Group #15749796 — subgroups

Subgroups — #15749796

Order 3328.

222 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₄×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#431	—	2	1	no	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88052	C₄	1	2	yes	—
7	C₂×C₄⋊#1055	—	2	1	no	—
8	C₁₃×#335	C₄	1	2	yes	—
9	C₁₃×#334	—	2	1	no	—
10	#11718053	—	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	C₄×C₈×C₁₃	—	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	#88052	—	2	2	no	—
26	#88052	C₈	1	4	yes	complemented
27	C₂×#6090	—	2	2	no	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	4	1	no	—
30	#7158	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	—	4	2	no	—
36	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
42	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
45	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
46	C₂×#1055	—	4	2	no	—
47	C₂×#1055	—	8	2	no	—
48	#6090	—	2	4	no	—
49	C₄×M₃₂	—	13	1	no	—
50	#1564	D₁₃	1	1	yes	characteristic
51	C₂×#334	—	13	1	no	—
52	C₂³×C₁₃	#107	1	1	yes	characteristic
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	4	1	no	—
55	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#108	1	2	yes	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	—	2	1	no	—
65	C₈×C₁₃	—	8	2	no	—
66	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
67	C₈×C₁₃	#97	1	1	yes	characteristic
68	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
69	#1055	—	8	2	no	—
70	C₂×M₃₂	—	13	2	no	—
71	#431	—	2	1	no	—
72	C₂×C₄⋊C₈	—	26	1	no	—
73	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
74	#346	—	26	1	no	—
75	C₄×C₁₆	—	26	1	no	—
76	#335	Dic₁₃	1	2	yes	—
77	#334	—	2	1	no	—
78	#334	—	13	2	no	—
79	#334	—	26	1	no	—
80	C₂²×C₁₃	—	2	1	no	—
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	#335	1	2	yes	—
83	C₂²×C₁₃	#340	1	1	yes	characteristic
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	—	2	1	no	—
87	C₄×C₁₃	#334	1	2	yes	—
88	C₄×C₁₃	#336	1	1	yes	characteristic
89	C₄×C₁₃	#341	1	1	yes	characteristic
90	C₄×C₁₃	#360	1	2	yes	—
91	C₂²×C₈	—	52	1	no	—
92	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
93	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
95	M₃₂	—	13	4	no	—
96	M₃₂	—	26	2	no	—
97	C₂×C₁₆	—	26	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	26	2	no	—
100	C₄⋊C₈	—	52	1	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
104	C₄×C₈	D₅₂	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
107	C₂×C₁₃	—	2	2	no	—
108	C₂×C₁₃	#1277	1	1	yes	characteristic
109	C₂×C₁₃	#1386	1	1	yes	characteristic
110	C₂×C₁₃	#1388	1	1	yes	characteristic
111	C₂²×C₄	#6101	1	1	yes	characteristic
112	C₂²×C₄	#6104	1	1	yes	characteristic
113	C₂²×C₄	#6107	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	52	2	no	—
117	C₂×C₈	—	104	2	no	—
118	C₂×C₈	C₈×D₁₃	1	2	yes	—
119	C₂×C₈	#6094	1	2	yes	—
120	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
121	C₂×C₈	#6101	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
123	C₂×C₈	#6103	1	1	yes	characteristic
124	C₄²	#6103	1	1	yes	characteristic
125	C₄²	#6105	1	1	yes	characteristic
126	C₄²	#6106	1	1	yes	characteristic
127	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
128	C₁₆	—	26	4	no	—
129	C₁₃	#7158	1	1	yes	characteristic, complemented, Sylow
130	C₂³	#88047	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	#88048	1	1	yes	characteristic
134	C₂×C₄	#88049	1	1	yes	characteristic
135	C₂×C₄	#88050	1	1	yes	characteristic
136	C₂×C₄	#88054	1	2	yes	—
137	C₂×C₄	#88062	1	2	yes	—
138	C₂×C₄	#88071	1	1	yes	characteristic
139	C₂×C₄	#88072	1	1	yes	characteristic
140	C₂×C₄	#88075	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	8	2	no	—
144	C₈	—	104	2	no	—
145	C₈	C₈×Dic₁₃	1	1	yes	characteristic
146	C₈	#88055	1	1	yes	characteristic
147	C₈	#88060	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#11718071	1	2	yes	—
151	C₂²	#11718121	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#11718072	1	2	yes	—
156	C₄	#11718142	1	1	yes	characteristic
157	C₄	#11718150	1	1	yes	characteristic
158	C₄	#11718158	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#13112670	1	1	yes	characteristic
161	C₂	#13112941	1	1	yes	characteristic
162	C₂	#13112947	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#88052

C₂

yes

maximal, characteristic

C₂×#11718049

C₂

yes

maximal, characteristic

C₁₃×#1564

C₂

yes

maximal, characteristic

C₁₃×#431

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88052

C₄

yes

—

C₂×C₄⋊#1055

—

C₁₃×#335

C₄

yes

—

C₁₃×#334

—

—

—

—

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

—

C₄×C₈×C₁₃

—

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

characteristic

#88052

—

#88052

C₈

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

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

M₁₆

yes

—

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

SD₁₆

yes

characteristic

C₄²×C₁₃

SD₁₆

yes

—

—

—

C₄×M₃₂

—

#1564

D₁₃

yes

characteristic

C₂×#334

—

C₂³×C₁₃

#107

yes

characteristic

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

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

#108

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₄×C₈

yes

characteristic

C₈×C₁₃

#97

yes

characteristic

C₈×C₁₃

C₂²⋊C₈

yes

—

#1055

—

C₂×M₃₂

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#346

—

C₄×C₁₆

—

#335

Dic₁₃

yes

—

#334

—

#334

—

#334

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#335

yes

—

C₂²×C₁₃

#340

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#334

yes

—

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

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₄×D₁₃

yes

characteristic

104

C₄×C₈

D₅₂

yes

characteristic

105

C₄×C₈

C₂×Dic₁₃

yes

characteristic

106

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

107

C₂×C₁₃

—

108

C₂×C₁₃

#1277

yes

characteristic

109

C₂×C₁₃

#1386

yes

characteristic

110

C₂×C₁₃

#1388

yes

characteristic

111

C₂²×C₄

#6101

yes

characteristic

112

C₂²×C₄

#6104

yes

characteristic

113

C₂²×C₄

#6107

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

104

—

118

C₂×C₈

C₈×D₁₃

yes

—

119

C₂×C₈

#6094

yes

—

120

C₂×C₈

C₄×Dic₁₃

yes

characteristic

121

C₂×C₈

#6101

yes

characteristic

122

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

123

C₂×C₈

#6103

yes

characteristic

124

C₄²

#6103

yes

characteristic

125

C₄²

#6105

yes

characteristic

126

C₄²

#6106

yes

characteristic

127

C₄²

C₂²⋊Dic₁₃

yes

characteristic

128

C₁₆

—

129

C₁₃

#7158

yes

characteristic, complemented, Sylow

130

C₂³

#88047

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#88048

yes

characteristic

134

C₂×C₄

#88049

yes

characteristic

135

C₂×C₄

#88050

yes

characteristic

136

C₂×C₄

#88054

yes

—

137

C₂×C₄

#88062

yes

—

138

C₂×C₄

#88071

yes

characteristic

139

C₂×C₄

#88072

yes

characteristic

140

C₂×C₄

#88075

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

104

—

145

C₈

C₈×Dic₁₃

yes

characteristic

146

C₈

#88055

yes

characteristic

147

C₈

#88060

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

#11718071

yes

—

151

C₂²

#11718121

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#11718072

yes

—

156

C₄

#11718142

yes

characteristic

157

C₄

#11718150

yes

characteristic

158

C₄

#11718158

yes

—

159

C₂

—

160

C₂

#13112670

yes

characteristic

161

C₂

#13112941

yes

characteristic

162

C₂

#13112947

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88072
commutator	C₈×C₁₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	#340
prime radical	C₂³×C₁₃	#107
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#7158
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

subgroup

G/H

center

C₂×C₄

#88072

commutator

C₈×C₁₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₃

Fitting

C₁₃×#1564

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄×C₈×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

N	H ≅ G/N	type
26. #88052	C₈	semidirect
129. C₁₃	#7158	semidirect

H ≅ G/N

type

26. #88052

C₈

semidirect

129. C₁₃

#7158

semidirect

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

subgroup	index
C₄×#88052	2
C₂×#11718049	2
C₁₃×#1564	2
#7158	13

subgroup

index

C₄×#88052

C₂×#11718049

C₁₃×#1564

#7158

Subgroups — #15749796

Conjugacy classes of subgroups

Complements

Maximal subgroups