Group #15779370 — subgroups

Subgroups — #15779370

Order 3328.

306 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₂×#11718085	C₂	1	4	yes	maximal
2	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
3	#13115008	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#108	—	2	1	no	—
6	C₂×C₁₃×#108	C₂²	1	2	yes	—
7	C₁₃×#423	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88059	—	2	1	no	—
10	C₂×#88059	C₂²	1	2	yes	—
11	C₂²×#6090	C₂²	1	2	yes	—
12	#11718585	—	2	1	no	—
13	#11718085	—	2	8	no	—
14	#11718053	C₄	1	4	yes	—
15	C₂×C₁₃×M₁₆	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×#6099	—	2	2	no	—
20	C₁₃×#108	—	2	4	no	—
21	C₁₃×#108	—	4	2	no	—
22	C₁₃×C₄⋊C₈	—	4	2	no	—
23	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₃	Q₈	1	2	yes	—
25	#88059	—	2	4	no	—
26	#88059	—	4	2	no	—
27	C₂×#6090	C₂×C₄	1	4	yes	—
28	C₂×#6090	D₄	1	4	yes	—
29	C₂×#6090	Q₈	1	4	yes	—
30	C₄⋊#1055	—	4	2	no	—
31	#12910	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₁₃×M₁₆	—	4	2	no	—
35	C₁₃×M₁₆	—	4	2	no	—
36	C₂×C₈×C₁₃	—	4	2	no	—
37	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₁₃	D₈	1	2	yes	—
39	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
40	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
43	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₄²×C₁₃	D₈	1	2	yes	—
46	#6099	—	4	2	no	—
47	#6099	—	4	2	no	—
48	C₂×#1055	—	4	2	no	—
49	#6090	C₄⋊C₄	1	8	yes	—
50	C₂×#368	—	13	4	no	—
51	C₂×#346	—	13	1	no	—
52	#1848	—	13	1	no	—
53	#1848	D₁₃	1	1	yes	characteristic
54	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	4	2	no	—
56	C₂×C₄×C₁₃	#107	1	4	yes	—
57	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₈×C₁₃	—	8	2	no	—
64	C₈×C₁₃	#107	1	4	yes	—
65	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
66	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
67	#1055	—	8	2	no	—
68	C₂²×C₁₆	—	13	2	no	—
69	C₂×#108	—	2	1	no	—
70	C₂×#108	—	13	2	no	—
71	C₂×#108	—	26	1	no	—
72	C₂×#108	C₂×D₁₃	1	2	yes	—
73	#423	—	2	1	no	—
74	#423	—	26	1	no	—
75	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
76	#368	—	26	8	no	—
77	#346	—	13	4	no	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₁₃	#494	1	1	yes	characteristic
81	C₂²×C₁₃	#500	1	1	yes	characteristic
82	C₄×C₁₃	—	2	4	no	—
83	C₄×C₁₃	#384	1	1	yes	characteristic
84	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
85	C₄×C₁₃	#493	1	1	yes	characteristic
86	C₄×C₁₃	#500	1	1	yes	characteristic
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	26	2	no	—
89	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
92	C₂×C₁₆	—	13	4	no	—
93	C₂×C₁₆	—	13	4	no	—
94	C₂×C₁₆	—	13	4	no	—
95	#108	—	2	4	no	—
96	#108	—	4	2	no	—
97	#108	—	26	4	no	—
98	#108	—	52	2	no	—
99	C₄⋊C₈	—	4	2	no	—
100	C₄⋊C₈	—	52	2	no	—
101	C₄×C₈	Dic₂₆	1	2	yes	—
102	C₄×C₈	C₄×D₁₃	1	2	yes	—
103	C₂×C₁₃	—	2	1	no	—
104	C₂×C₁₃	—	2	1	no	—
105	C₂×C₁₃	#1845	1	1	yes	characteristic
106	C₂×C₁₃	#2161	1	2	yes	—
107	C₂²×C₄	#6107	1	2	yes	—
108	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
109	M₁₆	—	4	2	no	—
110	M₁₆	—	4	2	no	—
111	M₁₆	—	52	2	no	—
112	M₁₆	—	52	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	52	2	no	—
115	C₂×C₈	#6101	1	2	yes	—
116	C₂×C₈	#6104	1	2	yes	—
117	C₂×C₈	#6107	1	2	yes	—
118	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
119	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
120	C₂×C₈	D₄×D₁₃	1	2	yes	—
121	C₂×C₈	Q₈×D₁₃	1	2	yes	—
122	C₄²	#6101	1	2	yes	—
123	C₄²	#6104	1	2	yes	—
124	C₁₆	—	13	8	no	—
125	C₁₃	#12910	1	1	yes	characteristic, complemented, Sylow
126	C₂³	C₂×#6107	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	#88047	1	4	yes	—
129	C₂×C₄	C₂×#6101	1	1	yes	characteristic
130	C₂×C₄	C₂×#6104	1	1	yes	characteristic
131	C₂×C₄	C₂×#6104	1	1	yes	characteristic
132	C₂×C₄	#88194	1	1	yes	characteristic
133	C₂×C₄	C₂×#6107	1	1	yes	characteristic
134	C₂×C₄	#88198	1	1	yes	characteristic
135	C₈	—	8	2	no	—
136	C₈	—	104	2	no	—
137	C₈	#88047	1	4	yes	—
138	C₈	#88142	1	2	yes	—
139	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
140	C₂²	—	2	2	no	—
141	C₂²	C₂×#88047	1	1	yes	characteristic
142	C₂²	#11718739	1	1	yes	characteristic
143	C₂²	#11718753	1	1	yes	characteristic
144	C₄	—	2	4	no	—
145	C₄	C₂×#88047	1	1	yes	characteristic
146	C₄	#11718637	1	1	yes	characteristic
147	C₄	#11718734	1	1	yes	characteristic
148	C₄	#11718739	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#13114995	1	1	yes	characteristic
152	C₂	#13116630	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718085

C₂

yes

maximal

C₂×#11718053

C₂

yes

maximal, characteristic

#13115008

C₂

yes

maximal, characteristic

C₁₃×#1848

C₂

yes

maximal, characteristic

C₂×C₁₃×#108

—

C₂×C₁₃×#108

C₂²

yes

—

C₁₃×#423

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

—

C₂×#88059

C₂²

yes

—

C₂²×#6090

C₂²

yes

—

#11718585

—

#11718085

—

#11718053

C₄

yes

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×#6099

—

C₁₃×#108

—

C₁₃×#108

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

#88059

—

#88059

—

C₂×#6090

C₂×C₄

yes

—

C₂×#6090

D₄

yes

—

C₂×#6090

Q₈

yes

—

C₄⋊#1055

—

#12910

—

maximal, Sylow

C₂²×C₄×C₁₃

Q₁₆

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

D₈

yes

—

—

—

—

C₄⋊C₄

yes

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#107

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

#107

yes

—

C₈×C₁₃

C₂×C₄⋊C₄

yes

—

C₈×C₁₃

C₄⋊Q₈

yes

—

#1055

—

C₂²×C₁₆

—

—

—

—

C₂×D₁₃

yes

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#368

—

#346

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄×C₈

Dic₂₆

yes

—

102

C₄×C₈

C₄×D₁₃

yes

—

103

C₂×C₁₃

—

104

C₂×C₁₃

—

105

C₂×C₁₃

#1845

yes

characteristic

106

C₂×C₁₃

#2161

yes

—

107

C₂²×C₄

#6107

yes

—

108

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

#6101

yes

—

116

C₂×C₈

#6104

yes

—

117

C₂×C₈

#6107

yes

—

118

C₂×C₈

C₂×Dic₂₆

yes

characteristic

119

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

120

C₂×C₈

D₄×D₁₃

yes

—

121

C₂×C₈

Q₈×D₁₃

yes

—

122

C₄²

#6101

yes

—

123

C₄²

#6104

yes

—

124

C₁₆

—

125

C₁₃

#12910

yes

characteristic, complemented, Sylow

126

C₂³

C₂×#6107

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

#88047

yes

—

129

C₂×C₄

C₂×#6101

yes

characteristic

130

C₂×C₄

C₂×#6104

yes

characteristic

131

C₂×C₄

C₂×#6104

yes

characteristic

132

C₂×C₄

#88194

yes

characteristic

133

C₂×C₄

C₂×#6107

yes

characteristic

134

C₂×C₄

#88198

yes

characteristic

135

C₈

—

136

C₈

—

104

—

137

C₈

#88047

yes

—

138

C₈

#88142

yes

—

139

C₈

C₄⋊C₄×D₁₃

yes

—

140

C₂²

—

141

C₂²

C₂×#88047

yes

characteristic

142

C₂²

#11718739

yes

characteristic

143

C₂²

#11718753

yes

characteristic

144

C₄

—

145

C₄

C₂×#88047

yes

characteristic

146

C₄

#11718637

yes

characteristic

147

C₄

#11718734

yes

characteristic

148

C₄

#11718739

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#13114995

yes

characteristic

152

C₂

#13116630

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#12910
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

subgroup

G/H

center

C₂×C₄

C₂×#6104

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1848

C₂

socle

C₂²×C₁₃

C₂×#107

prime radical

C₂³×C₁₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

N	H ≅ G/N	type
125. C₁₃	#12910	semidirect

H ≅ G/N

type

125. C₁₃

#12910

semidirect

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

subgroup	index
C₂×#11718085	2
C₂×#11718053	2
#13115008	2
C₁₃×#1848	2
#12910	13

subgroup

index

C₁₃×#1848

Subgroups — #15779370

Conjugacy classes of subgroups

Complements

Maximal subgroups