Subgroups — #28338714

Order 13696.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₅₆	C₂	1	4	yes	maximal
2	#20293007	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₈	—	2	2	no	—
5	C₂×Dic₄₂₈	C₂²	1	4	yes	—
6	#16812109	—	2	2	no	—
7	Dic₈₅₆	—	2	8	no	—
8	C₂×C₁₆×C₁₀₇	C₂²	1	2	yes	—
9	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁₄	—	2	4	no	—
11	Dic₄₂₈	—	2	8	no	—
12	Dic₄₂₈	—	4	4	no	—
13	C₄⋊Dic₁₀₇	—	4	4	no	—
14	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
16	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₀₇	D₄	1	4	yes	—
18	C₂×C₄×C₁₀₇	D₈	1	2	yes	—
19	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
20	C₂×Dic₁₀₇	—	4	4	no	—
21	Dic₂₁₄	—	4	4	no	—
22	Dic₂₁₄	—	4	4	no	—
23	C₈×C₁₀₇	D₈	1	2	yes	—
24	C₈×C₁₀₇	C₂×D₄	1	2	yes	—
25	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
26	C₄×C₁₀₇	Q₃₂	1	4	yes	complemented
27	C₄×C₁₀₇	#127	1	1	yes	characteristic
28	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
29	Dic₁₀₇	—	8	4	no	—
30	C₂×C₁₀₇	#493	1	1	yes	characteristic
31	C₂×C₁₀₇	C₂×Q₃₂	1	2	yes	—
32	#2247	—	107	1	no	maximal, Sylow
33	C₁₀₇	#2247	1	1	yes	characteristic, complemented, Sylow
34	C₂×Q₃₂	—	107	4	no	—
35	#494	—	107	2	no	—
36	C₄×C₁₆	D₁₀₇	1	1	yes	characteristic
37	C₂×Q₁₆	—	107	4	no	—
38	C₂×Q₁₆	—	214	2	no	—
39	C₄⋊Q₈	—	214	2	no	—
40	Q₃₂	—	214	8	no	—
41	C₂×C₁₆	C₂×D₁₀₇	1	2	yes	—
42	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
43	C₂×Q₈	—	214	4	no	—
44	Q₁₆	—	428	4	no	—
45	Q₁₆	—	214	8	no	—
46	C₂×C₈	D₄₂₈	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
48	C₄⋊C₄	—	428	4	no	—
49	C₄²	D₄₂₈	1	1	yes	characteristic
50	C₁₆	D₄₂₈	1	4	yes	—
51	Q₈	—	428	4	no	—
52	Q₈	—	428	4	no	—
53	C₂×C₄	—	428	4	no	—
54	C₂×C₄	D₈₅₆	1	2	yes	—
55	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
56	C₈	D₈₅₆	1	2	yes	—
57	C₈	C₂×D₄₂₈	1	2	yes	—
58	C₂²	C₂×D₈₅₆	1	1	yes	characteristic
59	C₄	—	856	4	no	—
60	C₄	Dic₈₅₆	1	4	yes	complemented
61	C₄	#16812106	1	1	yes	characteristic
62	C₄	C₂×D₈₅₆	1	1	yes	characteristic
63	C₂	#20293004	1	1	yes	characteristic
64	C₂	C₂×Dic₈₅₆	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₅₆
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	C₄×C₁₆×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₀₇
hypocenter	C₁₀₇	#2247
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₁₀₇	Q₃₂	semidirect
33. C₁₀₇	#2247	semidirect
60. C₄	Dic₈₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₅₆	2
#20293007	2
C₄×C₁₆×C₁₀₇	2
#2247	107