Subgroups — #89953

Order 448.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#6507	C₂	1	2	yes	maximal
3	C₂×#1144	C₂	1	2	yes	maximal
4	C₂×Dic₂₈	C₂	1	1	yes	maximal, characteristic
5	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×Dic₁₄	C₂²	1	2	yes	—
8	C₇×Q₁₆	—	2	2	no	—
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	#1144	—	2	4	no	—
11	Dic₇⋊C₄	—	2	4	no	—
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	Dic₂₈	—	2	2	no	—
15	#494	—	7	1	no	maximal, Sylow
16	C₇×Q₈	—	2	2	no	—
17	C₇×Q₈	—	2	2	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₇	—	2	2	no	—
21	C₂×Dic₇	D₄	1	2	yes	—
22	Dic₁₄	—	2	2	no	—
23	Dic₁₄	—	2	2	no	—
24	C₇×C₈	D₄	1	2	yes	—
25	C₇⋊C₈	D₄	1	2	yes	—
26	C₂×Q₁₆	—	7	2	no	—
27	C₂×Q₁₆	—	7	1	no	—
28	C₂×Q₁₆	D₇	1	1	yes	characteristic
29	C₄⋊Q₈	—	7	2	no	—
30	C₄×C₈	—	7	1	no	—
31	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₇	—	4	2	no	—
33	C₄×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₇	C₂×D₄	1	1	yes	characteristic
35	Dic₇	—	4	2	no	—
36	Dic₇	Q₁₆	1	4	yes	complemented
37	C₂×Q₈	—	7	2	no	—
38	C₂×Q₈	C₂×D₇	1	2	yes	—
39	Q₁₆	—	2	2	no	—
40	Q₁₆	—	14	4	no	—
41	Q₁₆	—	14	2	no	—
42	C₂×C₈	—	7	1	no	—
43	C₂×C₈	C₂×D₇	1	1	yes	characteristic
44	C₄⋊C₄	—	14	4	no	—
45	C₄²	—	7	1	no	—
46	C₂×C₇	#127	1	1	yes	characteristic
47	C₂×C₇	C₂×Q₁₆	1	2	yes	—
48	Q₈	—	2	2	no	—
49	Q₈	—	2	2	no	—
50	Q₈	—	14	2	no	—
51	Q₈	—	14	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	7	2	no	—
54	C₂×C₄	—	14	2	no	—
55	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
56	C₈	—	7	2	no	—
57	C₈	C₂²⋊D₇	1	2	yes	—
58	C₇	#494	1	1	yes	characteristic, complemented, Sylow
59	C₂²	D₄×D₇	1	1	yes	characteristic
60	C₄	—	4	2	no	—
61	C₄	—	7	4	no	—
62	C₄	—	28	2	no	—
63	C₄	D₄×D₇	1	1	yes	characteristic
64	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
65	C₂	D₇×Q₁₆	1	2	yes	—
66	C₂	#6503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	#494
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
36. Dic₇	Q₁₆	semidirect
58. C₇	#494	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₁₆	2
#6507	2
C₂×#1144	2
C₂×Dic₂₈	2
C₈×Dic₇	2
#494	7