Subgroups — #90404

Order 448.

236 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	#6499	C₂	1	4	yes	maximal, complemented
3	#6498	C₂	1	4	yes	maximal, complemented
4	D₄×Dic₇	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₁₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×D₄	C₂²	1	4	yes	—
7	C₂×C₇×D₄	C₂²	1	2	yes	—
8	C₂²×Dic₇	C₂²	1	4	yes	—
9	C₂×Dic₁₄	C₂²	1	2	yes	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₇	C₂²	1	8	yes	—
12	C₂²⋊Dic₇	C₂²	1	4	yes	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	Dic₇⋊C₄	C₂²	1	4	yes	—
15	C₄×Dic₇	C₂²	1	4	yes	—
16	#559	—	7	1	no	maximal, Sylow
17	C₂³×C₇	C₂³	1	4	yes	—
18	C₇×D₄	—	2	8	no	—
19	C₇×D₄	—	4	2	no	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₇	C₂³	1	2	yes	—
22	C₂×Dic₇	—	4	4	no	—
23	C₂×Dic₇	C₂³	1	4	yes	—
24	C₂×Dic₇	C₂³	1	4	yes	—
25	Dic₁₄	—	2	4	no	—
26	#127	D₇	1	1	yes	characteristic
27	#124	—	7	4	no	—
28	#123	—	7	4	no	—
29	C₄×Q₈	—	7	2	no	—
30	C₄×D₄	—	7	4	no	—
31	C₂²×C₇	—	2	8	no	—
32	C₂²×C₇	—	2	4	no	—
33	C₂²×C₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₇	—	2	1	no	—
35	C₄×C₇	#41	1	4	yes	—
36	Dic₇	—	2	4	no	—
37	Dic₇	—	2	4	no	—
38	C₂×Q₈	—	7	2	no	—
39	C₂×D₄	C₂×D₇	1	4	yes	—
40	C₂×D₄	C₂×D₇	1	2	yes	—
41	C₂²×C₄	—	7	4	no	—
42	C₄⋊C₄	—	7	4	no	—
43	C₄⋊C₄	—	7	2	no	—
44	C₂²⋊C₄	—	7	8	no	—
45	C₂²⋊C₄	—	7	4	no	—
46	C₄²	—	7	4	no	—
47	C₄²	C₂×D₇	1	1	yes	characteristic
48	C₂×C₇	—	4	4	no	—
49	C₂×C₇	C₂×#41	1	2	yes	—
50	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂³	C₂²×D₇	1	4	yes	—
52	Q₈	—	14	4	no	—
53	D₄	—	2	8	no	—
54	D₄	—	4	2	no	—
55	C₂×C₄	—	7	4	no	—
56	C₂×C₄	—	7	4	no	—
57	C₂×C₄	—	28	4	no	—
58	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₇	1	2	yes	—
60	C₇	#559	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₇	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	14	4	no	—
66	C₄	—	14	4	no	—
67	C₄	#1159	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#1159	1	2	yes	—
70	C₂	#6548	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting, prime radical	C₇×#127	C₂
hypercenter	#127	D₇
hypocenter	C₇	#559
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
2. #6499	C₂	semidirect
3. #6498	C₂	semidirect
4. D₄×Dic₇	C₂	semidirect
5. C₄×Dic₁₄	C₂	semidirect
60. C₇	#559	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
#6499	2
#6498	2
D₄×Dic₇	2
C₄×Dic₁₄	2
#559	7