Group #29567449 — subgroups

Subgroups — #29567449

Order 15616.

170 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 39 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	#21446038	C₂	1	1	yes	maximal, characteristic
2	#21443757	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1315	C₂	1	1	yes	maximal, characteristic
4	#16840529	C₄	1	1	yes	characteristic
5	#16839810	—	2	1	no	—
6	#16839746	C₄	1	1	yes	characteristic
7	#16839544	—	2	1	no	—
8	#16839527	—	2	1	no	—
9	C₆₁×#408	C₂²	1	1	yes	characteristic
10	C₆₁×C₂²⋊C₁₆	—	2	1	no	—
11	#14519784	—	4	2	no	—
12	C₂×#11758194	—	2	1	no	—
13	C₂×#11758194	—	2	1	no	—
14	C₂×#11758194	—	4	1	no	—
15	#14519717	—	4	1	no	—
16	#14519712	—	4	2	no	—
17	#14519696	C₈	1	2	yes	—
18	#14519694	—	2	1	no	—
19	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	—	4	1	no	—
23	C₂²⋊C₈×C₆₁	—	4	1	no	—
24	#41×C₆₁	—	4	1	no	—
25	#41×C₆₁	—	4	1	no	—
26	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
27	C₂×D₄×C₆₁	—	2	1	no	—
28	C₂×D₄×C₆₁	M₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	—	2	1	no	—
30	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
31	M₁₆×C₆₁	—	4	1	no	—
32	C₂×C₈×C₆₁	—	2	1	no	—
33	C₂×C₈×C₆₁	—	2	1	no	—
34	C₂×C₈×C₆₁	—	4	1	no	—
35	#11758194	—	2	2	no	—
36	#11758194	—	2	2	no	—
37	#11758194	—	4	1	no	—
38	#11758194	—	4	1	no	—
39	#11758194	—	4	2	no	—
40	C₂×#92418	—	4	1	no	—
41	C₂×#92418	—	4	1	no	—
42	C₂×#92418	—	4	2	no	—
43	C₁₆×C₆₁	—	8	1	no	—
44	#11758185	—	8	1	no	—
45	C₂³×C₆₁	—	2	1	no	—
46	C₂³×C₆₁	#101	1	1	yes	characteristic
47	Q₈×C₆₁	—	4	1	no	—
48	D₄×C₆₁	—	4	1	no	—
49	D₄×C₆₁	—	4	2	no	—
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	—	4	1	no	—
54	C₂×C₄×C₆₁	—	4	1	no	—
55	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
57	C₈×C₆₁	—	4	1	no	—
58	C₈×C₆₁	—	4	1	no	—
59	#92418	—	4	2	no	—
60	#92418	—	4	2	no	—
61	#7378	—	61	1	no	maximal, Sylow
62	C₂²×C₆₁	—	2	1	no	—
63	C₂²×C₆₁	—	4	1	no	—
64	C₂²×C₆₁	—	4	1	no	—
65	C₂²×C₆₁	—	8	1	no	—
66	C₂²×C₆₁	#324	1	1	yes	characteristic
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	4	1	no	—
69	C₄×C₆₁	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
70	C₄×C₆₁	#352	1	1	yes	characteristic
71	#1858	—	61	1	no	—
72	#1321	—	61	1	no	—
73	#1315	D₆₁	1	1	yes	characteristic
74	C₂×C₆₁	—	2	1	no	—
75	C₂×C₆₁	—	4	1	no	—
76	C₂×C₆₁	—	8	1	no	—
77	C₂×C₆₁	#1326	1	1	yes	characteristic
78	#568	—	61	1	no	—
79	#430	—	122	1	no	—
80	#413	—	61	1	no	—
81	#408	C₂×D₆₁	1	1	yes	characteristic
82	#349	—	122	1	no	—
83	C₂²⋊C₁₆	—	2	1	no	—
84	#341	—	122	1	no	—
85	C₆₁	#7378	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	C₄×D₆₁	1	1	yes	characteristic
87	#131	—	244	2	no	—
88	C₂×M₁₆	—	122	1	no	—
89	C₂×M₁₆	—	122	1	no	—
90	C₂×M₁₆	—	244	1	no	—
91	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
93	M₃₂	—	244	1	no	—
94	C₂×C₁₆	—	4	1	no	—
95	#108	—	244	2	no	—
96	#101	—	61	2	no	—
97	#100	—	122	1	no	—
98	C₂²⋊C₈	—	4	1	no	—
99	#41	—	4	1	no	—
100	#41	—	4	1	no	—
101	C₂×Q₈	C₈×D₆₁	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	#11758189	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	#11758198	1	1	yes	characteristic
106	M₁₆	—	4	1	no	—
107	M₁₆	—	122	2	no	—
108	M₁₆	—	122	2	no	—
109	M₁₆	—	244	1	no	—
110	M₁₆	—	244	1	no	—
111	M₁₆	—	244	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	244	1	no	—
116	C₂×C₈	—	244	1	no	—
117	C₂×C₈	—	244	2	no	—
118	C₁₆	—	8	1	no	—
119	C₁₆	—	488	1	no	—
120	C₂³	—	2	1	no	—
121	C₂³	#14519695	1	1	yes	characteristic
122	Q₈	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#14519689	1	1	yes	characteristic
131	C₂×C₄	#14519691	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	244	2	no	—
135	C₈	—	244	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#16839487	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	#16839550	1	1	yes	characteristic
144	C₄	#16839551	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	8	1	no	—
148	C₂	#21443768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21446038

C₂

yes

maximal, characteristic

#21443757

C₂

yes

maximal, characteristic

C₆₁×#1315

C₂

yes

maximal, characteristic

#16840529

C₄

yes

characteristic

#16839810

—

#16839746

C₄

yes

characteristic

#16839544

—

#16839527

—

C₆₁×#408

C₂²

yes

characteristic

C₆₁×C₂²⋊C₁₆

—

—

—

—

—

—

—

C₈

yes

—

#14519694

—

C₂×#41×C₆₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂²⋊C₈×C₆₁

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₈

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

M₁₆

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

—

—

—

—

—

—

—

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

—

C₂³×C₆₁

#101

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

#99

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#324

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

AΣL(1,𝔽₁₆)

yes

C₄×C₆₁

yes

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

—

—

122

—

#413

—

#408

C₂×D₆₁

yes

characteristic

#349

—

122

—

C₂²⋊C₁₆

—

#341

—

122

—

C₆₁

#7378

yes

characteristic, complemented, Sylow

C₂×#41

C₄×D₆₁

yes

characteristic

#131

—

244

—

C₂×M₁₆

—

122

—

C₂×M₁₆

—

122

—

C₂×M₁₆

—

244

—

C₂×M₁₆

D₂₄₄

yes

characteristic

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

M₃₂

—

244

—

C₂×C₁₆

—

#108

—

244

—

#101

—

#100

—

122

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

C₈×D₆₁

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

#11758189

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

#11758198

yes

characteristic

106

M₁₆

—

107

M₁₆

—

122

—

108

M₁₆

—

122

—

109

M₁₆

—

244

—

110

M₁₆

—

244

—

111

M₁₆

—

244

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

244

—

116

C₂×C₈

—

244

—

117

C₂×C₈

—

244

—

118

C₁₆

—

119

C₁₆

—

488

—

120

C₂³

—

121

C₂³

#14519695

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#14519689

yes

characteristic

131

C₂×C₄

#14519691

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

244

—

135

C₈

—

244

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#16839487

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#16839550

yes

characteristic

144

C₄

#16839551

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#21443768

yes

characteristic

	subgroup	G/H
center	C₄	#16839550
commutator	C₂×Q₈×C₆₁	C₂×C₈
Frattini	#408	C₂×D₆₁
Fitting	C₆₁×#1315	C₂
socle	C₂×C₆₁	#1326
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₆₁×#408	C₂²
hypercenter	#1315	D₆₁
hypocenter	C₆₁	#7378
LCS	G ▷ C₂×Q₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1315

subgroup

G/H

center

C₄

#16839550

commutator

C₂×Q₈×C₆₁

C₂×C₈

Frattini

#408

C₂×D₆₁

Fitting

C₆₁×#1315

C₂

socle

C₂×C₆₁

#1326

prime radical

C₂×#41×C₆₁

C₂×C₄

cosocle

C₆₁×#408

C₂²

D₆₁

C₆₁

G ▷ C₂×Q₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1315

N	H ≅ G/N	type
85. C₆₁	#7378	semidirect

H ≅ G/N

type

85. C₆₁

#7378

semidirect

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

subgroup	index
#21446038	2
#21443757	2
C₆₁×#1315	2
#7378	61

subgroup

index

#21446038

#21443757

C₆₁×#1315

#7378

Subgroups — #29567449

Conjugacy classes of subgroups

Complements

Maximal subgroups