Group #13107112 — subgroups

Subgroups — #13107112

Order 1600.

848 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 243 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	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717266	C₂	1	2	yes	maximal, complemented
3	#11717218	C₂	1	4	yes	maximal, complemented
4	#11717212	C₂	1	2	yes	maximal, complemented
5	#11717195	C₂	1	2	yes	maximal
6	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic
7	#11717189	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#6045	C₂	1	2	yes	maximal, complemented
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
12	C₂×#6045	C₂²	1	2	yes	—
13	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄²×C₅²	C₂²	1	2	yes	—
16	C₄²×C₅²	C₂²	1	1	yes	characteristic
17	#87808	C₂²	1	2	yes	—
18	#87808	C₂²	1	4	yes	—
19	#87808	C₂²	1	4	yes	—
20	C₄⋊#1039	C₂²	1	1	yes	characteristic
21	C₄⋊#1039	C₂²	1	2	yes	—
22	C₄⋊#1039	C₂²	1	2	yes	—
23	#87806	C₂²	1	4	yes	—
24	C₄×#1039	C₂²	1	4	yes	—
25	#65918	—	5	6	no	maximal
26	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
27	C₅²×Q₈	—	2	2	no	—
28	C₂×C₄×C₅²	C₂³	1	2	yes	—
29	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₅²	C₂³	1	2	yes	—
31	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
33	C₂×#1039	C₂³	1	4	yes	—
34	C₂×#1039	C₂³	1	2	yes	—
35	C₄⋊(C₅⋊D₅)	—	2	2	no	—
36	C₄×C₅⋊D₅	—	2	4	no	—
37	#6045	—	2	4	no	—
38	C₄×C₅×Q₈	D₅	1	6	yes	complemented
39	#4092	—	5	12	no	—
40	#4044	—	5	24	no	—
41	#4038	—	5	12	no	—
42	#4021	—	5	12	no	—
43	C₄×D₂₀	—	5	6	no	—
44	#4015	—	5	6	no	—
45	C₄×Dic₁₀	—	5	12	no	—
46	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
47	C₂×C₅⋊D₅	—	2	2	no	—
48	C₂×C₅⋊D₅	—	2	2	no	—
49	C₂×C₅⋊D₅	—	2	2	no	—
50	C₄×C₅²	—	2	2	no	—
51	C₄×C₅²	—	2	2	no	—
52	C₄×C₅²	—	2	1	no	—
53	C₄×C₅²	#41	1	2	yes	—
54	C₄×C₅²	#41	1	2	yes	—
55	#1039	—	2	4	no	—
56	#1039	—	2	2	no	—
57	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
58	C₂×D₂₀	—	5	6	no	—
59	C₂×C₄×D₅	—	5	12	no	—
60	C₂×Dic₁₀	—	5	12	no	—
61	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
62	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
63	C₄²×C₅	C₂×D₅	1	12	yes	—
64	C₄²×C₅	C₂×D₅	1	6	yes	—
65	#685	—	5	12	no	—
66	#685	—	5	24	no	—
67	#685	—	5	24	no	—
68	C₄⋊Dic₅	—	5	6	no	—
69	C₄⋊Dic₅	—	5	12	no	—
70	C₄⋊Dic₅	—	5	12	no	—
71	#683	—	5	24	no	—
72	C₄×Dic₅	—	5	24	no	—
73	#556	—	25	1	no	Sylow
74	C₂×C₅²	C₂×#41	1	1	yes	characteristic
75	C₂×C₅²	C₂×#41	1	1	yes	characteristic
76	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
77	C₅⋊D₅	—	4	2	no	—
78	C₂²×D₅	—	5	12	no	—
79	C₅×Q₈	—	2	12	no	—
80	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
81	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
82	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
83	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
84	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
85	C₂×Dic₅	—	5	12	no	—
86	C₂×Dic₅	—	5	24	no	—
87	D₂₀	—	10	12	no	—
88	C₄×D₅	—	10	24	no	—
89	Dic₁₀	—	10	24	no	—
90	C₄⋊Q₈	—	25	1	no	—
91	#126	—	25	4	no	—
92	#124	—	25	2	no	—
93	C₂²⋊Q₈	—	25	2	no	—
94	C₄×Q₈	—	25	2	no	—
95	C₄×Q₈	C₅⋊D₅	1	1	yes	characteristic, complemented
96	C₄×D₄	—	25	1	no	—
97	#117	—	25	2	no	—
98	C₅²	#556	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₅	C₂³×D₅	1	6	yes	—
100	C₂×D₅	—	10	12	no	—
101	C₂×D₅	—	10	12	no	—
102	C₂×D₅	—	10	12	no	—
103	C₄×C₅	—	2	12	no	—
104	C₄×C₅	—	2	12	no	—
105	C₄×C₅	—	2	6	no	—
106	C₄×C₅	#709	1	12	yes	—
107	C₄×C₅	#713	1	12	yes	—
108	Dic₅	—	10	12	no	—
109	Dic₅	—	10	24	no	—
110	C₂×Q₈	—	25	2	no	—
111	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
112	C₂×D₄	—	25	1	no	—
113	C₂²×C₄	—	25	2	no	—
114	C₄⋊C₄	—	25	2	no	—
115	C₄⋊C₄	—	25	1	no	—
116	C₄⋊C₄	—	25	4	no	—
117	C₄⋊C₄	—	25	2	no	—
118	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
119	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
120	C₂²⋊C₄	—	25	4	no	—
121	C₂²⋊C₄	—	25	2	no	—
122	C₂²⋊C₄	—	25	4	no	—
123	C₄²	—	25	4	no	—
124	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
125	C₄²	C₂×C₅⋊D₅	1	2	yes	—
126	C₂×C₅	C₂×#709	1	6	yes	—
127	C₂×C₅	C₂×#713	1	6	yes	—
128	C₂×C₅	#4150	1	6	yes	—
129	D₅	—	20	12	no	—
130	C₂³	—	25	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	—	50	4	no	—
133	D₄	—	50	2	no	—
134	C₂×C₄	—	25	2	no	—
135	C₂×C₄	—	25	4	no	—
136	C₂×C₄	—	50	4	no	—
137	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
138	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
140	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
141	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
142	C₅	#65918	1	6	yes	complemented
143	C₂²	—	50	2	no	—
144	C₂²	—	50	2	no	—
145	C₂²	—	50	2	no	—
146	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	1	no	—
150	C₄	—	50	2	no	—
151	C₄	—	50	4	no	—
152	C₄	#87900	1	2	yes	—
153	C₄	#87904	1	2	yes	—
154	C₂	—	100	2	no	—
155	C₂	C₂×#87900	1	1	yes	characteristic
156	C₂	C₂×#87904	1	1	yes	characteristic
157	C₂	#11717523	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅²×Q₈

C₂

yes

maximal, characteristic, complemented

#11717266

C₂

yes

maximal, complemented

#11717218

C₂

yes

maximal, complemented

#11717212

C₂

yes

maximal, complemented

#11717195

C₂

yes

maximal

C₄×C₄⋊(C₅⋊D₅)

C₂

yes

maximal, characteristic

#11717189

C₂

yes

maximal, characteristic, complemented

C₄×#6045

C₂

yes

maximal, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₂×C₄×C₅⋊D₅

C₂²

yes

—

C₂×#6045

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

characteristic

C₅²×C₄⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

#87808

C₂²

yes

—

#87808

C₂²

yes

—

#87808

C₂²

yes

—

C₄⋊#1039

C₂²

yes

characteristic

C₄⋊#1039

C₂²

yes

—

C₄⋊#1039

C₂²

yes

—

#87806

C₂²

yes

—

C₄×#1039

C₂²

yes

—

#65918

—

maximal

C₂²×C₅⋊D₅

C₂³

yes

—

C₅²×Q₈

—

C₂×C₄×C₅²

C₂³

yes

—

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

—

C₄⋊(C₅⋊D₅)

—

C₄×C₅⋊D₅

—

#6045

—

C₄×C₅×Q₈

D₅

yes

—

—

—

—

C₄×D₂₀

—

#4015

—

C₄×Dic₁₀

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

#41

yes

—

C₄×C₅²

#41

yes

—

#1039

—

#1039

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

#685

—

#685

—

#685

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

#556

—

Sylow

C₂×C₅²

C₂×#41

yes

characteristic

C₂×C₅²

C₂×#41

yes

characteristic

C₂×C₅²

2₋¹⁺⁴

yes

characteristic

C₅⋊D₅

—

C₂²×D₅

—

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×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

C₄⋊Q₈

—

#126

—

#124

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

C₅⋊D₅

yes

characteristic, complemented

C₄×D₄

—

#117

—

C₅²

#556

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂³×D₅

yes

—

100

C₂×D₅

—

101

C₂×D₅

—

102

C₂×D₅

—

103

C₄×C₅

—

104

C₄×C₅

—

105

C₄×C₅

—

106

C₄×C₅

#709

yes

—

107

C₄×C₅

#713

yes

—

108

Dic₅

—

109

Dic₅

—

110

C₂×Q₈

—

111

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

112

C₂×D₄

—

113

C₂²×C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×C₅⋊D₅

yes

characteristic

119

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

C₂×C₅⋊D₅

yes

characteristic

125

C₄²

C₂×C₅⋊D₅

yes

—

126

C₂×C₅

C₂×#709

yes

—

127

C₂×C₅

C₂×#713

yes

—

128

C₂×C₅

#4150

yes

—

129

D₅

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²×C₅⋊D₅

yes

—

138

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

139

C₂×C₄

C₂²×C₅⋊D₅

yes

—

140

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

141

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

142

C₅

#65918

yes

complemented

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂³×C₅⋊D₅

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#87900

yes

—

153

C₄

#87904

yes

—

154

C₂

—

100

—

155

C₂

C₂×#87900

yes

characteristic

156

C₂

C₂×#87904

yes

characteristic

157

C₂

#11717523

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄×C₅²×Q₈	C₂
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	C₄×Q₈	C₅⋊D₅
hypocenter	C₅²	#556
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×C₅⋊D₅

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₄×C₅²×Q₈

C₂

prime radical

C₂×C₄⋊(C₅⋊D₅)

C₂²

hypercenter

C₄×Q₈

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. C₄×C₅²×Q₈	C₂	semidirect
2. #11717266	C₂	semidirect
3. #11717218	C₂	semidirect
4. #11717212	C₂	semidirect
7. #11717189	C₂	semidirect
8. C₄×#6045	C₂	semidirect
38. C₄×C₅×Q₈	D₅	semidirect
95. C₄×Q₈	C₅⋊D₅	semidirect
98. C₅²	#556	semidirect
142. C₅	#65918	semidirect

H ≅ G/N

type

1. C₄×C₅²×Q₈

C₂

semidirect

2. #11717266

C₂

semidirect

3. #11717218

C₂

semidirect

4. #11717212

C₂

semidirect

7. #11717189

C₂

semidirect

8. C₄×#6045

C₂

semidirect

38. C₄×C₅×Q₈

D₅

semidirect

95. C₄×Q₈

C₅⋊D₅

semidirect

98. C₅²

#556

semidirect

142. C₅

#65918

semidirect

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

subgroup	index
C₄×C₅²×Q₈	2
#11717266	2
#11717218	2
#11717212	2
#11717195	2
C₄×C₄⋊(C₅⋊D₅)	2
#11717189	2
C₄×#6045	2
#65918	5

subgroup

index

C₄×C₅²×Q₈

C₄×C₄⋊(C₅⋊D₅)

Subgroups — #13107112

Conjugacy classes of subgroups

Complements

Maximal subgroups