Group #29594374 — subgroups

Subgroups — #29594374

Order 15616.

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

normal subgroups only

This group has 192 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	C₁₆×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#21445220	C₂	1	2	yes	maximal
3	#21443891	C₂	1	1	yes	maximal, characteristic
4	#21443877	C₂	1	2	yes	maximal
5	C₆₁×#2252	C₂	1	1	yes	maximal, characteristic
6	#16840054	—	2	2	no	—
7	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
8	C₄⋊#11758185	C₂²	1	1	yes	characteristic
9	#16839580	—	2	1	no	—
10	C₁₆×Dic₆₁	—	2	1	no	—
11	#16839515	C₂²	1	2	yes	—
12	#16839511	—	2	2	no	—
13	C₆₁×#500	C₂²	1	2	yes	—
14	C₆₁×#366	—	2	1	no	—
15	C₆₁×#366	—	2	1	no	—
16	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
18	C₂×#11758185	—	2	1	no	—
19	#14519710	—	4	2	no	—
20	C₄⋊#92418	—	2	2	no	—
21	C₄⋊#92418	D₄	1	1	yes	characteristic
22	#14519706	—	2	1	no	—
23	#14519700	—	4	2	no	—
24	C₈×Dic₆₁	—	2	1	no	—
25	C₄⋊Q₈×C₆₁	—	2	2	no	—
26	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
28	#107×C₆₁	—	2	2	no	—
29	#107×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₆₁	—	4	2	no	—
32	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	—	4	2	no	—
36	C₄⋊C₄×C₆₁	—	4	2	no	—
37	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
39	#11758196	—	2	1	no	—
40	C₄×Dic₆₁	—	2	1	no	—
41	C₂×#92418	—	2	1	no	—
42	C₂×#92418	—	4	2	no	—
43	C₁₆×C₆₁	—	2	1	no	—
44	C₁₆×C₆₁	Q₁₆	1	2	yes	—
45	#11758185	—	4	1	no	—
46	Q₈×C₆₁	—	16	2	no	—
47	C₂×C₄×C₆₁	—	8	2	no	—
48	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	1	no	—
52	C₂×Dic₆₁	—	2	1	no	—
53	Dic₁₂₂	—	2	1	no	—
54	Dic₁₂₂	Q₃₂	1	2	yes	—
55	C₈×C₆₁	—	2	1	no	—
56	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₆₁	#137	1	1	yes	characteristic
58	#92418	—	4	1	no	—
59	#92418	—	8	2	no	—
60	#12458	—	61	1	no	maximal, Sylow
61	C₂²×C₆₁	#466	1	1	yes	characteristic
62	C₄×C₆₁	—	2	1	no	—
63	C₄×C₆₁	—	16	2	no	—
64	C₄×C₆₁	#462	1	1	yes	characteristic
65	C₄×C₆₁	#473	1	1	yes	characteristic
66	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
67	C₄×C₆₁	#508	1	1	yes	characteristic
68	Dic₆₁	—	2	2	no	—
69	Dic₆₁	—	4	1	no	—
70	#2252	D₆₁	1	1	yes	characteristic
71	Q₈×C₁₆	—	61	1	no	—
72	#1676	—	61	2	no	—
73	#1371	—	61	1	no	—
74	#1365	—	61	2	no	—
75	C₂×C₆₁	#1674	1	1	yes	characteristic
76	C₂×C₆₁	#2216	1	1	yes	characteristic
77	C₂×C₆₁	#2221	1	1	yes	characteristic
78	#500	C₂×D₆₁	1	2	yes	—
79	#477	—	122	2	no	—
80	C₈×Q₈	—	61	1	no	—
81	#366	—	2	1	no	—
82	#366	—	2	1	no	—
83	C₄⋊C₁₆	—	61	1	no	—
84	C₄⋊C₁₆	—	122	1	no	—
85	C₄×C₁₆	—	122	1	no	—
86	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
87	#335	—	61	2	no	—
88	#333	—	122	2	no	—
89	C₆₁	#12458	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	61	1	no	—
92	C₂×C₁₆	—	122	1	no	—
93	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	244	2	no	—
98	C₄⋊C₈	—	61	1	no	—
99	C₄⋊C₈	—	122	1	no	—
100	C₄⋊C₈	—	122	2	no	—
101	#103	—	244	2	no	—
102	C₄×C₈	—	122	1	no	—
103	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	61	1	no	—
106	C₂×C₈	—	122	1	no	—
107	C₂×C₈	—	244	2	no	—
108	C₂×C₈	#11758224	1	1	yes	characteristic
109	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	61	1	no	—
113	C₄⋊C₄	—	122	1	no	—
114	C₄²	—	122	1	no	—
115	C₄²	D₄×D₆₁	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	244	1	no	—
118	C₁₆	#11758202	1	2	yes	—
119	Q₈	—	16	2	no	—
120	Q₈	—	61	2	no	—
121	Q₈	—	122	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	122	1	no	—
124	C₂×C₄	—	122	1	no	—
125	C₂×C₄	#14519748	1	1	yes	characteristic
126	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
127	C₂×C₄	#14519796	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	488	2	no	—
130	C₈	—	244	1	no	—
131	C₈	C₂×#11758202	1	1	yes	characteristic
132	C₈	#14519808	1	1	yes	characteristic
133	C₂²	#16839995	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	16	2	no	—
136	C₄	—	122	2	no	—
137	C₄	—	244	1	no	—
138	C₄	#16839969	1	1	yes	characteristic
139	C₄	#16840034	1	1	yes	characteristic
140	C₄	Q₃₂×D₆₁	1	1	yes	characteristic
141	C₄	#16840195	1	1	yes	characteristic
142	C₂	#21445210	1	1	yes	characteristic
143	C₂	#21447907	1	1	yes	characteristic
144	C₂	#21447936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₂₂

C₂

yes

maximal, characteristic

#21445220

C₂

yes

maximal

#21443891

C₂

yes

maximal, characteristic

#21443877

C₂

yes

maximal

C₆₁×#2252

C₂

yes

maximal, characteristic

#16840054

—

C₈×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊#11758185

C₂²

yes

characteristic

#16839580

—

C₁₆×Dic₆₁

—

#16839515

C₂²

yes

—

#16839511

—

C₆₁×#500

C₂²

yes

—

C₆₁×#366

—

C₆₁×#366

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₄×Dic₁₂₂

D₄

yes

—

—

—

D₄

yes

characteristic

#14519706

—

#14519700

—

C₈×Dic₆₁

—

C₄⋊Q₈×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

D₄

yes

characteristic

#107×C₆₁

—

#107×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

D₈

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

D₈

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₁₆×C₆₁

—

C₁₆×C₆₁

Q₁₆

yes

—

#11758185

—

Q₈×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

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

Q₃₂

yes

—

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₁₆

yes

C₈×C₆₁

yes

—

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₂×Q₃₂

yes

characteristic

C₄×C₆₁

#508

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2252

D₆₁

yes

characteristic

Q₈×C₁₆

—

#1676

—

#1371

—

#1365

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×D₆₁

yes

—

#477

—

122

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

122

—

C₄×C₁₆

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

#335

—

#333

—

122

—

C₆₁

#12458

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₁

yes

—

—

—

244

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

100

C₄⋊C₈

—

122

—

101

#103

—

244

—

102

C₄×C₈

—

122

—

103

C₄×C₈

C₂²×D₆₁

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×C₈

—

122

—

107

C₂×C₈

—

244

—

108

C₂×C₈

#11758224

yes

characteristic

109

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

122

—

114

C₄²

—

122

—

115

C₄²

D₄×D₆₁

yes

characteristic

116

C₁₆

—

117

C₁₆

—

244

—

118

C₁₆

#11758202

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

—

122

C₂×C₄

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

122

—

125

C₂×C₄

#14519748

yes

characteristic

126

C₂×C₄

D₈×D₆₁

yes

characteristic

127

C₂×C₄

#14519796

yes

characteristic

128

C₈

—

129

C₈

—

488

—

130

C₈

—

244

—

131

C₈

C₂×#11758202

yes

characteristic

132

C₈

#14519808

yes

characteristic

133

C₂²

#16839995

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

122

—

137

C₄

—

244

—

138

C₄

#16839969

yes

characteristic

139

C₄

#16840034

yes

characteristic

140

C₄

Q₃₂×D₆₁

yes

characteristic

141

C₄

#16840195

yes

characteristic

142

C₂

#21445210

yes

characteristic

143

C₂

#21447907

yes

characteristic

144

C₂

#21447936

yes

characteristic

	subgroup	G/H
center	C₂²	#16839995
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#2252	C₂
socle, prime radical	C₂²×C₆₁	#466
hypercenter	#2252	D₆₁
hypocenter	C₆₁	#12458
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16839995

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#2252

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
89. C₆₁	#12458	semidirect

H ≅ G/N

type

89. C₆₁

#12458

semidirect

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

subgroup	index
C₁₆×Dic₁₂₂	2
#21445220	2
#21443891	2
#21443877	2
C₆₁×#2252	2
#12458	61

subgroup

index

C₁₆×Dic₁₂₂

C₆₁×#2252

Subgroups — #29594374

Conjugacy classes of subgroups

Complements

Maximal subgroups