—	2	4	no	—
38	C₄×C₇	—	2	1	no	—
39	Dic₇	—	2	4	no	—
40	C₂×Q₈	—	7	2	no	—
41	C₂×D₄	—	7	4	no	—
42	C₂×D₄	—	7	2	no	—
43	C₂²×C₄	—	7	4	no	—
44	C₄⋊C₄	—	7	4	no	—
45	C₄⋊C₄	C₂×D₇	1	4	yes	—
46	C₄⋊C₄	C₂×D₇	1	2	yes	—
47	C₂²⋊C₄	—	7	4	no	—
48	C₂²⋊C₄	—	7	4	no	—
49	C₂²⋊C₄	—	7	4	no	—
50	C₄²	C₂×D₇	1	1	yes	characteristic
51	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
52	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
53	D₇	—	4	4	no	—
54	C₂³	—	7	4	no	—
55	Q₈	—	28	2	no	—
56	D₄	—	28	4	no	—
57	D₄	—	28	2	no	—
58	C₂×C₄	—	7	4	no	—
59	C₂×C₄	—	28	4	no	—
60	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₇	1	2	yes	—
62	C₂×C₄	C₂²×D₇	1	4	yes	—
63	C₇	#562	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	14	4	no	—
65	C₂²	—	14	4	no	—
66	C₂²	—	14	4	no	—
67	C₂²	C₂³×D₇	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	4	no	—
70	C₄	—	2	2	no	—
71	C₄	—	14	4	no	—
72	C₂	—	28	4	no	—
73	C₂	#6551	1	1	yes	characteristic
74	C₂	#6553	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₇×#125	C₂
hypercenter	#125	D₇
hypocenter	C₇	#562
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₇×#125	C₂	semidirect
2. #6459	C₂	semidirect
3. #6458	C₂	semidirect
4. #6457	C₂	semidirect
5. #6438	C₂	semidirect
27. #125	D₇	semidirect
63. C₇	#562	semidirect

Maximal subgroups

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

subgroup	index
C₇×#125	2
#6459	2
#6458	2
#6457	2
#6438	2
#562	7