Subgroups — #90268

Order 448.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1161	C₂	1	2	yes	maximal, complemented
2	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#6458	C₂	1	6	yes	maximal, complemented
4	#6437	C₂	1	3	yes	maximal, complemented
5	C₄×D₂₈	C₂	1	3	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	#1161	—	2	8	no	—
8	C₂×D₂₈	C₂²	1	3	yes	—
9	C₂×D₂₈	C₂²	1	6	yes	—
10	C₂×D₂₈	C₂²	1	6	yes	—
11	C₂×C₄×D₇	C₂²	1	6	yes	—
12	C₇×C₄⋊C₄	C₂²	1	3	yes	—
13	C₄²×C₇	C₂²	1	3	yes	—
14	(C₂×D₇)⋊C₄	C₂²	1	6	yes	—
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	#550	—	7	1	no	maximal, Sylow
17	C₂²×D₇	C₂³	1	6	yes	—
18	C₇×Q₈	D₄	1	4	yes	complemented
19	C₂×C₄×C₇	C₂³	1	3	yes	—
20	C₂×C₄×C₇	C₂³	1	3	yes	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₇	C₂³	1	2	yes	—
23	D₂₈	—	2	12	no	—
24	D₂₈	—	2	6	no	—
25	D₂₈	—	4	6	no	—
26	C₄×D₇	—	2	12	no	—
27	C₂×#41	—	7	2	no	—
28	#127	—	7	3	no	—
29	#121	—	7	6	no	—
30	C₄×Q₈	D₇	1	1	yes	characteristic, complemented
31	C₄×D₄	—	7	3	no	—
32	C₂²×C₇	C₂⁴	1	1	yes	characteristic
33	C₂×D₇	—	2	6	no	—
34	C₂×D₇	—	2	6	no	—
35	C₂×D₇	—	2	6	no	—
36	C₄×C₇	—	2	3	no	—
37	C₄×C₇	C₂×D₄	1	6	yes	—
38	C₄×C₇	#41	1	2	yes	complemented
39	Dic₇	—	2	2	no	—
40	#41	—	14	8	no	—
41	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
42	C₂×D₄	—	7	3	no	—
43	C₂×D₄	—	7	6	no	—
44	C₂×D₄	—	7	6	no	—
45	C₂²×C₄	—	7	6	no	—
46	C₄⋊C₄	—	7	1	no	—
47	C₄⋊C₄	C₂×D₇	1	3	yes	—
48	C₂²⋊C₄	—	7	6	no	—
49	C₄²	C₂×D₇	1	3	yes	—
50	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
51	C₂×C₇	C₂×#41	1	1	yes	characteristic
52	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
53	D₇	—	4	6	no	—
54	C₂³	—	7	6	no	—
55	Q₈	D₂₈	1	4	yes	complemented
56	D₄	—	14	12	no	—
57	D₄	—	14	6	no	—
58	D₄	—	28	6	no	—
59	C₂×C₄	—	7	2	no	—
60	C₂×C₄	—	14	12	no	—
61	C₂×C₄	C₂²×D₇	1	3	yes	—
62	C₂×C₄	C₂²×D₇	1	3	yes	—
63	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
64	C₇	#550	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	14	6	no	—
66	C₂²	—	14	6	no	—
67	C₂²	—	14	6	no	—
68	C₂²	C₂³×D₇	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	14	2	no	—
71	C₄	C₂×D₂₈	1	6	yes	—
72	C₄	#1161	1	2	yes	complemented
73	C₂	—	28	6	no	—
74	C₂	C₂²×D₂₈	1	1	yes	characteristic
75	C₂	C₂×#1161	1	1	yes	characteristic
76	C₂	#6553	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₄×C₇×Q₈	C₂
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	#550
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#1161	C₂	semidirect
2. C₄×C₇×Q₈	C₂	semidirect
3. #6458	C₂	semidirect
4. #6437	C₂	semidirect
5. C₄×D₂₈	C₂	semidirect
18. C₇×Q₈	D₄	semidirect
30. C₄×Q₈	D₇	semidirect
38. C₄×C₇	#41	semidirect
55. Q₈	D₂₈	semidirect
64. C₇	#550	semidirect
72. C₄	#1161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1161	2
C₄×C₇×Q₈	2
#6458	2
#6437	2
C₄×D₂₈	2
#550	7