Subgroups — #20295905

Order 6976.

98 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×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#89137	C₄	1	2	yes	—
5	#13182039	—	2	1	no	—
6	#13182038	C₄	1	2	yes	complemented
7	D₁₀₉⋊C₈	—	2	1	no	—
8	C₄⋊#89137	—	2	1	no	—
9	C₄⋊#89137	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
16	C₂×#89137	—	2	2	no	—
17	C₂×#89137	—	4	2	no	—
18	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₄×D₁₀₉	D₄	1	1	yes	characteristic
21	C₄×D₁₀₉	D₄	1	1	yes	characteristic
22	C₄×D₁₀₉	D₄	1	1	yes	characteristic
23	C₄×D₁₀₉	Q₈	1	1	yes	characteristic
24	#11724786	—	2	2	no	—
25	C₈×C₁₀₉	—	2	1	no	—
26	#11724784	—	2	1	no	—
27	C₂²×C₁₀₉	C₄²	1	1	yes	characteristic
28	C₂×D₁₀₉	—	2	1	no	—
29	C₂×D₁₀₉	—	2	1	no	—
30	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	#89137	—	4	2	no	—
33	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	Dic₁₀₉	C₂²⋊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	#100	1	1	yes	characteristic
41	D₁₀₉	#101	1	1	yes	characteristic
42	C₁₀₉	#343	1	1	yes	characteristic, complemented, Sylow
43	#343	—	109	1	no	maximal, Sylow
44	C₂×M₁₆	—	109	1	no	—
45	C₂×M₁₆	—	109	1	no	—
46	C₂×C₄⋊C₄	—	109	1	no	—
47	C₂²×C₄	—	109	1	no	—
48	C₂²×C₄	—	109	2	no	—
49	M₁₆	—	109	2	no	—
50	M₁₆	—	109	1	no	—
51	M₁₆	—	218	1	no	—
52	M₁₆	—	218	1	no	—
53	M₁₆	#89137	1	1	yes	characteristic, complemented
54	C₂×C₈	—	218	1	no	—
55	C₂×C₈	—	218	1	no	—
56	C₄⋊C₄	—	218	1	no	—
57	C₄⋊C₄	—	218	1	no	—
58	C₂³	—	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	—	109	1	no	—
64	C₂×C₄	—	436	2	no	—
65	C₂×C₄	—	218	2	no	—
66	C₂×C₄	C₂×#89137	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	218	2	no	—
69	C₈	—	218	1	no	—
70	C₂²	—	109	1	no	—
71	C₂²	—	109	1	no	—
72	C₂²	—	218	1	no	—
73	C₂²	—	218	1	no	—
74	C₂²	C₄×#89137	1	1	yes	characteristic
75	C₄	—	109	1	no	—
76	C₄	—	109	1	no	—
77	C₄	—	436	2	no	—
78	C₄	C₂²⋊#89137	1	1	yes	characteristic
79	C₄	C₄⋊#89137	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	109	1	no	—
82	C₂	—	109	1	no	—
83	C₂	—	218	1	no	—
84	C₂	(C₂×#89137)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#89137)⋊C₄
commutator	C₂²×C₁₀₉	C₄²
Frattini	C₂×C₄	C₂×#89137
Fitting	M₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#95
hypercenter	M₁₆	#89137
hypocenter	C₁₀₉	#343
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. #13182038	C₄	semidirect
11. #13182013	C₄	semidirect
14. M₁₆×C₁₀₉	C₄	semidirect
42. C₁₀₉	#343	semidirect
53. M₁₆	#89137	semidirect