Subgroups — #20295148

Order 6976.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812706	C₂	1	2	yes	maximal, complemented
2	#16812635	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812626	C₂	1	2	yes	maximal, complemented
4	#16812609	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#13182013	—	2	2	no	—
7	#13182013	C₂²	1	2	yes	—
8	C₂×#11724784	—	2	2	no	—
9	C₂×#11724784	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	#41×C₁₀₉	C₂²	1	2	yes	—
12	Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
13	SD₁₆×C₁₀₉	C₄	1	2	yes	complemented
14	D₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
17	D₄×C₁₀₉	—	2	2	no	—
18	D₄×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₀₉	—	2	2	no	—
20	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	2	1	no	—
22	C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	#11724784	—	2	2	no	—
25	#11724784	D₄	1	2	yes	—
26	C₂²×C₁₀₉	—	2	2	no	—
27	C₂²×C₁₀₉	#41	1	1	yes	characteristic
28	C₄×C₁₀₉	—	2	2	no	—
29	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	Dic₁₀₉	—	4	1	no	—
32	C₂×C₁₀₉	—	2	1	no	—
33	C₂×C₁₀₉	—	4	2	no	—
34	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
35	C₁₀₉	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	109	1	no	maximal, Sylow
37	#135	D₁₀₉	1	1	yes	characteristic
38	#131	—	109	2	no	—
39	#108	—	109	1	no	—
40	#104	—	109	2	no	—
41	#97	—	109	1	no	—
42	#41	C₂×D₁₀₉	1	2	yes	—
43	Q₁₆	Dic₁₀₉	1	1	yes	characteristic, complemented
44	SD₁₆	Dic₁₀₉	1	2	yes	complemented
45	D₈	Dic₁₀₉	1	1	yes	characteristic, complemented
46	M₁₆	—	109	2	no	—
47	M₁₆	—	218	2	no	—
48	C₂×C₈	—	109	1	no	—
49	C₂×C₈	—	218	2	no	—
50	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
51	C₄²	—	109	1	no	—
52	Q₈	C₂×Dic₁₀₉	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₁₀₉	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	218	1	no	—
57	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
58	C₈	—	109	2	no	—
59	C₈	—	218	2	no	—
60	C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
61	C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	#13182028	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	436	1	no	—
66	C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
67	C₄	D₄×D₁₀₉	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#135×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₄×D₄
hypercenter	#135	D₁₀₉
hypocenter	C₁₀₉	#444
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #16812706	C₂	semidirect
2. #16812635	C₂	semidirect
3. #16812626	C₂	semidirect
4. #16812609	C₂	semidirect
12. Q₁₆×C₁₀₉	C₄	semidirect
13. SD₁₆×C₁₀₉	C₄	semidirect
14. D₈×C₁₀₉	C₄	semidirect
35. C₁₀₉	#444	semidirect
43. Q₁₆	Dic₁₀₉	semidirect
44. SD₁₆	Dic₁₀₉	semidirect
45. D₈	Dic₁₀₉	semidirect