Subgroups — #27132784

Order 12224.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259344	C₂	1	1	yes	maximal, characteristic
2	#20259330	C₂	1	2	yes	maximal, complemented
3	#20259248	C₂	1	2	yes	maximal, complemented
4	#20259232	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690287	C₂²	1	2	yes	—
7	Dic₇₆₄	C₄	1	1	yes	characteristic, complemented
8	#15690276	C₄	1	2	yes	complemented
9	D₁₅₂₈	C₄	1	1	yes	characteristic, complemented
10	#15690270	—	2	2	no	—
11	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₁₉₁	—	2	2	no	—
13	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₉₁	C₂²	1	2	yes	—
15	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	Dic₃₈₂	C₂×C₄	1	2	yes	—
17	D₇₆₄	C₂×C₄	1	2	yes	—
18	C₂²⋊D₁₉₁	—	2	2	no	—
19	C₂×Dic₁₉₁	—	2	1	no	—
20	C₄×D₁₉₁	—	2	2	no	—
21	C₁₉₁⋊C₈	D₄	1	2	yes	—
22	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
23	C₈×C₁₉₁	—	2	2	no	—
24	C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₉₁	#41	1	1	yes	characteristic
27	C₂×D₁₉₁	—	2	2	no	—
28	C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
30	Dic₁₉₁	—	2	2	no	—
31	Dic₁₉₁	—	4	1	no	—
32	C₂×C₁₉₁	—	2	1	no	—
33	C₂×C₁₉₁	C₄×D₄	1	1	yes	characteristic
34	D₁₉₁	—	4	2	no	—
35	C₁₉₁	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	191	1	no	maximal, Sylow
37	#135	—	191	1	no	—
38	#131	—	191	2	no	—
39	#108	D₁₉₁	1	1	yes	characteristic, complemented
40	#104	—	191	2	no	—
41	#97	—	191	1	no	—
42	#41	—	191	2	no	—
43	Q₁₆	—	191	1	no	—
44	SD₁₆	—	191	2	no	—
45	D₈	—	191	1	no	—
46	M₁₆	—	382	2	no	—
47	M₁₆	C₂×D₁₉₁	1	2	yes	—
48	C₂×C₈	—	382	2	no	—
49	C₂×C₈	—	191	1	no	—
50	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
51	C₄²	—	191	1	no	—
52	Q₈	—	191	2	no	—
53	D₄	—	382	2	no	—
54	D₄	—	191	2	no	—
55	C₂×C₄	—	382	2	no	—
56	C₂×C₄	—	382	1	no	—
57	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	191	2	no	—
60	C₈	C₄×D₁₉₁	1	1	yes	characteristic
61	C₈	C₄×D₁₉₁	1	1	yes	characteristic
62	C₂²	—	382	2	no	—
63	C₂²	#15690288	1	1	yes	characteristic
64	C₄	—	382	2	no	—
65	C₄	—	764	1	no	—
66	C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
67	C₄	D₄×D₁₉₁	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	764	2	no	—
70	C₂	#20259284	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₉₁
commutator	C₄×C₁₉₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	#108×C₁₉₁	C₂
socle	C₂×C₁₉₁	C₄×D₄
prime radical	#20259344	C₂
cosocle	C₂×C₄×C₁₉₁	C₂³
hypercenter	#108	D₁₉₁
hypocenter	C₁₉₁	#444
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #20259330	C₂	semidirect
3. #20259248	C₂	semidirect
4. #20259232	C₂	semidirect
5. #108×C₁₉₁	C₂	semidirect
7. Dic₇₆₄	C₄	semidirect
8. #15690276	C₄	semidirect
9. D₁₅₂₈	C₄	semidirect
35. C₁₉₁	#444	semidirect
39. #108	D₁₉₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20259344	2
#20259330	2
#20259248	2
#20259232	2
#108×C₁₉₁	2
#444	191