Subgroups — #20292917

Order 6848.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812109	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812040	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812030	C₂	1	2	yes	maximal
5	C₄×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₄	—	2	1	no	—
7	C₂×C₁₀₇⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₀₇	—	2	1	no	—
9	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
10	C₂×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₁₀₇	—	2	1	no	—
12	C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
14	C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂³×C₁₀₇	—	2	1	no	—
16	D₄×C₁₀₇	—	2	1	no	—
17	D₄×C₁₀₇	Q₈	1	2	yes	complemented
18	C₂×C₄×C₁₀₇	—	2	1	no	—
19	C₂×C₄×C₁₀₇	—	4	2	no	—
20	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₁₀₇	—	2	2	no	—
24	Dic₂₁₄	—	4	2	no	—
25	C₁₀₇⋊C₈	—	2	2	no	—
26	C₂²×C₁₀₇	—	2	2	no	—
27	C₂²×C₁₀₇	—	4	1	no	—
28	C₂²×C₁₀₇	—	4	1	no	—
29	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₀₇	—	2	1	no	—
31	C₄×C₁₀₇	—	4	1	no	—
32	C₄×C₁₀₇	SD₁₆	1	2	yes	—
33	C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₀₇	#41	1	1	yes	characteristic
35	Dic₁₀₇	—	4	2	no	—
36	C₂×C₁₀₇	—	4	2	no	—
37	C₂×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₁₀₇	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	107	1	no	maximal, Sylow
42	C₄⋊Q₈	—	107	1	no	—
43	C₄×D₄	D₁₀₇	1	1	yes	characteristic
44	#106	—	107	2	no	—
45	C₄⋊C₈	—	107	1	no	—
46	#102	—	107	2	no	—
47	C₂×Q₈	—	214	1	no	—
48	C₂×D₄	C₂×D₁₀₇	1	1	yes	characteristic
49	C₂²×C₄	—	2	1	no	—
50	C₂×C₈	—	107	2	no	—
51	C₄⋊C₄	—	107	2	no	—
52	C₄⋊C₄	—	214	1	no	—
53	C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	C₂×D₁₀₇	1	1	yes	characteristic
56	C₂³	—	2	1	no	—
57	Q₈	—	428	2	no	—
58	D₄	—	2	1	no	—
59	D₄	Dic₂₁₄	1	2	yes	complemented
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	214	2	no	—
63	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
66	C₈	—	214	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	428	2	no	—
74	C₄	#13133806	1	2	yes	—
75	C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
76	C₄	#13133818	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	#16812088	1	1	yes	characteristic
79	C₂	C₂×#13133806	1	1	yes	characteristic
80	C₂	#16812134	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	C₄×D₄×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₄
hypercenter	C₄×D₄	D₁₀₇
hypocenter	C₁₀₇	#476
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #16812109	C₂	semidirect
2. #16812040	C₂	semidirect
3. C₄⋊(C₁₀₇⋊C₈)	C₂	semidirect
17. D₄×C₁₀₇	Q₈	semidirect
40. C₁₀₇	#476	semidirect
59. D₄	Dic₂₁₄	semidirect