Subgroups — #34091801

Order 19904.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₁₁	C₂	1	2	yes	maximal
2	#23755640	C₂	1	1	yes	maximal, characteristic
3	#23755632	C₂	1	2	yes	maximal
4	#23755618	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₃₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₃₁₁	—	2	2	no	—
8	C₄⋊Dic₃₁₁	C₂²	1	2	yes	—
9	C₄×Dic₃₁₁	—	2	2	no	—
10	C₄×Dic₃₁₁	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₃₁₁	C₂²	1	2	yes	—
12	Q₁₆×C₃₁₁	C₄	1	4	yes	complemented
13	C₂×C₈×C₃₁₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₁₁	—	2	2	no	—
15	C₂×Dic₃₁₁	D₄	1	2	yes	—
16	C₃₁₁⋊C₈	—	2	1	no	—
17	Q₈×C₃₁₁	—	2	2	no	—
18	Q₈×C₃₁₁	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₃₁₁	—	2	2	no	—
20	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
21	C₈×C₃₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₃₁₁	C₂×D₄	1	1	yes	characteristic
23	C₄×C₃₁₁	—	2	4	no	—
24	C₄×C₃₁₁	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₃₁₁	#41	1	1	yes	characteristic
26	Dic₃₁₁	—	2	2	no	—
27	Dic₃₁₁	—	4	2	no	—
28	C₂×C₃₁₁	C₄×D₄	1	1	yes	characteristic
29	C₂×C₃₁₁	#137	1	2	yes	—
30	C₃₁₁	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	311	1	no	maximal, Sylow
32	C₂×Q₁₆	D₃₁₁	1	1	yes	characteristic
33	C₄×Q₈	—	311	2	no	—
34	#106	—	311	1	no	—
35	#103	—	311	2	no	—
36	#97	—	311	1	no	—
37	C₂×Q₈	C₂×D₃₁₁	1	2	yes	—
38	Q₁₆	Dic₃₁₁	1	4	yes	complemented
39	C₂×C₈	—	311	1	no	—
40	C₂×C₈	C₂×D₃₁₁	1	1	yes	characteristic
41	C₄⋊C₄	—	311	2	no	—
42	C₄⋊C₄	—	622	2	no	—
43	C₄²	—	311	1	no	—
44	C₄²	—	622	2	no	—
45	Q₈	—	2	2	no	—
46	Q₈	C₂×Dic₃₁₁	1	4	yes	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	311	2	no	—
49	C₂×C₄	—	622	2	no	—
50	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
51	C₈	—	622	1	no	—
52	C₈	C₂×Dic₃₁₁	1	2	yes	—
53	C₂²	D₄×D₃₁₁	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	622	2	no	—
56	C₄	—	1244	2	no	—
57	C₄	C₂²×Dic₃₁₁	1	1	yes	characteristic
58	C₄	#19097314	1	1	yes	characteristic
59	C₂	D₄×Dic₃₁₁	1	1	yes	characteristic
60	C₂	#23755740	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₁₁
commutator	C₄×C₃₁₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₁
Fitting	C₂×Q₁₆×C₃₁₁	C₂
socle, prime radical	C₂²×C₃₁₁	C₂×D₄
cosocle	C₂×C₄×C₃₁₁	C₂³
hypercenter	C₂×Q₁₆	D₃₁₁
hypocenter	C₃₁₁	#441
LCS	G ▷ C₄×C₃₁₁ ▷ C₂×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
12. Q₁₆×C₃₁₁	C₄	semidirect
30. C₃₁₁	#441	semidirect
38. Q₁₆	Dic₃₁₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₁₁	2
#23755640	2
#23755632	2
#23755618	2
C₂×Q₁₆×C₃₁₁	2
#441	311