Group #15756592 — subgroups

Subgroups — #15756592

Order 3328.

322 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 139 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	#13114953	C₂	1	1	yes	maximal, characteristic
2	#13114582	C₂	1	1	yes	maximal, characteristic
3	#13112670	C₂	1	4	yes	maximal
4	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
6	C₄³×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#1055	C₂²	1	2	yes	—
8	C₂×C₄×#1055	C₂²	1	2	yes	—
9	C₁₃×#336	—	2	2	no	—
10	#11718116	—	2	2	no	—
11	#11718116	C₄	1	2	yes	—
12	#11718116	C₄	1	2	yes	—
13	C₂²×C₈×C₁₃	—	2	2	no	—
14	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×#1055	—	2	2	no	—
20	C₂²×#1055	C₂×C₄	1	4	yes	—
21	C₁₃×C₄⋊C₈	—	2	4	no	—
22	C₄⋊#1055	—	2	4	no	—
23	C₄×#1055	—	2	2	no	—
24	C₄×#1055	—	2	2	no	—
25	#8788	—	13	1	no	maximal, Sylow
26	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₁₃	—	4	2	no	—
32	C₂×C₈×C₁₃	—	4	4	no	—
33	C₄²×C₁₃	—	2	2	no	—
34	C₄²×C₁₃	—	2	2	no	—
35	C₄²×C₁₃	—	2	4	no	—
36	C₄²×C₁₃	—	4	2	no	—
37	C₄²×C₁₃	#41	1	2	yes	—
38	C₄²×C₁₃	#41	1	2	yes	—
39	C₂×#1055	—	2	4	no	—
40	C₂×#1055	—	2	4	no	—
41	C₂×#1055	—	2	4	no	—
42	C₂×#1055	—	4	2	no	—
43	C₂×#1055	—	4	4	no	—
44	#1831	—	13	1	no	—
45	#1831	D₁₃	1	1	yes	characteristic
46	#1744	—	13	1	no	—
47	#1277	—	13	4	no	—
48	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	—	2	2	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	#117	1	2	yes	—
57	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#125	1	2	yes	—
60	C₂×C₄×C₁₃	#131	1	2	yes	—
61	C₂×C₄×C₁₃	#131	1	2	yes	—
62	C₂×C₄×C₁₃	#135	1	4	yes	—
63	C₈×C₁₃	—	8	2	no	—
64	#1055	—	4	4	no	—
65	#1055	—	8	2	no	—
66	C₂×C₄⋊C₈	—	13	2	no	—
67	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
68	C₂×C₄×C₈	—	13	2	no	—
69	C₄³	C₂×D₁₃	1	1	yes	characteristic
70	#336	—	2	2	no	—
71	#336	—	13	2	no	—
72	#336	—	13	2	no	—
73	#336	—	26	2	no	—
74	C₂²×C₁₃	#382	1	1	yes	characteristic
75	C₂²×C₁₃	#424	1	1	yes	characteristic
76	C₂²×C₁₃	#424	1	1	yes	characteristic
77	C₂²×C₁₃	#427	1	2	yes	—
78	C₂²×C₁₃	#430	1	2	yes	—
79	C₄×C₁₃	—	2	2	no	—
80	C₄×C₁₃	—	4	2	no	—
81	C₄×C₁₃	—	4	2	no	—
82	C₄×C₁₃	#433	1	4	yes	—
83	C₄×C₁₃	#488	1	2	yes	—
84	C₄×C₁₃	#488	1	2	yes	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	13	4	no	—
87	C₂²×C₈	—	26	2	no	—
88	C₂×C₄²	Dic₂₆	1	2	yes	—
89	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
90	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
92	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
93	C₄⋊C₈	—	2	4	no	—
94	C₄⋊C₈	—	26	4	no	—
95	C₄×C₈	—	26	2	no	—
96	C₄×C₈	—	26	2	no	—
97	C₂×C₁₃	#1572	1	2	yes	—
98	C₂×C₁₃	#1572	1	2	yes	—
99	C₂×C₁₃	#1832	1	1	yes	characteristic
100	C₂×C₁₃	#1835	1	1	yes	characteristic
101	C₂×C₁₃	#1838	1	1	yes	characteristic
102	C₂²×C₄	#6101	1	2	yes	—
103	C₂²×C₄	#6101	1	2	yes	—
104	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	4	no	—
109	C₂×C₈	—	26	4	no	—
110	C₂×C₈	—	26	4	no	—
111	C₂×C₈	—	26	4	no	—
112	C₂×C₈	—	52	2	no	—
113	C₂×C₈	—	52	4	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	4	no	—
117	C₄²	—	4	2	no	—
118	C₄²	#6129	1	2	yes	—
119	C₄²	#6131	1	2	yes	—
120	C₁₃	#8788	1	1	yes	characteristic, complemented, Sylow
121	C₂³	C₂×#6101	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#88144	1	2	yes	—
130	C₂×C₄	#88146	1	2	yes	—
131	C₂×C₄	#88155	1	2	yes	—
132	C₂×C₄	#88161	1	2	yes	—
133	C₂×C₄	#88190	1	1	yes	characteristic
134	C₂×C₄	#88201	1	1	yes	characteristic
135	C₂×C₄	#88204	1	4	yes	—
136	C₈	—	8	2	no	—
137	C₈	—	52	4	no	—
138	C₈	—	104	2	no	—
139	C₂²	#11718638	1	1	yes	characteristic
140	C₂²	#11718659	1	2	yes	—
141	C₂²	#11718763	1	1	yes	characteristic
142	C₂²	#11718780	1	1	yes	characteristic
143	C₂²	#11718785	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	2	no	—
146	C₄	—	4	2	no	—
147	C₄	#11718501	1	4	yes	—
148	C₄	#11718717	1	2	yes	—
149	C₄	#11718729	1	2	yes	—
150	C₂	#13113614	1	2	yes	—
151	C₂	#13113617	1	2	yes	—
152	C₂	#13114955	1	1	yes	characteristic
153	C₂	#13114966	1	1	yes	characteristic
154	C₂	#13114976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114953

C₂

yes

maximal, characteristic

#13114582

C₂

yes

maximal, characteristic

#13112670

C₂

yes

maximal

C₁₃×#1831

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

C₁₃×#336

—

#11718116

—

#11718116

C₄

yes

—

#11718116

C₄

yes

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₂²×#1055

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

—

—

—

—

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

—

—

—

—

—

—

—

D₁₃

yes

characteristic

#1744

—

#1277

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₁₃

yes

—

—

—

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#430

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#433

yes

—

C₄×C₁₃

#488

yes

—

C₄×C₁₃

#488

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₂₆

yes

—

C₂×C₄²

C₄×D₁₃

yes

characteristic

C₂×C₄²

C₄×D₁₃

yes

characteristic

C₂×C₄²

C₂²⋊D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

#1572

yes

—

C₂×C₁₃

#1572

yes

—

C₂×C₁₃

#1832

yes

characteristic

100

C₂×C₁₃

#1835

yes

characteristic

101

C₂×C₁₃

#1838

yes

characteristic

102

C₂²×C₄

#6101

yes

—

103

C₂²×C₄

#6101

yes

—

104

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

105

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

106

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

#6129

yes

—

119

C₄²

#6131

yes

—

120

C₁₃

#8788

yes

characteristic, complemented, Sylow

121

C₂³

C₂×#6101

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#88144

yes

—

130

C₂×C₄

#88146

yes

—

131

C₂×C₄

#88155

yes

—

132

C₂×C₄

#88161

yes

—

133

C₂×C₄

#88190

yes

characteristic

134

C₂×C₄

#88201

yes

characteristic

135

C₂×C₄

#88204

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

104

—

139

C₂²

#11718638

yes

characteristic

140

C₂²

#11718659

yes

—

141

C₂²

#11718763

yes

characteristic

142

C₂²

#11718780

yes

characteristic

143

C₂²

#11718785

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#11718501

yes

—

148

C₄

#11718717

yes

—

149

C₄

#11718729

yes

—

150

C₂

#13113614

yes

—

151

C₂

#13113617

yes

—

152

C₂

#13114955

yes

characteristic

153

C₂

#13114966

yes

characteristic

154

C₂

#13114976

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#8788
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₁₃

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1831

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
120. C₁₃	#8788	semidirect

H ≅ G/N

type

120. C₁₃

#8788

semidirect

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

subgroup	index
#13114953	2
#13114582	2
#13112670	2
C₁₃×#1831	2
#8788	13

subgroup

index

C₁₃×#1831

Subgroups — #15756592

Conjugacy classes of subgroups

Complements

Maximal subgroups