Subgroups — #13098349

Order 1600.

165 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×#127	C₂	1	1	yes	maximal, characteristic
2	#11716432	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716381	C₂	1	4	yes	maximal, complemented
4	#11716351	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×C₂₅	—	2	2	no	—
6	C₂×D₄×C₂₅	C₂²	1	2	yes	—
7	C₄²×C₂₅	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₅	C₂²	1	2	yes	—
9	#87718	—	2	2	no	—
10	C₂×#6014	C₂²	1	2	yes	—
11	#65360	—	5	1	no	maximal
12	C₂³×C₂₅	—	2	2	no	—
13	D₄×C₂₅	—	2	4	no	—
14	D₄×C₂₅	—	4	4	no	—
15	C₂×C₄×C₂₅	D₄	1	2	yes	—
16	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
17	C₂×Dic₂₅	—	2	2	no	—
18	#6014	—	2	2	no	—
19	C₅×#127	D₅	1	1	yes	characteristic
20	#4016	—	5	1	no	—
21	#3964	—	5	4	no	—
22	#3935	—	5	1	no	—
23	C₂²×C₂₅	—	4	2	no	—
24	C₂²×C₂₅	—	4	4	no	—
25	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
26	C₄×C₂₅	—	2	2	no	—
27	C₄×C₂₅	#41	1	2	yes	—
28	Dic₂₅	—	4	2	no	—
29	C₂×C₅×D₄	—	2	2	no	—
30	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
31	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
32	C₄⋊Dic₅	—	5	2	no	—
33	#683	—	10	2	no	—
34	C₂×#168	—	5	2	no	—
35	#490	—	25	1	no	Sylow
36	C₂×C₂₅	—	8	2	no	—
37	C₂×C₂₅	#124	1	1	yes	characteristic
38	C₂×C₂₅	AGL(1,ℤ/8)	1	2	yes	—
39	C₂³×C₅	—	2	2	no	—
40	C₅×D₄	—	2	4	no	—
41	C₅×D₄	—	4	4	no	—
42	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
43	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
44	C₂×Dic₅	—	10	2	no	—
45	#168	—	10	2	no	—
46	#127	D₂₅	1	1	yes	characteristic
47	#125	—	25	1	no	—
48	#102	—	25	4	no	—
49	#97	—	25	1	no	—
50	C₂₅	#490	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	—	4	4	no	—
52	C₂²×C₅	—	4	2	no	—
53	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
54	C₄×C₅	—	2	2	no	—
55	C₄×C₅	#711	1	2	yes	—
56	Dic₅	—	20	2	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	C₂×D₂₅	1	2	yes	—
59	C₂×C₈	—	25	2	no	—
60	C₄⋊C₄	—	25	2	no	—
61	C₄⋊C₄	—	50	2	no	—
62	C₄²	C₂×D₂₅	1	1	yes	characteristic
63	C₂×C₅	—	8	2	no	—
64	C₂×C₅	#4078	1	2	yes	—
65	C₂×C₅	#4082	1	1	yes	characteristic
66	C₂³	—	2	2	no	—
67	D₄	—	2	4	no	—
68	D₄	—	4	4	no	—
69	C₂×C₄	—	50	2	no	—
70	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
71	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
72	C₈	—	50	2	no	—
73	C₅	#65360	1	1	yes	characteristic
74	C₂²	—	4	4	no	—
75	C₂²	—	4	2	no	—
76	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	100	2	no	—
79	C₄	#87746	1	2	yes	—
80	C₂	—	8	2	no	—
81	C₂	#11716495	1	2	yes	—
82	C₂	#11716498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×#127	C₂
socle	C₂²×C₅	C₂×C₂²⋊D₅
prime radical	C₅×#127	D₅
hypercenter	#127	D₂₅
hypocenter	C₂₅	#490
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. #11716432	C₂	semidirect
3. #11716381	C₂	semidirect
4. #11716351	C₂	semidirect
50. C₂₅	#490	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#127	2
#11716432	2
#11716381	2
#11716351	2
#65360	5