Subgroups — #34092442

Order 19904.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23755696	C₂	1	2	yes	maximal, complemented
2	#23755693	C₂	1	4	yes	maximal, complemented
3	#23755686	C₂	1	4	yes	maximal, complemented
4	#23755683	C₂	1	4	yes	maximal, complemented
5	#125×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₂₂	C₂²	1	2	yes	—
7	C₂×D₁₂₄₄	C₂²	1	4	yes	—
8	C₂×D₁₂₄₄	C₂²	1	2	yes	—
9	C₂×C₄×D₃₁₁	C₂²	1	4	yes	—
10	Dic₃₁₁⋊C₄	C₂²	1	4	yes	—
11	(C₂×D₃₁₁)⋊C₄	C₂²	1	4	yes	—
12	(C₂×D₃₁₁)⋊C₄	C₂²	1	4	yes	—
13	(C₂×D₃₁₁)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊C₄×C₃₁₁	C₂²	1	4	yes	—
15	C₄⋊C₄×C₃₁₁	C₂²	1	2	yes	—
16	C₄²×C₃₁₁	C₂²	1	1	yes	characteristic
17	C₂²×D₃₁₁	C₂³	1	4	yes	—
18	Dic₆₂₂	—	4	2	no	—
19	D₁₂₄₄	—	4	4	no	—
20	D₁₂₄₄	—	4	2	no	—
21	C₂×Dic₃₁₁	C₂³	1	4	yes	—
22	C₄×D₃₁₁	—	4	4	no	—
23	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₁₁	C₂³	1	2	yes	—
25	C₂×C₄×C₃₁₁	C₂³	1	4	yes	—
26	C₂²×C₃₁₁	C₂⁴	1	1	yes	characteristic
27	C₂×D₃₁₁	—	2	4	no	—
28	C₂×D₃₁₁	—	2	4	no	—
29	C₂×D₃₁₁	—	2	4	no	—
30	C₄×C₃₁₁	—	2	4	no	—
31	C₄×C₃₁₁	—	2	2	no	—
32	C₄×C₃₁₁	—	2	1	no	—
33	Dic₃₁₁	—	2	4	no	—
34	C₂×C₃₁₁	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₃₁₁	2₋¹⁺⁴	1	1	yes	characteristic
36	D₃₁₁	—	4	4	no	—
37	C₃₁₁	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	311	1	no	maximal, Sylow
39	#125	D₃₁₁	1	1	yes	characteristic, complemented
40	#124	—	311	2	no	—
41	#123	—	311	4	no	—
42	C₂²⋊Q₈	—	311	4	no	—
43	#121	—	311	4	no	—
44	C₂×Q₈	—	311	2	no	—
45	C₂×D₄	—	311	4	no	—
46	C₂×D₄	—	311	2	no	—
47	C₂²×C₄	—	311	4	no	—
48	C₄⋊C₄	—	311	4	no	—
49	C₄⋊C₄	C₂×D₃₁₁	1	4	yes	—
50	C₄⋊C₄	C₂×D₃₁₁	1	2	yes	—
51	C₂²⋊C₄	—	311	4	no	—
52	C₂²⋊C₄	—	311	4	no	—
53	C₂²⋊C₄	—	311	4	no	—
54	C₄²	C₂×D₃₁₁	1	1	yes	characteristic
55	C₂³	—	311	4	no	—
56	Q₈	—	1244	2	no	—
57	D₄	—	1244	4	no	—
58	D₄	—	1244	2	no	—
59	C₂×C₄	—	311	4	no	—
60	C₂×C₄	—	1244	4	no	—
61	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₁₁	1	2	yes	—
63	C₂×C₄	C₂²×D₃₁₁	1	4	yes	—
64	C₂²	—	622	4	no	—
65	C₂²	—	622	4	no	—
66	C₂²	—	622	4	no	—
67	C₂²	C₂³×D₃₁₁	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	622	4	no	—
72	C₂	—	1244	4	no	—
73	C₂	#23755754	1	2	yes	—
74	C₂	#23755756	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁₁
commutator, socle, cosocle	C₂²×C₃₁₁	C₂⁴
Fitting	#125×C₃₁₁	C₂
hypercenter	#125	D₃₁₁
hypocenter	C₃₁₁	#562
LCS	G ▷ C₂²×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #23755696	C₂	semidirect
2. #23755693	C₂	semidirect
3. #23755686	C₂	semidirect
4. #23755683	C₂	semidirect
5. #125×C₃₁₁	C₂	semidirect
37. C₃₁₁	#562	semidirect
39. #125	D₃₁₁	semidirect

Maximal subgroups

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

subgroup	index
#23755696	2
#23755693	2
#23755686	2
#23755683	2
#125×C₃₁₁	2
#562	311