Subgroups — #20295103

Order 6976.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13182012	C₂	1	2	yes	maximal, complemented
2	C₄×D₄₃₆	C₂	1	1	yes	maximal, characteristic
3	#16812611	C₂	1	2	yes	maximal, complemented
4	C₄×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₃₆	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
8	#13182012	—	2	4	no	—
9	C₂×#11724784	C₂²	1	1	yes	characteristic
10	C₂×#11724784	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
12	#13182006	C₄	1	2	yes	—
13	C₂×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₀₉	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
20	D₄₃₆	—	2	2	no	—
21	C₄×D₁₀₉	—	2	2	no	—
22	C₄×D₁₀₉	—	2	2	no	—
23	C₈×C₁₀₉	—	2	2	no	—
24	#11724784	—	2	1	no	—
25	#11724784	D₄	1	2	yes	complemented
26	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₁₀₉	—	2	2	no	—
28	C₂×D₁₀₉	—	2	2	no	—
29	C₂×D₁₀₉	—	2	2	no	—
30	C₄×C₁₀₉	—	2	1	no	—
31	C₄×C₁₀₉	M₁₆	1	2	yes	complemented
32	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₀₉	#41	1	1	yes	characteristic
34	Dic₁₀₉	—	2	2	no	—
35	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
36	C₂×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₁₀₉	#131	1	1	yes	characteristic
38	D₁₀₉	—	4	2	no	—
39	C₁₀₉	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	109	1	no	maximal, Sylow
41	C₂×M₁₆	—	109	2	no	—
42	C₄×D₄	—	109	1	no	—
43	C₄⋊C₈	D₁₀₉	1	1	yes	characteristic, complemented
44	C₂²⋊C₈	—	109	2	no	—
45	C₄×C₈	—	109	1	no	—
46	C₂×D₄	—	109	1	no	—
47	C₂²×C₄	—	109	2	no	—
48	M₁₆	—	218	4	no	—
49	C₂×C₈	—	109	1	no	—
50	C₂×C₈	—	109	1	no	—
51	C₂×C₈	C₂×D₁₀₉	1	2	yes	—
52	C₄⋊C₄	—	109	1	no	—
53	C₂²⋊C₄	—	109	2	no	—
54	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
55	C₂³	—	109	2	no	—
56	D₄	—	218	2	no	—
57	C₂×C₄	—	109	2	no	—
58	C₂×C₄	—	218	2	no	—
59	C₂×C₄	—	218	2	no	—
60	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
61	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	109	2	no	—
65	C₈	—	218	1	no	—
66	C₂²	—	218	2	no	—
67	C₂²	—	218	2	no	—
68	C₂²	—	218	2	no	—
69	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	218	2	no	—
72	C₄	#13182012	1	2	yes	complemented
73	C₄	D₄×D₁₀₉	1	1	yes	characteristic
74	C₄	#13182030	1	1	yes	characteristic
75	C₂	—	436	2	no	—
76	C₂	#16812661	1	1	yes	characteristic
77	C₂	C₂×#13182012	1	1	yes	characteristic
78	C₂	#16812707	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₄⋊C₈×C₁₀₉	C₂
hypercenter	C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#436
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#13182012	C₂	semidirect
3. #16812611	C₂	semidirect
4. C₄×#11724784	C₂	semidirect
5. C₄⋊C₈×C₁₀₉	C₂	semidirect
25. #11724784	D₄	semidirect
31. C₄×C₁₀₉	M₁₆	semidirect
39. C₁₀₉	#436	semidirect
43. C₄⋊C₈	D₁₀₉	semidirect
72. C₄	#13182012	semidirect