Subgroups — #11726820

Order 896.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₅₆	C₂	1	4	yes	maximal
2	#89471	C₂	1	2	yes	maximal
3	C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₈	—	2	2	no	—
5	C₂×Dic₂₈	C₂²	1	4	yes	—
6	#6432	—	2	2	no	—
7	C₂×C₇×C₁₆	C₂²	1	2	yes	—
8	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	Dic₅₆	—	2	8	no	—
10	#2247	—	7	1	no	maximal, Sylow
11	C₂×Dic₁₄	—	2	4	no	—
12	C₂×C₇×C₈	D₄	1	1	yes	characteristic
13	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
14	C₄²×C₇	D₄	1	1	yes	characteristic
15	C₄⋊Dic₇	—	4	4	no	—
16	Dic₂₈	—	2	8	no	—
17	Dic₂₈	—	4	4	no	—
18	C₇×C₁₆	D₄	1	4	yes	—
19	C₂×Q₃₂	—	7	4	no	—
20	#494	—	7	2	no	—
21	C₄×C₁₆	D₇	1	1	yes	characteristic
22	C₂×C₄×C₇	D₈	1	2	yes	—
23	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₇	—	4	4	no	—
25	Dic₁₄	—	4	4	no	—
26	Dic₁₄	—	4	4	no	—
27	C₇×C₈	D₈	1	2	yes	—
28	C₇×C₈	C₂×D₄	1	2	yes	—
29	C₂×Q₁₆	—	7	4	no	—
30	C₂×Q₁₆	—	14	2	no	—
31	C₄⋊Q₈	—	14	2	no	—
32	Q₃₂	—	14	8	no	—
33	C₂×C₁₆	C₂×D₇	1	2	yes	—
34	C₄×C₈	C₂×D₇	1	1	yes	characteristic
35	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₇	Q₃₂	1	4	yes	complemented
37	C₄×C₇	#127	1	1	yes	characteristic
38	C₄×C₇	C₂×D₈	1	1	yes	characteristic
39	Dic₇	—	8	4	no	—
40	C₂×Q₈	—	14	4	no	—
41	Q₁₆	—	14	8	no	—
42	Q₁₆	—	28	4	no	—
43	C₂×C₈	D₂₈	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
45	C₄⋊C₄	—	28	4	no	—
46	C₄²	D₂₈	1	1	yes	characteristic
47	C₁₆	D₂₈	1	4	yes	—
48	C₂×C₇	#493	1	1	yes	characteristic
49	C₂×C₇	C₂×Q₃₂	1	2	yes	—
50	Q₈	—	28	4	no	—
51	Q₈	—	28	4	no	—
52	C₂×C₄	—	28	4	no	—
53	C₂×C₄	D₅₆	1	2	yes	—
54	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
55	C₈	D₅₆	1	2	yes	—
56	C₈	C₂×D₂₈	1	2	yes	—
57	C₇	#2247	1	1	yes	characteristic, complemented, Sylow
58	C₂²	C₂×D₅₆	1	1	yes	characteristic
59	C₄	—	56	4	no	—
60	C₄	Dic₅₆	1	4	yes	complemented
61	C₄	#6437	1	1	yes	characteristic
62	C₄	C₂×D₅₆	1	1	yes	characteristic
63	C₂	#89467	1	1	yes	characteristic
64	C₂	C₂×Dic₅₆	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₄×C₇×C₁₆	C₂
socle, prime radical	C₂²×C₇	C₂×D₈
hypercenter	C₄×C₁₆	D₇
hypocenter	C₇	#2247
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
36. C₄×C₇	Q₃₂	semidirect
57. C₇	#2247	semidirect
60. C₄	Dic₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₅₆	2
#89471	2
C₄×C₇×C₁₆	2
#2247	7