Group #10614209 — subgroups

Subgroups — #10614209

Order 640.

500 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 197 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×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4015	C₂	1	1	yes	maximal, characteristic
3	#65519	C₂	1	2	yes	maximal
4	#65268	C₂	1	4	yes	maximal
5	#65265	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
7	#65217	C₂	1	4	yes	maximal
8	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀	C₂²	1	2	yes	—
10	C₄×C₅×Q₈	—	2	4	no	—
11	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₅	C₂²	1	2	yes	—
14	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
18	C₂×#683	C₂²	1	8	yes	—
19	C₂×C₄×Dic₅	C₂²	1	2	yes	—
20	#4015	—	2	4	no	—
21	#3963	C₂²	1	8	yes	—
22	#3963	C₂²	1	2	yes	—
23	#2567	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₈	—	2	2	no	—
25	C₂×C₅×Q₈	—	2	2	no	—
26	C₂×C₅×Q₈	Q₈	1	4	yes	complemented
27	C₂²×C₄×C₅	C₂³	1	2	yes	—
28	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅	C₂³	1	2	yes	—
30	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅	C₂³	1	4	yes	—
33	C₂²×Dic₅	C₂³	1	4	yes	—
34	C₂×Dic₁₀	—	2	4	no	—
35	C₂×Dic₁₀	—	4	4	no	—
36	C₅×C₄⋊C₄	—	2	4	no	—
37	C₅×C₄⋊C₄	—	2	2	no	—
38	C₄²×C₅	—	2	4	no	—
39	C₄²×C₅	D₄	1	4	yes	—
40	C₄⋊Dic₅	—	2	4	no	—
41	C₄⋊Dic₅	—	2	4	no	—
42	C₄⋊Dic₅	Q₈	1	8	yes	complemented
43	#683	—	4	8	no	—
44	C₄×Dic₅	—	2	4	no	—
45	C₂×C₄⋊Q₈	—	5	1	no	—
46	C₂×C₄×Q₈	D₅	1	1	yes	characteristic
47	#395	—	5	4	no	—
48	#391	—	5	2	no	—
49	#389	—	5	4	no	—
50	#384	—	5	2	no	—
51	C₄×C₄⋊C₄	—	5	1	no	—
52	C₂³×C₅	C₂⁴	1	1	yes	characteristic
53	C₅×Q₈	—	2	8	no	—
54	C₂×C₄×C₅	—	2	2	no	—
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	2	no	—
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	—	2	4	no	—
59	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₅	#41	1	2	yes	—
66	C₂×Dic₅	—	2	4	no	—
67	C₂×Dic₅	—	2	4	no	—
68	C₂×Dic₅	—	2	4	no	—
69	C₂×Dic₅	—	2	8	no	—
70	C₂×Dic₅	C₂×Q₈	1	8	yes	—
71	Dic₁₀	—	4	8	no	—
72	C₂²×Q₈	—	5	2	no	—
73	C₂²×Q₈	C₂×D₅	1	1	yes	characteristic
74	C₄⋊Q₈	—	10	4	no	—
75	C₄×Q₈	—	2	4	no	—
76	C₂×C₄⋊C₄	—	5	2	no	—
77	C₂×C₄⋊C₄	—	5	2	no	—
78	C₂×C₄⋊C₄	—	5	2	no	—
79	C₂×C₄⋊C₄	—	5	8	no	—
80	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
82	C₂×C₄²	—	5	2	no	—
83	C₂×C₄²	C₂×D₅	1	2	yes	—
84	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
85	#95	—	5	2	no	—
86	#95	—	5	8	no	—
87	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×C₅	C₂²×Q₈	1	2	yes	—
90	C₂²×C₅	C₂×#41	1	1	yes	characteristic
91	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
92	C₄×C₅	—	2	4	no	—
93	C₄×C₅	—	2	2	no	—
94	C₄×C₅	—	4	2	no	—
95	C₄×C₅	C₂²⋊Q₈	1	4	yes	—
96	C₄×C₅	C₄⋊Q₈	1	4	yes	—
97	Dic₅	—	2	8	no	—
98	Dic₅	—	4	4	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	10	4	no	—
102	C₂×Q₈	—	20	4	no	—
103	C₂×Q₈	Dic₁₀	1	4	yes	complemented
104	C₂²×C₄	—	5	4	no	—
105	C₂²×C₄	—	5	4	no	—
106	C₂²×C₄	C₂²×D₅	1	2	yes	—
107	C₂²×C₄	C₂²×D₅	1	2	yes	—
108	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	5	8	no	—
114	C₄⋊C₄	—	10	4	no	—
115	C₄⋊C₄	—	10	4	no	—
116	C₄⋊C₄	—	20	8	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	10	4	no	—
119	C₄²	C₂²⋊D₅	1	4	yes	—
120	C₂×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₅	#535	1	1	yes	characteristic
123	C₂×C₅	#554	1	2	yes	—
124	C₂×C₅	Q₈²	1	2	yes	—
125	C₂³	C₂³×D₅	1	1	yes	characteristic
126	Q₈	—	2	8	no	—
127	Q₈	—	20	8	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	5	8	no	—
134	C₂×C₄	—	10	4	no	—
135	C₂×C₄	—	10	4	no	—
136	C₂×C₄	—	10	8	no	—
137	C₂×C₄	—	10	4	no	—
138	C₂×C₄	C₂×Dic₁₀	1	4	yes	—
139	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
140	C₂×C₄	#709	1	2	yes	—
141	C₂×C₄	Q₈×D₅	1	4	yes	—
142	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
145	C₅	#2567	1	1	yes	characteristic, complemented, Sylow
146	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
147	C₂²	C₂×#709	1	1	yes	characteristic
148	C₂²	C₂×Q₈×D₅	1	2	yes	—
149	C₂²	#4149	1	2	yes	—
150	C₂²	C₂²×C₂²⋊D₅	1	1	yes	characteristic
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	10	8	no	—
155	C₄	—	20	4	no	—
156	C₄	#4070	1	4	yes	—
157	C₄	#4090	1	4	yes	—
158	C₂	Q₈×Dic₁₀	1	2	yes	—
159	C₂	#65913	1	2	yes	—
160	C₂	C₂×#4070	1	1	yes	characteristic
161	C₂	C₂×#4090	1	1	yes	characteristic
162	C₂	#66164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₅×Q₈

C₂

yes

maximal, characteristic

C₂×#4015

C₂

yes

maximal, characteristic

#65519

C₂

yes

maximal

#65268

C₂

yes

maximal

#65265

C₂

yes

maximal

C₄×C₄⋊Dic₅

C₂

yes

maximal, characteristic

#65217

C₂

yes

maximal

C₂²×C₅×Q₈

C₂²

yes

characteristic

C₂²×Dic₁₀

C₂²

yes

—

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

#4015

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2567

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

Q₈

yes

complemented

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

D₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

Q₈

yes

complemented

#683

—

C₄×Dic₅

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₅

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₂³×C₅

C₂⁴

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×Q₈

yes

—

Dic₁₀

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

—

C₂×C₄²

C₂×D₅

yes

characteristic

#95

—

#95

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂²×Q₈

yes

characteristic

C₂²×C₅

C₂²×Q₈

yes

—

C₂²×C₅

C₂×#41

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂²⋊Q₈

yes

—

C₄×C₅

C₄⋊Q₈

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

Dic₁₀

yes

complemented

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₅

yes

—

107

C₂²×C₄

C₂²×D₅

yes

—

108

C₂²×C₄

C₂²×D₅

yes

characteristic

109

C₂²×C₄

C₂²×D₅

yes

characteristic

110

C₂²×C₄

C₂²×D₅

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₂²⋊D₅

yes

—

120

C₂×C₅

C₂×C₂²⋊Q₈

yes

characteristic

121

C₂×C₅

C₂×C₄⋊Q₈

yes

characteristic

122

C₂×C₅

#535

yes

characteristic

123

C₂×C₅

#554

yes

—

124

C₂×C₅

Q₈²

yes

—

125

C₂³

C₂³×D₅

yes

characteristic

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×Dic₁₀

yes

—

139

C₂×C₄

C₂×Dic₁₀

yes

—

140

C₂×C₄

#709

yes

—

141

C₂×C₄

Q₈×D₅

yes

—

142

C₂×C₄

C₂×C₂²⋊D₅

yes

—

143

C₂×C₄

C₂×C₂²⋊D₅

yes

—

144

C₂×C₄

C₂×C₂²⋊D₅

yes

—

145

C₅

#2567

yes

characteristic, complemented, Sylow

146

C₂²

C₂²×Dic₁₀

yes

characteristic

147

C₂²

C₂×#709

yes

characteristic

148

C₂²

C₂×Q₈×D₅

yes

—

149

C₂²

#4149

yes

—

150

C₂²

C₂²×C₂²⋊D₅

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#4070

yes

—

157

C₄

#4090

yes

—

158

C₂

Q₈×Dic₁₀

yes

—

159

C₂

#65913

yes

—

160

C₂

C₂×#4070

yes

characteristic

161

C₂

C₂×#4090

yes

characteristic

162

C₂

#66164

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₂×C₄×C₅×Q₈	C₂
hypercenter	C₂×C₄×Q₈	D₅
hypocenter	C₅	#2567
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, prime radical, cosocle

C₂³×C₅

C₂⁴

Fitting

C₂×C₄×C₅×Q₈

C₂

hypercenter

C₂×C₄×Q₈

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
26. C₂×C₅×Q₈	Q₈	semidirect
42. C₄⋊Dic₅	Q₈	semidirect
103. C₂×Q₈	Dic₁₀	semidirect
145. C₅	#2567	semidirect

H ≅ G/N

type

26. C₂×C₅×Q₈

Q₈

semidirect

42. C₄⋊Dic₅

Q₈

semidirect

103. C₂×Q₈

Dic₁₀

semidirect

145. C₅

#2567

semidirect

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

subgroup	index
C₂×C₄×C₅×Q₈	2
C₂×#4015	2
#65519	2
#65268	2
#65265	2
C₄×C₄⋊Dic₅	2
#65217	2
#2567	5

subgroup

index

C₂×C₄×C₅×Q₈

C₄×C₄⋊Dic₅

Subgroups — #10614209

Conjugacy classes of subgroups

Complements

Maximal subgroups