Group #29596194 — subgroups

Subgroups — #29596194

Order 15616.

394 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16839592	C₂	1	2	yes	maximal, complemented
2	#21445857	C₂	1	2	yes	maximal
3	#21443913	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519713	C₂²	1	4	yes	—
6	#16839726	—	2	2	no	—
7	#16839592	—	2	4	no	—
8	#16839527	—	2	2	no	—
9	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₆₁	—	2	4	no	—
11	C₂²×C₁₆×C₆₁	C₄	1	4	yes	—
12	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#11758194	—	2	4	no	—
15	#14519713	—	2	4	no	—
16	#14519713	—	2	4	no	—
17	C₂²⋊#92418	—	4	4	no	—
18	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	—	2	2	no	—
20	C₂²×C₈×C₆₁	—	2	2	no	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
24	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
25	C₂×C₁₆×C₆₁	—	2	2	no	—
26	C₂×C₁₆×C₆₁	—	2	8	no	—
27	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
30	C₂⁴×C₆₁	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	—	2	2	no	—
33	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₆₁	D₈	1	2	yes	—
35	C₂²×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	—	2	2	no	—
38	C₂×C₈×C₆₁	—	2	2	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	—	2	4	no	—
41	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₁	#41	1	2	yes	—
46	#11758194	—	4	4	no	—
47	#11758194	—	4	4	no	—
48	C₂×#92418	—	4	4	no	—
49	C₁₆×C₆₁	—	2	2	no	—
50	C₁₆×C₆₁	C₂²⋊C₄	1	4	yes	—
51	C₂³×C₆₁	—	2	4	no	—
52	C₂³×C₆₁	—	2	2	no	—
53	C₂³×C₆₁	#107	1	2	yes	—
54	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	2	2	no	—
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	4	no	—
59	C₂×C₄×C₆₁	#107	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#121	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#135	1	2	yes	—
64	C₈×C₆₁	—	2	2	no	—
65	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₆₁	#117	1	1	yes	characteristic
67	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
68	#92418	—	8	4	no	—
69	#12837	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	8	no	—
71	C₂²×C₆₁	—	2	2	no	—
72	C₂²×C₆₁	—	2	2	no	—
73	C₂²×C₆₁	—	2	2	no	—
74	C₂²×C₆₁	#368	1	4	yes	complemented
75	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
76	C₂²×C₆₁	#467	1	2	yes	—
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	#380	1	1	yes	characteristic
79	C₄×C₆₁	#427	1	1	yes	characteristic
80	C₄×C₆₁	#466	1	2	yes	—
81	C₂³×C₁₆	D₆₁	1	1	yes	characteristic
82	C₂×#368	—	61	2	no	—
83	#1810	—	61	2	no	—
84	#1381	—	61	2	no	—
85	C₂×C₆₁	—	2	4	no	—
86	C₂×C₆₁	—	2	2	no	—
87	C₂×C₆₁	#1808	1	1	yes	characteristic
88	C₂×C₆₁	C₂×#368	1	2	yes	—
89	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
90	C₂²×C₁₆	—	2	4	no	—
91	C₂²×C₁₆	Dic₆₁	1	4	yes	—
92	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
93	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
94	C₂×#108	—	61	4	no	—
95	#407	—	122	2	no	—
96	#368	—	122	4	no	—
97	#341	—	122	2	no	—
98	C₆₁	#12837	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
100	C₂×M₁₆	—	122	4	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	Dic₁₂₂	1	2	yes	—
104	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
106	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	8	no	—
109	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
110	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
111	C₂×C₁₆	C₂²⋊D₆₁	1	4	yes	—
112	#108	—	122	4	no	—
113	#108	—	122	4	no	—
114	C₂²⋊C₈	—	244	4	no	—
115	C₂⁴	Dic₂₄₄	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	D₄₈₈	1	2	yes	—
119	C₂²×C₄	Dic₂₄₄	1	1	yes	characteristic
120	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
121	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
122	M₁₆	—	244	4	no	—
123	M₁₆	—	244	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	244	4	no	—
129	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
130	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
131	C₂×C₈	#11758222	1	2	yes	—
132	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
133	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
134	C₁₆	—	2	2	no	—
135	C₁₆	C₂²⋊Dic₆₁	1	4	yes	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	#14519711	1	2	yes	—
139	C₂³	C₂×Dic₂₄₄	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	#14519711	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
146	C₂×C₄	#14519751	1	2	yes	—
147	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
148	C₂×C₄	#14519799	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	488	4	no	—
151	C₈	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
152	C₈	#14519736	1	1	yes	characteristic
153	C₈	#14519756	1	2	yes	—
154	C₂²	—	2	8	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	#16839592	1	4	yes	complemented
159	C₂²	C₂×#14519711	1	1	yes	characteristic
160	C₂²	#16840003	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	#16839622	1	1	yes	characteristic
163	C₄	#16839800	1	1	yes	characteristic
164	C₄	#16839998	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	2	4	no	—
167	C₂	#21445848	1	1	yes	characteristic
168	C₂	C₂×#16839592	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839592

C₂

yes

maximal, complemented

#21445857

C₂

yes

maximal

#21443913

C₂

yes

maximal, complemented

C₂³×C₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519713

C₂²

yes

—

#16839726

—

#16839592

—

#16839527

—

C₂³×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₁₆×C₆₁

—

C₂²×C₁₆×C₆₁

C₄

yes

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂²×C₁₆×C₆₁

C₂²

yes

—

—

—

—

C₂³×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

Q₈

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₂⁴×C₆₁

Q₁₆

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

D₈

yes

—

C₂²×C₄×C₆₁

Q₁₆

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

#11758194

—

#11758194

—

C₂×#92418

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂²⋊C₄

yes

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

#107

yes

—

C₂³×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#107

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₁

#117

yes

characteristic

C₈×C₆₁

C₂²⋊Q₈

yes

—

#92418

—

#12837

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂³×C₁₆

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1808

yes

characteristic

C₂×C₆₁

C₂×#368

yes

—

C₂³×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

Dic₆₁

yes

—

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

C₂×#108

—

#407

—

122

—

#368

—

122

—

#341

—

122

—

C₆₁

#12837

yes

characteristic, complemented, Sylow

C₂³×C₄

D₂₄₄

yes

characteristic

100

C₂×M₁₆

—

122

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

Dic₁₂₂

yes

—

104

C₂²×C₈

D₂₄₄

yes

characteristic

105

C₂²×C₈

C₂×Dic₆₁

yes

—

106

C₂²×C₈

C₂²×D₆₁

yes

characteristic

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₂×Dic₆₁

yes

—

110

C₂×C₁₆

C₂×Dic₆₁

yes

—

111

C₂×C₁₆

C₂²⋊D₆₁

yes

—

112

#108

—

122

—

113

#108

—

122

—

114

C₂²⋊C₈

—

244

—

115

C₂⁴

Dic₂₄₄

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

D₄₈₈

yes

—

119

C₂²×C₄

Dic₂₄₄

yes

characteristic

120

C₂²×C₄

C₄⋊Dic₆₁

yes

—

121

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

122

M₁₆

—

244

—

123

M₁₆

—

244

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

244

—

129

C₂×C₈

C₄⋊Dic₆₁

yes

—

130

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

131

C₂×C₈

#11758222

yes

—

132

C₂×C₈

C₂²×Dic₆₁

yes

characteristic

133

C₂×C₈

C₂×C₂²⋊D₆₁

yes

—

134

C₁₆

—

135

C₁₆

C₂²⋊Dic₆₁

yes

—

136

C₂³

—

137

C₂³

—

138

C₂³

#14519711

yes

—

139

C₂³

C₂×Dic₂₄₄

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#14519711

yes

—

145

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

146

C₂×C₄

#14519751

yes

—

147

C₂×C₄

C₂×D₄₈₈

yes

characteristic

148

C₂×C₄

#14519799

yes

—

149

C₈

—

150

C₈

—

488

—

151

C₈

C₂×C₂²⋊Dic₆₁

yes

characteristic

152

C₈

#14519736

yes

characteristic

153

C₈

#14519756

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#16839592

yes

complemented

159

C₂²

C₂×#14519711

yes

characteristic

160

C₂²

#16840003

yes

—

161

C₄

—

162

C₄

#16839622

yes

characteristic

163

C₄

#16839800

yes

characteristic

164

C₄

#16839998

yes

—

165

C₂

—

166

C₂

—

167

C₂

#21445848

yes

characteristic

168

C₂

C₂×#16839592

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂³×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂⁴×C₆₁	Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂³×C₁₆	D₆₁
hypocenter	C₆₁	#12837
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₄₈₈

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₂³×C₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂×#107

prime radical

C₂⁴×C₆₁

Q₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

hypercenter

C₂³×C₁₆

D₆₁

hypocenter

C₆₁

#12837

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

N	H ≅ G/N	type
1. C₂×#16839592	C₂	semidirect
3. #21443913	C₂	semidirect
74. C₂²×C₆₁	#368	semidirect
98. C₆₁	#12837	semidirect
158. C₂²	#16839592	semidirect

H ≅ G/N

type

1. C₂×#16839592

C₂

semidirect

3. #21443913

C₂

semidirect

74. C₂²×C₆₁

#368

semidirect

98. C₆₁

#12837

semidirect

158. C₂²

#16839592

semidirect

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

subgroup	index
C₂×#16839592	2
#21445857	2
#21443913	2
C₂³×C₁₆×C₆₁	2
#12837	61

subgroup

index

C₂³×C₁₆×C₆₁

Subgroups — #29596194

Conjugacy classes of subgroups

Complements

Maximal subgroups