Group #13098645 — subgroups

Subgroups — C₂₅×#496 #13098645

Order 1600.

216 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×C₂₅	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×C₂₅	C₂	1	2	yes	maximal, complemented
3	C₂₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₂₅	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₂₅	—	2	2	no	—
8	C₂×D₄×C₂₅	C₂²	1	1	yes	characteristic
9	C₂×D₄×C₂₅	C₂²	1	2	yes	—
10	SD₁₆×C₂₅	—	2	4	no	—
11	D₈×C₂₅	—	2	4	no	—
12	C₂×C₈×C₂₅	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₂₅	—	2	2	no	—
14	C₄²×C₂₅	C₂²	1	1	yes	characteristic
15	C₅×#496	C₅	1	1	yes	maximal, characteristic
16	C₂³×C₂₅	—	2	1	no	—
17	C₂³×C₂₅	—	2	2	no	—
18	Q₈×C₂₅	—	2	2	no	—
19	D₄×C₂₅	—	2	2	no	—
20	D₄×C₂₅	—	2	2	no	—
21	D₄×C₂₅	—	2	2	no	—
22	D₄×C₂₅	—	4	4	no	—
23	C₂×C₄×C₂₅	—	2	1	no	—
24	C₂×C₄×C₂₅	D₄	1	2	yes	—
25	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
26	C₈×C₂₅	D₄	1	4	yes	complemented
27	C₂×C₅×SD₁₆	C₂×C₅	1	2	yes	—
28	C₂×C₅×D₈	C₂×C₅	1	2	yes	—
29	C₅×#127	C₂×C₅	1	1	yes	characteristic
30	C₅×#124	C₂×C₅	1	1	yes	characteristic
31	C₅×#97	C₂×C₅	1	1	yes	characteristic
32	C₂²×C₂₅	—	4	1	no	—
33	C₂²×C₂₅	—	4	2	no	—
34	C₂²×C₂₅	—	4	4	no	—
35	C₂²×C₂₅	—	4	2	no	—
36	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
37	C₄×C₂₅	—	2	2	no	—
38	C₄×C₂₅	—	4	1	no	—
39	C₄×C₂₅	C₂×D₄	1	2	yes	—
40	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
41	C₂×C₅×D₄	—	2	2	no	—
42	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
43	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
44	C₅×SD₁₆	—	2	4	no	—
45	C₅×D₈	—	2	4	no	—
46	C₂×C₅×C₈	C₂²×C₅	1	2	yes	—
47	C₅×C₂²⋊C₄	—	2	2	no	—
48	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
49	#496	C₂₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₂₅	—	8	2	no	—
51	C₂×C₂₅	—	8	1	no	—
52	C₂×C₂₅	#127	1	1	yes	characteristic
53	C₂×C₂₅	AGL(1,ℤ/8)	1	2	yes	—
54	C₂³×C₅	—	2	1	no	—
55	C₂³×C₅	—	2	2	no	—
56	C₅×Q₈	—	2	2	no	—
57	C₅×D₄	—	2	2	no	—
58	C₅×D₄	—	2	2	no	—
59	C₅×D₄	—	2	2	no	—
60	C₅×D₄	—	4	4	no	—
61	C₂×C₄×C₅	—	2	1	no	—
62	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
63	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
64	C₅×C₈	C₅×D₄	1	4	yes	—
65	C₂×SD₁₆	C₂×C₂₅	1	2	yes	complemented
66	C₂×D₈	C₂×C₂₅	1	2	yes	complemented
67	#127	C₂×C₂₅	1	1	yes	characteristic, complemented
68	#124	C₂×C₂₅	1	1	yes	characteristic, complemented
69	#97	C₂×C₂₅	1	1	yes	characteristic, complemented
70	C₂₅	#496	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×C₅	—	4	1	no	—
72	C₂²×C₅	—	4	2	no	—
73	C₂²×C₅	—	4	4	no	—
74	C₂²×C₅	—	4	2	no	—
75	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
76	C₄×C₅	—	2	2	no	—
77	C₄×C₅	—	4	1	no	—
78	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
79	C₂×Q₈	C₂²×C₂₅	1	1	yes	characteristic
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂²×C₂₅	1	1	yes	characteristic
82	C₂×D₄	C₂²×C₂₅	1	2	yes	—
83	SD₁₆	—	2	4	no	—
84	D₈	—	2	4	no	—
85	C₂×C₈	C₂²×C₂₅	1	2	yes	—
86	C₂²⋊C₄	—	2	2	no	—
87	C₄²	C₂²×C₂₅	1	1	yes	characteristic
88	C₂×C₅	—	8	2	no	—
89	C₂×C₅	—	8	1	no	—
90	C₂×C₅	C₅×#127	1	1	yes	characteristic
91	C₂×C₅	C₅×AGL(1,ℤ/8)	1	2	yes	—
92	C₂³	—	2	1	no	—
93	C₂³	—	2	2	no	—
94	Q₈	—	2	2	no	—
95	D₄	—	2	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	4	4	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	D₄×C₂₅	1	2	yes	—
101	C₂×C₄	C₂³×C₂₅	1	1	yes	characteristic
102	C₈	D₄×C₂₅	1	4	yes	complemented
103	C₅	C₅×#496	1	1	yes	characteristic
104	C₂²	—	4	1	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	—	4	2	no	—
108	C₂²	C₂×D₄×C₂₅	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	4	1	no	—
111	C₄	C₂×D₄×C₂₅	1	2	yes	—
112	C₂	—	8	2	no	—
113	C₂	—	8	1	no	—
114	C₂	C₂₅×#127	1	1	yes	characteristic
115	C₂	C₂₅×AGL(1,ℤ/8)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×SD₁₆×C₂₅

C₂

yes

maximal, complemented

C₂×D₈×C₂₅

C₂

yes

maximal, complemented

C₂₅×#127

C₂

yes

maximal, characteristic, complemented

C₂₅×#124

C₂

yes

maximal, characteristic, complemented

C₂₅×#97

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₂₅

C₂²

yes

characteristic

C₂×D₄×C₂₅

—

C₂×D₄×C₂₅

C₂²

yes

characteristic

C₂×D₄×C₂₅

C₂²

yes

—

SD₁₆×C₂₅

—

D₈×C₂₅

—

C₂×C₈×C₂₅

C₂²

yes

—

C₂²⋊C₄×C₂₅

—

C₄²×C₂₅

C₂²

yes

characteristic

C₅×#496

C₅

yes

maximal, characteristic

C₂³×C₂₅

—

C₂³×C₂₅

—

Q₈×C₂₅

—

D₄×C₂₅

—

D₄×C₂₅

—

D₄×C₂₅

—

D₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

D₄

yes

—

C₂×C₄×C₂₅

C₂³

yes

characteristic

C₈×C₂₅

D₄

yes

complemented

C₂×C₅×SD₁₆

C₂×C₅

yes

—

C₂×C₅×D₈

C₂×C₅

yes

—

C₅×#127

C₂×C₅

yes

characteristic

C₅×#124

C₂×C₅

yes

characteristic

C₅×#97

C₂×C₅

yes

characteristic

C₂²×C₂₅

—

C₂²×C₂₅

—

C₂²×C₂₅

—

C₂²×C₂₅

—

C₂²×C₂₅

C₂×D₄

yes

characteristic

C₄×C₂₅

—

C₄×C₂₅

—

C₄×C₂₅

C₂×D₄

yes

—

C₂×C₅×Q₈

C₂²×C₅

yes

characteristic

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂²×C₅

yes

characteristic

C₂×C₅×D₄

C₂²×C₅

yes

—

C₅×SD₁₆

—

C₅×D₈

—

C₂×C₅×C₈

C₂²×C₅

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂²×C₅

yes

characteristic

#496

C₂₅

yes

characteristic, direct product factor, Sylow

C₂×C₂₅

—

C₂×C₂₅

—

C₂×C₂₅

#127

yes

characteristic

C₂×C₂₅

AGL(1,ℤ/8)

yes

—

C₂³×C₅

—

C₂³×C₅

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₅×D₄

yes

—

C₂×C₄×C₅

C₂³×C₅

yes

characteristic

C₅×C₈

C₅×D₄

yes

—

C₂×SD₁₆

C₂×C₂₅

yes

complemented

C₂×D₈

C₂×C₂₅

yes

complemented

#127

C₂×C₂₅

yes

characteristic, complemented

#124

C₂×C₂₅

yes

characteristic, complemented

#97

C₂×C₂₅

yes

characteristic, complemented

C₂₅

#496

yes

characteristic, direct product factor, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₅×D₄

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₅×D₄

yes

—

C₂×Q₈

C₂²×C₂₅

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²×C₂₅

yes

characteristic

C₂×D₄

C₂²×C₂₅

yes

—

SD₁₆

—

D₈

—

C₂×C₈

C₂²×C₂₅

yes

—

C₂²⋊C₄

—

C₄²

C₂²×C₂₅

yes

characteristic

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₅×#127

yes

characteristic

C₂×C₅

C₅×AGL(1,ℤ/8)

yes

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

D₄×C₂₅

yes

—

101

C₂×C₄

C₂³×C₂₅

yes

characteristic

102

C₈

D₄×C₂₅

yes

complemented

103

C₅

C₅×#496

yes

characteristic

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂×D₄×C₂₅

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

C₂×D₄×C₂₅

yes

—

112

C₂

—

113

C₂

—

114

C₂

C₂₅×#127

yes

characteristic

115

C₂

C₂₅×AGL(1,ℤ/8)

yes

—

	subgroup	G/H
center	C₂²×C₂₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₂₅
Frattini, cosocle	C₂×C₄×C₅	C₂³×C₅
socle	C₂²×C₅	C₂×C₅×D₄
prime radical	C₅×#496	C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅ ◁ C₄²×C₂₅ ◁ G

subgroup

G/H

center

C₂²×C₂₅

C₂×D₄

commutator

C₂×C₄

C₂³×C₂₅

Frattini, cosocle

C₂×C₄×C₅

C₂³×C₅

socle

C₂²×C₅

C₂×C₅×D₄

C₅

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₂₅ ◁ C₄²×C₂₅ ◁ G

N	H ≅ G/N	type
1. C₂×SD₁₆×C₂₅	C₂	semidirect
2. C₂×D₈×C₂₅	C₂	semidirect
3. C₂₅×#127	C₂	semidirect
4. C₂₅×#124	C₂	semidirect
5. C₂₅×#97	C₂	semidirect
26. C₈×C₂₅	D₄	semidirect
49. #496	C₂₅	direct (N × H)
65. C₂×SD₁₆	C₂×C₂₅	semidirect
66. C₂×D₈	C₂×C₂₅	semidirect
67. #127	C₂×C₂₅	semidirect
68. #124	C₂×C₂₅	semidirect
69. #97	C₂×C₂₅	semidirect
70. C₂₅	#496	direct (N × H)
102. C₈	D₄×C₂₅	semidirect

H ≅ G/N

type

1. C₂×SD₁₆×C₂₅

C₂

semidirect

2. C₂×D₈×C₂₅

C₂

semidirect

3. C₂₅×#127

C₂

semidirect

4. C₂₅×#124

C₂

semidirect

5. C₂₅×#97

C₂

semidirect

26. C₈×C₂₅

D₄

semidirect

49. #496

C₂₅

direct (N × H)

65. C₂×SD₁₆

C₂×C₂₅

semidirect

66. C₂×D₈

C₂×C₂₅

semidirect

67. #127

C₂×C₂₅

semidirect

68. #124

C₂×C₂₅

semidirect

69. #97

C₂×C₂₅

semidirect

70. C₂₅

#496

direct (N × H)

102. C₈

D₄×C₂₅

semidirect

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

subgroup	index
C₂×SD₁₆×C₂₅	2
C₂×D₈×C₂₅	2
C₂₅×#127	2
C₂₅×#124	2
C₂₅×#97	2
C₅×#496	5

subgroup

index

C₂×SD₁₆×C₂₅

C₂×D₈×C₂₅

C₂₅×#127

C₂₅×#124

C₂₅×#97

C₅×#496

Subgroups — C₂₅×#496 #13098645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C25×#496 #13098645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₅×#496 #13098645