Subgroups — #34091479

Order 19904.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23755714	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097315	C₄	1	1	yes	characteristic
5	#19097299	C₄	1	1	yes	characteristic
6	#19097298	—	2	1	no	—
7	C₈×D₃₁₁	—	2	1	no	—
8	C₃₁₁⋊C₁₆	—	2	1	no	—
9	C₂×C₈×C₃₁₁	C₂²	1	1	yes	characteristic
10	C₁₆×C₃₁₁	—	2	1	no	—
11	Dic₆₂₂	C₈	1	1	yes	characteristic
12	D₁₂₄₄	C₈	1	1	yes	characteristic
13	C₂²⋊D₃₁₁	—	2	1	no	—
14	C₄×D₃₁₁	—	2	1	no	—
15	C₃₁₁⋊C₈	—	2	1	no	—
16	C₂×C₄×C₃₁₁	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₁₁	D₄	1	1	yes	characteristic
18	C₈×C₃₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₃₁₁	M₁₆	1	1	yes	characteristic
20	C₂×D₃₁₁	—	2	1	no	—
21	C₄×C₃₁₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₁₁	C₂×C₈	1	1	yes	characteristic
23	Dic₃₁₁	—	2	1	no	—
24	C₂×C₃₁₁	—	2	1	no	—
25	C₂×C₃₁₁	C₂²⋊C₈	1	1	yes	characteristic
26	D₃₁₁	—	4	1	no	—
27	C₃₁₁	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	311	1	no	maximal, Sylow
29	#131	—	311	1	no	—
30	M₃₂	D₃₁₁	1	1	yes	characteristic, complemented
31	C₂×C₁₆	—	311	1	no	—
32	#41	—	311	1	no	—
33	M₁₆	—	311	1	no	—
34	M₁₆	—	622	1	no	—
35	C₂×C₈	—	622	1	no	—
36	C₂×C₈	C₂×D₃₁₁	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	622	1	no	—
39	Q₈	—	311	1	no	—
40	D₄	—	311	1	no	—
41	D₄	—	622	1	no	—
42	C₂×C₄	—	622	1	no	—
43	C₂×C₄	C₄×D₃₁₁	1	1	yes	characteristic
44	C₈	—	622	1	no	—
45	C₈	C₂²⋊D₃₁₁	1	1	yes	characteristic
46	C₈	D₁₂₄₄	1	1	yes	characteristic
47	C₂²	—	622	1	no	—
48	C₂²	#19097298	1	1	yes	characteristic
49	C₄	—	622	1	no	—
50	C₄	(C₂×D₃₁₁)⋊C₄	1	1	yes	characteristic
51	C₄	C₈×D₃₁₁	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1244	1	no	—
54	C₂	(C₂×D₃₁₁)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₃₁₁
commutator	C₄×C₃₁₁	C₂×C₈
Frattini	C₂×C₈	C₂×D₃₁₁
Fitting	M₃₂×C₃₁₁	C₂
socle	C₂×C₃₁₁	C₂²⋊C₈
prime radical	#19097315	C₄
cosocle	C₂×C₈×C₃₁₁	C₂²
hypercenter	M₃₂	D₃₁₁
hypocenter	C₃₁₁	#350
LCS	G ▷ C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁₁⋊C₁₆	C₂	semidirect
3. M₃₂×C₃₁₁	C₂	semidirect
27. C₃₁₁	#350	semidirect
30. M₃₂	D₃₁₁	semidirect

Maximal subgroups

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

subgroup	index
#23755714	2
C₂×C₃₁₁⋊C₁₆	2
M₃₂×C₃₁₁	2
#350	311