Group #29596408 — subgroups

Subgroups — #29596408

Order 15616.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#21447418	C₂	1	1	yes	maximal, characteristic
2	#21445942	C₂	1	2	yes	maximal
3	#21443923	C₂	1	1	yes	maximal, characteristic
4	#21443914	C₂	1	2	yes	maximal
5	C₆₁×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	C₂×#14519713	C₂²	1	2	yes	—
8	#16839717	C₂²	1	1	yes	characteristic
9	#16839537	C₄	1	2	yes	complemented
10	C₁₆×Dic₆₁	C₄	1	2	yes	complemented
11	#16839532	—	2	2	no	—
12	#16839518	—	2	2	no	—
13	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#427	C₂²	1	2	yes	—
15	C₂×#11758194	—	2	2	no	—
16	C₂×#11758194	D₄	1	1	yes	characteristic
17	#14519736	D₄	1	1	yes	characteristic
18	C₂×#11758185	—	2	2	no	—
19	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
20	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
21	#14519713	—	2	2	no	—
22	#14519713	—	2	2	no	—
23	#14519687	Q₈	1	2	yes	—
24	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	#117×C₆₁	—	2	2	no	—
27	M₃₂×C₆₁	—	2	2	no	—
28	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
29	#107×C₆₁	—	2	2	no	—
30	#106×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₆₁	#41	1	2	yes	—
35	C₂×C₈×C₆₁	#41	1	2	yes	—
36	C₄⋊C₄×C₆₁	—	4	2	no	—
37	C₂²⋊C₄×C₆₁	—	4	2	no	—
38	C₄²×C₆₁	—	4	2	no	—
39	C₂²⋊Dic₆₁	—	2	1	no	—
40	C₄⋊Dic₆₁	—	2	1	no	—
41	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
42	#11758194	—	2	2	no	—
43	#11758194	—	4	2	no	—
44	#11758194	—	4	2	no	—
45	C₂×#92418	—	4	2	no	—
46	C₂×#92418	C₄⋊C₄	1	2	yes	—
47	C₁₆×C₆₁	—	2	2	no	—
48	#11758185	—	2	1	no	—
49	#11758185	—	2	1	no	—
50	C₂³×C₆₁	#135	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	8	2	no	—
52	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#124	1	2	yes	—
54	C₂×C₄×C₆₁	#135	1	2	yes	—
55	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
56	C₂×Dic₆₁	—	2	2	no	—
57	C₈×C₆₁	#117	1	1	yes	characteristic
58	C₈×C₆₁	#117	1	1	yes	characteristic
59	C₈×C₆₁	#125	1	2	yes	—
60	#92418	—	2	2	no	—
61	#92418	—	8	2	no	—
62	#12893	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	—	4	1	no	—
64	C₂²×C₆₁	#427	1	1	yes	characteristic
65	C₂²×C₆₁	#488	1	2	yes	—
66	C₄×C₆₁	—	16	2	no	—
67	C₄×C₆₁	#382	1	1	yes	characteristic
68	C₄×C₆₁	#427	1	1	yes	characteristic
69	C₄×C₆₁	#488	1	2	yes	—
70	Dic₆₁	—	4	2	no	—
71	#2108	—	61	1	no	—
72	#1833	—	61	2	no	—
73	#1385	—	61	1	no	—
74	#1384	D₆₁	1	1	yes	characteristic
75	#1382	—	61	2	no	—
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	—	2	2	no	—
78	C₂×C₆₁	#1835	1	1	yes	characteristic
79	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
80	C₂²×C₁₆	—	61	1	no	—
81	C₂×#108	—	61	2	no	—
82	#427	C₂×D₆₁	1	2	yes	—
83	#405	—	61	1	no	—
84	#346	—	61	2	no	—
85	C₄×C₁₆	—	61	2	no	—
86	#344	—	122	2	no	—
87	H(3,ℤ/4)	—	122	2	no	—
88	C₆₁	#12893	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₁₆	—	61	1	no	—
90	C₂×M₁₆	—	122	2	no	—
91	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
92	#117	—	2	2	no	—
93	#117	—	61	1	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	61	1	no	—
96	C₂×C₁₆	—	61	1	no	—
97	C₂×C₁₆	—	122	2	no	—
98	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
99	#108	—	122	2	no	—
100	#108	—	122	2	no	—
101	#107	—	2	2	no	—
102	#106	—	2	2	no	—
103	#97	—	61	2	no	—
104	C₄×C₈	—	61	2	no	—
105	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
106	M₁₆	—	122	2	no	—
107	M₁₆	—	244	2	no	—
108	M₁₆	—	244	2	no	—
109	C₂×C₈	—	61	2	no	—
110	C₂×C₈	—	244	2	no	—
111	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
112	C₂×C₈	#11758222	1	2	yes	—
113	C₂×C₈	#11758224	1	2	yes	—
114	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	122	1	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	122	1	no	—
119	C₄²	—	4	2	no	—
120	C₄²	—	61	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	122	1	no	—
123	C₁₆	—	122	1	no	—
124	C₂³	#14519796	1	1	yes	characteristic
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	—	122	2	no	—
127	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
128	C₂×C₄	#14519763	1	2	yes	—
129	C₂×C₄	#14519798	1	2	yes	—
130	C₂×C₄	#14519800	1	1	yes	characteristic
131	C₈	—	122	2	no	—
132	C₈	—	488	2	no	—
133	C₈	#14519733	1	1	yes	characteristic
134	C₈	#14519734	1	1	yes	characteristic
135	C₈	#14519767	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#16839799	1	1	yes	characteristic
138	C₂²	#16840115	1	2	yes	—
139	C₄	—	16	2	no	—
140	C₄	—	244	2	no	—
141	C₄	#16839626	1	1	yes	characteristic
142	C₄	#16839797	1	1	yes	characteristic
143	C₄	#16840118	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	2	1	no	—
146	C₂	#21445948	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21447418

C₂

yes

maximal, characteristic

#21445942

C₂

yes

maximal

#21443923

C₂

yes

maximal, characteristic

#21443914

C₂

yes

maximal

C₆₁×#1384

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

#16839717

C₂²

yes

characteristic

#16839537

C₄

yes

complemented

C₁₆×Dic₆₁

C₄

yes

complemented

#16839532

—

#16839518

—

C₂×M₃₂×C₆₁

C₂²

yes

characteristic

C₆₁×#427

C₂²

yes

—

C₂×#11758194

—

C₂×#11758194

D₄

yes

characteristic

#14519736

D₄

yes

characteristic

C₂×#11758185

—

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

—

—

Q₈

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

—

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

#107×C₆₁

—

#106×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

C₄⋊C₄

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₁₆×C₆₁

—

#11758185

—

#11758185

—

C₂³×C₆₁

#135

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#124

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

C₈×C₆₁

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#427

yes

characteristic

C₂²×C₆₁

#488

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic

#1382

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1835

yes

characteristic

C₂×M₃₂

C₂×D₆₁

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

#427

C₂×D₆₁

yes

—

#405

—

#346

—

C₄×C₁₆

—

#344

—

122

—

H(3,ℤ/4)

—

122

—

C₆₁

#12893

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

122

—

C₂²×C₈

C₂²×D₆₁

yes

characteristic

#117

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₄×D₆₁

yes

—

#108

—

122

—

100

#108

—

122

—

101

#107

—

102

#106

—

103

#97

—

104

C₄×C₈

—

105

C₂²×C₄

D₄×D₆₁

yes

characteristic

106

M₁₆

—

122

—

107

M₁₆

—

244

—

108

M₁₆

—

244

—

109

C₂×C₈

—

110

C₂×C₈

—

244

—

111

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

112

C₂×C₈

#11758222

yes

—

113

C₂×C₈

#11758224

yes

—

114

C₂×C₈

Q₈×D₆₁

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

122

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

122

—

119

C₄²

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

122

—

123

C₁₆

—

122

—

124

C₂³

#14519796

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

122

—

127

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

128

C₂×C₄

#14519763

yes

—

129

C₂×C₄

#14519798

yes

—

130

C₂×C₄

#14519800

yes

characteristic

131

C₈

—

122

—

132

C₈

—

488

—

133

C₈

#14519733

yes

characteristic

134

C₈

#14519734

yes

characteristic

135

C₈

#14519767

yes

—

136

C₂²

—

137

C₂²

#16839799

yes

characteristic

138

C₂²

#16840115

yes

—

139

C₄

—

140

C₄

—

244

—

141

C₄

#16839626

yes

characteristic

142

C₄

#16839797

yes

characteristic

143

C₄

#16840118

yes

—

144

C₂

—

145

C₂

—

146

C₂

#21445948

yes

characteristic

	subgroup	G/H
center	C₄	#16839797
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1384	C₂
socle	C₂×C₆₁	#1835
prime radical	C₂³×C₆₁	#135
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1384	D₆₁
hypocenter	C₆₁	#12893
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#16839797

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1384

C₂

C₂×C₆₁

C₂³×C₆₁

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
9. #16839537	C₄	semidirect
10. C₁₆×Dic₆₁	C₄	semidirect
88. C₆₁	#12893	semidirect

H ≅ G/N

type

9. #16839537

C₄

semidirect

10. C₁₆×Dic₆₁

C₄

semidirect

88. C₆₁

#12893

semidirect

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

subgroup	index
#21447418	2
#21445942	2
#21443923	2
#21443914	2
C₆₁×#1384	2
#12893	61

subgroup

index

C₆₁×#1384

Subgroups — #29596408

Conjugacy classes of subgroups

Complements

Maximal subgroups