Subgroups — #34091812

Order 19904.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23755730	C₂	1	1	yes	maximal, characteristic
2	#23755714	C₂	1	2	yes	maximal, complemented
3	#23755644	C₂	1	1	yes	maximal, characteristic, complemented
4	#23755636	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097315	C₂²	1	2	yes	—
7	Dic₁₂₄₄	C₄	1	1	yes	characteristic, complemented
8	#19097304	C₄	1	2	yes	complemented
9	D₂₄₈₈	C₄	1	1	yes	characteristic, complemented
10	#19097299	C₂²	1	2	yes	—
11	#19097298	—	2	2	no	—
12	C₈×D₃₁₁	—	2	2	no	—
13	C₂×C₈×C₃₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₁₁	C₂²	1	1	yes	characteristic
15	C₄²×C₃₁₁	C₂²	1	1	yes	characteristic
16	Dic₆₂₂	C₂×C₄	1	2	yes	—
17	D₁₂₄₄	C₂×C₄	1	2	yes	—
18	C₂²⋊D₃₁₁	—	2	2	no	—
19	C₄×D₃₁₁	—	2	2	no	—
20	C₃₁₁⋊C₈	—	2	2	no	—
21	C₂×C₄×C₃₁₁	—	2	1	no	—
22	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
23	C₈×C₃₁₁	C₂×C₄	1	1	yes	characteristic
24	C₈×C₃₁₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃₁₁	D₄	1	1	yes	characteristic
26	C₈×C₃₁₁	D₄	1	1	yes	characteristic
27	C₂²×C₃₁₁	#41	1	1	yes	characteristic
28	C₂×D₃₁₁	—	2	2	no	—
29	C₄×C₃₁₁	—	2	2	no	—
30	C₄×C₃₁₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₃₁₁	C₂×D₄	1	1	yes	characteristic
32	Dic₃₁₁	—	2	2	no	—
33	C₂×C₃₁₁	—	2	1	no	—
34	C₂×C₃₁₁	C₄×D₄	1	1	yes	characteristic
35	D₃₁₁	—	4	2	no	—
36	C₃₁₁	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	311	1	no	maximal, Sylow
38	#135	—	311	1	no	—
39	#131	—	311	2	no	—
40	#108	—	311	1	no	—
41	#104	—	311	2	no	—
42	C₄×C₈	D₃₁₁	1	1	yes	characteristic, complemented
43	#41	—	311	2	no	—
44	Q₁₆	—	311	1	no	—
45	SD₁₆	—	311	2	no	—
46	D₈	—	311	1	no	—
47	M₁₆	—	311	2	no	—
48	M₁₆	—	622	2	no	—
49	C₂×C₈	—	622	2	no	—
50	C₂×C₈	C₂×D₃₁₁	1	1	yes	characteristic
51	C₂×C₈	C₂×D₃₁₁	1	1	yes	characteristic
52	C₄²	C₂×D₃₁₁	1	1	yes	characteristic
53	Q₈	—	311	2	no	—
54	D₄	—	311	2	no	—
55	D₄	—	622	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	622	2	no	—
58	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
59	C₈	—	622	2	no	—
60	C₈	C₄×D₃₁₁	1	1	yes	characteristic
61	C₈	C₄×D₃₁₁	1	1	yes	characteristic
62	C₈	D₁₂₄₄	1	1	yes	characteristic
63	C₈	D₁₂₄₄	1	1	yes	characteristic
64	C₂²	—	622	2	no	—
65	C₂²	#19097315	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	622	2	no	—
68	C₄	C₂×C₄×D₃₁₁	1	1	yes	characteristic
69	C₄	C₂×D₁₂₄₄	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1244	2	no	—
72	C₂	C₄×D₁₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₂₄₄
commutator	C₄×C₃₁₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₁
Fitting	C₄×C₈×C₃₁₁	C₂
socle	C₂×C₃₁₁	C₄×D₄
prime radical	#23755730	C₂
cosocle	C₂×C₄×C₃₁₁	C₂³
hypercenter	C₄×C₈	D₃₁₁
hypocenter	C₃₁₁	#443
LCS	G ▷ C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #23755714	C₂	semidirect
3. #23755644	C₂	semidirect
4. #23755636	C₂	semidirect
5. C₄×C₈×C₃₁₁	C₂	semidirect
7. Dic₁₂₄₄	C₄	semidirect
8. #19097304	C₄	semidirect
9. D₂₄₈₈	C₄	semidirect
36. C₃₁₁	#443	semidirect
42. C₄×C₈	D₃₁₁	semidirect

Maximal subgroups

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

subgroup	index
#23755730	2
#23755714	2
#23755644	2
#23755636	2
C₄×C₈×C₃₁₁	2
#443	311