Group #29600149 — subgroups

Subgroups — #29600149

Order 15616.

234 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 89 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	#21446572	C₂	1	1	yes	maximal, characteristic
2	#21445994	C₂	1	1	yes	maximal, characteristic
3	#21443934	C₂	1	2	yes	maximal
4	#21443931	C₂	1	2	yes	maximal
5	C₆₁×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519713	—	2	1	no	—
7	C₂×#14519713	C₂²	1	1	yes	characteristic
8	C₂×#14519713	C₂²	1	2	yes	—
9	#16839800	C₂²	1	1	yes	characteristic
10	#16839788	—	2	1	no	—
11	#16839786	—	2	1	no	—
12	#16839522	—	2	1	no	—
13	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
14	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#11758194	—	2	1	no	—
16	C₂×#11758194	—	2	2	no	—
17	#14519736	—	2	1	no	—
18	#14519713	—	2	4	no	—
19	#14519713	—	2	2	no	—
20	#14519713	—	4	2	no	—
21	#14519711	—	2	1	no	—
22	#14519709	Q₈	1	2	yes	—
23	C₄⋊#92418	—	4	2	no	—
24	#14519706	—	4	2	no	—
25	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
28	M₃₂×C₆₁	—	2	2	no	—
29	M₃₂×C₆₁	—	2	2	no	—
30	C₂×C₁₆×C₆₁	—	2	2	no	—
31	#97×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	—	2	1	no	—
33	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	—	2	1	no	—
35	C₂×C₈×C₆₁	—	2	1	no	—
36	C₂×C₈×C₆₁	D₈	1	2	yes	—
37	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₆₁	#41	1	2	yes	—
39	C₂×C₈×C₆₁	#41	1	2	yes	—
40	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
41	C₄²×C₆₁	—	2	1	no	—
42	C₄²×C₆₁	SD₁₆	1	2	yes	—
43	C₂²⋊Dic₆₁	—	4	1	no	—
44	C₄⋊Dic₆₁	—	2	2	no	—
45	C₄×Dic₆₁	—	4	1	no	—
46	#11758194	—	4	1	no	—
47	#11758194	—	4	2	no	—
48	#11758194	—	4	2	no	—
49	#11758194	—	4	1	no	—
50	C₂×#92418	—	4	1	no	—
51	C₂×#92418	—	4	2	no	—
52	C₁₆×C₆₁	—	4	2	no	—
53	C₂³×C₆₁	#137	1	1	yes	characteristic
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	8	1	no	—
56	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
62	C₂×Dic₆₁	—	4	2	no	—
63	C₈×C₆₁	—	2	2	no	—
64	C₈×C₆₁	#125	1	2	yes	—
65	C₈×C₆₁	#126	1	2	yes	—
66	#92418	—	8	1	no	—
67	#92418	—	8	2	no	—
68	#13540	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	—	4	1	no	—
70	C₂²×C₆₁	#474	1	1	yes	characteristic
71	C₂²×C₆₁	#483	1	1	yes	characteristic
72	C₂²×C₆₁	#491	1	1	yes	characteristic
73	C₄×C₆₁	—	4	2	no	—
74	C₄×C₆₁	#400	1	1	yes	characteristic
75	C₄×C₆₁	#475	1	1	yes	characteristic
76	C₄×C₆₁	#480	1	1	yes	characteristic
77	C₄×C₆₁	#486	1	1	yes	characteristic
78	Dic₆₁	—	8	2	no	—
79	#1946	—	61	1	no	—
80	#1849	—	61	1	no	—
81	#1389	—	61	2	no	—
82	#1388	—	61	2	no	—
83	#1375	D₆₁	1	1	yes	characteristic
84	C₂×C₆₁	—	2	1	no	—
85	C₂×C₆₁	—	2	1	no	—
86	C₂×C₆₁	—	2	1	no	—
87	C₂×C₆₁	#2099	1	1	yes	characteristic
88	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
89	C₂×#108	—	61	2	no	—
90	C₂×#108	—	61	1	no	—
91	C₂×#108	—	122	1	no	—
92	#427	—	61	1	no	—
93	#424	—	122	1	no	—
94	#423	—	122	1	no	—
95	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
96	#339	—	122	1	no	—
97	C₆₁	#13540	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	122	1	no	—
99	C₂×M₁₆	—	122	2	no	—
100	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
102	#117	—	122	1	no	—
103	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	#108	—	122	4	no	—
108	#108	—	122	2	no	—
109	#108	—	244	2	no	—
110	#107	—	122	1	no	—
111	#106	—	61	2	no	—
112	C₄⋊C₈	—	244	2	no	—
113	C₄⋊C₈	—	244	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
117	M₁₆	—	244	1	no	—
118	M₁₆	—	244	1	no	—
119	M₁₆	—	244	2	no	—
120	M₁₆	—	244	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	244	1	no	—
124	C₂×C₈	—	244	2	no	—
125	C₂×C₈	D₄₈₈	1	2	yes	—
126	C₂×C₈	#11758222	1	2	yes	—
127	C₂×C₈	#11758224	1	2	yes	—
128	C₂×C₈	#11758224	1	1	yes	characteristic
129	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
130	C₄⋊C₄	—	122	2	no	—
131	C₂²⋊C₄	—	244	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	244	1	no	—
134	C₄²	#11758190	1	2	yes	—
135	C₁₆	—	4	2	no	—
136	C₂³	#14519810	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	244	2	no	—
140	C₂×C₄	#14519757	1	1	yes	characteristic
141	C₂×C₄	#14519761	1	1	yes	characteristic
142	C₂×C₄	#14519765	1	1	yes	characteristic
143	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
144	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
145	C₂×C₄	#14519810	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	488	1	no	—
148	C₈	—	488	2	no	—
149	C₈	#14519767	1	2	yes	—
150	C₈	#14519770	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#16840043	1	1	yes	characteristic
153	C₂²	#16840093	1	1	yes	characteristic
154	C₂²	#16840131	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	488	2	no	—
157	C₄	#16839702	1	1	yes	characteristic
158	C₄	#16840048	1	1	yes	characteristic
159	C₄	#16840076	1	1	yes	characteristic
160	C₄	#16840109	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#21447378	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21446572

C₂

yes

maximal, characteristic

#21445994

C₂

yes

maximal, characteristic

#21443934

C₂

yes

maximal

#21443931

C₂

yes

maximal

C₆₁×#1375

C₂

yes

maximal, characteristic

C₂×#14519713

—

C₂×#14519713

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

#16839800

C₂²

yes

—

—

—

C₂×M₃₂×C₆₁

C₂²

yes

—

C₂×#97×C₆₁

C₂²

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₄⋊#92418

—

#14519706

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

M₃₂×C₆₁

—

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

#97×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

—

—

—

—

—

—

C₁₆×C₆₁

—

C₂³×C₆₁

#137

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

#137

yes

characteristic

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#125

yes

—

C₈×C₆₁

#126

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2099

yes

characteristic

C₂×M₃₂

C₂×D₆₁

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

—

122

—

#427

—

#424

—

122

—

#423

—

122

—

C₂×#97

C₂×D₆₁

yes

characteristic

#339

—

122

—

C₆₁

#13540

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂×M₁₆

—

122

—

100

C₂²×C₈

D₂₄₄

yes

characteristic

101

C₂²×C₈

C₂²×D₆₁

yes

characteristic

102

#117

—

122

—

103

C₂×C₄²

D₂₄₄

yes

characteristic

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

122

—

108

#108

—

122

—

109

#108

—

244

—

110

#107

—

122

—

111

#106

—

112

C₄⋊C₈

—

244

—

113

C₄⋊C₈

—

244

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

117

M₁₆

—

244

—

118

M₁₆

—

244

—

119

M₁₆

—

244

—

120

M₁₆

—

244

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

244

—

124

C₂×C₈

—

244

—

125

C₂×C₈

D₄₈₈

yes

—

126

C₂×C₈

#11758222

yes

—

127

C₂×C₈

#11758224

yes

—

128

C₂×C₈

#11758224

yes

characteristic

129

C₂×C₈

Q₈×D₆₁

yes

characteristic

130

C₄⋊C₄

—

122

—

131

C₂²⋊C₄

—

244

—

132

C₄²

—

133

C₄²

—

244

—

134

C₄²

#11758190

yes

—

135

C₁₆

—

136

C₂³

#14519810

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

244

—

140

C₂×C₄

#14519757

yes

characteristic

141

C₂×C₄

#14519761

yes

characteristic

142

C₂×C₄

#14519765

yes

characteristic

143

C₂×C₄

C₂×D₄₈₈

yes

characteristic

144

C₂×C₄

C₂×#11758190

yes

characteristic

145

C₂×C₄

#14519810

yes

characteristic

146

C₈

—

147

C₈

—

488

—

148

C₈

—

488

—

149

C₈

#14519767

yes

—

150

C₈

#14519770

yes

—

151

C₂²

—

152

C₂²

#16840043

yes

characteristic

153

C₂²

#16840093

yes

characteristic

154

C₂²

#16840131

yes

characteristic

155

C₄

—

156

C₄

—

488

—

157

C₄

#16839702

yes

characteristic

158

C₄

#16840048

yes

characteristic

159

C₄

#16840076

yes

characteristic

160

C₄

#16840109

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#21447378

yes

characteristic

	subgroup	G/H
center	C₄	#16840109
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1375	C₂
socle	C₂×C₆₁	#2099
prime radical	C₂³×C₆₁	#137
hypercenter	#1375	D₆₁
hypocenter	C₆₁	#13540
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

subgroup

G/H

center

C₄

#16840109

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1375

C₂

C₂×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

N	H ≅ G/N	type
97. C₆₁	#13540	semidirect

H ≅ G/N

type

97. C₆₁

#13540

semidirect

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

subgroup	index
#21446572	2
#21445994	2
#21443934	2
#21443931	2
C₆₁×#1375	2
#13540	61

subgroup

index

C₆₁×#1375

Subgroups — #29600149

Conjugacy classes of subgroups

Complements

Maximal subgroups