Subgroups — #90191

Order 448.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#6489	C₂	1	2	yes	maximal
3	#6487	C₂	1	4	yes	maximal
4	#6453	C₂	1	4	yes	maximal, complemented
5	#6451	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	2	yes	—
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₄	C₂²	1	4	yes	—
9	C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂²⋊Dic₇	C₂²	1	4	yes	—
11	C₄⋊Dic₇	C₂²	1	2	yes	—
12	C₄⋊Dic₇	C₂²	1	4	yes	—
13	C₄⋊Dic₇	C₂²	1	2	yes	—
14	Dic₇⋊C₄	C₂²	1	8	yes	—
15	C₄×Dic₇	C₂²	1	4	yes	—
16	#544	—	7	1	no	maximal, Sylow
17	C₂³×C₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₇	—	4	2	no	—
19	C₂×C₄×C₇	Q₈	1	4	yes	complemented
20	C₂×C₄×C₇	C₂³	1	4	yes	—
21	C₂×C₄×C₇	C₂³	1	2	yes	—
22	C₂×Dic₇	C₂³	1	4	yes	—
23	C₂×Dic₇	C₂³	1	4	yes	—
24	Dic₁₄	—	4	4	no	—
25	C₄⋊Q₈	—	7	4	no	—
26	#125	—	7	4	no	—
27	C₂²⋊Q₈	—	7	4	no	—
28	#117	—	7	2	no	—
29	C₂×C₄⋊C₄	D₇	1	1	yes	characteristic
30	C₂²×C₇	—	2	1	no	—
31	C₂²×C₇	—	2	1	no	—
32	C₂²×C₇	C₂×Q₈	1	2	yes	—
33	C₂²×C₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₇	—	2	4	no	—
35	C₄×C₇	C₂×Q₈	1	4	yes	—
36	Dic₇	—	2	4	no	—
37	Dic₇	—	2	4	no	—
38	C₂×Q₈	—	7	4	no	—
39	C₂²×C₄	C₂×D₇	1	2	yes	—
40	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
41	C₄⋊C₄	—	7	8	no	—
42	C₄⋊C₄	—	7	2	no	—
43	C₄⋊C₄	—	7	2	no	—
44	C₄⋊C₄	—	7	4	no	—
45	C₄⋊C₄	C₂×D₇	1	4	yes	—
46	C₂²⋊C₄	—	7	4	no	—
47	C₄²	—	7	4	no	—
48	C₂×C₇	—	2	2	no	—
49	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
52	C₂³	C₂²×D₇	1	1	yes	characteristic
53	Q₈	—	28	4	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	—	7	4	no	—
56	C₂×C₄	—	7	4	no	—
57	C₂×C₄	Dic₁₄	1	4	yes	complemented
58	C₂×C₄	C₂²×D₇	1	4	yes	—
59	C₂×C₄	C₂²×D₇	1	2	yes	—
60	C₇	#544	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Dic₁₄	1	2	yes	—
64	C₂²	C₂³×D₇	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	14	4	no	—
67	C₄	—	14	4	no	—
68	C₄	C₂×Dic₁₄	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×Dic₁₄	1	1	yes	characteristic
71	C₂	#6548	1	1	yes	characteristic
72	C₂	#6551	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₂×C₇×C₄⋊C₄	C₂
prime radical	C₂³×C₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	#544
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #6453	C₂	semidirect
5. #6451	C₂	semidirect
19. C₂×C₄×C₇	Q₈	semidirect
57. C₂×C₄	Dic₁₄	semidirect
60. C₇	#544	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊C₄	2
#6489	2
#6487	2
#6453	2
#6451	2
#544	7