Group #29585046 — subgroups

Subgroups — #29585046

Order 15616.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#21446800	C₂	1	1	yes	maximal, characteristic
2	#21445973	C₂	1	1	yes	maximal, characteristic
3	#21444590	C₂	1	2	yes	maximal
4	#21444223	C₂	1	2	yes	maximal
5	C₆₁×#1274	C₂	1	1	yes	maximal, characteristic
6	#16840319	C₂²	1	1	yes	characteristic
7	#16839786	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#92418	—	2	2	no	—
9	C₂×C₄⋊#92418	C₂²	1	2	yes	—
10	#16839522	—	2	2	no	—
11	#16839513	—	2	4	no	—
12	#16839494	—	2	4	no	—
13	C₆₁×#431	C₂²	1	2	yes	—
14	C₆₁×#376	C₂²	1	1	yes	characteristic
15	C₂×#11758194	—	2	1	no	—
16	C₂²×#92418	—	2	2	no	—
17	#14519777	Q₈	1	2	yes	—
18	#14519769	—	2	1	no	—
19	#14519756	—	2	2	no	—
20	C₄×Dic₁₂₂	—	2	2	no	—
21	#14519736	—	2	1	no	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	—	4	4	no	—
26	C₂×C₄²×C₆₁	D₄	1	2	yes	—
27	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	—	2	2	no	—
29	C₂²⋊C₈×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	D₄	1	4	yes	—
31	#95×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	—	2	2	no	—
33	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	#41	1	2	yes	—
35	C₂×Dic₁₂₂	—	2	2	no	—
36	C₂×C₈×C₆₁	—	2	4	no	—
37	C₄²×C₆₁	—	2	2	no	—
38	C₄²×C₆₁	SD₁₆	1	4	yes	—
39	C₄²×C₆₁	C₂×D₄	1	2	yes	—
40	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₆₁	#41	1	1	yes	characteristic
42	C₂²⋊Dic₆₁	—	4	1	no	—
43	C₄⋊Dic₆₁	—	2	2	no	—
44	C₄⋊Dic₆₁	—	2	2	no	—
45	#11758196	—	4	2	no	—
46	C₄×Dic₆₁	—	4	1	no	—
47	#11758194	—	4	2	no	—
48	C₂×#92418	—	4	2	no	—
49	C₂×#92418	—	4	1	no	—
50	C₂×#92418	—	4	4	no	—
51	C₂³×C₆₁	#123	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	2	4	no	—
53	C₂×C₄×C₆₁	—	4	2	no	—
54	C₂×C₄×C₆₁	—	4	4	no	—
55	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
56	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#123	1	2	yes	—
58	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₆₁	#137	1	2	yes	—
61	C₂×Dic₆₁	—	4	2	no	—
62	Dic₁₂₂	—	4	4	no	—
63	C₈×C₆₁	—	4	4	no	—
64	#92418	—	8	2	no	—
65	#92418	—	8	1	no	—
66	#10670	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	—	2	2	no	—
68	C₂²×C₆₁	#399	1	1	yes	characteristic
69	C₂²×C₆₁	#483	1	2	yes	—
70	C₄×C₆₁	—	2	2	no	—
71	C₄×C₆₁	—	2	2	no	—
72	C₄×C₆₁	—	4	4	no	—
73	C₄×C₆₁	#475	1	2	yes	—
74	C₄×C₆₁	#492	1	1	yes	characteristic
75	C₄×C₆₁	#497	1	1	yes	characteristic
76	Dic₆₁	—	8	2	no	—
77	#1995	—	61	1	no	—
78	#1843	—	61	1	no	—
79	#1570	—	61	2	no	—
80	#1481	—	61	2	no	—
81	#1274	D₆₁	1	1	yes	characteristic
82	C₂×C₆₁	—	2	2	no	—
83	C₂×C₆₁	#2096	1	1	yes	characteristic
84	C₂×C₆₁	#2098	1	2	yes	—
85	#533	—	61	1	no	—
86	#431	C₂×D₆₁	1	2	yes	—
87	#423	—	61	1	no	—
88	C₂×C₄⋊C₈	—	61	2	no	—
89	C₂×C₄⋊C₈	—	122	2	no	—
90	#376	C₂×D₆₁	1	1	yes	characteristic
91	#339	—	122	2	no	—
92	#334	—	122	4	no	—
93	Q₈⋊C₈	—	122	4	no	—
94	C₆₁	#10670	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	122	1	no	—
96	C₂²×C₈	—	122	2	no	—
97	C₄⋊Q₈	—	61	2	no	—
98	#125	—	122	1	no	—
99	C₂²⋊Q₈	—	122	2	no	—
100	C₄×Q₈	—	122	2	no	—
101	#117	—	122	1	no	—
102	C₂×C₄²	D₂₄₄	1	2	yes	—
103	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	122	2	no	—
106	C₄⋊C₈	—	122	2	no	—
107	C₄⋊C₈	—	122	2	no	—
108	C₄⋊C₈	—	244	4	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₄×C₈	D₂₄₄	1	4	yes	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	122	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
115	C₂²×C₄	#11758224	1	2	yes	—
116	M₁₆	—	244	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	244	2	no	—
119	C₂×C₈	—	244	1	no	—
120	C₂×C₈	—	244	4	no	—
121	C₄⋊C₄	—	122	2	no	—
122	C₄⋊C₄	—	122	2	no	—
123	C₄⋊C₄	—	244	2	no	—
124	C₂²⋊C₄	—	244	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	244	1	no	—
127	C₄²	#11758190	1	4	yes	—
128	C₄²	C₂×D₂₄₄	1	2	yes	—
129	C₄²	#11758224	1	1	yes	characteristic
130	C₄²	Q₈×D₆₁	1	1	yes	characteristic
131	C₂³	#14519761	1	1	yes	characteristic
132	Q₈	—	244	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	244	2	no	—
137	C₂×C₄	#14519757	1	2	yes	—
138	C₂×C₄	#14519757	1	1	yes	characteristic
139	C₂×C₄	#14519761	1	2	yes	—
140	C₂×C₄	#14519774	1	1	yes	characteristic
141	C₂×C₄	C₂×#11758190	1	2	yes	—
142	C₂×C₄	#14519810	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	488	2	no	—
145	C₈	—	488	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#16839699	1	1	yes	characteristic
148	C₂²	#16840093	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	488	2	no	—
153	C₄	#16840048	1	2	yes	—
154	C₄	#16840135	1	1	yes	characteristic
155	C₄	#16840156	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#21447361	1	1	yes	characteristic
158	C₂	#21447373	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446800

C₂

yes

maximal, characteristic

#21445973

C₂

yes

maximal, characteristic

#21444590

C₂

yes

maximal

#21444223

C₂

yes

maximal

C₆₁×#1274

C₂

yes

maximal, characteristic

#16840319

C₂²

yes

characteristic

#16839786

C₂²

yes

characteristic

C₂×C₄⋊#92418

—

C₂×C₄⋊#92418

C₂²

yes

—

#16839522

—

#16839513

—

#16839494

—

C₆₁×#431

C₂²

yes

—

C₆₁×#376

C₂²

yes

characteristic

C₂×#11758194

—

C₂²×#92418

—

#14519777

Q₈

yes

—

#14519769

—

#14519756

—

C₄×Dic₁₂₂

—

—

—

—

—

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

—

—

—

—

C₂³×C₆₁

#123

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#399

yes

characteristic

C₂²×C₆₁

#483

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#475

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#2096

yes

characteristic

C₂×C₆₁

#2098

yes

—

#533

—

#431

C₂×D₆₁

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

122

—

#376

C₂×D₆₁

yes

characteristic

#339

—

122

—

#334

—

122

—

Q₈⋊C₈

—

122

—

C₆₁

#10670

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

—

122

—

C₄⋊Q₈

—

#125

—

122

—

C₂²⋊Q₈

—

122

—

100

C₄×Q₈

—

122

—

101

#117

—

122

—

102

C₂×C₄²

D₂₄₄

yes

—

103

C₂×C₄²

C₂²×D₆₁

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

122

—

106

C₄⋊C₈

—

122

—

107

C₄⋊C₈

—

122

—

108

C₄⋊C₈

—

244

—

109

C₂²⋊C₈

—

110

C₄×C₈

D₂₄₄

yes

—

111

#95

—

112

C₂×Q₈

—

122

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

115

C₂²×C₄

#11758224

yes

—

116

M₁₆

—

244

—

117

C₂×C₈

—

118

C₂×C₈

—

244

—

119

C₂×C₈

—

244

—

120

C₂×C₈

—

244

—

121

C₄⋊C₄

—

122

—

122

C₄⋊C₄

—

122

—

123

C₄⋊C₄

—

244

—

124

C₂²⋊C₄

—

244

—

125

C₄²

—

126

C₄²

—

244

—

127

C₄²

#11758190

yes

—

128

C₄²

C₂×D₂₄₄

yes

—

129

C₄²

#11758224

yes

characteristic

130

C₄²

Q₈×D₆₁

yes

characteristic

131

C₂³

#14519761

yes

characteristic

132

Q₈

—

244

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

244

—

137

C₂×C₄

#14519757

yes

—

138

C₂×C₄

#14519757

yes

characteristic

139

C₂×C₄

#14519761

yes

—

140

C₂×C₄

#14519774

yes

characteristic

141

C₂×C₄

C₂×#11758190

yes

—

142

C₂×C₄

#14519810

yes

—

143

C₈

—

144

C₈

—

488

—

145

C₈

—

488

—

146

C₂²

—

147

C₂²

#16839699

yes

characteristic

148

C₂²

#16840093

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

488

—

153

C₄

#16840048

yes

—

154

C₄

#16840135

yes

characteristic

155

C₄

#16840156

yes

characteristic

156

C₂

—

157

C₂

#21447361

yes

characteristic

158

C₂

#21447373

yes

—

	subgroup	G/H
center	C₂×C₄	#14519774
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1274	C₂
socle	C₂²×C₆₁	#399
prime radical	C₂³×C₆₁	#123
hypercenter	#1274	D₆₁
hypocenter	C₆₁	#10670
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#14519774

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1274

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
94. C₆₁	#10670	semidirect

H ≅ G/N

type

94. C₆₁

#10670

semidirect

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

subgroup	index
#21446800	2
#21445973	2
#21444590	2
#21444223	2
C₆₁×#1274	2
#10670	61

subgroup

index

C₆₁×#1274

Subgroups — #29585046

Conjugacy classes of subgroups

Complements

Maximal subgroups