Subgroups — #20295901

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	C₂×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#13182039	C₄	1	2	yes	—
5	#13182038	—	2	1	no	—
6	D₁₀₉⋊C₈	—	2	1	no	—
7	D₁₀₉⋊C₈	—	2	1	no	—
8	D₁₀₉⋊C₈	C₄	1	2	yes	—
9	C₂×#11724786	—	2	1	no	—
10	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
11	#13182013	C₄	1	1	yes	characteristic
12	#13182012	—	2	1	no	—
13	C₈×D₁₀₉	—	2	1	no	—
14	M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂²×D₁₀₉	Q₈	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
18	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₄×D₁₀₉	D₄	1	1	yes	characteristic
21	C₄×D₁₀₉	D₄	1	1	yes	characteristic
22	#11724786	—	2	2	no	—
23	#11724786	—	2	2	no	—
24	C₈×C₁₀₉	—	2	1	no	—
25	#11724784	—	2	1	no	—
26	C₂²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂×D₁₀₉	—	2	2	no	—
28	C₂×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₁₀₉	C₄²	1	1	yes	characteristic
31	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₁₀₉	—	2	1	no	—
35	C₂×C₁₀₉	#95	1	1	yes	characteristic
36	D₁₀₉	—	2	1	no	—
37	D₁₀₉	#108	1	2	yes	—
38	C₁₀₉	#341	1	1	yes	characteristic, complemented, Sylow
39	#341	—	109	1	no	maximal, Sylow
40	C₂×M₁₆	—	109	1	no	—
41	C₂×M₁₆	—	109	1	no	—
42	C₂²×C₈	—	109	1	no	—
43	C₂²×C₄	—	109	1	no	—
44	M₁₆	—	109	2	no	—
45	M₁₆	—	109	1	no	—
46	M₁₆	—	218	1	no	—
47	M₁₆	—	218	1	no	—
48	M₁₆	#89137	1	1	yes	characteristic
49	C₂×C₈	—	109	2	no	—
50	C₂×C₈	—	218	1	no	—
51	C₂×C₈	—	218	1	no	—
52	C₂×C₈	—	218	1	no	—
53	C₂×C₈	—	218	1	no	—
54	C₂³	—	109	1	no	—
55	C₂×C₄	—	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	C₂×#89137	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	218	2	no	—
63	C₈	—	218	2	no	—
64	C₈	—	218	1	no	—
65	C₂²	—	109	1	no	—
66	C₂²	—	109	1	no	—
67	C₂²	—	218	2	no	—
68	C₂²	C₄⋊#89137	1	1	yes	characteristic
69	C₄	—	109	1	no	—
70	C₄	—	109	1	no	—
71	C₄	C₄×#89137	1	1	yes	characteristic
72	C₄	C₂²⋊#89137	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	109	2	no	—
75	C₂	—	218	1	no	—
76	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	#341
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
38. C₁₀₉	#341	semidirect