Subgroups — C₄×Dic₈₅₆ #28338321

Order 13696.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₅₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₄₂₈	C₂	1	2	yes	maximal
3	#20292453	C₂	1	1	yes	maximal, characteristic
4	#20292432	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₈	C₂²	1	2	yes	—
7	C₄×Dic₂₁₄	—	2	2	no	—
8	Dic₈₅₆	C₄	1	4	yes	direct product factor
9	#16812043	C₂²	1	2	yes	—
10	#16812034	—	2	2	no	—
11	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₁₄	—	2	2	no	—
15	Dic₄₂₈	—	2	2	no	—
16	Dic₄₂₈	C₂×C₄	1	4	yes	—
17	C₄⋊Dic₁₀₇	—	2	2	no	—
18	Dic₁₀₇⋊C₄	—	4	2	no	—
19	C₄×Dic₁₀₇	—	4	2	no	—
20	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
22	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₀₇	—	2	1	no	—
24	C₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₁₀₇	—	4	2	no	—
29	C₂×Dic₁₀₇	—	4	2	no	—
30	Dic₂₁₄	—	2	4	no	—
31	Dic₂₁₄	—	4	2	no	—
32	C₈×C₁₀₇	—	2	1	no	—
33	C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₀₇	#41	1	1	yes	characteristic
35	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₀₇	—	2	1	no	—
37	C₄×C₁₀₇	Q₃₂	1	2	yes	complemented
38	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
39	C₄×C₁₀₇	#135	1	1	yes	characteristic
40	Dic₁₀₇	—	4	4	no	—
41	Dic₁₀₇	—	8	2	no	—
42	C₂×C₁₀₇	C₄×D₈	1	1	yes	characteristic
43	C₂×C₁₀₇	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₁₀₇	#508	1	1	yes	characteristic
45	C₄×Q₃₂	—	107	1	no	maximal, Sylow
46	C₁₀₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₃₂	—	107	1	no	—
48	C₄×Q₁₆	—	107	2	no	—
49	#366	—	107	1	no	—
50	#358	—	107	2	no	—
51	C₄×C₁₆	D₁₀₇	1	1	yes	characteristic
52	C₂×Q₁₆	—	107	2	no	—
53	C₄×Q₈	—	214	2	no	—
54	Q₃₂	—	107	4	no	—
55	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
56	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
57	#107	—	107	2	no	—
58	#103	—	214	2	no	—
59	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
60	C₂×Q₈	—	214	2	no	—
61	Q₁₆	—	107	4	no	—
62	Q₁₆	—	214	2	no	—
63	C₂×C₈	D₄₂₈	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
65	C₄⋊C₄	—	428	2	no	—
66	C₄⋊C₄	—	214	2	no	—
67	C₄²	—	428	2	no	—
68	C₄²	D₄₂₈	1	1	yes	characteristic
69	C₁₆	—	2	1	no	—
70	C₁₆	C₄×D₁₀₇	1	2	yes	—
71	Q₈	—	428	2	no	—
72	Q₈	—	214	4	no	—
73	C₂×C₄	—	428	2	no	—
74	C₂×C₄	—	428	2	no	—
75	C₂×C₄	D₈₅₆	1	1	yes	characteristic
76	C₂×C₄	D₈₅₆	1	1	yes	characteristic
77	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	C₂×C₄×D₁₀₇	1	1	yes	characteristic
80	C₈	#13133818	1	1	yes	characteristic
81	C₂²	C₂×D₈₅₆	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	856	2	no	—
84	C₄	—	428	4	no	—
85	C₄	Dic₈₅₆	1	2	yes	direct product factor
86	C₄	C₄×D₄₂₈	1	1	yes	characteristic
87	C₄	#16812131	1	1	yes	characteristic
88	C₂	C₄×D₈₅₆	1	1	yes	characteristic
89	C₂	C₂×Dic₈₅₆	1	1	yes	characteristic
90	C₂	#20293059	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈₅₆
commutator	C₈×C₁₀₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	C₄×C₁₆×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	C₄×C₁₆	D₁₀₇
hypocenter	C₁₀₇	C₄×Q₃₂
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
8. Dic₈₅₆	C₄	direct (N × H)
37. C₄×C₁₀₇	Q₃₂	semidirect
46. C₁₀₇	C₄×Q₃₂	semidirect
85. C₄	Dic₈₅₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Dic₈₅₆	2
C₄×Dic₄₂₈	2
#20292453	2
#20292432	2
C₄×C₁₆×C₁₀₇	2
C₄×Q₃₂	107

Subgroups — C4×Dic856 #28338321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₈₅₆ #28338321