Group #15750148 — subgroups

Subgroups — #15750148

Order 3328.

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

normal subgroups only

This group has 194 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	#13116635	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#432	—	2	1	no	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊#1055	—	2	1	no	—
7	C₁₃×#335	C₄	1	2	yes	—
8	C₁₃×#334	—	2	1	no	—
9	C₂²⋊#6090	C₄	1	2	yes	—
10	C₄⋊#6090	—	2	1	no	—
11	C₄×#6090	—	2	1	no	—
12	#11718049	—	2	1	no	—
13	#11718049	C₄	1	2	yes	—
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×#1055	—	4	1	no	—
18	C₁₃×C₄⋊C₈	—	2	2	no	—
19	C₁₃×C₂²⋊C₈	—	4	1	no	—
20	C₁₃×#97	—	4	1	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂×#6090	—	2	2	no	—
26	C₄⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	4	1	no	—
28	#7290	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	—	4	1	no	—
34	C₂×C₈×C₁₃	—	4	2	no	—
35	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
42	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
43	C₂×#1055	—	4	2	no	—
44	C₂×#1055	—	8	2	no	—
45	#6090	—	4	2	no	—
46	#2162	—	13	1	no	—
47	#1568	D₁₃	1	1	yes	characteristic
48	C₂×#334	—	13	1	no	—
49	C₂³×C₁₃	#107	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	1	no	—
52	C₂×C₄×C₁₃	—	2	1	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
60	C₈×C₁₃	—	2	1	no	—
61	C₈×C₁₃	—	2	1	no	—
62	C₈×C₁₃	—	2	1	no	—
63	C₈×C₁₃	—	8	2	no	—
64	C₈×C₁₃	#108	1	2	yes	—
65	#1055	—	8	2	no	—
66	#432	—	2	1	no	—
67	C₂×C₄⋊C₈	—	26	1	no	—
68	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
69	C₄⋊C₁₆	—	26	1	no	—
70	C₂²⋊C₁₆	—	13	2	no	—
71	C₄×C₁₆	—	26	1	no	—
72	#335	Dic₁₃	1	2	yes	—
73	#334	—	2	1	no	—
74	#334	—	13	2	no	—
75	#334	—	26	1	no	—
76	C₂²×C₁₃	—	2	1	no	—
77	C₂²×C₁₃	—	2	1	no	—
78	C₂²×C₁₃	#340	1	1	yes	characteristic
79	C₂²×C₁₃	#368	1	2	yes	—
80	C₄×C₁₃	—	2	1	no	—
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	—	2	1	no	—
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	#341	1	1	yes	characteristic
85	C₄×C₁₃	#344	1	1	yes	characteristic
86	C₄×C₁₃	#360	1	2	yes	—
87	C₂²×C₈	—	52	1	no	—
88	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
89	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
90	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
91	C₂×C₁₆	—	26	2	no	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	—	26	2	no	—
94	C₄⋊C₈	—	52	1	no	—
95	C₂²⋊C₈	—	4	1	no	—
96	#97	—	4	1	no	—
97	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
98	C₄×C₈	D₅₂	1	1	yes	characteristic
99	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
100	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
101	C₂×C₁₃	—	2	2	no	—
102	C₂×C₁₃	#1301	1	1	yes	characteristic
103	C₂×C₁₃	#1381	1	1	yes	characteristic
104	C₂×C₁₃	#1386	1	1	yes	characteristic
105	C₂²×C₄	#6101	1	1	yes	characteristic
106	C₂²×C₄	#6104	1	1	yes	characteristic
107	C₂²×C₄	#6107	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	52	2	no	—
112	C₂×C₈	—	104	2	no	—
113	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
114	C₂×C₈	#6101	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
116	C₂×C₈	#6103	1	1	yes	characteristic
117	C₄²	#6103	1	1	yes	characteristic
118	C₄²	#6105	1	1	yes	characteristic
119	C₄²	#6106	1	1	yes	characteristic
120	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
121	C₁₆	—	52	2	no	—
122	C₁₃	#7290	1	1	yes	characteristic, complemented, Sylow
123	C₂³	#88047	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	#88048	1	1	yes	characteristic
129	C₂×C₄	#88049	1	1	yes	characteristic
130	C₂×C₄	#88050	1	1	yes	characteristic
131	C₂×C₄	#88071	1	1	yes	characteristic
132	C₂×C₄	#88072	1	1	yes	characteristic
133	C₂×C₄	#88075	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	2	no	—
138	C₈	—	104	2	no	—
139	C₈	#88062	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#11718085	1	2	yes	—
143	C₂²	#11718121	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	#11718148	1	1	yes	characteristic
149	C₄	#11718150	1	1	yes	characteristic
150	C₄	#11718158	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	#13112731	1	1	yes	characteristic
153	C₂	#13112929	1	1	yes	characteristic
154	C₂	#13112941	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116635

C₂

yes

maximal, characteristic

C₂×#11718049

C₂

yes

maximal, characteristic

C₁₃×#1568

C₂

yes

maximal, characteristic

C₁₃×#432

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

—

C₁₃×#335

C₄

yes

—

C₁₃×#334

—

C₂²⋊#6090

C₄

yes

—

—

—

—

C₄

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

Q₈

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂²×#1055

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

D₈

yes

characteristic

C₂²×C₄×C₁₃

Q₁₆

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

SD₁₆

yes

characteristic

C₄²×C₁₃

SD₁₆

yes

—

—

—

—

D₁₃

yes

characteristic

C₂×#334

—

C₂³×C₁₃

#107

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

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

—

C₈×C₁₃

—

C₈×C₁₃

#108

yes

—

#1055

—

#432

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#335

Dic₁₃

yes

—

#334

—

#334

—

#334

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#340

yes

characteristic

C₂²×C₁₃

#368

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂²×C₈

—

C₂²×C₈

Dic₂₆

yes

characteristic

C₂²×C₈

C₄×D₁₃

yes

characteristic

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₄×D₁₃

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

100

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

101

C₂×C₁₃

—

102

C₂×C₁₃

#1301

yes

characteristic

103

C₂×C₁₃

#1381

yes

characteristic

104

C₂×C₁₃

#1386

yes

characteristic

105

C₂²×C₄

#6101

yes

characteristic

106

C₂²×C₄

#6104

yes

characteristic

107

C₂²×C₄

#6107

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

104

—

113

C₂×C₈

C₄×Dic₁₃

yes

characteristic

114

C₂×C₈

#6101

yes

characteristic

115

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

116

C₂×C₈

#6103

yes

characteristic

117

C₄²

#6103

yes

characteristic

118

C₄²

#6105

yes

characteristic

119

C₄²

#6106

yes

characteristic

120

C₄²

C₂²⋊Dic₁₃

yes

characteristic

121

C₁₆

—

122

C₁₃

#7290

yes

characteristic, complemented, Sylow

123

C₂³

#88047

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#88048

yes

characteristic

129

C₂×C₄

#88049

yes

characteristic

130

C₂×C₄

#88050

yes

characteristic

131

C₂×C₄

#88071

yes

characteristic

132

C₂×C₄

#88072

yes

characteristic

133

C₂×C₄

#88075

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

104

—

139

C₈

#88062

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

#11718085

yes

—

143

C₂²

#11718121

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#11718148

yes

characteristic

149

C₄

#11718150

yes

characteristic

150

C₄

#11718158

yes

—

151

C₂

—

152

C₂

#13112731

yes

characteristic

153

C₂

#13112929

yes

characteristic

154

C₂

#13112941

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88072
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₁₃×#1568	C₂
socle	C₂²×C₁₃	#340
prime radical	C₂³×C₁₃	#107
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	#1568	D₁₃
hypocenter	C₁₃	#7290
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

subgroup

G/H

center

C₂×C₄

#88072

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₁₃

Fitting

C₁₃×#1568

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄×C₈×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

N	H ≅ G/N	type
122. C₁₃	#7290	semidirect

H ≅ G/N

type

122. C₁₃

#7290

semidirect

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

subgroup	index
#13116635	2
C₂×#11718049	2
C₁₃×#1568	2
#7290	13

subgroup

index

#13116635

C₂×#11718049

C₁₃×#1568

#7290

Subgroups — #15750148

Conjugacy classes of subgroups

Complements

Maximal subgroups