Subgroups — #28336077

Order 13696.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293030	C₂	1	4	yes	maximal
2	#20293007	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#494×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₂₈	—	2	2	no	—
6	#16812109	C₂²	1	2	yes	—
7	#16812042	—	2	8	no	—
8	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₀₇	—	2	2	no	—
10	C₄⋊Q₈×C₁₀₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₁₄	—	2	2	no	—
13	Dic₄₂₈	—	4	4	no	—
14	C₂×C₁₀₇⋊C₈	D₄	1	4	yes	—
15	C₄⋊Dic₁₀₇	—	2	4	no	—
16	C₂×Q₈×C₁₀₇	—	2	2	no	—
17	Q₁₆×C₁₀₇	—	4	4	no	—
18	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₀₇	—	2	4	no	—
20	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
21	Q₈×C₁₀₇	—	4	4	no	—
22	C₂×C₄×C₁₀₇	—	4	2	no	—
23	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
24	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
25	C₂×Dic₁₀₇	—	4	2	no	—
26	Dic₂₁₄	—	4	4	no	—
27	C₈×C₁₀₇	D₈	1	4	yes	—
28	C₁₀₇⋊C₈	Q₁₆	1	8	yes	complemented
29	C₂²×C₁₀₇	#127	1	1	yes	characteristic
30	C₄×C₁₀₇	—	8	2	no	—
31	C₄×C₁₀₇	C₂×D₈	1	2	yes	—
32	C₄×C₁₀₇	C₂×Q₁₆	1	4	yes	—
33	Dic₁₀₇	—	8	2	no	—
34	C₂×C₁₀₇	#493	1	1	yes	characteristic
35	C₂×C₁₀₇	#494	1	2	yes	—
36	#1713	—	107	1	no	maximal, Sylow
37	C₁₀₇	#1713	1	1	yes	characteristic, complemented, Sylow
38	#500	—	107	4	no	—
39	#494	—	107	1	no	—
40	#494	D₁₀₇	1	1	yes	characteristic
41	C₈²	—	107	1	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₂×Q₁₆	—	214	2	no	—
44	C₄⋊Q₈	—	107	2	no	—
45	C₄⋊Q₈	C₂×D₁₀₇	1	2	yes	—
46	#107	—	214	8	no	—
47	C₄×C₈	—	107	2	no	—
48	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	214	2	no	—
51	Q₁₆	—	4	4	no	—
52	Q₁₆	—	428	4	no	—
53	C₂×C₈	—	107	4	no	—
54	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	214	4	no	—
57	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	428	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	428	2	no	—
62	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
64	C₈	—	107	8	no	—
65	C₈	#13133803	1	4	yes	—
66	C₂²	#16812105	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	856	2	no	—
69	C₄	C₂×#13133803	1	2	yes	—
70	C₄	Q₁₆×D₁₀₇	1	4	yes	—
71	C₂	#20293003	1	1	yes	characteristic
72	C₂	#20293005	1	2	yes	—

	subgroup	G/H
center	C₂²	#16812105
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#494×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	#127
hypercenter	#494	D₁₀₇
hypocenter	C₁₀₇	#1713
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
28. C₁₀₇⋊C₈	Q₁₆	semidirect
37. C₁₀₇	#1713	semidirect

Maximal subgroups

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

subgroup	index
#20293030	2
#20293007	2
C₈×C₁₀₇⋊C₈	2
#494×C₁₀₇	2
#1713	107