Group #29594275 — subgroups

Subgroups — #29594275

Order 15616.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#21448072	C₂	1	1	yes	maximal, characteristic
2	#21445189	C₂	1	2	yes	maximal
3	#21443891	C₂	1	1	yes	maximal, characteristic
4	#21443869	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₈₈	—	2	1	no	—
7	C₂×Dic₄₈₈	—	2	1	no	—
8	#16840142	C₂²	1	2	yes	—
9	#16840047	—	2	2	no	—
10	#16839515	C₂²	1	2	yes	—
11	#16839509	—	2	2	no	—
12	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×C₄⋊C₁₆	—	2	1	no	—
14	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₁	—	2	1	no	—
16	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₄₄	—	2	2	no	—
18	C₂×Dic₂₄₄	—	2	2	no	—
19	#14519777	—	2	2	no	—
20	Dic₄₈₈	—	4	2	no	—
21	Dic₄₈₈	—	4	2	no	—
22	#14519708	—	4	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	#14519701	—	4	2	no	—
25	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₁	—	2	1	no	—
27	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₆₁	—	2	1	no	—
30	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	—	2	1	no	—
32	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
34	C₂×Dic₁₂₂	—	4	2	no	—
35	C₂×C₈×C₆₁	—	2	1	no	—
36	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₆₁	—	2	1	no	—
39	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
40	C₄²×C₆₁	—	2	1	no	—
41	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₆₁	—	4	2	no	—
43	C₄⋊Dic₆₁	—	4	2	no	—
44	C₂×#92418	—	4	2	no	—
45	Dic₂₄₄	—	4	2	no	—
46	Dic₂₄₄	—	4	2	no	—
47	Dic₂₄₄	—	8	2	no	—
48	C₁₆×C₆₁	—	2	1	no	—
49	C₁₆×C₆₁	—	4	1	no	—
50	C₁₆×C₆₁	SD₁₆	1	2	yes	—
51	Q₈×C₆₁	—	2	1	no	—
52	Q₈×C₆₁	Q₃₂	1	2	yes	complemented
53	C₂×C₄×C₆₁	—	2	1	no	—
54	C₂×C₄×C₆₁	—	2	1	no	—
55	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
58	C₂×Dic₆₁	—	8	2	no	—
59	Dic₁₂₂	—	8	2	no	—
60	Dic₁₂₂	—	8	2	no	—
61	C₈×C₆₁	—	2	1	no	—
62	C₈×C₆₁	—	4	1	no	—
63	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	#92418	—	8	2	no	—
66	#12440	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	#466	1	1	yes	characteristic
68	C₄×C₆₁	—	2	2	no	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	4	1	no	—
71	C₄×C₆₁	#460	1	1	yes	characteristic
72	C₄×C₆₁	#472	1	1	yes	characteristic
73	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
74	C₄×C₆₁	#508	1	1	yes	characteristic
75	Dic₆₁	—	16	2	no	—
76	#2247	—	61	1	no	—
77	Q₈×C₁₆	D₆₁	1	1	yes	characteristic
78	#1670	—	61	2	no	—
79	#1371	—	61	1	no	—
80	#1362	—	61	2	no	—
81	C₂×C₆₁	#1668	1	1	yes	characteristic
82	C₂×C₆₁	#2216	1	1	yes	characteristic
83	C₂×C₆₁	#2222	1	1	yes	characteristic
84	C₂×Q₃₂	—	122	1	no	—
85	C₂×Q₃₂	—	122	1	no	—
86	#494	—	61	2	no	—
87	#475	—	122	2	no	—
88	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
91	C₄×C₁₆	—	2	1	no	—
92	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
93	#335	—	61	2	no	—
94	#332	—	122	2	no	—
95	C₆₁	#12440	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	122	2	no	—
97	C₂×Q₁₆	—	122	2	no	—
98	C₄⋊Q₈	—	122	2	no	—
99	C₄×Q₈	D₂₄₄	1	1	yes	characteristic
100	Q₃₂	—	244	2	no	—
101	Q₃₂	—	244	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
105	#106	—	244	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	122	2	no	—
108	C₄⋊C₈	D₂₄₄	1	1	yes	characteristic
109	#103	—	244	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
112	C₂×Q₈	—	244	2	no	—
113	C₂×Q₈	D₄₈₈	1	1	yes	characteristic
114	Q₁₆	—	488	2	no	—
115	Q₁₆	—	244	2	no	—
116	Q₁₆	—	244	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	244	2	no	—
119	C₂×C₈	#11758222	1	1	yes	characteristic
120	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	244	2	no	—
123	C₄⋊C₄	—	244	2	no	—
124	C₄⋊C₄	D₄₈₈	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	#11758201	1	2	yes	—
130	Q₈	—	2	1	no	—
131	Q₈	—	488	2	no	—
132	Q₈	—	488	2	no	—
133	Q₈	Dic₄₈₈	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	488	2	no	—
137	C₂×C₄	#14519751	1	1	yes	characteristic
138	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
139	C₂×C₄	#14519799	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	488	2	no	—
143	C₈	C₂×#11758201	1	1	yes	characteristic
144	C₈	#14519802	1	1	yes	characteristic
145	C₂²	#16839998	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	976	2	no	—
150	C₄	#16839962	1	1	yes	characteristic
151	C₄	#16840032	1	1	yes	characteristic
152	C₄	C₂×Dic₄₈₈	1	1	yes	characteristic
153	C₄	#16840194	1	1	yes	characteristic
154	C₂	#21445179	1	1	yes	characteristic
155	C₂	#21447910	1	1	yes	characteristic
156	C₂	#21447944	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448072

C₂

yes

maximal, characteristic

#21445189

C₂

yes

maximal

#21443891

C₂

yes

maximal, characteristic

#21443869

C₂

yes

maximal

Q₈×C₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

—

C₂×Dic₄₈₈

—

#16840142

C₂²

yes

—

#16840047

—

#16839515

C₂²

yes

—

#16839509

—

C₈×Q₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₆₁×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

Dic₄₈₈

—

Dic₄₈₈

—

#14519708

—

C₄⋊#92418

—

#14519701

—

C₄×Q₈×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

D₈

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₈

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

SD₁₆

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

Q₃₂

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×SD₁₆

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×Q₃₂

yes

characteristic

C₄×C₆₁

#508

yes

characteristic

Dic₆₁

—

#2247

—

Q₈×C₁₆

D₆₁

yes

—

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×Q₃₂

—

122

—

C₂×Q₃₂

—

122

—

#494

—

#475

—

122

—

C₈×Q₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₆₁

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

#335

—

#332

—

122

—

C₆₁

#12440

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

122

—

C₄×Q₈

D₂₄₄

yes

characteristic

100

Q₃₂

—

244

—

101

Q₃₂

—

244

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

105

#106

—

244

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

122

—

108

C₄⋊C₈

D₂₄₄

yes

characteristic

109

#103

—

244

—

110

C₄×C₈

—

111

C₄×C₈

C₂²×D₆₁

yes

characteristic

112

C₂×Q₈

—

244

—

113

C₂×Q₈

D₄₈₈

yes

characteristic

114

Q₁₆

—

488

—

115

Q₁₆

—

244

—

116

Q₁₆

—

244

—

117

C₂×C₈

—

118

C₂×C₈

—

244

—

119

C₂×C₈

#11758222

yes

characteristic

120

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

244

—

123

C₄⋊C₄

—

244

—

124

C₄⋊C₄

D₄₈₈

yes

characteristic

125

C₄²

—

126

C₄²

C₂×D₂₄₄

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

#11758201

yes

—

130

Q₈

—

131

Q₈

—

488

—

132

Q₈

—

488

—

133

Q₈

Dic₄₈₈

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

488

—

137

C₂×C₄

#14519751

yes

characteristic

138

C₂×C₄

C₂×D₄₈₈

yes

characteristic

139

C₂×C₄

#14519799

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

488

—

143

C₈

C₂×#11758201

yes

characteristic

144

C₈

#14519802

yes

characteristic

145

C₂²

#16839998

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

976

—

150

C₄

#16839962

yes

characteristic

151

C₄

#16840032

yes

characteristic

152

C₄

C₂×Dic₄₈₈

yes

characteristic

153

C₄

#16840194

yes

characteristic

154

C₂

#21445179

yes

characteristic

155

C₂

#21447910

yes

characteristic

156

C₂

#21447944

yes

characteristic

	subgroup	G/H
center	C₂²	#16839998
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	Q₈×C₁₆×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	#466
hypercenter	Q₈×C₁₆	D₆₁
hypocenter	C₆₁	#12440
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#16839998

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

Q₈×C₁₆×C₆₁

C₂

socle, prime radical

C₂²×C₆₁

#466

hypercenter

Q₈×C₁₆

D₆₁

hypocenter

C₆₁

#12440

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
52. Q₈×C₆₁	Q₃₂	semidirect
95. C₆₁	#12440	semidirect
133. Q₈	Dic₄₈₈	semidirect

H ≅ G/N

type

52. Q₈×C₆₁

Q₃₂

semidirect

95. C₆₁

#12440

semidirect

133. Q₈

Dic₄₈₈

semidirect

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

subgroup	index
#21448072	2
#21445189	2
#21443891	2
#21443869	2
Q₈×C₁₆×C₆₁	2
#12440	61

subgroup

index

Q₈×C₁₆×C₆₁

Subgroups — #29594275

Conjugacy classes of subgroups

Complements

Maximal subgroups