Subgroups — #20295908

Order 6976.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#16812778	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182012	C₂	1	1	yes	maximal, characteristic
4	#13182039	—	2	1	no	—
5	#13182038	C₄	1	2	yes	complemented
6	D₁₀₉⋊C₈	—	2	1	no	—
7	C₄⋊#89137	—	2	1	no	—
8	C₂²⋊#89137	C₄	1	2	yes	—
9	C₄×#89137	—	2	1	no	—
10	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
11	#13182012	—	2	1	no	—
12	#13182012	—	2	1	no	—
13	C₂×#11724784	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×#89137	—	2	2	no	—
17	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₀₉	D₄	1	1	yes	characteristic
20	C₄×D₁₀₉	D₄	1	1	yes	characteristic
21	C₄×D₁₀₉	D₄	1	1	yes	characteristic
22	C₄×D₁₀₉	Q₈	1	1	yes	characteristic
23	#11724786	—	2	2	no	—
24	C₈×C₁₀₉	—	2	1	no	—
25	#11724784	—	2	1	no	—
26	C₂²×C₁₀₉	C₄²	1	1	yes	characteristic
27	C₂×D₁₀₉	—	4	1	no	—
28	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
30	#89137	—	4	2	no	—
31	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₁₀₉	—	2	1	no	—
36	C₂×C₁₀₉	#95	1	1	yes	characteristic
37	D₁₀₉	—	2	1	no	—
38	D₁₀₉	—	2	1	no	—
39	C₁₀₉	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	109	1	no	maximal, Sylow
41	C₂×M₁₆	—	109	1	no	—
42	C₂×M₁₆	—	109	1	no	—
43	#117	—	109	1	no	—
44	C₂²×C₄	—	109	1	no	—
45	M₁₆	—	109	2	no	—
46	M₁₆	—	218	1	no	—
47	M₁₆	—	218	1	no	—
48	M₁₆	—	218	1	no	—
49	C₂×C₈	—	109	1	no	—
50	C₂×C₈	—	218	1	no	—
51	C₂×C₈	#89137	1	1	yes	characteristic, complemented
52	C₄⋊C₄	—	218	1	no	—
53	C₂²⋊C₄	—	109	2	no	—
54	C₄²	—	218	1	no	—
55	C₂³	—	109	1	no	—
56	C₂×C₄	—	109	1	no	—
57	C₂×C₄	—	109	1	no	—
58	C₂×C₄	—	109	1	no	—
59	C₂×C₄	—	109	1	no	—
60	C₂×C₄	—	109	1	no	—
61	C₂×C₄	—	218	2	no	—
62	C₂×C₄	C₂×#89137	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	218	2	no	—
65	C₈	—	218	1	no	—
66	C₂²	—	109	1	no	—
67	C₂²	—	109	1	no	—
68	C₂²	—	436	1	no	—
69	C₂²	C₄×#89137	1	1	yes	characteristic
70	C₄	—	109	1	no	—
71	C₄	—	109	1	no	—
72	C₄	—	436	2	no	—
73	C₄	C₂²⋊#89137	1	1	yes	characteristic
74	C₄	C₄⋊#89137	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	218	1	no	—
77	C₂	—	218	1	no	—
78	C₂	(C₂×#89137)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#89137
commutator	C₂²×C₁₀₉	C₄²
Frattini	C₂×C₄	C₂×#89137
Fitting	C₂×C₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	#95
hypercenter	C₂×C₈	#89137
hypocenter	C₁₀₉	#344
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. #13182038	C₄	semidirect
13. C₂×#11724784	C₄	semidirect
14. C₂×C₈×C₁₀₉	C₄	semidirect
39. C₁₀₉	#344	semidirect
51. C₂×C₈	#89137	semidirect