Subgroups — #89963

Order 448.

130 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, complemented
2	#6510	C₂	1	2	yes	maximal, complemented
3	C₂×#1143	C₂	1	2	yes	maximal, complemented
4	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic
5	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×D₂₈	C₂²	1	2	yes	—
8	C₇×Q₁₆	—	2	2	no	—
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	#1143	—	2	4	no	—
11	(C₂×D₇)⋊C₄	—	2	4	no	—
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	D₅₆	—	2	2	no	—
15	#495	—	7	1	no	maximal, Sylow
16	C₂²×D₇	—	2	2	no	—
17	C₇×Q₈	—	2	2	no	—
18	C₇×Q₈	—	2	2	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₇	D₄	1	2	yes	—
22	D₂₈	—	2	2	no	—
23	D₂₈	—	2	2	no	—
24	C₇×C₈	D₄	1	2	yes	—
25	C₇⋊C₈	D₄	1	2	yes	—
26	C₂×Q₁₆	D₇	1	1	yes	characteristic, complemented
27	C₂×SD₁₆	—	7	2	no	—
28	C₂×D₈	—	7	1	no	—
29	#124	—	7	2	no	—
30	C₄×C₈	—	7	1	no	—
31	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂×D₇	—	4	2	no	—
33	C₂×D₇	—	4	4	no	—
34	C₄×C₇	—	4	2	no	—
35	C₄×C₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	Dic₇	—	2	2	no	—
38	C₂×Q₈	C₂×D₇	1	2	yes	—
39	C₂×D₄	—	7	2	no	—
40	Q₁₆	—	2	2	no	—
41	SD₁₆	—	14	4	no	—
42	D₈	—	14	2	no	—
43	C₂×C₈	—	7	1	no	—
44	C₂×C₈	C₂×D₇	1	1	yes	characteristic
45	C₂²⋊C₄	—	14	4	no	—
46	C₄²	—	7	1	no	—
47	C₂×C₇	#127	1	1	yes	characteristic
48	C₂×C₇	#135	1	2	yes	—
49	D₇	—	8	2	no	—
50	C₂³	—	14	2	no	—
51	Q₈	—	2	2	no	—
52	Q₈	—	2	2	no	—
53	D₄	—	14	2	no	—
54	D₄	—	14	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	7	2	no	—
57	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
58	C₈	—	7	2	no	—
59	C₈	C₂²⋊D₇	1	2	yes	—
60	C₇	#495	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	28	4	no	—
62	C₂²	—	28	2	no	—
63	C₂²	D₄×D₇	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	14	2	no	—
66	C₄	D₄×D₇	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
68	C₂	—	56	2	no	—
69	C₂	#6482	1	2	yes	—
70	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	C₂²×C₇	C₂×D₄
prime radical	C₂×D₅₆	C₂
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	#495
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
1. C₂×C₇×Q₁₆	C₂	semidirect
2. #6510	C₂	semidirect
3. C₂×#1143	C₂	semidirect
5. C₈×Dic₇	C₂	semidirect
26. C₂×Q₁₆	D₇	semidirect
60. C₇	#495	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
#6510	2
C₂×#1143	2
C₂×D₅₆	2
C₈×Dic₇	2
#495	7