Group #15767942 — subgroups

Subgroups — #15767942

Order 3328.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13114990	C₂	1	1	yes	maximal, characteristic
2	#13113612	C₂	1	2	yes	maximal
3	#13113239	C₂	1	2	yes	maximal
4	#13112665	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1995	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	1	yes	characteristic
7	C₁₃×#423	C₂²	1	1	yes	characteristic
8	C₁₃×#376	C₂²	1	1	yes	characteristic
9	#11718586	C₂²	1	2	yes	—
10	C₂×C₄⋊#1055	—	2	2	no	—
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₁₃×#339	—	2	2	no	—
13	Q₈⋊#1055	—	2	4	no	—
14	#11718071	—	2	4	no	—
15	C₂×C₁₃×M₁₆	—	2	1	no	—
16	C₁₃×C₄⋊Q₈	Q₈	1	2	yes	—
17	C₁₃×#125	—	2	1	no	—
18	C₁₃×C₂²⋊Q₈	—	2	2	no	—
19	C₄×Q₈×C₁₃	—	2	2	no	—
20	C₁₃×#117	—	2	1	no	—
21	C₂×C₄²×C₁₃	D₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×#1055	—	2	2	no	—
24	C₁₃×C₄⋊C₈	—	2	2	no	—
25	C₁₃×#95	—	2	2	no	—
26	C₂²⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	2	2	no	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄⋊#1055	—	4	4	no	—
31	C₄×#1055	D₄	1	4	yes	—
32	#10668	—	13	1	no	maximal, Sylow
33	C₂×Q₈×C₁₃	—	2	2	no	—
34	C₂²×C₄×C₁₃	—	2	2	no	—
35	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	#41	1	2	yes	—
37	C₁₃×M₁₆	—	4	2	no	—
38	C₂×C₈×C₁₃	—	4	1	no	—
39	C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₁₃×C₄⋊C₄	—	4	2	no	—
42	C₁₃×C₂²⋊C₄	—	4	1	no	—
43	C₄²×C₁₃	—	2	2	no	—
44	C₄²×C₁₃	—	4	1	no	—
45	C₄²×C₁₃	Q₁₆	1	4	yes	—
46	C₄²×C₁₃	C₂×D₄	1	2	yes	—
47	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₁₃	#41	1	1	yes	characteristic
49	C₂×#1055	—	2	4	no	—
50	C₂×#1055	—	4	2	no	—
51	C₂×#1055	—	4	4	no	—
52	#1995	D₁₃	1	1	yes	characteristic
53	#1843	—	13	1	no	—
54	#1571	—	13	2	no	—
55	#1481	—	13	2	no	—
56	#1274	—	13	1	no	—
57	C₂³×C₁₃	#123	1	1	yes	characteristic
58	Q₈×C₁₃	—	4	4	no	—
59	C₂×C₄×C₁₃	—	2	4	no	—
60	C₂×C₄×C₁₃	—	4	2	no	—
61	C₂×C₄×C₁₃	—	4	4	no	—
62	C₂×C₄×C₁₃	—	4	2	no	—
63	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	#123	1	2	yes	—
66	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₈×C₁₃	—	8	1	no	—
70	#1055	—	4	4	no	—
71	#1055	—	8	2	no	—
72	#533	C₂×D₁₃	1	1	yes	characteristic
73	#431	—	13	2	no	—
74	#423	C₂×D₁₃	1	1	yes	characteristic
75	C₂×C₄⋊C₈	—	13	2	no	—
76	C₂×C₄⋊C₈	—	26	2	no	—
77	#376	C₂×D₁₃	1	1	yes	characteristic
78	#339	—	2	2	no	—
79	#335	—	26	4	no	—
80	Q₈⋊C₈	—	26	4	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	#399	1	1	yes	characteristic
83	C₂²×C₁₃	#484	1	2	yes	—
84	C₄×C₁₃	—	2	2	no	—
85	C₄×C₁₃	—	2	2	no	—
86	C₄×C₁₃	—	4	4	no	—
87	C₄×C₁₃	—	8	2	no	—
88	C₄×C₁₃	#477	1	2	yes	—
89	C₄×C₁₃	#494	1	1	yes	characteristic
90	C₄×C₁₃	#496	1	1	yes	characteristic
91	C₂×M₁₆	—	2	1	no	—
92	C₂²×C₈	—	26	2	no	—
93	C₄⋊Q₈	Dic₂₆	1	2	yes	—
94	#125	—	2	1	no	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	—	2	1	no	—
98	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
99	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	26	2	no	—
102	C₄⋊C₈	—	26	2	no	—
103	C₄⋊C₈	—	26	2	no	—
104	C₄⋊C₈	—	52	4	no	—
105	C₂²⋊C₈	—	26	2	no	—
106	C₄×C₈	—	13	4	no	—
107	#95	—	2	2	no	—
108	C₂×C₁₃	—	2	2	no	—
109	C₂×C₁₃	#2095	1	1	yes	characteristic
110	C₂×C₁₃	#2098	1	2	yes	—
111	C₂×Q₈	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	#6129	1	2	yes	—
114	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
115	M₁₆	—	4	2	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	26	4	no	—
118	C₂×C₈	—	52	2	no	—
119	C₂×C₈	—	52	4	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₂²⋊C₄	—	4	1	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	4	1	no	—
126	C₄²	#6107	1	4	yes	—
127	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
128	C₄²	#6127	1	1	yes	characteristic
129	C₄²	D₄×D₁₃	1	2	yes	—
130	C₁₃	#10668	1	1	yes	characteristic, complemented, Sylow
131	C₂³	#88189	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#88132	1	2	yes	—
138	C₂×C₄	#88137	1	2	yes	—
139	C₂×C₄	#88178	1	1	yes	characteristic
140	C₂×C₄	#88194	1	1	yes	characteristic
141	C₂×C₄	C₂×#6107	1	2	yes	—
142	C₂×C₄	#88203	1	2	yes	—
143	C₈	—	8	1	no	—
144	C₈	—	52	4	no	—
145	C₈	—	104	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#11718610	1	1	yes	characteristic
148	C₂²	#11718705	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	2	no	—
153	C₄	#11718682	1	2	yes	—
154	C₄	#11718719	1	1	yes	characteristic
155	C₄	#11718753	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#13116374	1	1	yes	characteristic
158	C₂	#13116391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114990

C₂

yes

maximal, characteristic

#13113612

C₂

yes

maximal

#13113239

C₂

yes

maximal

#13112665

C₂

yes

maximal, characteristic

C₁₃×#1995

C₂

yes

maximal, characteristic

C₁₃×#533

C₂²

yes

characteristic

C₁₃×#423

C₂²

yes

characteristic

C₁₃×#376

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

C₂²

yes

—

C₁₃×#339

—

Q₈⋊#1055

—

#11718071

—

C₂×C₁₃×M₁₆

—

C₁₃×C₄⋊Q₈

Q₈

yes

—

C₁₃×#125

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₁₃×C₄⋊C₈

—

C₁₃×#95

—

C₂²⋊#1055

—

—

—

—

—

D₄

yes

—

#10668

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

—

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

#123

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₈×C₁₃

—

#1055

—

#1055

—

#533

C₂×D₁₃

yes

characteristic

#431

—

#423

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#376

C₂×D₁₃

yes

characteristic

#339

—

#335

—

Q₈⋊C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

#399

yes

characteristic

C₂²×C₁₃

#484

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#477

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₂₆

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²⋊D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

C₄×C₈

—

107

#95

—

108

C₂×C₁₃

—

109

C₂×C₁₃

#2095

yes

characteristic

110

C₂×C₁₃

#2098

yes

—

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

#6129

yes

—

114

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

#6107

yes

—

127

C₄²

C₂×Dic₂₆

yes

characteristic

128

C₄²

#6127

yes

characteristic

129

C₄²

D₄×D₁₃

yes

—

130

C₁₃

#10668

yes

characteristic, complemented, Sylow

131

C₂³

#88189

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#88132

yes

—

138

C₂×C₄

#88137

yes

—

139

C₂×C₄

#88178

yes

characteristic

140

C₂×C₄

#88194

yes

characteristic

141

C₂×C₄

C₂×#6107

yes

—

142

C₂×C₄

#88203

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

104

—

146

C₂²

—

147

C₂²

#11718610

yes

characteristic

148

C₂²

#11718705

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#11718682

yes

—

154

C₄

#11718719

yes

characteristic

155

C₄

#11718753

yes

characteristic

156

C₂

—

157

C₂

#13116374

yes

characteristic

158

C₂

#13116391

yes

—

	subgroup	G/H
center	C₂×C₄	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1995	C₂
socle	C₂²×C₁₃	#399
prime radical	C₂³×C₁₃	#123
hypercenter	#1995	D₁₃
hypocenter	C₁₃	#10668
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#88194

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1995

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

N	H ≅ G/N	type
130. C₁₃	#10668	semidirect

H ≅ G/N

type

130. C₁₃

#10668

semidirect

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

subgroup	index
#13114990	2
#13113612	2
#13113239	2
#13112665	2
C₁₃×#1995	2
#10668	13

subgroup

index

C₁₃×#1995

Subgroups — #15767942

Conjugacy classes of subgroups

Complements

Maximal subgroups