Subgroups — #28416801

Order 14912.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21418532	C₂	1	1	yes	maximal, characteristic
2	#21418455	C₂	1	4	yes	maximal, complemented
3	#21418446	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₉₃₂	—	2	2	no	—
6	C₂×D₉₃₂	C₂²	1	2	yes	—
7	C₂×#14272338	C₂²	1	2	yes	—
8	C₄⋊C₄×C₂₃₃	—	2	2	no	—
9	C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
10	C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
11	C₂²×D₂₃₃	—	2	2	no	—
12	D₉₃₂	—	2	4	no	—
13	D₉₃₂	—	4	4	no	—
14	#14272338	—	2	2	no	—
15	C₂×C₄×C₂₃₃	—	2	2	no	—
16	C₂×C₄×C₂₃₃	D₄	1	2	yes	—
17	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
18	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
19	C₂×D₂₃₃	—	4	2	no	—
20	C₂×D₂₃₃	—	4	4	no	—
21	C₄×C₂₃₃	—	2	2	no	—
22	C₄×C₂₃₃	—	4	2	no	—
23	C₄×C₂₃₃	#41	1	2	yes	—
24	C₂×C₂₃₃	#124	1	1	yes	characteristic
25	C₂×C₂₃₃	AGL(1,ℤ/8)	1	2	yes	—
26	D₂₃₃	—	8	2	no	—
27	C₂₃₃	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	233	1	no	maximal, Sylow
29	#127	—	233	1	no	—
30	#125	D₂₃₃	1	1	yes	characteristic, complemented
31	#102	—	233	4	no	—
32	#97	—	233	1	no	—
33	C₂×D₄	—	466	2	no	—
34	C₂×D₄	—	233	2	no	—
35	C₂×C₈	—	233	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₂₃₃	1	2	yes	—
38	C₄²	C₂×D₂₃₃	1	1	yes	characteristic
39	C₂³	—	466	2	no	—
40	D₄	—	932	4	no	—
41	D₄	—	466	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₂₃₃	1	2	yes	—
44	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
45	C₈	—	466	2	no	—
46	C₂²	—	932	2	no	—
47	C₂²	—	932	4	no	—
48	C₂²	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#16836595	1	2	yes	—
52	C₂	—	1864	2	no	—
53	C₂	#21418524	1	1	yes	characteristic
54	C₂	#21418560	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃₃
commutator, cosocle	C₂×C₄×C₂₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₃
Fitting	#125×C₂₃₃	C₂
socle	C₂²×C₂₃₃	C₂×D₄
prime radical	#21418532	C₂
hypercenter	#125	D₂₃₃
hypocenter	C₂₃₃	#490
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #21418455	C₂	semidirect
3. #21418446	C₂	semidirect
4. #125×C₂₃₃	C₂	semidirect
27. C₂₃₃	#490	semidirect
30. #125	D₂₃₃	semidirect

Maximal subgroups

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

subgroup	index
#21418532	2
#21418455	2
#21418446	2
#125×C₂₃₃	2
#490	233