Group #15779288 — subgroups

Subgroups — #15779288

Order 3328.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#13116595	C₂	1	1	yes	maximal, characteristic
2	#13114591	C₂	1	1	yes	maximal, characteristic
3	#13112926	C₂	1	2	yes	maximal
4	#13112921	C₂	1	2	yes	maximal
5	C₁₃×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₁₃×#405	C₂²	1	1	yes	characteristic
8	#11718762	C₂²	1	2	yes	—
9	C₂×#88052	C₂²	1	1	yes	characteristic
10	C₂²×#6090	C₂²	1	1	yes	characteristic
11	C₁₃×#364	—	2	2	no	—
12	#11718152	C₄	1	1	yes	characteristic
13	#11718149	C₄	1	1	yes	characteristic
14	#11718123	C₄	1	2	yes	—
15	C₄⋊#6090	—	2	2	no	—
16	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
19	#88180	C₂×C₄	1	2	yes	—
20	C₂×#6099	C₂×C₄	1	2	yes	—
21	C₁₃×M₃₂	—	2	2	no	—
22	C₁₃×M₃₂	—	2	2	no	—
23	C₂×C₁₃×C₁₆	—	2	2	no	—
24	C₁₃×#97	Q₈	1	2	yes	—
25	C₄×C₈×C₁₃	D₄	1	2	yes	—
26	#88055	—	2	2	no	—
27	C₈×Dic₁₃	—	2	2	no	—
28	#88052	—	2	2	no	—
29	C₂×#6090	—	2	1	no	—
30	C₂×#6090	—	2	1	no	—
31	C₂×#6090	—	2	2	no	—
32	#12885	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₁₃	#41	1	2	yes	—
39	C₂×C₈×C₁₃	#41	1	2	yes	—
40	C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₁₃×C₂²⋊C₄	—	2	1	no	—
42	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂²⋊Dic₁₃	—	2	2	no	—
44	C₄⋊Dic₁₃	—	2	2	no	—
45	C₄×Dic₁₃	—	2	2	no	—
46	#6099	—	2	2	no	—
47	#6099	—	2	2	no	—
48	C₂×#1055	—	2	2	no	—
49	C₁₃×C₁₆	—	4	2	no	—
50	#6090	—	4	1	no	—
51	#6090	—	4	1	no	—
52	#2153	D₁₃	1	1	yes	characteristic
53	#2151	—	13	1	no	—
54	#1747	—	13	1	no	—
55	#1379	—	13	2	no	—
56	#1378	—	13	2	no	—
57	C₂³×C₁₃	#131	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#117	1	2	yes	—
61	C₂×C₄×C₁₃	#131	1	2	yes	—
62	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
63	C₂×Dic₁₃	—	4	2	no	—
64	C₈×C₁₃	—	2	2	no	—
65	C₈×C₁₃	#124	1	1	yes	characteristic
66	C₈×C₁₃	#124	1	1	yes	characteristic
67	C₈×C₁₃	#125	1	2	yes	—
68	#1055	—	4	2	no	—
69	C₂×M₃₂	—	13	1	no	—
70	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
71	C₂²×C₁₆	—	13	1	no	—
72	#405	—	13	2	no	—
73	#405	C₂×D₁₃	1	1	yes	characteristic
74	#364	—	2	2	no	—
75	C₄⋊C₁₆	—	26	2	no	—
76	#338	—	13	1	no	—
77	H(3,ℤ/4)	—	13	1	no	—
78	H(3,ℤ/4)	—	13	2	no	—
79	C₂²×C₁₃	—	4	1	no	—
80	C₂²×C₁₃	#424	1	1	yes	characteristic
81	C₂²×C₁₃	#433	1	2	yes	—
82	C₄×C₁₃	—	4	2	no	—
83	C₄×C₁₃	#382	1	1	yes	characteristic
84	C₄×C₁₃	#424	1	1	yes	characteristic
85	C₄×C₁₃	#433	1	2	yes	—
86	Dic₁₃	—	8	2	no	—
87	C₂×M₁₆	—	13	2	no	—
88	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
89	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
90	#117	—	13	2	no	—
91	#117	C₄×D₁₃	1	1	yes	characteristic
92	M₃₂	—	2	2	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	26	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	26	2	no	—
97	C₂×C₁₆	—	26	1	no	—
98	C₂×C₁₆	—	26	1	no	—
99	#97	—	26	2	no	—
100	#97	Dic₂₆	1	2	yes	—
101	C₄×C₈	—	26	2	no	—
102	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
103	C₂×C₁₃	—	2	2	no	—
104	C₂×C₁₃	—	2	1	no	—
105	C₂×C₁₃	#1832	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
107	M₁₆	—	2	2	no	—
108	M₁₆	—	26	2	no	—
109	M₁₆	—	26	2	no	—
110	C₂×C₈	—	26	2	no	—
111	C₂×C₈	#6101	1	2	yes	—
112	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
113	C₂×C₈	#6129	1	2	yes	—
114	C₂×C₈	#6131	1	2	yes	—
115	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	26	2	no	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₂²⋊C₄	—	26	2	no	—
120	C₄²	—	26	2	no	—
121	C₄²	#6101	1	2	yes	—
122	C₁₆	—	4	2	no	—
123	C₁₆	—	52	1	no	—
124	C₁₆	—	52	1	no	—
125	C₁₃	#12885	1	1	yes	characteristic, complemented, Sylow
126	C₂³	#88161	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	52	2	no	—
129	C₂×C₄	#88146	1	2	yes	—
130	C₂×C₄	#88155	1	2	yes	—
131	C₂×C₄	#88161	1	1	yes	characteristic
132	C₂×C₄	C₂×#6101	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	52	2	no	—
135	C₈	#88144	1	2	yes	—
136	C₈	#88190	1	1	yes	characteristic
137	C₈	#88201	1	1	yes	characteristic
138	C₂²	—	4	1	no	—
139	C₂²	#11718501	1	2	yes	—
140	C₂²	#11718780	1	1	yes	characteristic
141	C₄	—	4	2	no	—
142	C₄	—	104	2	no	—
143	C₄	#11718501	1	2	yes	—
144	C₄	#11718638	1	1	yes	characteristic
145	C₄	#11718763	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#13114955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116595

C₂

yes

maximal, characteristic

#13114591

C₂

yes

maximal, characteristic

#13112926

C₂

yes

maximal

#13112921

C₂

yes

maximal

C₁₃×#2153

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₂²

yes

—

C₁₃×#405

C₂²

yes

characteristic

#11718762

C₂²

yes

—

C₂×#88052

C₂²

yes

characteristic

C₂²×#6090

C₂²

yes

characteristic

C₁₃×#364

—

#11718152

C₄

yes

characteristic

#11718149

C₄

yes

characteristic

#11718123

C₄

yes

—

C₄⋊#6090

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

C₂×C₄

yes

characteristic

#88180

C₂×C₄

yes

—

C₂×#6099

C₂×C₄

yes

—

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₁₃×#97

Q₈

yes

—

C₄×C₈×C₁₃

D₄

yes

—

#88055

—

C₈×Dic₁₃

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#6099

—

#6099

—

C₂×#1055

—

C₁₃×C₁₆

—

#6090

—

#6090

—

#2153

D₁₃

yes

—

—

—

—

C₂³×C₁₃

#131

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

characteristic

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

C₈×C₁₃

yes

—

#1055

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₁₃

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₁₃

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂²×C₁₃

—

C₂²×C₁₃

#424

yes

characteristic

C₂²×C₁₃

#433

yes

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₁₃

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

#117

—

#117

C₄×D₁₃

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

100

#97

Dic₂₆

yes

—

101

C₄×C₈

—

102

C₄×C₈

C₂²⋊D₁₃

yes

—

103

C₂×C₁₃

—

104

C₂×C₁₃

—

105

C₂×C₁₃

#1832

yes

characteristic

106

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

#6101

yes

—

112

C₂×C₈

C₂×Dic₂₆

yes

characteristic

113

C₂×C₈

#6129

yes

—

114

C₂×C₈

#6131

yes

—

115

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

#6101

yes

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₁₃

#12885

yes

characteristic, complemented, Sylow

126

C₂³

#88161

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#88146

yes

—

130

C₂×C₄

#88155

yes

—

131

C₂×C₄

#88161

yes

characteristic

132

C₂×C₄

C₂×#6101

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

#88144

yes

—

136

C₈

#88190

yes

characteristic

137

C₈

#88201

yes

characteristic

138

C₂²

—

139

C₂²

#11718501

yes

—

140

C₂²

#11718780

yes

characteristic

141

C₄

—

142

C₄

—

104

—

143

C₄

#11718501

yes

—

144

C₄

#11718638

yes

characteristic

145

C₄

#11718763

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

#13114955

yes

characteristic

	subgroup	G/H
center	C₈	#88190
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2153	C₂
socle	C₂×C₁₃	#1832
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2153	D₁₃
hypocenter	C₁₃	#12885
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#88190

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#2153

C₂

C₂×C₁₃

C₂³×C₁₃

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
125. C₁₃	#12885	semidirect

H ≅ G/N

type

125. C₁₃

#12885

semidirect

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

subgroup	index
#13116595	2
#13114591	2
#13112926	2
#13112921	2
C₁₃×#2153	2
#12885	13

subgroup

index

C₁₃×#2153

Subgroups — #15779288

Conjugacy classes of subgroups

Complements

Maximal subgroups