Subgroups — #28334629

Order 13696.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293006	C₂	1	1	yes	maximal, characteristic
2	#20292426	C₂	1	1	yes	maximal, characteristic
3	#356×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133808	—	2	1	no	—
5	C₂×#13133808	—	2	1	no	—
6	#16812109	C₄	1	1	yes	characteristic
7	#16812028	—	2	1	no	—
8	C₄×C₁₀₇⋊C₈	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
10	#101×C₁₀₇	—	2	1	no	—
11	C₂×Dic₂₁₄	—	2	1	no	—
12	#13133808	—	4	2	no	—
13	#13133808	—	4	2	no	—
14	#13133802	—	4	1	no	—
15	C₂×C₁₀₇⋊C₈	—	2	1	no	—
16	C₄⋊Dic₁₀₇	—	4	1	no	—
17	C₂×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
19	M₁₆×C₁₀₇	—	4	1	no	—
20	C₄⋊C₄×C₁₀₇	—	4	1	no	—
21	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	Q₈×C₁₀₇	—	4	1	no	—
23	Q₈×C₁₀₇	—	4	1	no	—
24	C₂×C₄×C₁₀₇	—	2	1	no	—
25	C₂×C₄×C₁₀₇	—	2	1	no	—
26	C₂×C₄×C₁₀₇	—	2	1	no	—
27	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₁₀₇	—	4	1	no	—
29	Dic₂₁₄	—	4	1	no	—
30	Dic₂₁₄	—	4	1	no	—
31	C₈×C₁₀₇	—	8	1	no	—
32	C₁₀₇⋊C₈	—	2	2	no	—
33	C₁₀₇⋊C₈	—	8	1	no	—
34	C₂²×C₁₀₇	#99	1	1	yes	characteristic
35	C₄×C₁₀₇	—	2	2	no	—
36	C₄×C₁₀₇	—	2	1	no	—
37	C₄×C₁₀₇	—	4	1	no	—
38	C₄×C₁₀₇	—	4	1	no	—
39	Dic₁₀₇	—	8	1	no	—
40	C₂×C₁₀₇	—	2	1	no	—
41	C₂×C₁₀₇	#353	1	1	yes	characteristic
42	#1411	—	107	1	no	maximal, Sylow
43	C₁₀₇	#1411	1	1	yes	characteristic, complemented, Sylow
44	#494	—	107	1	no	—
45	#356	—	107	1	no	—
46	#356	D₁₀₇	1	1	yes	characteristic
47	C₂×Q₁₆	—	214	1	no	—
48	C₂×Q₁₆	—	214	1	no	—
49	C₄⋊Q₈	—	107	1	no	—
50	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
51	#101	—	2	1	no	—
52	#101	—	214	1	no	—
53	C₄×C₈	—	107	1	no	—
54	C₂×Q₈	—	214	1	no	—
55	C₂×Q₈	D₄₂₈	1	1	yes	characteristic
56	C₂×Q₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
57	Q₁₆	—	428	2	no	—
58	Q₁₆	—	428	2	no	—
59	M₁₆	—	4	1	no	—
60	M₁₆	—	428	1	no	—
61	C₂×C₈	—	214	1	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	428	1	no	—
64	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
65	Q₈	—	4	1	no	—
66	Q₈	—	4	1	no	—
67	Q₈	—	428	1	no	—
68	Q₈	—	428	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	428	1	no	—
73	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
74	C₈	—	8	1	no	—
75	C₈	—	856	1	no	—
76	C₈	—	214	2	no	—
77	C₂²	#16812023	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	856	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#20292419	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292419
commutator	C₄²×C₁₀₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₀₇
Fitting	#356×C₁₀₇	C₂
socle	C₂×C₁₀₇	#353
prime radical	C₂²×C₁₀₇	#99
cosocle	C₄⋊Q₈×C₁₀₇	C₂²
hypercenter	#356	D₁₀₇
hypocenter	C₁₀₇	#1411
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂×C₄×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
43. C₁₀₇	#1411	semidirect

Maximal subgroups

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

subgroup	index
#20293006	2
#20292426	2
#356×C₁₀₇	2
#1411	107