Group #29585048 — subgroups

Subgroups — #29585048

Order 15616.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21445973	C₂	1	1	yes	maximal, characteristic
2	#21444591	C₂	1	2	yes	maximal
3	#21444222	C₂	1	2	yes	maximal
4	#21443648	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1995	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×C₄⋊#92418	—	2	2	no	—
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	#16839512	—	2	4	no	—
10	Q₈⋊#92418	—	2	4	no	—
11	C₆₁×#533	C₂²	1	1	yes	characteristic
12	C₆₁×#423	C₂²	1	1	yes	characteristic
13	C₆₁×#376	C₂²	1	1	yes	characteristic
14	C₆₁×#339	—	2	2	no	—
15	C₂²×#92418	—	2	2	no	—
16	C₄⋊#92418	—	2	2	no	—
17	C₄⋊#92418	—	2	2	no	—
18	C₄⋊#92418	—	2	2	no	—
19	C₄⋊#92418	—	4	4	no	—
20	C₂²⋊#92418	—	2	2	no	—
21	C₄×#92418	D₄	1	4	yes	—
22	C₂×M₁₆×C₆₁	—	2	1	no	—
23	C₄⋊Q₈×C₆₁	Q₈	1	2	yes	—
24	#125×C₆₁	—	2	1	no	—
25	C₂²⋊Q₈×C₆₁	—	2	2	no	—
26	C₄×Q₈×C₆₁	—	2	2	no	—
27	#117×C₆₁	—	2	1	no	—
28	C₂×C₄²×C₆₁	D₄	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	—	2	2	no	—
31	#95×C₆₁	—	2	2	no	—
32	C₂×Q₈×C₆₁	—	2	2	no	—
33	C₂²×C₄×C₆₁	—	2	2	no	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	#41	1	2	yes	—
36	M₁₆×C₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	—	4	1	no	—
38	C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₄⋊C₄×C₆₁	—	2	2	no	—
40	C₄⋊C₄×C₆₁	—	4	2	no	—
41	C₂²⋊C₄×C₆₁	—	4	1	no	—
42	C₄²×C₆₁	—	2	2	no	—
43	C₄²×C₆₁	—	4	1	no	—
44	C₄²×C₆₁	SD₁₆	1	4	yes	—
45	C₄²×C₆₁	C₂×D₄	1	2	yes	—
46	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₆₁	#41	1	1	yes	characteristic
48	C₂×#92418	—	2	4	no	—
49	C₂×#92418	—	4	2	no	—
50	C₂×#92418	—	4	4	no	—
51	C₂³×C₆₁	#123	1	1	yes	characteristic
52	Q₈×C₆₁	—	4	4	no	—
53	C₂×C₄×C₆₁	—	2	4	no	—
54	C₂×C₄×C₆₁	—	4	2	no	—
55	C₂×C₄×C₆₁	—	4	4	no	—
56	C₂×C₄×C₆₁	—	4	2	no	—
57	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#123	1	2	yes	—
60	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₆₁	#137	1	2	yes	—
63	C₈×C₆₁	—	8	1	no	—
64	#92418	—	4	4	no	—
65	#92418	—	8	2	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	—	8	2	no	—
74	C₄×C₆₁	#475	1	2	yes	—
75	C₄×C₆₁	#492	1	1	yes	characteristic
76	C₄×C₆₁	#497	1	1	yes	characteristic
77	#1995	D₆₁	1	1	yes	characteristic
78	#1843	—	61	1	no	—
79	#1570	—	61	2	no	—
80	#1481	—	61	2	no	—
81	#1274	—	61	1	no	—
82	C₂×C₆₁	—	2	2	no	—
83	C₂×C₆₁	#2096	1	1	yes	characteristic
84	C₂×C₆₁	#2098	1	2	yes	—
85	#533	C₂×D₆₁	1	1	yes	characteristic
86	#431	—	61	2	no	—
87	#423	C₂×D₆₁	1	1	yes	characteristic
88	C₂×C₄⋊C₈	—	61	2	no	—
89	C₂×C₄⋊C₈	—	122	2	no	—
90	#376	C₂×D₆₁	1	1	yes	characteristic
91	#339	—	2	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	—	2	1	no	—
96	C₂²×C₈	—	122	2	no	—
97	C₄⋊Q₈	Dic₁₂₂	1	2	yes	—
98	#125	—	2	1	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	#117	—	2	1	no	—
102	C₂×C₄²	C₂²⋊D₆₁	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	—	122	2	no	—
110	C₄×C₈	—	61	4	no	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	#11758224	1	2	yes	—
115	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	122	4	no	—
119	C₂×C₈	—	244	2	no	—
120	C₂×C₈	—	244	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	1	no	—
127	C₄²	#11758201	1	4	yes	—
128	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
129	C₄²	#11758222	1	1	yes	characteristic
130	C₄²	D₄×D₆₁	1	2	yes	—
131	C₂³	#14519758	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#14519754	1	2	yes	—
138	C₂×C₄	#14519756	1	1	yes	characteristic
139	C₂×C₄	#14519759	1	2	yes	—
140	C₂×C₄	#14519773	1	1	yes	characteristic
141	C₂×C₄	C₂×#11758201	1	2	yes	—
142	C₂×C₄	#14519807	1	2	yes	—
143	C₈	—	8	1	no	—
144	C₈	—	488	2	no	—
145	C₈	—	244	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#16839698	1	1	yes	characteristic
148	C₂²	#16840090	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	2	no	—
153	C₄	#16840047	1	2	yes	—
154	C₄	#16840136	1	1	yes	characteristic
155	C₄	#16840157	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#21447362	1	1	yes	characteristic
158	C₂	#21447374	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445973

C₂

yes

maximal, characteristic

#21444591

C₂

yes

maximal

#21444222

C₂

yes

maximal

#21443648

C₂

yes

maximal, characteristic

C₆₁×#1995

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₂×C₄⋊#92418

—

C₂×C₄⋊#92418

C₂²

yes

—

#16839512

—

Q₈⋊#92418

—

C₆₁×#533

C₂²

yes

characteristic

C₆₁×#423

C₂²

yes

characteristic

C₆₁×#376

C₂²

yes

characteristic

C₆₁×#339

—

C₂²×#92418

—

—

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

Q₈

yes

—

#125×C₆₁

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

#95×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×Q₈

yes

C₄²×C₆₁

yes

—

—

—

C₂³×C₆₁

#123

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

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

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#399

yes

characteristic

C₂²×C₆₁

#483

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#475

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#2096

yes

characteristic

C₂×C₆₁

#2098

yes

—

#533

C₂×D₆₁

yes

characteristic

#431

—

#423

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

122

—

#376

C₂×D₆₁

yes

characteristic

#339

—

#334

—

122

—

Q₈⋊C₈

—

122

—

C₆₁

#10670

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

122

—

C₄⋊Q₈

Dic₁₂₂

yes

—

#125

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

—

102

C₂×C₄²

C₂²⋊D₆₁

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

—

122

—

110

C₄×C₈

—

111

#95

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

#11758224

yes

—

115

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

244

—

120

C₂×C₈

—

244

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11758201

yes

—

128

C₄²

C₂×Dic₁₂₂

yes

characteristic

129

C₄²

#11758222

yes

characteristic

130

C₄²

D₄×D₆₁

yes

—

131

C₂³

#14519758

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#14519754

yes

—

138

C₂×C₄

#14519756

yes

characteristic

139

C₂×C₄

#14519759

yes

—

140

C₂×C₄

#14519773

yes

characteristic

141

C₂×C₄

C₂×#11758201

yes

—

142

C₂×C₄

#14519807

yes

—

143

C₈

—

144

C₈

—

488

—

145

C₈

—

244

—

146

C₂²

—

147

C₂²

#16839698

yes

characteristic

148

C₂²

#16840090

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#16840047

yes

—

154

C₄

#16840136

yes

characteristic

155

C₄

#16840157

yes

characteristic

156

C₂

—

157

C₂

#21447362

yes

characteristic

158

C₂

#21447374

yes

—

	subgroup	G/H
center	C₂×C₄	#14519773
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1995	C₂
socle	C₂²×C₆₁	#399
prime radical	C₂³×C₆₁	#123
hypercenter	#1995	D₆₁
hypocenter	C₆₁	#10670
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#14519773

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1995

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

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
#21445973	2
#21444591	2
#21444222	2
#21443648	2
C₆₁×#1995	2
#10670	61

subgroup

index

C₆₁×#1995

Subgroups — #29585048

Conjugacy classes of subgroups

Complements

Maximal subgroups