Subgroups — #28338333

Order 13696.

148 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	#20292455	C₂	1	1	yes	maximal, characteristic
4	#20292432	C₂	1	2	yes	maximal
5	#346×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	complemented
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	2	yes	—
26	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₁₀₇	—	4	2	no	—
28	C₂×Dic₁₀₇	—	4	2	no	—
29	Dic₂₁₄	—	2	4	no	—
30	Dic₂₁₄	—	4	2	no	—
31	C₈×C₁₀₇	—	2	1	no	—
32	C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₀₇	#41	1	1	yes	characteristic
34	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₀₇	—	2	2	no	—
36	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₀₇	#135	1	1	yes	characteristic
38	Dic₁₀₇	—	4	4	no	—
39	Dic₁₀₇	—	8	2	no	—
40	C₂×C₁₀₇	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₀₇	#510	1	2	yes	—
42	#2176	—	107	1	no	maximal, Sylow
43	C₁₀₇	#2176	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	107	1	no	—
45	C₄×Q₁₆	—	107	2	no	—
46	#367	—	107	1	no	—
47	#358	—	107	2	no	—
48	#346	D₁₀₇	1	1	yes	characteristic
49	C₂×Q₁₆	—	107	2	no	—
50	C₄×Q₈	—	214	2	no	—
51	Q₃₂	—	107	4	no	—
52	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
54	#107	—	107	2	no	—
55	#103	—	214	2	no	—
56	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
57	C₂×Q₈	—	214	2	no	—
58	Q₁₆	—	107	4	no	—
59	Q₁₆	—	214	2	no	—
60	C₂×C₈	D₄₂₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
62	C₄⋊C₄	—	428	2	no	—
63	C₄⋊C₄	—	214	2	no	—
64	C₄²	—	428	2	no	—
65	C₄²	D₄₂₈	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₁₀₇	1	2	yes	—
68	Q₈	—	428	2	no	—
69	Q₈	—	214	4	no	—
70	C₂×C₄	—	428	2	no	—
71	C₂×C₄	—	428	2	no	—
72	C₂×C₄	D₈₅₆	1	2	yes	—
73	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₁₀₇	1	1	yes	characteristic
76	C₈	#13133818	1	1	yes	characteristic
77	C₂²	C₂×D₈₅₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	856	2	no	—
80	C₄	—	428	4	no	—
81	C₄	C₄×D₄₂₈	1	1	yes	characteristic
82	C₄	#16812131	1	1	yes	characteristic
83	C₂	C₄×D₈₅₆	1	1	yes	characteristic
84	C₂	#20293070	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₅₆
commutator	C₈×C₁₀₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	#346×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	#346	D₁₀₇
hypocenter	C₁₀₇	#2176
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₈₅₆	C₄	semidirect
43. C₁₀₇	#2176	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
C₄×Dic₄₂₈	2
#20292455	2
#20292432	2
#346×C₁₀₇	2
#2176	107