Group #29594638 — subgroups

Subgroups — #29594638

Order 15616.

212 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 69 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	#21447962	C₂	1	2	yes	maximal
2	#21445437	C₂	1	1	yes	maximal, characteristic
3	#21444717	C₂	1	1	yes	maximal, characteristic
4	#21443873	C₂	1	2	yes	maximal
5	C₆₁×#1367	C₂	1	1	yes	maximal, characteristic
6	#16840167	—	2	1	no	—
7	#16840167	C₂²	1	2	yes	—
8	#16840142	—	2	1	no	—
9	#16839892	—	2	1	no	—
10	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
11	#16839590	—	2	2	no	—
12	#16839567	—	2	2	no	—
13	#16839515	C₂²	1	1	yes	characteristic
14	#16839494	—	2	1	no	—
15	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₄₄	—	4	1	no	—
18	C₂×Dic₂₄₄	—	4	1	no	—
19	C₂×Dic₂₄₄	D₄	1	1	yes	characteristic
20	#14519777	—	2	2	no	—
21	C₄×Dic₁₂₂	—	2	1	no	—
22	#14519711	—	2	2	no	—
23	#14519711	—	2	2	no	—
24	#14519711	—	4	2	no	—
25	#14519711	D₄	1	1	yes	characteristic
26	C₄⋊#92418	—	2	1	no	—
27	#14519706	—	4	1	no	—
28	#14519702	—	2	1	no	—
29	#14519687	—	4	1	no	—
30	C₂×C₁₆×C₆₁	—	2	2	no	—
31	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×Dic₁₂₂	—	2	1	no	—
35	C₂×Dic₁₂₂	—	4	2	no	—
36	C₂×C₈×C₆₁	—	2	1	no	—
37	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
38	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
41	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₆₁	—	2	1	no	—
43	C₄⋊Dic₆₁	—	4	2	no	—
44	C₄⋊Dic₆₁	—	4	2	no	—
45	#11758196	—	4	1	no	—
46	C₄×Dic₆₁	—	4	1	no	—
47	C₂×#92418	—	4	1	no	—
48	Dic₂₄₄	—	2	2	no	—
49	Dic₂₄₄	—	8	2	no	—
50	Dic₂₄₄	—	8	2	no	—
51	C₁₆×C₆₁	—	4	2	no	—
52	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×Dic₆₁	—	4	1	no	—
56	C₂×Dic₆₁	—	4	1	no	—
57	C₂×Dic₆₁	—	8	2	no	—
58	Dic₁₂₂	—	4	1	no	—
59	Dic₁₂₂	—	4	1	no	—
60	Dic₁₂₂	—	8	2	no	—
61	Dic₁₂₂	—	8	2	no	—
62	C₈×C₆₁	—	2	2	no	—
63	C₈×C₆₁	—	2	1	no	—
64	C₈×C₆₁	—	2	1	no	—
65	C₈×C₆₁	SD₃₂	1	2	yes	—
66	C₈×C₆₁	#137	1	2	yes	—
67	#92418	—	8	1	no	—
68	#12498	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	#459	1	1	yes	characteristic
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	#470	1	1	yes	characteristic
72	C₄×C₆₁	#473	1	1	yes	characteristic
73	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
74	C₄×C₆₁	#509	1	1	yes	characteristic
75	Dic₆₁	—	8	1	no	—
76	Dic₆₁	—	8	1	no	—
77	Dic₆₁	—	16	2	no	—
78	#2225	—	61	2	no	—
79	#1713	—	61	1	no	—
80	#1591	—	61	1	no	—
81	#1367	D₆₁	1	1	yes	characteristic
82	#1364	—	61	2	no	—
83	C₂×C₆₁	#1689	1	1	yes	characteristic
84	C₂×C₆₁	#2207	1	1	yes	characteristic
85	C₂×C₆₁	#2214	1	1	yes	characteristic
86	#500	—	61	2	no	—
87	#500	—	122	1	no	—
88	#494	—	122	1	no	—
89	#446	—	122	1	no	—
90	C₄×Q₁₆	—	61	1	no	—
91	#367	—	122	2	no	—
92	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
93	#358	—	122	2	no	—
94	#335	—	61	1	no	—
95	Q₈⋊C₈	—	122	1	no	—
96	C₈²	C₂×D₆₁	1	1	yes	characteristic
97	C₆₁	#12498	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	61	1	no	—
99	C₂×Q₁₆	—	244	1	no	—
100	C₂×Q₁₆	—	244	1	no	—
101	C₄⋊Q₈	—	122	2	no	—
102	C₄×Q₈	—	122	1	no	—
103	C₂×C₁₆	—	2	2	no	—
104	#107	—	61	1	no	—
105	#107	—	122	2	no	—
106	#107	—	122	2	no	—
107	#107	—	244	2	no	—
108	C₄⋊C₈	—	122	1	no	—
109	C₄⋊C₈	—	244	1	no	—
110	#103	—	122	1	no	—
111	#97	—	244	1	no	—
112	C₄×C₈	D₂₄₄	1	1	yes	characteristic
113	C₄×C₈	D₂₄₄	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
115	C₂×Q₈	—	122	1	no	—
116	C₂×Q₈	—	244	2	no	—
117	Q₁₆	—	122	2	no	—
118	Q₁₆	—	488	2	no	—
119	Q₁₆	—	488	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	244	1	no	—
122	C₂×C₈	D₄₈₈	1	1	yes	characteristic
123	C₂×C₈	D₄₈₈	1	1	yes	characteristic
124	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
125	C₂×C₈	#11758226	1	1	yes	characteristic
126	C₄⋊C₄	—	122	1	no	—
127	C₄⋊C₄	—	244	2	no	—
128	C₄⋊C₄	—	244	2	no	—
129	C₄⋊C₄	—	244	1	no	—
130	C₄²	—	244	1	no	—
131	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	Q₈	—	488	2	no	—
134	Q₈	—	488	2	no	—
135	Q₈	—	244	1	no	—
136	Q₈	—	244	1	no	—
137	C₂×C₄	—	488	2	no	—
138	C₂×C₄	—	244	1	no	—
139	C₂×C₄	—	244	1	no	—
140	C₂×C₄	#14519752	1	1	yes	characteristic
141	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
142	C₂×C₄	#14519805	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	488	1	no	—
147	C₈	#14519722	1	2	yes	—
148	C₈	#14519806	1	2	yes	—
149	C₂²	#16839958	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	976	2	no	—
152	C₄	—	488	1	no	—
153	C₄	—	488	1	no	—
154	C₄	#16840021	1	1	yes	characteristic
155	C₄	#16840038	1	1	yes	characteristic
156	C₄	C₂×#14519722	1	1	yes	characteristic
157	C₄	#16840200	1	1	yes	characteristic
158	C₂	#21445309	1	1	yes	characteristic
159	C₂	#21447860	1	1	yes	characteristic
160	C₂	#21447901	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447962

C₂

yes

maximal

#21445437

C₂

yes

maximal, characteristic

#21444717

C₂

yes

maximal, characteristic

#21443873

C₂

yes

maximal

C₆₁×#1367

C₂

yes

maximal, characteristic

#16840167

—

#16840167

C₂²

yes

—

#16840142

—

#16839892

—

C₄×Dic₂₄₄

C₂²

yes

—

—

C₂²

yes

characteristic

#16839494

—

C₆₁×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

D₄

yes

characteristic

#14519777

—

C₄×Dic₁₂₂

—

—

—

—

D₄

yes

—

—

—

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

characteristic

C₂×C₈×C₆₁

D₈

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

SD₃₂

yes

—

C₈×C₆₁

#137

yes

—

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×SD₃₂

yes

characteristic

C₄×C₆₁

#509

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic

#1364

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

122

—

#494

—

122

—

#446

—

122

—

C₄×Q₁₆

—

#367

—

122

—

C₄⋊C₁₆

C₂×D₆₁

yes

—

#358

—

122

—

#335

—

Q₈⋊C₈

—

122

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12498

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

244

—

100

C₂×Q₁₆

—

244

—

101

C₄⋊Q₈

—

122

—

102

C₄×Q₈

—

122

—

103

C₂×C₁₆

—

104

#107

—

105

#107

—

122

—

106

#107

—

122

—

107

#107

—

244

—

108

C₄⋊C₈

—

122

—

109

C₄⋊C₈

—

244

—

110

#103

—

122

—

111

#97

—

244

—

112

C₄×C₈

D₂₄₄

yes

characteristic

113

C₄×C₈

D₂₄₄

yes

characteristic

114

C₄×C₈

C₂²×D₆₁

yes

characteristic

115

C₂×Q₈

—

122

—

116

C₂×Q₈

—

244

—

117

Q₁₆

—

122

—

118

Q₁₆

—

488

—

119

Q₁₆

—

488

—

120

C₂×C₈

—

121

C₂×C₈

—

244

—

122

C₂×C₈

D₄₈₈

yes

characteristic

123

C₂×C₈

D₄₈₈

yes

characteristic

124

C₂×C₈

D₄×D₆₁

yes

characteristic

125

C₂×C₈

#11758226

yes

characteristic

126

C₄⋊C₄

—

122

—

127

C₄⋊C₄

—

244

—

128

C₄⋊C₄

—

244

—

129

C₄⋊C₄

—

244

—

130

C₄²

—

244

—

131

C₄²

C₂×D₂₄₄

yes

characteristic

132

C₁₆

—

133

Q₈

—

488

—

134

Q₈

—

488

—

135

Q₈

—

244

—

136

Q₈

—

244

—

137

C₂×C₄

—

488

—

138

C₂×C₄

—

244

—

139

C₂×C₄

—

244

—

140

C₂×C₄

#14519752

yes

characteristic

141

C₂×C₄

C₂×D₄₈₈

yes

characteristic

142

C₂×C₄

#14519805

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

488

—

147

C₈

#14519722

yes

—

148

C₈

#14519806

yes

—

149

C₂²

#16839958

yes

characteristic

150

C₄

—

151

C₄

—

976

—

152

C₄

—

488

—

153

C₄

—

488

—

154

C₄

#16840021

yes

characteristic

155

C₄

#16840038

yes

characteristic

156

C₄

C₂×#14519722

yes

characteristic

157

C₄

#16840200

yes

characteristic

158

C₂

#21445309

yes

characteristic

159

C₂

#21447860

yes

characteristic

160

C₂

#21447901

yes

characteristic

	subgroup	G/H
center	C₂²	#16839958
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1367	C₂
socle, prime radical	C₂²×C₆₁	#459
hypercenter	#1367	D₆₁
hypocenter	C₆₁	#12498
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#16839958

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1367

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
97. C₆₁	#12498	semidirect

H ≅ G/N

type

97. C₆₁

#12498

semidirect

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

subgroup	index
#21447962	2
#21445437	2
#21444717	2
#21443873	2
C₆₁×#1367	2
#12498	61

subgroup

index

C₆₁×#1367

Subgroups — #29594638

Conjugacy classes of subgroups

Complements

Maximal subgroups