Group #29592003 — subgroups

Subgroups — #29592003

Order 15616.

210 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₄×Dic₄₈₈	C₂	1	1	yes	maximal, characteristic
2	#21444715	C₂	1	2	yes	maximal
3	#21443782	C₂	1	2	yes	maximal
4	C₁₆×#92418	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₈₈	C₄	1	1	yes	characteristic
7	#16839889	—	2	2	no	—
8	C₄×Dic₂₄₄	C₂²	1	2	yes	—
9	#16839588	C₄	1	1	yes	characteristic
10	#16839567	C₄	1	2	yes	—
11	C₄×#11758185	C₂²	1	1	yes	characteristic
12	#16839492	—	2	2	no	—
13	C₈×#92418	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#335	C₂²	1	2	yes	—
16	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₂₂	—	2	2	no	—
18	Dic₄₈₈	—	2	2	no	—
19	C₂×#11758185	—	2	1	no	—
20	#14519711	C₂×C₄	1	2	yes	—
21	#14519706	—	4	2	no	—
22	#14519702	—	2	2	no	—
23	#14519687	—	4	2	no	—
24	C₄×#92418	D₄	1	1	yes	characteristic
25	C₄×#92418	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	—	2	2	no	—
29	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₂₂	—	2	2	no	—
31	C₂×C₈×C₆₁	—	4	2	no	—
32	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
34	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₆₁	—	2	2	no	—
36	#11758196	—	4	2	no	—
37	C₄×Dic₆₁	—	4	2	no	—
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	D₈	1	1	yes	characteristic
40	C₂×#92418	D₈	1	1	yes	characteristic
41	Dic₂₄₄	—	2	2	no	—
42	Dic₂₄₄	—	2	2	no	—
43	C₁₆×C₆₁	—	2	1	no	—
44	C₁₆×C₆₁	M₁₆	1	2	yes	—
45	#11758185	—	2	2	no	—
46	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
49	C₂×Dic₆₁	—	4	2	no	—
50	C₂×Dic₆₁	—	4	2	no	—
51	Dic₁₂₂	—	4	2	no	—
52	Dic₁₂₂	—	4	2	no	—
53	C₈×C₆₁	—	2	1	no	—
54	C₈×C₆₁	—	8	2	no	—
55	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₆₁	#131	1	1	yes	characteristic
57	#92418	—	2	2	no	—
58	#92418	—	2	1	no	—
59	#92418	Q₃₂	1	2	yes	complemented
60	#12083	—	61	1	no	maximal, Sylow
61	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
62	C₄×C₆₁	—	2	1	no	—
63	C₄×C₆₁	#436	1	1	yes	characteristic
64	C₄×C₆₁	#443	1	1	yes	characteristic
65	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
66	C₄×C₆₁	#508	1	1	yes	characteristic
67	Dic₆₁	—	8	2	no	—
68	Dic₆₁	—	8	2	no	—
69	C₄×Q₃₂	—	61	1	no	—
70	#1591	—	61	2	no	—
71	#1371	D₆₁	1	1	yes	characteristic
72	#1332	—	61	2	no	—
73	C₈×C₁₆	—	61	1	no	—
74	C₂×C₆₁	#1589	1	1	yes	characteristic
75	C₂×C₆₁	C₄×Q₃₂	1	1	yes	characteristic
76	C₂×C₆₁	#2178	1	1	yes	characteristic
77	C₂×Q₃₂	—	61	1	no	—
78	#446	—	122	2	no	—
79	C₄×Q₁₆	—	61	2	no	—
80	#366	—	61	1	no	—
81	#358	—	61	2	no	—
82	C₄×C₁₆	—	61	1	no	—
83	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
84	#335	C₂×D₆₁	1	2	yes	—
85	Q₈⋊C₈	—	122	2	no	—
86	C₈²	—	61	1	no	—
87	C₆₁	#12083	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	61	2	no	—
89	C₄×Q₈	—	122	2	no	—
90	Q₃₂	—	122	2	no	—
91	C₂×C₁₆	—	122	1	no	—
92	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
94	#107	—	61	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	244	2	no	—
97	#103	—	122	2	no	—
98	#97	—	244	2	no	—
99	C₄×C₈	—	61	1	no	—
100	C₄×C₈	—	61	1	no	—
101	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
102	C₂×Q₈	—	122	2	no	—
103	Q₁₆	—	122	2	no	—
104	Q₁₆	—	122	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	61	1	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	122	1	no	—
109	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
110	C₂×C₈	#11758226	1	1	yes	characteristic
111	C₄⋊C₄	—	122	2	no	—
112	C₄⋊C₄	—	244	2	no	—
113	C₄²	—	244	2	no	—
114	C₄²	D₄×D₆₁	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	122	2	no	—
117	C₁₆	#11758189	1	2	yes	—
118	Q₈	—	244	2	no	—
119	Q₈	—	244	2	no	—
120	C₂×C₄	—	244	2	no	—
121	C₂×C₄	—	244	2	no	—
122	C₂×C₄	#14519739	1	1	yes	characteristic
123	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
124	C₂×C₄	#14519800	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	61	2	no	—
128	C₈	—	122	2	no	—
129	C₈	—	122	1	no	—
130	C₈	C₂×#11758189	1	1	yes	characteristic
131	C₈	#14519785	1	1	yes	characteristic
132	C₂²	#16839844	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	488	2	no	—
135	C₄	—	488	2	no	—
136	C₄	#16839839	1	1	yes	characteristic
137	C₄	#16839878	1	1	yes	characteristic
138	C₄	Q₃₂×D₆₁	1	1	yes	characteristic
139	C₄	#16840191	1	1	yes	characteristic
140	C₂	#21444703	1	1	yes	characteristic
141	C₂	#21447677	1	1	yes	characteristic
142	C₂	#21447699	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₄₈₈

C₂

yes

maximal, characteristic

#21444715

C₂

yes

maximal

#21443782

C₂

yes

maximal

C₁₆×#92418

C₂

yes

maximal, characteristic

C₆₁×#1371

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

C₄

yes

characteristic

#16839889

—

C₄×Dic₂₄₄

C₂²

yes

—

#16839588

C₄

yes

characteristic

#16839567

C₄

yes

—

C₄×#11758185

C₂²

yes

characteristic

#16839492

—

C₈×#92418

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×#335

C₂²

yes

—

C₂×Dic₂₄₄

C₂×C₄

yes

—

C₄×Dic₁₂₂

—

Dic₄₈₈

—

C₂×#11758185

—

#14519711

C₂×C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#92418

D₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

D₈

yes

characteristic

C₂×#92418

D₈

yes

characteristic

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

—

C₁₆×C₆₁

M₁₆

yes

—

#11758185

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×M₁₆

yes

C₈×C₆₁

yes

—

—

Q₃₂

yes

complemented

#12083

—

maximal, Sylow

C₂²×C₆₁

C₄×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×Q₃₂

yes

characteristic

C₄×C₆₁

#508

yes

characteristic

Dic₆₁

—

Dic₆₁

—

C₄×Q₃₂

—

#1591

—

#1371

D₆₁

yes

characteristic

#1332

—

C₈×C₁₆

—

C₂×C₆₁

#1589

yes

characteristic

C₂×C₆₁

C₄×Q₃₂

yes

characteristic

C₂×C₆₁

#2178

yes

characteristic

C₂×Q₃₂

—

#446

—

122

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

#335

C₂×D₆₁

yes

—

Q₈⋊C₈

—

122

—

C₈²

—

C₆₁

#12083

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

122

—

Q₃₂

—

122

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₄×D₆₁

yes

characteristic

C₂×C₁₆

C₄×D₆₁

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

244

—

#103

—

122

—

#97

—

244

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₆₁

yes

characteristic

102

C₂×Q₈

—

122

—

103

Q₁₆

—

122

—

104

Q₁₆

—

122

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

122

—

109

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

110

C₂×C₈

#11758226

yes

characteristic

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

244

—

113

C₄²

—

244

—

114

C₄²

D₄×D₆₁

yes

characteristic

115

C₁₆

—

116

C₁₆

—

122

—

117

C₁₆

#11758189

yes

—

118

Q₈

—

244

—

119

Q₈

—

244

—

120

C₂×C₄

—

244

—

121

C₂×C₄

—

244

—

122

C₂×C₄

#14519739

yes

characteristic

123

C₂×C₄

D₈×D₆₁

yes

characteristic

124

C₂×C₄

#14519800

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

122

—

129

C₈

—

122

—

130

C₈

C₂×#11758189

yes

characteristic

131

C₈

#14519785

yes

characteristic

132

C₂²

#16839844

yes

characteristic

133

C₄

—

134

C₄

—

488

—

135

C₄

—

488

—

136

C₄

#16839839

yes

characteristic

137

C₄

#16839878

yes

characteristic

138

C₄

Q₃₂×D₆₁

yes

characteristic

139

C₄

#16840191

yes

characteristic

140

C₂

#21444703

yes

characteristic

141

C₂

#21447677

yes

characteristic

142

C₂

#21447699

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1371	C₂
socle, prime radical	C₂²×C₆₁	C₄×D₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1371	D₆₁
hypocenter	C₆₁	#12083
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂×C₄

D₈×D₆₁

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1371

C₂

socle, prime radical

C₂²×C₆₁

C₄×D₈

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
59. #92418	Q₃₂	semidirect
87. C₆₁	#12083	semidirect

H ≅ G/N

type

59. #92418

Q₃₂

semidirect

87. C₆₁

#12083

semidirect

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

subgroup	index
C₄×Dic₄₈₈	2
#21444715	2
#21443782	2
C₁₆×#92418	2
C₆₁×#1371	2
#12083	61

subgroup

index

C₄×Dic₄₈₈

#21444715

#21443782

C₁₆×#92418

C₆₁×#1371

#12083

Subgroups — #29592003

Conjugacy classes of subgroups

Complements

Maximal subgroups