Subgroups — #34091965

Order 19904.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23755736	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#23755714	C₂	1	1	yes	maximal, characteristic, complemented
4	#23755624	C₂	1	2	yes	maximal, complemented
5	#108×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097315	C₂²	1	1	yes	characteristic
7	C₂×D₁₂₄₄	C₂²	1	2	yes	—
8	#19097304	—	2	2	no	—
9	D₂₄₈₈	—	2	2	no	—
10	D₂₄₈₈	—	2	1	no	—
11	D₂₄₈₈	—	2	1	no	—
12	#19097299	C₂²	1	1	yes	characteristic
13	#19097298	—	2	1	no	—
14	C₈×D₃₁₁	—	2	1	no	—
15	M₁₆×C₃₁₁	C₂²	1	2	yes	—
16	C₂×C₈×C₃₁₁	C₂²	1	1	yes	characteristic
17	C₂²×D₃₁₁	—	2	2	no	—
18	Dic₆₂₂	D₄	1	1	yes	characteristic
19	D₁₂₄₄	—	2	2	no	—
20	D₁₂₄₄	—	2	2	no	—
21	D₁₂₄₄	D₄	1	1	yes	characteristic
22	C₂²⋊D₃₁₁	—	2	1	no	—
23	C₄×D₃₁₁	—	2	1	no	—
24	C₃₁₁⋊C₈	—	2	1	no	—
25	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
26	C₈×C₃₁₁	—	2	2	no	—
27	C₈×C₃₁₁	D₄	1	1	yes	characteristic
28	C₈×C₃₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₃₁₁	#41	1	1	yes	characteristic
30	C₂×D₃₁₁	—	2	1	no	—
31	C₂×D₃₁₁	—	4	2	no	—
32	C₂×D₃₁₁	—	8	2	no	—
33	C₄×C₃₁₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₁₁	C₂×D₄	1	1	yes	characteristic
35	Dic₃₁₁	—	2	1	no	—
36	C₂×C₃₁₁	—	2	1	no	—
37	C₂×C₃₁₁	#121	1	1	yes	characteristic
38	D₃₁₁	—	4	1	no	—
39	D₃₁₁	—	8	2	no	—
40	C₃₁₁	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	311	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	311	2	no	—
43	C₂×D₈	—	311	1	no	—
44	#131	—	311	1	no	—
45	#108	D₃₁₁	1	1	yes	characteristic, complemented
46	#100	—	311	2	no	—
47	#41	—	311	1	no	—
48	C₂×D₄	—	311	2	no	—
49	SD₁₆	—	622	2	no	—
50	D₈	—	622	2	no	—
51	D₈	—	622	1	no	—
52	D₈	—	622	1	no	—
53	M₁₆	—	311	1	no	—
54	M₁₆	—	622	1	no	—
55	M₁₆	C₂×D₃₁₁	1	2	yes	—
56	C₂×C₈	—	622	1	no	—
57	C₂×C₈	C₂×D₃₁₁	1	1	yes	characteristic
58	C₂³	—	622	2	no	—
59	Q₈	—	311	1	no	—
60	D₄	—	311	1	no	—
61	D₄	—	622	1	no	—
62	D₄	—	622	2	no	—
63	D₄	—	622	2	no	—
64	C₂×C₄	—	622	1	no	—
65	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	622	1	no	—
68	C₈	D₁₂₄₄	1	1	yes	characteristic
69	C₈	D₁₂₄₄	1	1	yes	characteristic
70	C₂²	—	622	1	no	—
71	C₂²	—	2488	2	no	—
72	C₂²	—	1244	2	no	—
73	C₂²	#19097316	1	1	yes	characteristic
74	C₄	—	622	1	no	—
75	C₄	D₄×D₃₁₁	1	1	yes	characteristic
76	C₄	C₂×D₁₂₄₄	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	2488	2	no	—
79	C₂	—	1244	1	no	—
80	C₂	#23755683	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23755683
commutator, cosocle	C₂×C₄×C₃₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₁
Fitting	#108×C₃₁₁	C₂
socle	C₂×C₃₁₁	#121
hypercenter	#108	D₃₁₁
hypocenter	C₃₁₁	#472
LCS	G ▷ C₂×C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
1. #23755736	C₂	semidirect
2. C₂×D₂₄₈₈	C₂	semidirect
3. #23755714	C₂	semidirect
4. #23755624	C₂	semidirect
5. #108×C₃₁₁	C₂	semidirect
40. C₃₁₁	#472	semidirect
45. #108	D₃₁₁	semidirect

Maximal subgroups

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

subgroup	index
#23755736	2
C₂×D₂₄₈₈	2
#23755714	2
#23755624	2
#108×C₃₁₁	2
#472	311