Subgroups — #20295898

Order 6976.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812778	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#89137	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#89137	—	2	1	no	—
5	C₄⋊#89137	C₄	1	2	yes	complemented
6	C₂²⋊#89137	—	2	1	no	—
7	C₄×#89137	—	2	1	no	—
8	C₄×#89137	—	2	1	no	—
9	C₄×#89137	C₄	1	2	yes	complemented
10	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
11	#13182013	C₄	1	1	yes	characteristic, complemented
12	#13182012	—	2	1	no	—
13	C₈×D₁₀₉	—	2	1	no	—
14	M₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
16	C₂×#89137	—	2	4	no	—
17	C₂×#89137	—	2	2	no	—
18	C₂×#89137	—	2	2	no	—
19	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₀₉	Q₈	1	1	yes	characteristic
21	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄×D₁₀₉	D₄	1	1	yes	characteristic
24	C₄×D₁₀₉	D₄	1	1	yes	characteristic
25	C₈×C₁₀₉	—	2	1	no	—
26	#11724784	—	2	1	no	—
27	C₂²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₁₀₉	—	2	2	no	—
29	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
31	#89137	—	2	4	no	—
32	#89137	—	4	2	no	—
33	C₄×C₁₀₉	C₄²	1	1	yes	characteristic
34	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₁₀₉	—	2	1	no	—
38	C₂×C₁₀₉	#95	1	1	yes	characteristic
39	D₁₀₉	—	2	1	no	—
40	D₁₀₉	#104	1	2	yes	—
41	C₁₀₉	#339	1	1	yes	characteristic, complemented, Sylow
42	#339	—	109	1	no	maximal, Sylow
43	C₂×M₁₆	—	109	1	no	—
44	#117	—	109	1	no	—
45	C₂×C₄²	—	109	1	no	—
46	C₂²×C₄	—	109	1	no	—
47	C₂²×C₄	—	218	1	no	—
48	M₁₆	—	109	1	no	—
49	M₁₆	—	218	1	no	—
50	M₁₆	#89137	1	1	yes	characteristic, complemented
51	C₂×C₈	—	218	1	no	—
52	C₄⋊C₄	—	109	2	no	—
53	C₂²⋊C₄	—	218	1	no	—
54	C₄²	—	109	2	no	—
55	C₄²	—	218	1	no	—
56	C₄²	—	218	1	no	—
57	C₂³	—	109	1	no	—
58	C₂×C₄	—	109	1	no	—
59	C₂×C₄	—	109	1	no	—
60	C₂×C₄	—	109	1	no	—
61	C₂×C₄	—	109	1	no	—
62	C₂×C₄	—	109	1	no	—
63	C₂×C₄	—	218	2	no	—
64	C₂×C₄	—	218	2	no	—
65	C₂×C₄	—	218	4	no	—
66	C₂×C₄	C₂×#89137	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	218	1	no	—
69	C₂²	—	109	1	no	—
70	C₂²	—	109	1	no	—
71	C₂²	—	218	2	no	—
72	C₂²	C₄⋊#89137	1	1	yes	characteristic
73	C₄	—	109	1	no	—
74	C₄	—	109	1	no	—
75	C₄	—	436	2	no	—
76	C₄	—	218	4	no	—
77	C₄	C₄×#89137	1	1	yes	characteristic
78	C₄	C₂²⋊#89137	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	—	109	2	no	—
81	C₂	—	218	1	no	—
82	C₂	(C₂×#89137)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₄²
Frattini	C₂×C₄	C₂×#89137
Fitting	M₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#95
hypercenter	M₁₆	#89137
hypocenter	C₁₀₉	#339
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄⋊#89137	C₄	semidirect
9. C₄×#89137	C₄	semidirect
11. #13182013	C₄	semidirect
14. M₁₆×C₁₀₉	C₄	semidirect
41. C₁₀₉	#339	semidirect
50. M₁₆	#89137	semidirect