Subgroups — #89645

Order 448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₈	C₂	1	1	yes	maximal, characteristic
2	#6450	C₂	1	2	yes	maximal
3	C₇×#107	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
5	#6384	C₂	1	2	yes	maximal
6	C₂×Dic₁₄	C₂²	1	2	yes	—
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	Dic₇⋊C₄	—	2	2	no	—
10	C₄×Dic₇	—	2	2	no	—
11	C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	Dic₂₈	C₄	1	4	yes	complemented
14	C₄×Q₁₆	—	7	1	no	maximal, Sylow
15	C₂×C₄×C₇	—	2	2	no	—
16	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₇	—	2	2	no	—
18	C₂×Dic₇	D₄	1	1	yes	characteristic
19	C₂×Dic₇	D₄	1	1	yes	characteristic
20	Dic₁₄	—	2	2	no	—
21	Dic₁₄	C₂×C₄	1	4	yes	—
22	C₇×C₈	C₂×C₄	1	2	yes	—
23	C₇⋊C₈	—	2	1	no	—
24	C₂×Q₁₆	—	7	1	no	—
25	C₄×Q₈	—	7	2	no	—
26	#107	D₇	1	1	yes	characteristic
27	#103	—	7	2	no	—
28	C₄×C₈	—	7	1	no	—
29	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₇	—	4	2	no	—
31	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₇	#41	1	1	yes	characteristic
33	Dic₇	—	2	4	no	—
34	Dic₇	—	2	1	no	—
35	Dic₇	Q₁₆	1	2	yes	complemented
36	C₂×Q₈	—	7	2	no	—
37	Q₁₆	—	7	4	no	—
38	C₂×C₈	—	7	1	no	—
39	C₂×C₈	C₂×D₇	1	1	yes	characteristic
40	C₄⋊C₄	—	14	2	no	—
41	C₄⋊C₄	C₂×D₇	1	2	yes	—
42	C₄²	—	7	1	no	—
43	C₄²	—	14	2	no	—
44	C₂×C₇	C₄×D₄	1	1	yes	characteristic
45	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₇	#135	1	1	yes	characteristic
47	Q₈	—	7	4	no	—
48	Q₈	—	14	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	7	1	no	—
51	C₂×C₄	—	7	1	no	—
52	C₂×C₄	—	14	2	no	—
53	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
54	C₈	—	14	1	no	—
55	C₈	C₄×D₇	1	2	yes	—
56	C₇	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂²	D₄×D₇	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	7	2	no	—
60	C₄	—	14	4	no	—
61	C₄	—	14	1	no	—
62	C₄	C₂×C₄×D₇	1	1	yes	characteristic
63	C₄	#1161	1	1	yes	characteristic
64	C₂	#6456	1	1	yes	characteristic
65	C₂	#6475	1	1	yes	characteristic
66	C₂	D₇×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#107	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#107	D₇
hypocenter	C₇	C₄×Q₁₆
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
13. Dic₂₈	C₄	semidirect
35. Dic₇	Q₁₆	semidirect
56. C₇	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₈	2
#6450	2
C₇×#107	2
C₈×Dic₇	2
#6384	2
C₄×Q₁₆	7