Group #15756634 — subgroups

Subgroups — #15756634

Order 3328.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13113599	C₂	1	1	yes	maximal, characteristic
2	#13113128	C₂	1	1	yes	maximal, characteristic
3	#13112699	C₂	1	2	yes	maximal
4	#13112697	C₂	1	2	yes	maximal
5	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#432	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718587	C₂²	1	2	yes	—
9	#11718586	C₂²	1	1	yes	characteristic
10	#11718586	C₂²	1	1	yes	characteristic
11	C₁₃×#335	—	2	1	no	—
12	C₁₃×#334	—	2	1	no	—
13	#11718050	—	2	1	no	—
14	#11718049	—	2	1	no	—
15	#11718046	C₄	1	4	yes	—
16	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₁₃×C₄⋊C₈	—	2	2	no	—
19	C₁₃×C₂²⋊C₈	—	2	2	no	—
20	C₁₃×#97	—	2	2	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²⋊#1055	—	2	1	no	—
25	C₂²⋊#1055	—	2	1	no	—
26	C₂²⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	2	1	no	—
28	C₄⋊#1055	—	2	1	no	—
29	C₄⋊#1055	—	2	2	no	—
30	#88043	—	2	2	no	—
31	C₄×#1055	C₂×C₄	1	2	yes	—
32	C₄×#1055	C₂×C₄	1	2	yes	—
33	#8799	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	4	2	no	—
38	C₂×C₈×C₁₃	—	4	2	no	—
39	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₁₃	#41	1	1	yes	characteristic
44	C₄²×C₁₃	#41	1	1	yes	characteristic
45	C₂×#1055	—	2	2	no	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	4	2	no	—
48	#1568	—	13	1	no	—
49	#1568	D₁₃	1	1	yes	characteristic
50	#1452	—	13	1	no	—
51	#1287	—	13	2	no	—
52	#1287	—	13	2	no	—
53	C₂³×C₁₃	#125	1	1	yes	characteristic
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	4	1	no	—
56	C₂×C₄×C₁₃	—	4	1	no	—
57	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
63	C₈×C₁₃	—	2	2	no	—
64	C₈×C₁₃	—	2	2	no	—
65	C₈×C₁₃	—	8	2	no	—
66	#1055	—	4	2	no	—
67	#1055	—	4	2	no	—
68	#1055	—	8	2	no	—
69	#432	—	13	2	no	—
70	#432	C₂×D₁₃	1	2	yes	—
71	#431	—	13	1	no	—
72	#431	—	13	1	no	—
73	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
74	#335	—	2	1	no	—
75	#335	—	26	1	no	—
76	#334	—	2	1	no	—
77	#334	—	26	1	no	—
78	#322	—	13	4	no	—
79	C₂²×C₁₃	—	2	1	no	—
80	C₂²×C₁₃	—	2	1	no	—
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	#382	1	1	yes	characteristic
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	—	2	1	no	—
87	C₄×C₁₃	—	2	1	no	—
88	C₄×C₁₃	#430	1	2	yes	—
89	C₂²×C₈	Dic₂₆	1	2	yes	—
90	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	26	1	no	—
93	C₄⋊C₈	—	26	1	no	—
94	C₄⋊C₈	—	26	2	no	—
95	C₂²⋊C₈	—	2	2	no	—
96	C₂²⋊C₈	—	26	1	no	—
97	C₂²⋊C₈	—	26	1	no	—
98	C₂²⋊C₈	—	26	2	no	—
99	#97	—	2	2	no	—
100	#97	—	26	2	no	—
101	C₄×C₈	—	13	2	no	—
102	C₄×C₈	—	13	2	no	—
103	C₄×C₈	C₄×D₁₃	1	2	yes	—
104	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
106	C₂×C₁₃	—	4	1	no	—
107	C₂×C₁₃	#1833	1	1	yes	characteristic
108	C₂×C₁₃	#1833	1	1	yes	characteristic
109	C₂×C₁₃	#1838	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
111	C₂²×C₄	#6129	1	1	yes	characteristic
112	C₂²×C₄	#6131	1	1	yes	characteristic
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	26	2	no	—
116	C₂×C₈	—	26	2	no	—
117	C₂×C₈	—	52	2	no	—
118	C₂×C₈	#6101	1	2	yes	—
119	C₂×C₈	#6101	1	2	yes	—
120	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
121	C₄²	#6129	1	1	yes	characteristic
122	C₄²	#6131	1	1	yes	characteristic
123	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
124	C₁₃	#8799	1	1	yes	characteristic, complemented, Sylow
125	C₂³	#88144	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	#88144	1	1	yes	characteristic
130	C₂×C₄	#88146	1	1	yes	characteristic
131	C₂×C₄	#88146	1	1	yes	characteristic
132	C₂×C₄	C₂×#6101	1	1	yes	characteristic
133	C₂×C₄	#88190	1	1	yes	characteristic
134	C₂×C₄	#88201	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	2	no	—
138	C₈	—	52	2	no	—
139	C₈	—	52	2	no	—
140	C₈	—	104	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#11718638	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#11718785	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	#13114959	1	1	yes	characteristic
153	C₂	#13114961	1	1	yes	characteristic
154	C₂	#13114976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13113599

C₂

yes

maximal, characteristic

#13113128

C₂

yes

maximal, characteristic

#13112699

C₂

yes

maximal

#13112697

C₂

yes

maximal

C₁₃×#1568

C₂

yes

maximal, characteristic

C₁₃×#432

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718587

C₂²

yes

—

#11718586

C₂²

yes

characteristic

#11718586

C₂²

yes

characteristic

C₁₃×#335

—

C₁₃×#334

—

#11718050

—

#11718049

—

#11718046

C₄

yes

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₂²⋊#1055

—

C₂²⋊#1055

—

C₂²⋊#1055

—

—

—

—

—

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

#8799

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×Q₈

yes

C₂²×C₄×C₁₃

yes

C₂²×C₄×C₁₃

yes

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

—

—

—

D₁₃

yes

—

—

—

C₂³×C₁₃

#125

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

C₂×D₁₃

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#382

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#430

yes

—

C₂²×C₈

Dic₂₆

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₄×D₁₃

yes

—

104

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

105

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

106

C₂×C₁₃

—

107

C₂×C₁₃

#1833

yes

characteristic

108

C₂×C₁₃

#1833

yes

characteristic

109

C₂×C₁₃

#1838

yes

characteristic

110

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

111

C₂²×C₄

#6129

yes

characteristic

112

C₂²×C₄

#6131

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

#6101

yes

—

119

C₂×C₈

#6101

yes

—

120

C₄²

C₂×C₄×D₁₃

yes

characteristic

121

C₄²

#6129

yes

characteristic

122

C₄²

#6131

yes

characteristic

123

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

124

C₁₃

#8799

yes

characteristic, complemented, Sylow

125

C₂³

#88144

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#88144

yes

characteristic

130

C₂×C₄

#88146

yes

characteristic

131

C₂×C₄

#88146

yes

characteristic

132

C₂×C₄

C₂×#6101

yes

characteristic

133

C₂×C₄

#88190

yes

characteristic

134

C₂×C₄

#88201

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

104

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#11718638

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#11718785

yes

—

151

C₂

—

152

C₂

#13114959

yes

characteristic

153

C₂

#13114961

yes

characteristic

154

C₂

#13114976

yes

characteristic

	subgroup	G/H
center	C₂²	#11718638
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1568	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#125
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1568	D₁₃
hypocenter	C₁₃	#8799
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

subgroup

G/H

center

C₂²

#11718638

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1568

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

N	H ≅ G/N	type
124. C₁₃	#8799	semidirect

H ≅ G/N

type

124. C₁₃

#8799

semidirect

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

subgroup	index
#13113599	2
#13113128	2
#13112699	2
#13112697	2
C₁₃×#1568	2
#8799	13

subgroup

index

C₁₃×#1568

Subgroups — #15756634

Conjugacy classes of subgroups

Complements

Maximal subgroups