Group #13107250 — subgroups

Subgroups — #13107250

Order 1600.

800 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 251 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	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1039	C₂	1	2	yes	maximal
3	#11717264	C₂	1	2	yes	maximal
4	#11717210	C₂	1	2	yes	maximal
5	#11717209	C₂	1	4	yes	maximal
6	#11717207	C₂	1	2	yes	maximal
7	C₄×#6045	C₂	1	1	yes	maximal, characteristic
8	C₄×#6045	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅²×Q₈	C₂²	1	2	yes	—
10	C₂×#6045	C₂²	1	1	yes	characteristic
11	C₂×#6045	C₂²	1	1	yes	characteristic
12	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
13	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
14	C₄²×C₅²	C₂²	1	1	yes	characteristic
15	C₄⋊#1039	C₂²	1	1	yes	characteristic
16	C₄⋊#1039	C₂²	1	1	yes	characteristic
17	C₄⋊#1039	C₂²	1	4	yes	—
18	#87806	C₂²	1	2	yes	—
19	#87806	C₂²	1	2	yes	—
20	#87806	C₂²	1	4	yes	—
21	#87806	C₂²	1	4	yes	—
22	C₄×#1039	C₂²	1	2	yes	—
23	C₄×#1039	C₂²	1	2	yes	—
24	C₄×#1039	C₂²	1	4	yes	—
25	#66055	—	5	6	no	maximal
26	C₅²×Q₈	—	2	4	no	—
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₅²	C₂³	1	4	yes	—
29	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
31	C₂×#1039	C₂³	1	2	yes	—
32	C₂×#1039	C₂³	1	2	yes	—
33	C₂×#1039	C₂³	1	2	yes	—
34	C₂×#1039	C₂³	1	2	yes	—
35	#6045	—	2	2	no	—
36	#6045	Q₈	1	4	yes	complemented
37	C₅×C₄⋊Q₈	D₅	1	6	yes	—
38	Q₈×Dic₅	—	5	12	no	—
39	#4090	—	5	12	no	—
40	#4036	—	5	12	no	—
41	#4035	—	5	24	no	—
42	#4033	—	5	12	no	—
43	C₄×Dic₁₀	—	5	6	no	—
44	C₄×Dic₁₀	—	5	6	no	—
45	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
46	C₄×C₅²	—	2	4	no	—
47	C₄×C₅²	—	2	1	no	—
48	C₄×C₅²	C₂×Q₈	1	2	yes	—
49	C₄×C₅²	#41	1	2	yes	—
50	#1039	—	2	2	no	—
51	#1039	—	2	2	no	—
52	#1039	—	2	2	no	—
53	#1039	C₂×Q₈	1	4	yes	—
54	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
55	C₂×Dic₁₀	—	5	6	no	—
56	C₂×Dic₁₀	—	5	6	no	—
57	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
58	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
59	C₄²×C₅	C₂×D₅	1	6	yes	—
60	C₄⋊Dic₅	—	5	6	no	—
61	C₄⋊Dic₅	—	5	6	no	—
62	C₄⋊Dic₅	—	5	24	no	—
63	#683	—	5	12	no	—
64	#683	—	5	12	no	—
65	#683	—	5	24	no	—
66	#683	—	5	24	no	—
67	C₄×Dic₅	—	5	12	no	—
68	C₄×Dic₅	—	5	12	no	—
69	C₄×Dic₅	—	5	24	no	—
70	#557	—	25	1	no	Sylow
71	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₅²	C₂×#41	1	1	yes	characteristic
73	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
74	C₅×Q₈	—	2	24	no	—
75	C₂×C₄×C₅	C₂²×D₅	1	24	yes	—
76	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
77	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
78	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
79	C₂×Dic₅	—	5	12	no	—
80	C₂×Dic₅	—	5	12	no	—
81	C₂×Dic₅	—	5	12	no	—
82	C₂×Dic₅	—	5	12	no	—
83	Dic₁₀	—	5	24	no	—
84	Dic₁₀	—	10	12	no	—
85	C₄⋊Q₈	—	25	2	no	—
86	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
87	#125	—	25	4	no	—
88	#125	—	25	2	no	—
89	C₄×Q₈	—	25	2	no	—
90	C₄×Q₈	—	25	2	no	—
91	C₄×Q₈	—	25	1	no	—
92	C₄×Q₈	—	25	1	no	—
93	C₅²	#557	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₅	C₂³×D₅	1	6	yes	—
95	C₄×C₅	—	2	24	no	—
96	C₄×C₅	—	2	6	no	—
97	C₄×C₅	#711	1	12	yes	—
98	C₄×C₅	Q₈×D₅	1	12	yes	—
99	Dic₅	—	5	24	no	—
100	Dic₅	—	10	12	no	—
101	Dic₅	—	10	12	no	—
102	Dic₅	—	10	12	no	—
103	C₂×Q₈	—	25	1	no	—
104	C₂×Q₈	—	25	1	no	—
105	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
106	C₄⋊C₄	—	25	1	no	—
107	C₄⋊C₄	—	25	2	no	—
108	C₄⋊C₄	—	25	2	no	—
109	C₄⋊C₄	—	25	4	no	—
110	C₄⋊C₄	—	25	1	no	—
111	C₄⋊C₄	—	25	4	no	—
112	C₄⋊C₄	—	25	4	no	—
113	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
114	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
115	C₄²	—	25	4	no	—
116	C₄²	—	25	2	no	—
117	C₄²	—	25	2	no	—
118	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
119	C₂×C₅	C₂×#711	1	6	yes	—
120	C₂×C₅	C₂×Q₈×D₅	1	6	yes	—
121	C₂×C₅	#4147	1	6	yes	—
122	Q₈	—	2	4	no	—
123	Q₈	—	25	4	no	—
124	Q₈	—	50	2	no	—
125	C₂×C₄	—	25	2	no	—
126	C₂×C₄	—	25	2	no	—
127	C₂×C₄	—	25	2	no	—
128	C₂×C₄	—	25	2	no	—
129	C₂×C₄	C₂²×C₅⋊D₅	1	4	yes	—
130	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
133	C₅	#66055	1	6	yes	complemented
134	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	2	1	no	—
137	C₄	—	25	4	no	—
138	C₄	—	50	2	no	—
139	C₄	—	50	2	no	—
140	C₄	—	50	2	no	—
141	C₄	#87902	1	2	yes	—
142	C₄	Q₈×C₅⋊D₅	1	2	yes	—
143	C₂	C₂×#87902	1	1	yes	characteristic
144	C₂	C₂×Q₈×C₅⋊D₅	1	1	yes	characteristic
145	C₂	#11717520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×C₄⋊Q₈

C₂

yes

maximal, characteristic

Q₈×#1039

C₂

yes

maximal

#11717264

C₂

yes

maximal

#11717210

C₂

yes

maximal

#11717209

C₂

yes

maximal

#11717207

C₂

yes

maximal

C₄×#6045

C₂

yes

maximal, characteristic

C₄×#6045

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

C₂²

yes

—

C₂×#6045

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₅²×C₄⋊C₄

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₄⋊#1039

C₂²

yes

characteristic

C₄⋊#1039

C₂²

yes

characteristic

C₄⋊#1039

C₂²

yes

—

#87806

C₂²

yes

—

#87806

C₂²

yes

—

#87806

C₂²

yes

—

#87806

C₂²

yes

—

C₄×#1039

C₂²

yes

—

C₄×#1039

C₂²

yes

—

C₄×#1039

C₂²

yes

—

#66055

—

maximal

C₅²×Q₈

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

—

C₂×#1039

C₂³

yes

—

C₂×#1039

C₂³

yes

—

C₂×#1039

C₂³

yes

—

#6045

—

#6045

Q₈

yes

complemented

C₅×C₄⋊Q₈

D₅

yes

—

Q₈×Dic₅

—

—

—

—

—

C₄×Dic₁₀

—

C₄×Dic₁₀

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

—

C₄×C₅²

#41

yes

—

—

—

—

C₂×Q₈

yes

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

#557

—

Sylow

C₂×C₅²

C₂²×Q₈

yes

characteristic

C₂×C₅²

C₂×#41

yes

characteristic

C₂×C₅²

2₋¹⁺⁴

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₄⋊Q₈

—

C₄⋊Q₈

C₅⋊D₅

yes

characteristic

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₅²

#557

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂³×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#711

yes

—

C₄×C₅

Q₈×D₅

yes

—

Dic₅

—

100

Dic₅

—

101

Dic₅

—

102

Dic₅

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×C₅⋊D₅

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

114

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×C₅⋊D₅

yes

characteristic

119

C₂×C₅

C₂×#711

yes

—

120

C₂×C₅

C₂×Q₈×D₅

yes

—

121

C₂×C₅

#4147

yes

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²×C₅⋊D₅

yes

—

130

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

131

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

132

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

133

C₅

#66055

yes

complemented

134

C₂²

C₂³×C₅⋊D₅

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#87902

yes

—

142

C₄

Q₈×C₅⋊D₅

yes

—

143

C₂

C₂×#87902

yes

characteristic

144

C₂

C₂×Q₈×C₅⋊D₅

yes

characteristic

145

C₂

#11717520

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, prime radical, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×C₄⋊Q₈	C₂
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#557
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×C₅⋊D₅

commutator, socle, prime radical, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₅²×C₄⋊Q₈

C₂

hypercenter

C₄⋊Q₈

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
36. #6045	Q₈	semidirect
93. C₅²	#557	semidirect
133. C₅	#66055	semidirect

H ≅ G/N

type

36. #6045

Q₈

semidirect

93. C₅²

#557

semidirect

133. C₅

#66055

semidirect

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

subgroup	index
C₅²×C₄⋊Q₈	2
Q₈×#1039	2
#11717264	2
#11717210	2
#11717209	2
#11717207	2
C₄×#6045	2
#66055	5

subgroup

index

C₅²×C₄⋊Q₈

Subgroups — #13107250

Conjugacy classes of subgroups

Complements

Maximal subgroups