Subgroups — #20296037

Order 6976.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812778	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic
3	#16812768	C₂	1	2	yes	maximal, complemented
4	#16812767	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#89137	—	2	1	no	—
7	C₄⋊#89137	—	2	1	no	—
8	C₄⋊#89137	C₂²	1	1	yes	characteristic
9	C₄⋊#89137	C₂²	1	2	yes	—
10	C₄⋊#89137	C₂²	1	1	yes	characteristic
11	C₂²⋊#89137	—	2	1	no	—
12	C₄×#89137	—	2	1	no	—
13	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	#13182013	C₄	1	1	yes	characteristic, complemented
15	#13182012	C₄	1	2	yes	complemented
16	C₈×D₁₀₉	C₂²	1	2	yes	—
17	M₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
18	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
19	C₂×#89137	—	2	1	no	—
20	C₂×#89137	—	2	1	no	—
21	C₂×#89137	—	2	2	no	—
22	C₂×#89137	—	4	2	no	—
23	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₀₉	Q₈	1	1	yes	characteristic
25	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₀₉	D₄	1	1	yes	characteristic
27	C₄×D₁₀₉	Q₈	1	1	yes	characteristic
28	C₄×D₁₀₉	C₂³	1	1	yes	characteristic
29	C₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
30	#11724784	C₂×C₄	1	2	yes	complemented
31	C₂²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×D₁₀₉	—	2	1	no	—
33	C₂×D₁₀₉	—	2	1	no	—
34	C₂×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×D₁₀₉	C₂×D₄	1	1	yes	characteristic
36	#89137	—	4	1	no	—
37	#89137	—	4	2	no	—
38	#89137	—	4	1	no	—
39	C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₁₀₉	—	2	1	no	—
44	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	D₁₀₉	—	2	1	no	—
46	D₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₁₀₉	#137	1	1	yes	characteristic
48	C₁₀₉	#428	1	1	yes	characteristic, complemented, Sylow
49	#428	—	109	1	no	maximal, Sylow
50	C₂×M₁₆	—	109	1	no	—
51	#117	—	109	1	no	—
52	C₂×C₄⋊C₄	—	109	1	no	—
53	#107	—	109	2	no	—
54	#106	—	109	2	no	—
55	C₂²×C₄	—	109	1	no	—
56	C₂²×C₄	—	218	1	no	—
57	M₁₆	—	109	1	no	—
58	M₁₆	—	109	2	no	—
59	M₁₆	#89137	1	1	yes	characteristic, complemented
60	C₂×C₈	—	109	2	no	—
61	C₄⋊C₄	—	109	1	no	—
62	C₄⋊C₄	—	109	2	no	—
63	C₄⋊C₄	—	109	1	no	—
64	C₄⋊C₄	—	218	1	no	—
65	C₂²⋊C₄	—	218	1	no	—
66	C₄²	—	218	1	no	—
67	C₂³	—	109	1	no	—
68	C₂×C₄	—	109	1	no	—
69	C₂×C₄	—	109	1	no	—
70	C₂×C₄	—	109	1	no	—
71	C₂×C₄	—	109	1	no	—
72	C₂×C₄	—	109	1	no	—
73	C₂×C₄	—	436	2	no	—
74	C₂×C₄	—	218	1	no	—
75	C₂×C₄	—	218	1	no	—
76	C₂×C₄	—	218	2	no	—
77	C₂×C₄	C₂×#89137	1	1	yes	characteristic
78	C₈	—	109	2	no	—
79	C₈	C₂×#89137	1	2	yes	complemented
80	C₂²	—	109	1	no	—
81	C₂²	—	109	1	no	—
82	C₂²	—	218	1	no	—
83	C₂²	—	218	1	no	—
84	C₂²	C₄⋊#89137	1	1	yes	characteristic
85	C₄	—	109	1	no	—
86	C₄	—	109	1	no	—
87	C₄	—	436	2	no	—
88	C₄	—	436	1	no	—
89	C₄	—	436	1	no	—
90	C₄	C₄⋊#89137	1	1	yes	characteristic
91	C₄	C₂²×#89137	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	109	1	no	—
94	C₂	—	109	1	no	—
95	C₂	—	218	1	no	—
96	C₂	C₂×C₄⋊#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₄	C₂²×#89137
Fitting	M₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₄⋊C₄
hypercenter	M₁₆	#89137
hypocenter	C₁₀₉	#428
LCS	G ▷ C₄×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
3. #16812768	C₂	semidirect
4. #16812767	C₂	semidirect
14. #13182013	C₄	semidirect
15. #13182012	C₄	semidirect
17. M₁₆×C₁₀₉	C₄	semidirect
29. C₈×C₁₀₉	C₂×C₄	semidirect
30. #11724784	C₂×C₄	semidirect
48. C₁₀₉	#428	semidirect
59. M₁₆	#89137	semidirect
79. C₈	C₂×#89137	semidirect