Subgroups — #20295938

Order 6976.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812768	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#89137	—	2	1	no	—
5	#13182032	—	2	1	no	—
6	D₄×D₁₀₉	—	2	1	no	—
7	D₈₇₂	C₄	1	1	yes	characteristic, complemented
8	#13182014	—	2	1	no	—
9	C₈×D₁₀₉	C₂²	1	1	yes	characteristic
10	D₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
11	C₂²×D₁₀₉	—	4	1	no	—
12	C₂×#89137	—	4	1	no	—
13	D₄×C₁₀₉	—	2	1	no	—
14	C₂²⋊D₁₀₉	—	4	1	no	—
15	D₄₃₆	—	2	1	no	—
16	C₄×D₁₀₉	D₄	1	1	yes	characteristic
17	C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	#11724784	D₄	1	1	yes	characteristic
19	C₂²×C₁₀₉	—	4	1	no	—
20	C₂×D₁₀₉	—	4	1	no	—
21	C₂×D₁₀₉	—	8	1	no	—
22	C₂×D₁₀₉	—	8	1	no	—
23	C₂×D₁₀₉	D₈	1	1	yes	characteristic
24	#89137	—	8	1	no	—
25	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	Dic₁₀₉	SD₁₆	1	1	yes	characteristic
27	C₂×C₁₀₉	—	8	1	no	—
28	C₂×C₁₀₉	#102	1	1	yes	characteristic
29	D₁₀₉	—	8	1	no	—
30	D₁₀₉	D₁₆	1	1	yes	characteristic
31	D₁₀₉	SD₃₂	1	1	yes	characteristic
32	C₁₀₉	#357	1	1	yes	characteristic, complemented, Sylow
33	#357	—	109	1	no	maximal, Sylow
34	C₂×D₈	—	109	1	no	—
35	C₂×C₁₆	—	109	1	no	—
36	#107	—	109	1	no	—
37	C₂×D₄	—	218	1	no	—
38	D₈	—	109	1	no	—
39	D₈	—	218	1	no	—
40	D₈	#89137	1	1	yes	characteristic, complemented
41	C₂×C₈	—	109	1	no	—
42	C₄⋊C₄	—	218	1	no	—
43	C₁₆	—	218	1	no	—
44	C₂³	—	436	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	436	1	no	—
47	D₄	—	218	1	no	—
48	C₂×C₄	—	109	1	no	—
49	C₂×C₄	—	436	1	no	—
50	C₈	—	109	1	no	—
51	C₈	C₂×#89137	1	1	yes	characteristic
52	C₂²	—	4	1	no	—
53	C₂²	—	872	1	no	—
54	C₂²	—	872	1	no	—
55	C₂²	—	109	1	no	—
56	C₂²	—	436	1	no	—
57	C₄	—	872	1	no	—
58	C₄	—	109	1	no	—
59	C₄	C₂²⋊#89137	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	872	1	no	—
62	C₂	—	109	1	no	—
63	C₂	—	109	1	no	—
64	C₂	#16812761	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16812761
commutator	C₈×C₁₀₉	C₂×C₄
Frattini	C₈	C₂×#89137
Fitting	D₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	#102
hypercenter	D₈	#89137
hypocenter	C₁₀₉	#357
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₁₀₉⋊C₁₆	C₂	semidirect
2. #16812768	C₂	semidirect
7. D₈₇₂	C₄	semidirect
10. D₈×C₁₀₉	C₄	semidirect
32. C₁₀₉	#357	semidirect
40. D₈	#89137	semidirect