Group #29596395 — subgroups

Subgroups — #29596395

Order 15616.

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

normal subgroups only

This group has 310 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×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
2	#21445983	C₂	1	1	yes	maximal, characteristic
3	#21443910	C₂	1	4	yes	maximal
4	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₂²	1	2	yes	—
6	C₂×#14519711	C₂²	1	2	yes	—
7	C₂×#14519709	—	2	1	no	—
8	#16839628	—	2	1	no	—
9	C₄×#11758185	C₄	1	4	yes	complemented
10	C₂×#107×C₆₁	C₂²	1	2	yes	—
11	C₂×#106×C₆₁	—	2	1	no	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#382	—	2	1	no	—
14	C₂×C₄⋊Dic₆₁	—	2	2	no	—
15	C₂×#11758185	—	2	2	no	—
16	C₂×#11758185	—	2	2	no	—
17	C₂×#11758185	C₂×C₄	1	4	yes	—
18	#14519711	—	2	4	no	—
19	#14519709	—	4	2	no	—
20	#14519684	—	4	2	no	—
21	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
25	#107×C₆₁	—	2	4	no	—
26	#106×C₆₁	—	4	2	no	—
27	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₁	Q₈	1	2	yes	—
29	#95×C₆₁	—	4	2	no	—
30	C₂²×C₄×C₆₁	—	4	2	no	—
31	C₂²×C₄×C₆₁	D₈	1	2	yes	—
32	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₆₁	—	4	2	no	—
34	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
36	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₆₁	#41	1	2	yes	—
40	C₂×C₈×C₆₁	#41	1	2	yes	—
41	C₄⋊C₄×C₆₁	—	4	2	no	—
42	C₄⋊C₄×C₆₁	—	4	2	no	—
43	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₄²×C₆₁	Q₁₆	1	2	yes	—
45	C₄⋊Dic₆₁	—	4	2	no	—
46	C₄⋊Dic₆₁	—	4	2	no	—
47	#11758185	—	2	4	no	—
48	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	2	4	no	—
50	C₂×C₄×C₆₁	—	8	2	no	—
51	C₂×C₄×C₆₁	—	8	4	no	—
52	C₂×C₄×C₆₁	#107	1	4	yes	—
53	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₆₁	—	8	2	no	—
60	C₂×Dic₆₁	—	8	4	no	—
61	C₈×C₆₁	#106	1	4	yes	—
62	C₈×C₆₁	#117	1	2	yes	—
63	C₈×C₆₁	#125	1	2	yes	—
64	#12889	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
66	C₂²×C₆₁	#486	1	1	yes	characteristic
67	C₂²×C₆₁	#486	1	1	yes	characteristic
68	C₂²×C₆₁	#508	1	4	yes	—
69	C₄×C₆₁	—	2	4	no	—
70	C₄×C₆₁	—	16	2	no	—
71	C₄×C₆₁	#382	1	1	yes	characteristic
72	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
73	C₄×C₆₁	#487	1	1	yes	characteristic
74	C₄×C₆₁	#487	1	1	yes	characteristic
75	Dic₆₁	—	16	2	no	—
76	C₂×C₄×C₁₆	—	61	1	no	—
77	#1846	—	61	1	no	—
78	#1846	D₆₁	1	1	yes	characteristic
79	#1380	—	61	4	no	—
80	C₂×C₆₁	#1834	1	1	yes	characteristic
81	C₂×C₆₁	#2158	1	2	yes	—
82	C₂×C₆₁	#2242	1	2	yes	—
83	C₂×C₆₁	#2242	1	2	yes	—
84	C₂²×C₁₆	—	61	2	no	—
85	C₂×#107	—	61	2	no	—
86	C₂×#107	C₂×D₆₁	1	2	yes	—
87	C₂×#106	—	2	1	no	—
88	C₂×#106	—	122	1	no	—
89	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
90	#382	—	2	1	no	—
91	#382	—	122	1	no	—
92	C₄×C₁₆	—	61	4	no	—
93	C₆₁	#12889	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	122	2	no	—
98	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
99	C₂×C₁₆	—	61	4	no	—
100	C₂×C₁₆	—	122	2	no	—
101	C₂×C₁₆	—	122	2	no	—
102	#107	—	2	4	no	—
103	#107	—	122	4	no	—
104	#106	—	4	2	no	—
105	#106	—	244	2	no	—
106	C₄×C₈	Dic₁₂₂	1	2	yes	—
107	C₄×C₈	C₄×D₆₁	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	244	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	244	2	no	—
112	C₂²×C₄	#11758199	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
114	C₂×C₈	#11758196	1	2	yes	—
115	C₂×C₈	#11758200	1	2	yes	—
116	C₂×C₈	#11758201	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
119	C₂×C₈	#11758224	1	2	yes	—
120	C₂×C₈	#11758226	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	244	2	no	—
124	C₄⋊C₄	—	244	2	no	—
125	C₄²	#11758196	1	2	yes	—
126	C₄²	#11758202	1	2	yes	—
127	C₁₆	—	122	4	no	—
128	C₂³	C₂×#11758199	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	488	4	no	—
133	C₂×C₄	—	488	2	no	—
134	C₂×C₄	#14519710	1	4	yes	—
135	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
136	C₂×C₄	#14519764	1	1	yes	characteristic
137	C₂×C₄	#14519766	1	1	yes	characteristic
138	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
139	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
140	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
141	C₈	#14519708	1	4	yes	—
142	C₈	#14519735	1	2	yes	—
143	C₈	#14519768	1	2	yes	—
144	C₂²	C₂×#14519710	1	1	yes	characteristic
145	C₂²	#16840108	1	1	yes	characteristic
146	C₂²	#16840110	1	1	yes	characteristic
147	C₂²	#16840192	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	976	2	no	—
151	C₄	#16839627	1	1	yes	characteristic
152	C₄	C₂×#14519708	1	1	yes	characteristic
153	C₄	#16840112	1	1	yes	characteristic
154	C₄	#16840114	1	1	yes	characteristic
155	C₂	#21445946	1	1	yes	characteristic
156	C₂	#21447601	1	2	yes	—
157	C₂	#21448050	1	2	yes	—
158	C₂	#21448053	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11758185

C₂

yes

maximal, characteristic

#21445983

C₂

yes

maximal, characteristic

#21443910

C₂

yes

maximal

C₆₁×#1846

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

—

C₂×#14519711

C₂²

yes

—

C₂×#14519709

—

#16839628

—

C₄×#11758185

C₄

yes

complemented

C₂×#107×C₆₁

C₂²

yes

—

C₂×#106×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#382

—

C₂×C₄⋊Dic₆₁

—

C₂×#11758185

—

C₂×#11758185

—

C₂×#11758185

C₂×C₄

yes

—

#14519711

—

#14519709

—

#14519684

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

#107×C₆₁

—

#106×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

Q₁₆

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758185

—

C₂³×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#107

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

#117

yes

—

C₈×C₆₁

#125

yes

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₆₁

yes

characteristic

#1380

—

C₂×C₆₁

#1834

yes

characteristic

C₂×C₆₁

#2158

yes

—

C₂×C₆₁

#2242

yes

—

C₂×C₆₁

#2242

yes

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₆₁

yes

—

C₂×#106

—

C₂×#106

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#382

—

#382

—

122

—

C₄×C₁₆

—

C₆₁

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

C₂²×C₈

C₂²×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

122

—

101

C₂×C₁₆

—

122

—

102

#107

—

103

#107

—

122

—

104

#106

—

105

#106

—

244

—

106

C₄×C₈

Dic₁₂₂

yes

—

107

C₄×C₈

C₄×D₆₁

yes

—

108

#95

—

109

#95

—

244

—

110

C₂²×C₄

—

111

C₂²×C₄

—

244

—

112

C₂²×C₄

#11758199

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

114

C₂×C₈

#11758196

yes

—

115

C₂×C₈

#11758200

yes

—

116

C₂×C₈

#11758201

yes

—

117

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

118

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

119

C₂×C₈

#11758224

yes

—

120

C₂×C₈

#11758226

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

244

—

124

C₄⋊C₄

—

244

—

125

C₄²

#11758196

yes

—

126

C₄²

#11758202

yes

—

127

C₁₆

—

122

—

128

C₂³

C₂×#11758199

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

488

—

133

C₂×C₄

—

488

—

134

C₂×C₄

#14519710

yes

—

135

C₂×C₄

C₂×#11758196

yes

characteristic

136

C₂×C₄

#14519764

yes

characteristic

137

C₂×C₄

#14519766

yes

characteristic

138

C₂×C₄

C₂×#11758201

yes

characteristic

139

C₂×C₄

C₂×#11758200

yes

characteristic

140

C₂×C₄

C₂×#11758202

yes

characteristic

141

C₈

#14519708

yes

—

142

C₈

#14519735

yes

—

143

C₈

#14519768

yes

—

144

C₂²

C₂×#14519710

yes

characteristic

145

C₂²

#16840108

yes

characteristic

146

C₂²

#16840110

yes

characteristic

147

C₂²

#16840192

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

976

—

151

C₄

#16839627

yes

characteristic

152

C₄

C₂×#14519708

yes

characteristic

153

C₄

#16840112

yes

characteristic

154

C₄

#16840114

yes

characteristic

155

C₂

#21445946

yes

characteristic

156

C₂

#21447601

yes

—

157

C₂

#21448050

yes

—

158

C₂

#21448053

yes

—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1846	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1846	D₆₁
hypocenter	C₆₁	#12889
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#11758199

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1846

C₂

socle, prime radical

C₂³×C₆₁

C₂×D₈

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×#11758185	C₄	semidirect
93. C₆₁	#12889	semidirect

H ≅ G/N

type

9. C₄×#11758185

C₄

semidirect

93. C₆₁

#12889

semidirect

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

subgroup	index
C₂×C₄×#11758185	2
#21445983	2
#21443910	2
C₆₁×#1846	2
#12889	61

subgroup

index

C₆₁×#1846

Subgroups — #29596395

Conjugacy classes of subgroups

Complements

Maximal subgroups