Subgroups — #20295108

Order 6976.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₇₂	C₂	1	1	yes	maximal, characteristic
2	#16812661	C₂	1	2	yes	maximal, complemented
3	#16812619	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₃₆	C₂²	1	2	yes	—
7	C₂×C₄×D₁₀₉	—	2	2	no	—
8	D₈₇₂	C₄	1	4	yes	complemented
9	C₂×#11724784	C₂²	1	1	yes	characteristic
10	#13182006	—	2	2	no	—
11	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×D₁₀₉	—	2	2	no	—
15	C₂×C₄×C₁₀₉	—	2	2	no	—
16	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
18	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
19	D₄₃₆	—	2	2	no	—
20	D₄₃₆	C₂×C₄	1	4	yes	complemented
21	C₄×D₁₀₉	—	4	4	no	—
22	C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
23	#11724784	—	2	1	no	—
24	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
25	C₂×D₁₀₉	—	2	4	no	—
26	C₂×D₁₀₉	—	4	2	no	—
27	C₂×D₁₀₉	—	4	2	no	—
28	C₄×C₁₀₉	—	4	2	no	—
29	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₁₀₉	#41	1	1	yes	characteristic
31	Dic₁₀₉	—	2	1	no	—
32	Dic₁₀₉	D₈	1	2	yes	complemented
33	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
34	C₂×C₁₀₉	C₂×D₈	1	1	yes	characteristic
35	C₂×C₁₀₉	#135	1	1	yes	characteristic
36	D₁₀₉	—	4	4	no	—
37	C₁₀₉	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₈	—	109	1	no	maximal, Sylow
39	C₂×D₈	—	109	1	no	—
40	C₄×D₄	—	109	2	no	—
41	#107	D₁₀₉	1	1	yes	characteristic, complemented
42	#102	—	109	2	no	—
43	C₄×C₈	—	109	1	no	—
44	C₂×D₄	—	109	2	no	—
45	C₂²×C₄	—	218	2	no	—
46	D₈	—	109	4	no	—
47	C₂×C₈	—	109	1	no	—
48	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
49	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
50	C₂²⋊C₄	—	218	2	no	—
51	C₄²	—	109	1	no	—
52	C₂³	—	218	2	no	—
53	D₄	—	109	4	no	—
54	D₄	—	218	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	109	1	no	—
57	C₂×C₄	—	109	1	no	—
58	C₂×C₄	—	436	4	no	—
59	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
60	C₈	—	218	1	no	—
61	C₈	C₄×D₁₀₉	1	2	yes	complemented
62	C₂²	—	436	2	no	—
63	C₂²	—	436	2	no	—
64	C₂²	—	218	4	no	—
65	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	109	2	no	—
68	C₄	—	218	1	no	—
69	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
70	C₄	#13182030	1	1	yes	characteristic
71	C₂	—	436	4	no	—
72	C₂	#16812661	1	1	yes	characteristic
73	C₂	D₈×D₁₀₉	1	1	yes	characteristic
74	C₂	#16812722	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#107×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#107	D₁₀₉
hypocenter	C₁₀₉	C₄×D₈
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
2. #16812661	C₂	semidirect
3. #16812619	C₂	semidirect
4. C₈×Dic₁₀₉	C₂	semidirect
5. #107×C₁₀₉	C₂	semidirect
8. D₈₇₂	C₄	semidirect
20. D₄₃₆	C₂×C₄	semidirect
22. C₈×C₁₀₉	C₂×C₄	semidirect
32. Dic₁₀₉	D₈	semidirect
37. C₁₀₉	C₄×D₈	semidirect
41. #107	D₁₀₉	semidirect
61. C₈	C₄×D₁₀₉	semidirect