Subgroups — #20295896

Order 6976.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13182039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#89137	C₂	1	1	yes	maximal, characteristic
3	#16812657	C₂	1	1	yes	maximal, characteristic
4	C₂²×#89137	—	2	1	no	—
5	#13182039	—	2	1	no	—
6	#13182039	C₄	1	2	yes	complemented
7	C₂×#11724786	—	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	C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic, complemented
12	#13182006	—	2	1	no	—
13	C₄×Dic₁₀₉	—	2	1	no	—
14	C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
16	C₂×#89137	—	2	2	no	—
17	C₂×#89137	—	2	2	no	—
18	C₂×#89137	—	2	2	no	—
19	C₂×C₄×C₁₀₉	—	2	1	no	—
20	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	2	1	no	—
22	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
23	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	C₄×D₁₀₉	Q₈	1	1	yes	characteristic
27	#11724786	—	2	2	no	—
28	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₁₀₉	—	2	1	no	—
30	C₂×D₁₀₉	—	2	1	no	—
31	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	#89137	—	2	4	no	—
34	C₄×C₁₀₉	—	4	1	no	—
35	C₄×C₁₀₉	C₄²	1	1	yes	characteristic
36	C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	Dic₁₀₉	—	4	1	no	—
38	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₁₀₉	#95	1	1	yes	characteristic
41	C₂×C₁₀₉	#104	1	1	yes	characteristic
42	C₂×C₁₀₉	#104	1	1	yes	characteristic
43	D₁₀₉	—	2	1	no	—
44	D₁₀₉	—	2	1	no	—
45	C₁₀₉	#339	1	1	yes	characteristic, complemented, Sylow
46	#339	—	109	1	no	maximal, Sylow
47	C₂×M₁₆	—	109	1	no	—
48	#117	—	109	1	no	—
49	C₂×C₄²	—	109	1	no	—
50	C₂²×C₄	—	109	1	no	—
51	C₂²×C₄	—	218	1	no	—
52	M₁₆	—	109	2	no	—
53	M₁₆	—	218	1	no	—
54	C₂×C₈	—	218	1	no	—
55	C₄⋊C₄	—	109	1	no	—
56	C₄⋊C₄	#89137	1	1	yes	characteristic, complemented
57	C₂²⋊C₄	—	218	1	no	—
58	C₄²	—	109	2	no	—
59	C₄²	—	218	1	no	—
60	C₄²	—	218	1	no	—
61	C₂³	—	109	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	109	1	no	—
64	C₂×C₄	—	109	1	no	—
65	C₂×C₄	—	109	1	no	—
66	C₂×C₄	—	109	1	no	—
67	C₂×C₄	—	109	1	no	—
68	C₂×C₄	—	218	1	no	—
69	C₂×C₄	—	218	2	no	—
70	C₂×C₄	—	218	2	no	—
71	C₂×C₄	—	218	2	no	—
72	C₂×C₄	C₂×#89137	1	1	yes	characteristic
73	C₈	—	218	2	no	—
74	C₂²	—	109	1	no	—
75	C₂²	—	109	1	no	—
76	C₂²	—	218	1	no	—
77	C₂²	—	218	1	no	—
78	C₂²	C₂²⋊#89137	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	—	109	1	no	—
81	C₄	—	109	1	no	—
82	C₄	—	436	1	no	—
83	C₄	—	218	4	no	—
84	C₄	C₄×#89137	1	1	yes	characteristic
85	C₄	C₄⋊#89137	1	1	yes	characteristic
86	C₂	—	218	1	no	—
87	C₂	—	218	1	no	—
88	C₂	(C₂×#89137)⋊C₄	1	1	yes	characteristic
89	C₂	#16812763	1	1	yes	characteristic
90	C₂	#16812764	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
6. #13182039	C₄	semidirect
9. C₄×#89137	C₄	semidirect
11. C₄⋊Dic₁₀₉	C₄	semidirect
14. C₄⋊C₄×C₁₀₉	C₄	semidirect
45. C₁₀₉	#339	semidirect
56. C₄⋊C₄	#89137	semidirect