Subgroups — #20296067

Order 6976.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#89137	C₂	1	2	yes	maximal, complemented
2	#16812767	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812761	C₂	1	2	yes	maximal, complemented
4	#16812753	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#89137	—	2	2	no	—
7	D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊#89137	C₂²	1	2	yes	—
9	C₂²⋊#89137	—	2	2	no	—
10	C₄×#89137	C₂²	1	1	yes	characteristic
11	D₄×D₁₀₉	C₂²	1	2	yes	—
12	D₈₇₂	C₄	1	1	yes	characteristic, complemented
13	#13182014	C₄	1	2	yes	complemented
14	C₈×D₁₀₉	C₂²	1	1	yes	characteristic
15	D₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂²×D₁₀₉	—	2	2	no	—
17	C₂×#89137	—	2	2	no	—
18	C₂×#89137	—	4	4	no	—
19	C₂×#89137	D₄	1	2	yes	—
20	D₄×C₁₀₉	C₂×C₄	1	2	yes	complemented
21	C₂²⋊D₁₀₉	—	2	2	no	—
22	D₄₃₆	C₂×C₄	1	2	yes	complemented
23	C₄×D₁₀₉	C₂³	1	1	yes	characteristic
24	#11724786	—	2	1	no	—
25	C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic, complemented
26	#11724784	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂²×C₁₀₉	—	2	2	no	—
28	C₂×D₁₀₉	—	2	2	no	—
29	C₂×D₁₀₉	—	4	2	no	—
30	C₂×D₁₀₉	—	4	2	no	—
31	C₂×D₁₀₉	C₂×D₄	1	1	yes	characteristic
32	#89137	—	2	2	no	—
33	#89137	—	4	2	no	—
34	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	Dic₁₀₉	#41	1	1	yes	characteristic
36	C₂×C₁₀₉	—	4	2	no	—
37	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
38	D₁₀₉	—	4	2	no	—
39	D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
40	C₁₀₉	#442	1	1	yes	characteristic, complemented, Sylow
41	#442	—	109	1	no	maximal, Sylow
42	C₂×D₈	—	109	1	no	—
43	C₄×D₄	—	109	2	no	—
44	#106	—	109	1	no	—
45	#102	—	109	2	no	—
46	#97	—	109	1	no	—
47	C₂×D₄	—	109	2	no	—
48	C₂²×C₄	—	218	2	no	—
49	D₈	—	109	2	no	—
50	D₈	—	109	1	no	—
51	D₈	#89137	1	1	yes	characteristic, complemented
52	C₂×C₈	—	109	1	no	—
53	C₂×C₈	—	109	1	no	—
54	C₄⋊C₄	—	109	2	no	—
55	C₂²⋊C₄	—	218	2	no	—
56	C₄²	—	109	1	no	—
57	C₂³	—	218	2	no	—
58	D₄	—	109	2	no	—
59	D₄	—	218	2	no	—
60	D₄	C₂×#89137	1	2	yes	complemented
61	C₂×C₄	—	109	1	no	—
62	C₂×C₄	—	109	2	no	—
63	C₂×C₄	—	436	4	no	—
64	C₂×C₄	—	218	2	no	—
65	C₈	—	109	1	no	—
66	C₈	—	218	1	no	—
67	C₈	C₂×#89137	1	1	yes	characteristic, complemented
68	C₂²	—	2	2	no	—
69	C₂²	—	109	1	no	—
70	C₂²	—	436	2	no	—
71	C₂²	—	436	2	no	—
72	C₂²	—	218	2	no	—
73	C₄	—	109	1	no	—
74	C₄	—	436	2	no	—
75	C₄	—	218	2	no	—
76	C₄	C₂²×#89137	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	—	109	2	no	—
79	C₂	—	436	2	no	—
80	C₂	D₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#89137
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₄	C₂²×#89137
Fitting	D₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₄×D₄
hypercenter	D₈	#89137
hypocenter	C₁₀₉	#442
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#89137	C₂	semidirect
2. #16812767	C₂	semidirect
3. #16812761	C₂	semidirect
4. #16812753	C₂	semidirect
12. D₈₇₂	C₄	semidirect
13. #13182014	C₄	semidirect
15. D₈×C₁₀₉	C₄	semidirect
20. D₄×C₁₀₉	C₂×C₄	semidirect
22. D₄₃₆	C₂×C₄	semidirect
25. C₈×C₁₀₉	C₂×C₄	semidirect
26. #11724784	C₂×C₄	semidirect
40. C₁₀₉	#442	semidirect
51. D₈	#89137	semidirect
60. D₄	C₂×#89137	semidirect
67. C₈	C₂×#89137	semidirect