Group #29594276 — subgroups

Subgroups — #29594276

Order 15616.

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

normal subgroups only

This group has 216 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	#21448072	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
3	#21445190	C₂	1	2	yes	maximal
4	#21443867	C₂	1	2	yes	maximal
5	C₆₁×#1371	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	#16840048	—	2	2	no	—
10	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
11	C₄⋊#11758185	C₂²	1	1	yes	characteristic
12	#16839580	—	2	1	no	—
13	C₁₆×Dic₆₁	—	2	1	no	—
14	#16839507	—	2	2	no	—
15	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#335	C₂²	1	2	yes	—
17	C₂×Dic₂₄₄	—	2	2	no	—
18	C₂×Dic₂₄₄	—	2	2	no	—
19	#14519777	—	2	2	no	—
20	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
21	Dic₄₈₈	—	4	2	no	—
22	Dic₄₈₈	—	4	2	no	—
23	C₂×#11758185	—	2	1	no	—
24	#14519709	—	4	2	no	—
25	C₄⋊#92418	D₄	1	1	yes	characteristic
26	#14519706	—	2	1	no	—
27	#14519702	—	4	2	no	—
28	C₈×Dic₆₁	—	2	1	no	—
29	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₆₁	—	2	2	no	—
32	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₂₂	—	4	2	no	—
34	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
35	C₂×C₈×C₆₁	—	4	2	no	—
36	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
38	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₆₁	—	4	2	no	—
40	C₄⋊Dic₆₁	—	4	2	no	—
41	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
42	#11758196	—	2	1	no	—
43	C₄×Dic₆₁	—	2	1	no	—
44	C₂×#92418	—	2	1	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	SD₁₆	1	2	yes	—
50	#11758185	—	4	1	no	—
51	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
54	C₂×Dic₆₁	—	2	1	no	—
55	C₂×Dic₆₁	—	2	1	no	—
56	C₂×Dic₆₁	—	8	2	no	—
57	Dic₁₂₂	—	2	1	no	—
58	Dic₁₂₂	—	8	2	no	—
59	Dic₁₂₂	—	8	2	no	—
60	Dic₁₂₂	Q₃₂	1	2	yes	complemented
61	C₈×C₆₁	—	2	1	no	—
62	C₈×C₆₁	—	8	2	no	—
63	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	#92418	—	4	1	no	—
66	#12440	—	61	1	no	maximal, Sylow
67	C₂²×C₆₁	#466	1	1	yes	characteristic
68	C₄×C₆₁	—	2	1	no	—
69	C₄×C₆₁	#460	1	1	yes	characteristic
70	C₄×C₆₁	#472	1	1	yes	characteristic
71	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
72	C₄×C₆₁	#508	1	1	yes	characteristic
73	Dic₆₁	—	2	2	no	—
74	Dic₆₁	—	4	1	no	—
75	Dic₆₁	—	16	2	no	—
76	#2247	—	61	1	no	—
77	Q₈×C₁₆	—	61	1	no	—
78	#1670	—	61	2	no	—
79	#1371	D₆₁	1	1	yes	characteristic
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	—	61	1	no	—
89	C₄⋊C₁₆	—	61	1	no	—
90	C₄⋊C₁₆	—	122	1	no	—
91	C₄×C₁₆	—	122	1	no	—
92	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
93	#335	C₂×D₆₁	1	2	yes	—
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	—	61	1	no	—
100	Q₃₂	—	244	2	no	—
101	Q₃₂	—	244	2	no	—
102	C₂×C₁₆	—	122	1	no	—
103	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
104	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
105	#106	—	244	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	61	1	no	—
108	C₄⋊C₈	—	122	1	no	—
109	#103	—	244	2	no	—
110	C₄×C₈	—	122	1	no	—
111	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
112	C₂×Q₈	—	61	1	no	—
113	C₂×Q₈	—	244	2	no	—
114	Q₁₆	—	488	2	no	—
115	Q₁₆	—	244	2	no	—
116	Q₁₆	—	244	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	122	1	no	—
119	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
120	C₂×C₈	#11758226	1	1	yes	characteristic
121	C₄⋊C₄	—	61	1	no	—
122	C₄⋊C₄	—	122	1	no	—
123	C₄⋊C₄	—	244	2	no	—
124	C₄⋊C₄	—	244	2	no	—
125	C₄²	—	122	1	no	—
126	C₄²	D₄×D₆₁	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	244	1	no	—
129	C₁₆	#11758190	1	2	yes	—
130	Q₈	—	61	2	no	—
131	Q₈	—	122	1	no	—
132	Q₈	—	488	2	no	—
133	Q₈	—	488	2	no	—
134	C₂×C₄	—	122	1	no	—
135	C₂×C₄	—	122	1	no	—
136	C₂×C₄	—	488	2	no	—
137	C₂×C₄	#14519752	1	1	yes	characteristic
138	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
139	C₂×C₄	#14519800	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	244	1	no	—
143	C₈	C₂×#11758190	1	1	yes	characteristic
144	C₈	#14519805	1	1	yes	characteristic
145	C₂²	#16839999	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	122	2	no	—
148	C₄	—	976	2	no	—
149	C₄	—	244	1	no	—
150	C₄	#16839963	1	1	yes	characteristic
151	C₄	#16840033	1	1	yes	characteristic
152	C₄	Q₃₂×D₆₁	1	1	yes	characteristic
153	C₄	#16840191	1	1	yes	characteristic
154	C₂	#21445180	1	1	yes	characteristic
155	C₂	#21447911	1	1	yes	characteristic
156	C₂	#21447945	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448072

C₂

yes

maximal, characteristic

C₁₆×Dic₁₂₂

C₂

yes

maximal, characteristic

#21445190

C₂

yes

maximal

#21443867

C₂

yes

maximal

C₆₁×#1371

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

—

C₂×Dic₄₈₈

—

#16840142

C₂²

yes

—

#16840048

—

C₈×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊#11758185

C₂²

yes

characteristic

#16839580

—

C₁₆×Dic₆₁

—

#16839507

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×#335

C₂²

yes

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

C₄×Dic₁₂₂

D₄

yes

characteristic

Dic₄₈₈

—

Dic₄₈₈

—

C₂×#11758185

—

#14519709

—

C₄⋊#92418

D₄

yes

characteristic

#14519706

—

#14519702

—

C₈×Dic₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₈

yes

characteristic

C₂×C₈×C₆₁

—

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

D₈

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

—

C₁₆×C₆₁

SD₁₆

yes

—

#11758185

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

Q₃₂

yes

complemented

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×SD₁₆

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×Q₃₂

yes

characteristic

C₄×C₆₁

#508

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2247

—

Q₈×C₁₆

—

#1670

—

#1371

D₆₁

yes

characteristic

#1362

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×Q₃₂

—

122

—

C₂×Q₃₂

—

122

—

#494

—

#475

—

122

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

122

—

C₄×C₁₆

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

#335

C₂×D₆₁

yes

—

#332

—

122

—

C₆₁

#12440

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

122

—

C₄×Q₈

—

100

Q₃₂

—

244

—

101

Q₃₂

—

244

—

102

C₂×C₁₆

—

122

—

103

C₂×C₁₆

D₂₄₄

yes

characteristic

104

C₂×C₁₆

D₂₄₄

yes

characteristic

105

#106

—

244

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

122

—

109

#103

—

244

—

110

C₄×C₈

—

122

—

111

C₄×C₈

C₂²×D₆₁

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

244

—

114

Q₁₆

—

488

—

115

Q₁₆

—

244

—

116

Q₁₆

—

244

—

117

C₂×C₈

—

118

C₂×C₈

—

122

—

119

C₂×C₈

C₂×D₂₄₄

yes

characteristic

120

C₂×C₈

#11758226

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

122

—

123

C₄⋊C₄

—

244

—

124

C₄⋊C₄

—

244

—

125

C₄²

—

122

—

126

C₄²

D₄×D₆₁

yes

characteristic

127

C₁₆

—

128

C₁₆

—

244

—

129

C₁₆

#11758190

yes

—

130

Q₈

—

131

Q₈

—

122

—

132

Q₈

—

488

—

133

Q₈

—

488

—

134

C₂×C₄

—

122

—

135

C₂×C₄

—

122

—

136

C₂×C₄

—

488

—

137

C₂×C₄

#14519752

yes

characteristic

138

C₂×C₄

D₈×D₆₁

yes

characteristic

139

C₂×C₄

#14519800

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

244

—

143

C₈

C₂×#11758190

yes

characteristic

144

C₈

#14519805

yes

characteristic

145

C₂²

#16839999

yes

characteristic

146

C₄

—

147

C₄

—

122

—

148

C₄

—

976

—

149

C₄

—

244

—

150

C₄

#16839963

yes

characteristic

151

C₄

#16840033

yes

characteristic

152

C₄

Q₃₂×D₆₁

yes

characteristic

153

C₄

#16840191

yes

characteristic

154

C₂

#21445180

yes

characteristic

155

C₂

#21447911

yes

characteristic

156

C₂

#21447945

yes

characteristic

	subgroup	G/H
center	C₂²	#16839999
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1371	C₂
socle, prime radical	C₂²×C₆₁	#466
hypercenter	#1371	D₆₁
hypocenter	C₆₁	#12440
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#16839999

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1371

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
60. Dic₁₂₂	Q₃₂	semidirect
95. C₆₁	#12440	semidirect

H ≅ G/N

type

60. Dic₁₂₂

Q₃₂

semidirect

95. C₆₁

#12440

semidirect

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

subgroup	index
#21448072	2
C₁₆×Dic₁₂₂	2
#21445190	2
#21443867	2
C₆₁×#1371	2
#12440	61

subgroup

index

C₁₆×Dic₁₂₂

C₆₁×#1371

Subgroups — #29594276

Conjugacy classes of subgroups

Complements

Maximal subgroups