Group #15756571 — subgroups

Subgroups — #15756571

Order 3328.

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

normal subgroups only

This group has 262 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	#13114588	C₂	1	1	yes	maximal, characteristic
2	#13113582	C₂	1	2	yes	maximal
3	#13112680	C₂	1	2	yes	maximal
4	#13112665	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1274	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#431	C₂²	1	2	yes	—
7	C₁₃×#376	C₂²	1	1	yes	characteristic
8	#11718586	C₂²	1	2	yes	—
9	C₂×#88043	C₂²	1	1	yes	characteristic
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	#11718116	C₄	1	2	yes	—
12	#11718116	C₄	1	2	yes	—
13	#11718072	—	2	2	no	—
14	#11718071	—	2	2	no	—
15	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
17	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
18	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
19	C₂²×#1055	C₂×C₄	1	2	yes	—
20	C₁₃×C₄⋊C₈	—	2	2	no	—
21	C₁₃×C₂²⋊C₈	—	2	2	no	—
22	C₄×C₈×C₁₃	—	2	2	no	—
23	C₁₃×#95	—	2	2	no	—
24	C₂²⋊#1055	—	2	2	no	—
25	C₄⋊#1055	—	2	2	no	—
26	#88043	—	2	2	no	—
27	C₄×#1055	—	2	2	no	—
28	C₄×#1055	D₄	1	1	yes	characteristic
29	C₄×#1055	D₄	1	1	yes	characteristic
30	C₄×#1055	Q₈	1	2	yes	—
31	#8782	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	#41	1	2	yes	—
35	C₂×C₈×C₁₃	—	4	2	no	—
36	C₄²×C₁₃	—	2	2	no	—
37	C₄²×C₁₃	—	2	2	no	—
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₄²×C₁₃	#41	1	2	yes	—
41	C₂×#1055	—	2	2	no	—
42	C₂×#1055	—	2	2	no	—
43	C₂×#1055	—	2	1	no	—
44	C₂×#1055	—	2	1	no	—
45	C₂×#1055	—	2	2	no	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	4	2	no	—
48	C₂×#1055	C₄⋊C₄	1	2	yes	—
49	C₂×#1055	C₄⋊C₄	1	2	yes	—
50	#1746	—	13	1	no	—
51	#1564	—	13	2	no	—
52	#1281	—	13	2	no	—
53	#1274	—	13	1	no	—
54	#1274	D₁₃	1	1	yes	characteristic
55	C₂³×C₁₃	#117	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	4	2	no	—
57	C₂×C₄×C₁₃	—	4	4	no	—
58	C₂×C₄×C₁₃	—	4	2	no	—
59	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#124	1	2	yes	—
62	C₂×C₄×C₁₃	#125	1	2	yes	—
63	C₂×C₄×C₁₃	#131	1	2	yes	—
64	C₂×C₄×C₁₃	#131	1	2	yes	—
65	C₈×C₁₃	—	8	2	no	—
66	#1055	—	2	2	no	—
67	#1055	—	2	2	no	—
68	#1055	—	4	1	no	—
69	#1055	—	4	1	no	—
70	#1055	—	8	2	no	—
71	#431	—	13	2	no	—
72	#431	C₂×D₁₃	1	2	yes	—
73	C₂×#97	—	13	1	no	—
74	C₂×C₄×C₈	—	13	1	no	—
75	#376	C₂×D₁₃	1	1	yes	characteristic
76	#336	—	13	2	no	—
77	#336	—	13	2	no	—
78	#335	—	26	2	no	—
79	#334	—	26	2	no	—
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	#382	1	1	yes	characteristic
82	C₂²×C₁₃	#433	1	2	yes	—
83	C₄×C₁₃	—	2	2	no	—
84	C₄×C₁₃	—	2	2	no	—
85	C₄×C₁₃	—	8	2	no	—
86	C₄×C₁₃	#424	1	2	yes	—
87	C₄×C₁₃	#433	1	1	yes	characteristic
88	C₄×C₁₃	#433	1	1	yes	characteristic
89	C₂²×C₈	—	13	2	no	—
90	C₂²×C₈	—	13	1	no	—
91	C₂²×C₈	—	13	1	no	—
92	C₂×C₄²	C₄×D₁₃	1	2	yes	—
93	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	26	2	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	26	2	no	—
98	#97	—	26	2	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	13	1	no	—
101	C₄×C₈	—	13	1	no	—
102	C₄×C₈	—	13	2	no	—
103	C₄×C₈	—	26	2	no	—
104	#95	—	2	2	no	—
105	C₂×C₁₃	—	2	2	no	—
106	C₂×C₁₃	#1832	1	1	yes	characteristic
107	C₂×C₁₃	#1833	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
110	C₂²×C₄	#6129	1	2	yes	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	13	2	no	—
113	C₂×C₈	—	13	2	no	—
114	C₂×C₈	—	26	2	no	—
115	C₂×C₈	—	26	2	no	—
116	C₂×C₈	—	26	1	no	—
117	C₂×C₈	—	26	1	no	—
118	C₂×C₈	—	26	2	no	—
119	C₂×C₈	—	26	2	no	—
120	C₂×C₈	—	52	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	#6127	1	2	yes	—
124	C₄²	D₄×D₁₃	1	1	yes	characteristic
125	C₄²	Q₈×D₁₃	1	1	yes	characteristic
126	C₁₃	#8782	1	1	yes	characteristic, complemented, Sylow
127	C₂³	#88131	1	1	yes	characteristic
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	#88126	1	1	yes	characteristic
132	C₂×C₄	#88139	1	2	yes	—
133	C₂×C₄	#88143	1	2	yes	—
134	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
135	C₂×C₄	#88155	1	2	yes	—
136	C₂×C₄	#88161	1	2	yes	—
137	C₈	—	8	2	no	—
138	C₈	—	26	2	no	—
139	C₈	—	26	2	no	—
140	C₈	—	52	1	no	—
141	C₈	—	52	1	no	—
142	C₈	—	104	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	#11718311	1	1	yes	characteristic
145	C₂²	#11718381	1	2	yes	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	8	2	no	—
149	C₄	#11718351	1	1	yes	characteristic
150	C₄	#11718370	1	1	yes	characteristic
151	C₄	#11718500	1	2	yes	—
152	C₂	—	2	2	no	—
153	C₂	#13114954	1	1	yes	characteristic
154	C₂	#13114958	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114588

C₂

yes

maximal, characteristic

#13113582

C₂

yes

maximal

#13112680

C₂

yes

maximal

#13112665

C₂

yes

maximal, characteristic

C₁₃×#1274

C₂

yes

maximal, characteristic

C₁₃×#431

C₂²

yes

—

C₁₃×#376

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₂×#88043

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

#11718116

C₄

yes

—

#11718116

C₄

yes

—

#11718072

—

#11718071

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

—

C₁₃×#95

—

C₂²⋊#1055

—

—

—

—

D₄

yes

characteristic

C₄×#1055

D₄

yes

characteristic

C₄×#1055

Q₈

yes

—

#8782

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

#117

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

#125

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₈×C₁₃

—

—

—

—

—

—

—

C₂×D₁₃

yes

—

C₂×#97

—

C₂×C₄×C₈

—

#376

C₂×D₁₃

yes

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

#433

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#424

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₄×D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

#95

—

105

C₂×C₁₃

—

106

C₂×C₁₃

#1832

yes

characteristic

107

C₂×C₁₃

#1833

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

110

C₂²×C₄

#6129

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄²

—

122

C₄²

—

123

C₄²

#6127

yes

—

124

C₄²

D₄×D₁₃

yes

characteristic

125

C₄²

Q₈×D₁₃

yes

characteristic

126

C₁₃

#8782

yes

characteristic, complemented, Sylow

127

C₂³

#88131

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#88126

yes

characteristic

132

C₂×C₄

#88139

yes

—

133

C₂×C₄

#88143

yes

—

134

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

135

C₂×C₄

#88155

yes

—

136

C₂×C₄

#88161

yes

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

104

—

143

C₂²

—

144

C₂²

#11718311

yes

characteristic

145

C₂²

#11718381

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#11718351

yes

characteristic

150

C₄

#11718370

yes

characteristic

151

C₄

#11718500

yes

—

152

C₂

—

153

C₂

#13114954

yes

characteristic

154

C₂

#13114958

yes

—

	subgroup	G/H
center	C₂×C₄	#88126
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1274	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#117
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1274	D₁₃
hypocenter	C₁₃	#8782
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₂×C₄

#88126

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1274

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

N	H ≅ G/N	type
126. C₁₃	#8782	semidirect

H ≅ G/N

type

126. C₁₃

#8782

semidirect

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

subgroup	index
#13114588	2
#13113582	2
#13112680	2
#13112665	2
C₁₃×#1274	2
#8782	13

subgroup

index

C₁₃×#1274

Subgroups — #15756571

Conjugacy classes of subgroups

Complements

Maximal subgroups