Group #13098549 — subgroups

Subgroups — C₂₅×#400 #13098549

Order 1600.

201 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₄⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
3	C₂₅×#95	C₂	1	1	yes	maximal, characteristic
4	C₂₅×#95	C₂	1	2	yes	maximal
5	C₂₅×#95	C₂	1	2	yes	maximal
6	C₂²×C₄×C₂₅	C₂²	1	2	yes	—
7	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₂₅	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₂₅	—	2	2	no	—
12	C₄⋊C₄×C₂₅	—	2	2	no	—
13	C₅×#400	C₅	1	1	yes	maximal, characteristic
14	C₂³×C₂₅	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₂₅	—	2	1	no	—
16	C₂×C₄×C₂₅	—	2	1	no	—
17	C₂×C₄×C₂₅	—	2	2	no	—
18	C₂×C₄×C₂₅	—	2	1	no	—
19	C₂×C₄×C₂₅	—	2	2	no	—
20	C₂×C₄×C₂₅	—	2	2	no	—
21	C₂×C₄×C₂₅	—	2	4	no	—
22	C₂×C₄×C₂₅	—	2	2	no	—
23	C₂×C₄×C₂₅	—	2	2	no	—
24	C₂×C₄×C₂₅	—	2	2	no	—
25	C₂×C₄×C₂₅	D₄	1	2	yes	—
26	C₂×C₄×C₂₅	Q₈	1	2	yes	—
27	C₂×C₅×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
28	C₂×C₅×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
29	C₅×#95	C₂×C₅	1	2	yes	—
30	C₅×#95	C₂×C₅	1	2	yes	—
31	C₅×#95	C₂×C₅	1	1	yes	characteristic
32	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₂₅	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₂₅	#41	1	1	yes	characteristic
35	C₂²×C₂₅	#41	1	2	yes	—
36	C₂²×C₂₅	#41	1	2	yes	—
37	C₄×C₂₅	—	2	2	no	—
38	C₄×C₂₅	—	2	2	no	—
39	C₄×C₂₅	—	4	1	no	—
40	C₄×C₂₅	—	4	2	no	—
41	C₄×C₂₅	—	4	2	no	—
42	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
45	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
47	C₅×C₄⋊C₄	—	2	2	no	—
48	C₅×C₄⋊C₄	—	2	2	no	—
49	#400	C₂₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₂₅	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂×C₂₅	#123	1	1	yes	characteristic
52	C₂×C₂₅	#124	1	1	yes	characteristic
53	C₂×C₂₅	#125	1	2	yes	—
54	C₂×C₂₅	#126	1	2	yes	—
55	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
56	C₂×C₄×C₅	—	2	1	no	—
57	C₂×C₄×C₅	—	2	1	no	—
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	2	1	no	—
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	4	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
67	C₂×C₄×C₅	C₅×Q₈	1	2	yes	—
68	C₂×C₄⋊C₄	C₂×C₂₅	1	1	yes	characteristic
69	C₂×C₄⋊C₄	C₂×C₂₅	1	1	yes	characteristic
70	#95	C₂×C₂₅	1	1	yes	characteristic
71	#95	C₂×C₂₅	1	2	yes	—
72	#95	C₂×C₂₅	1	2	yes	—
73	C₂₅	#400	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
75	C₂²×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
76	C₂²×C₅	C₅×#41	1	1	yes	characteristic
77	C₂²×C₅	C₅×#41	1	2	yes	—
78	C₂²×C₅	C₅×#41	1	2	yes	—
79	C₄×C₅	—	2	2	no	—
80	C₄×C₅	—	2	2	no	—
81	C₄×C₅	—	4	1	no	—
82	C₄×C₅	—	4	2	no	—
83	C₄×C₅	—	4	2	no	—
84	C₂²×C₄	C₂²×C₂₅	1	2	yes	—
85	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₂₅	1	2	yes	—
87	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
88	C₂²×C₄	C₂²×C₂₅	1	1	yes	characteristic
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₂×C₅	C₅×C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₅	C₅×#123	1	1	yes	characteristic
93	C₂×C₅	C₅×#124	1	1	yes	characteristic
94	C₂×C₅	C₅×#125	1	2	yes	—
95	C₂×C₅	C₅×#126	1	2	yes	—
96	C₂³	C₂³×C₂₅	1	1	yes	characteristic
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	D₄×C₂₅	1	2	yes	—
108	C₂×C₄	Q₈×C₂₅	1	2	yes	—
109	C₅	C₅×#400	1	1	yes	characteristic
110	C₂²	C₂×D₄×C₂₅	1	1	yes	characteristic
111	C₂²	C₂×Q₈×C₂₅	1	1	yes	characteristic
112	C₂²	#41×C₂₅	1	1	yes	characteristic
113	C₂²	#41×C₂₅	1	2	yes	—
114	C₂²	#41×C₂₅	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	1	no	—
118	C₄	—	4	2	no	—
119	C₄	—	4	2	no	—
120	C₂	C₂₅×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂	C₂₅×#123	1	1	yes	characteristic
122	C₂	C₂₅×#124	1	1	yes	characteristic
123	C₂	C₂₅×#125	1	2	yes	—
124	C₂	C₂₅×#126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₄×C₂₅

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₅

C₂

yes

maximal, characteristic

C₂₅×#95

C₂

yes

maximal, characteristic

C₂₅×#95

C₂

yes

maximal

C₂₅×#95

C₂

yes

maximal

C₂²×C₄×C₂₅

C₂²

yes

—

C₂²×C₄×C₂₅

C₂²

yes

characteristic

C₂²×C₄×C₂₅

C₂²

yes

—

C₂²×C₄×C₂₅

C₂²

yes

characteristic

C₂²×C₄×C₂₅

C₂²

yes

characteristic

C₄⋊C₄×C₂₅

—

C₄⋊C₄×C₂₅

—

C₅×#400

C₅

yes

maximal, characteristic

C₂³×C₂₅

C₂³

yes

characteristic

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

D₄

yes

—

C₂×C₄×C₂₅

Q₈

yes

—

C₂×C₅×C₄⋊C₄

C₂×C₅

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂×C₅

yes

characteristic

C₅×#95

C₂×C₅

yes

—

C₅×#95

C₂×C₅

yes

—

C₅×#95

C₂×C₅

yes

characteristic

C₂²×C₂₅

C₂×D₄

yes

characteristic

C₂²×C₂₅

C₂×Q₈

yes

C₂²×C₂₅

yes

C₂²×C₂₅

yes

—

C₂²×C₂₅

#41

yes

—

C₄×C₂₅

—

C₄×C₂₅

—

C₄×C₂₅

—

C₄×C₂₅

—

C₄×C₂₅

—

C₂²×C₄×C₅

C₂²×C₅

yes

characteristic

C₂²×C₄×C₅

C₂²×C₅

yes

—

C₂²×C₄×C₅

C₂²×C₅

yes

—

C₂²×C₄×C₅

C₂²×C₅

yes

characteristic

C₂²×C₄×C₅

C₂²×C₅

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

#400

C₂₅

yes

characteristic, direct product factor, Sylow

C₂×C₂₅

C₂²⋊Q₈

yes

C₂×C₂₅

yes

C₂×C₂₅

yes

C₂×C₂₅

yes

—

C₂×C₂₅

#126

yes

—

C₂³×C₅

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₅×D₄

yes

—

C₂×C₄×C₅

C₅×Q₈

yes

—

C₂×C₄⋊C₄

C₂×C₂₅

yes

characteristic

C₂×C₄⋊C₄

C₂×C₂₅

yes

characteristic

#95

C₂×C₂₅

yes

characteristic

#95

C₂×C₂₅

yes

—

#95

C₂×C₂₅

yes

—

C₂₅

#400

yes

characteristic, direct product factor, Sylow

C₂²×C₅

C₂×C₅×D₄

yes

characteristic

C₂²×C₅

C₂×C₅×Q₈

yes

characteristic

C₂²×C₅

C₅×#41

yes

characteristic

C₂²×C₅

C₅×#41

yes

—

C₂²×C₅

C₅×#41

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₂²×C₄

C₂²×C₂₅

yes

—

C₂²×C₄

C₂²×C₂₅

yes

characteristic

C₂²×C₄

C₂²×C₂₅

yes

—

C₂²×C₄

C₂²×C₂₅

yes

characteristic

C₂²×C₄

C₂²×C₂₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂×C₅

C₅×C₂²⋊Q₈

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

C₂×C₅

C₅×#126

yes

—

C₂³

C₂³×C₂₅

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×C₂₅

yes

—

108

C₂×C₄

Q₈×C₂₅

yes

—

109

C₅

C₅×#400

yes

characteristic

110

C₂²

C₂×D₄×C₂₅

yes

characteristic

111

C₂²

C₂×Q₈×C₂₅

yes

characteristic

112

C₂²

#41×C₂₅

yes

characteristic

113

C₂²

#41×C₂₅

yes

—

114

C₂²

#41×C₂₅

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₂

C₂₅×C₂²⋊Q₈

yes

characteristic

121

C₂

C₂₅×#123

yes

characteristic

122

C₂

C₂₅×#124

yes

characteristic

123

C₂

C₂₅×#125

yes

—

124

C₂

C₂₅×#126

yes

—

	subgroup	G/H
center	C₂³×C₂₅	C₂³
commutator	C₂³	C₂³×C₂₅
Frattini, socle, prime radical, cosocle	C₂³×C₅	C₂³×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₅ ◁ G

subgroup

G/H

center

C₂³×C₂₅

C₂³

commutator

C₂³

C₂³×C₂₅

Frattini, socle, prime radical, cosocle

C₂³×C₅

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₅ ◁ G

N	H ≅ G/N	type
49. #400	C₂₅	direct (N × H)
73. C₂₅	#400	direct (N × H)

H ≅ G/N

type

49. #400

C₂₅

direct (N × H)

73. C₂₅

#400

direct (N × H)

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

subgroup	index
C₂×C₄⋊C₄×C₂₅	2
C₂₅×#95	2
C₅×#400	5

subgroup

index

C₂×C₄⋊C₄×C₂₅

C₂₅×#95

C₅×#400

Subgroups — C₂₅×#400 #13098549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C25×#400 #13098549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₅×#400 #13098549