Subgroups — #20295897

Order 6976.

110 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	C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#13182039	C₄	1	2	yes	complemented
5	#13182038	—	2	1	no	—
6	D₁₀₉⋊C₈	—	2	1	no	—
7	C₄⋊#89137	C₄	1	2	yes	complemented
8	C₂²⋊#89137	—	2	1	no	—
9	C₄×#89137	—	2	1	no	—
10	C₂×C₄×D₁₀₉	—	2	1	no	—
11	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₀₉	—	2	1	no	—
13	C₄×Dic₁₀₉	C₄	1	1	yes	characteristic, complemented
14	C₄²×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
16	C₂×#89137	—	2	2	no	—
17	C₂×C₄×C₁₀₉	—	2	1	no	—
18	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₁₀₉	—	2	1	no	—
20	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
21	C₄×D₁₀₉	—	2	4	no	—
22	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₁₀₉	D₄	1	1	yes	characteristic
25	C₄×D₁₀₉	Q₈	1	1	yes	characteristic
26	#11724786	—	2	2	no	—
27	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×D₁₀₉	—	2	2	no	—
29	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	#89137	—	4	2	no	—
32	C₄×C₁₀₉	—	2	2	no	—
33	C₄×C₁₀₉	C₄²	1	1	yes	characteristic
34	C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	Dic₁₀₉	—	2	2	no	—
36	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₀₉	—	2	1	no	—
39	C₂×C₁₀₉	#95	1	1	yes	characteristic
40	D₁₀₉	—	2	1	no	—
41	D₁₀₉	#104	1	2	yes	—
42	C₁₀₉	#339	1	1	yes	characteristic, complemented, Sylow
43	#339	—	109	1	no	maximal, Sylow
44	C₂×M₁₆	—	109	1	no	—
45	#117	—	109	1	no	—
46	C₂×C₄²	—	109	1	no	—
47	C₂²×C₄	—	109	1	no	—
48	C₂²×C₄	—	218	1	no	—
49	M₁₆	—	109	2	no	—
50	M₁₆	—	218	1	no	—
51	C₂×C₈	—	218	1	no	—
52	C₄⋊C₄	—	109	2	no	—
53	C₂²⋊C₄	—	218	1	no	—
54	C₄²	—	109	1	no	—
55	C₄²	—	218	1	no	—
56	C₄²	—	218	1	no	—
57	C₄²	#89137	1	1	yes	characteristic, complemented
58	C₂³	—	109	1	no	—
59	C₂×C₄	—	2	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	—	109	1	no	—
65	C₂×C₄	—	218	4	no	—
66	C₂×C₄	—	218	2	no	—
67	C₂×C₄	—	218	1	no	—
68	C₂×C₄	C₂×#89137	1	1	yes	characteristic
69	C₈	—	218	2	no	—
70	C₂²	—	109	1	no	—
71	C₂²	—	109	1	no	—
72	C₂²	—	218	2	no	—
73	C₂²	C₂²⋊#89137	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	109	1	no	—
76	C₄	—	109	1	no	—
77	C₄	—	436	2	no	—
78	C₄	—	218	2	no	—
79	C₄	C₄×#89137	1	1	yes	characteristic
80	C₄	C₄⋊#89137	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	109	2	no	—
83	C₂	—	218	1	no	—
84	C₂	(C₂×#89137)⋊C₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #13182039	C₄	semidirect
7. C₄⋊#89137	C₄	semidirect
13. C₄×Dic₁₀₉	C₄	semidirect
14. C₄²×C₁₀₉	C₄	semidirect
42. C₁₀₉	#339	semidirect
57. C₄²	#89137	semidirect