Group #15756718 — subgroups

Subgroups — #15756718

Order 3328.

354 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 171 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	#13115008	C₂	1	1	yes	maximal, characteristic
2	#13113591	C₂	1	4	yes	maximal
3	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#108	—	2	2	no	—
6	C₁₃×#423	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88059	—	2	2	no	—
9	#11718585	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	2	yes	—
11	#11718072	—	2	8	no	—
12	C₈×#1055	C₄	1	1	yes	characteristic
13	C₈×#1055	C₄	1	2	yes	—
14	C₈×#1055	C₄	1	1	yes	characteristic
15	C₂×C₁₃×M₁₆	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×#6099	—	2	2	no	—
19	C₂²×#1055	Q₈	1	4	yes	—
20	C₁₃×#108	—	4	4	no	—
21	C₁₃×C₄⋊C₈	—	2	4	no	—
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	Q₈	1	2	yes	—
25	#88059	—	4	4	no	—
26	C₄⋊#1055	—	2	4	no	—
27	C₄×#1055	C₂×C₄	1	2	yes	—
28	C₄×#1055	C₂×C₄	1	2	yes	—
29	C₄×#1055	D₄	1	4	yes	—
30	#8822	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
33	C₁₃×M₁₆	—	4	4	no	—
34	C₂×C₈×C₁₃	—	4	2	no	—
35	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
38	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
39	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₁₃	C₂×D₄	1	2	yes	—
41	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	#6099	—	4	4	no	—
43	C₂×#1055	—	2	8	no	—
44	C₂×#1055	—	4	2	no	—
45	C₂×#1055	C₄⋊C₄	1	4	yes	—
46	C₂×#1055	C₄⋊C₄	1	4	yes	—
47	#1848	—	13	1	no	—
48	#1848	D₁₃	1	1	yes	characteristic
49	#1566	—	13	4	no	—
50	C₂×C₈²	—	13	1	no	—
51	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
59	C₈×C₁₃	—	8	2	no	—
60	C₈×C₁₃	#106	1	4	yes	—
61	C₈×C₁₃	#106	1	4	yes	—
62	#1055	—	2	4	no	—
63	#1055	—	8	2	no	—
64	#1055	#108	1	8	yes	complemented
65	C₂×#108	—	2	2	no	—
66	C₂×#108	—	26	2	no	—
67	#423	—	13	2	no	—
68	#423	C₂×D₁₃	1	2	yes	—
69	C₂×C₄×C₈	—	13	2	no	—
70	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
71	#334	—	26	8	no	—
72	C₈²	—	13	1	no	—
73	C₈²	—	13	2	no	—
74	C₈²	—	13	1	no	—
75	C₂²×C₁₃	—	2	2	no	—
76	C₂²×C₁₃	#384	1	1	yes	characteristic
77	C₂²×C₁₃	#498	1	2	yes	—
78	C₄×C₁₃	—	2	2	no	—
79	C₄×C₁₃	C₂×#106	1	2	yes	—
80	C₄×C₁₃	C₂×#108	1	4	yes	—
81	C₄×C₁₃	#492	1	1	yes	characteristic
82	C₄×C₁₃	#492	1	1	yes	characteristic
83	C₂×M₁₆	—	2	2	no	—
84	C₂×M₁₆	—	26	2	no	—
85	C₂²×C₈	—	13	4	no	—
86	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
87	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
88	#108	—	4	4	no	—
89	#108	—	52	4	no	—
90	C₄⋊C₈	—	2	4	no	—
91	C₄⋊C₈	—	26	4	no	—
92	C₄×C₈	—	13	2	no	—
93	C₄×C₈	—	13	2	no	—
94	C₄×C₈	—	13	4	no	—
95	C₄×C₈	Dic₂₆	1	2	yes	—
96	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
97	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
98	C₂×C₁₃	—	2	2	no	—
99	C₂×C₁₃	#1844	1	1	yes	characteristic
100	C₂×C₁₃	#1848	1	2	yes	—
101	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
102	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
103	M₁₆	—	4	4	no	—
104	M₁₆	—	52	4	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	13	4	no	—
107	C₂×C₈	—	13	4	no	—
108	C₂×C₈	—	26	8	no	—
109	C₂×C₈	—	52	2	no	—
110	C₂×C₈	#6101	1	2	yes	—
111	C₂×C₈	#6101	1	2	yes	—
112	C₂×C₈	#6105	1	4	yes	—
113	C₂×C₈	#6106	1	4	yes	—
114	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
115	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
116	C₄²	D₄×D₁₃	1	2	yes	—
117	C₁₃	#8822	1	1	yes	characteristic, complemented, Sylow
118	C₂³	#88198	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	#88142	1	2	yes	—
121	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
122	C₂×C₄	C₂×#6101	1	1	yes	characteristic
123	C₂×C₄	C₂×#6105	1	2	yes	—
124	C₂×C₄	#88194	1	1	yes	characteristic
125	C₂×C₄	C₂×#6106	1	2	yes	—
126	C₈	—	8	2	no	—
127	C₈	—	13	8	no	—
128	C₈	—	26	4	no	—
129	C₈	—	104	2	no	—
130	C₈	#88048	1	4	yes	—
131	C₈	#88048	1	4	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	#11718637	1	1	yes	characteristic
134	C₂²	#11718740	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	D₁₃×#108	1	4	yes	—
137	C₄	C₂×#88048	1	2	yes	—
138	C₄	#11718737	1	1	yes	characteristic
139	C₄	#11718746	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	#13114992	1	1	yes	characteristic
142	C₂	#13115006	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115008

C₂

yes

maximal, characteristic

#13113591

C₂

yes

maximal

C₂×C₈×#1055

C₂

yes

maximal, characteristic

C₁₃×#1848

C₂

yes

maximal, characteristic

C₂×C₁₃×#108

—

C₁₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

—

#11718585

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

#11718072

—

C₈×#1055

C₄

yes

characteristic

C₈×#1055

C₄

yes

—

C₈×#1055

C₄

yes

characteristic

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₂²×#1055

Q₈

yes

—

C₁₃×#108

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

Q₈

yes

—

#88059

—

C₄⋊#1055

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

D₄

yes

—

#8822

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

#1848

—

#1848

D₁₃

yes

characteristic

#1566

—

C₂×C₈²

—

C₂³×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

#106

yes

—

C₈×C₁₃

#106

yes

—

—

—

yes

—

—

—

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#334

—

C₈²

—

C₈²

—

C₈²

—

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#498

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#106

yes

—

C₄×C₁₃

C₂×#108

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

C₄×D₁₃

yes

characteristic

C₄×C₈

C₄×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1844

yes

characteristic

100

C₂×C₁₃

#1848

yes

—

101

C₂²×C₄

Q₈×D₁₃

yes

—

102

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

#6101

yes

—

111

C₂×C₈

#6101

yes

—

112

C₂×C₈

#6105

yes

—

113

C₂×C₈

#6106

yes

—

114

C₄²

C₂×Dic₂₆

yes

characteristic

115

C₄²

C₂×C₄×D₁₃

yes

characteristic

116

C₄²

D₄×D₁₃

yes

—

117

C₁₃

#8822

yes

characteristic, complemented, Sylow

118

C₂³

#88198

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

#88142

yes

—

121

C₂×C₄

C₄⋊C₄×D₁₃

yes

—

122

C₂×C₄

C₂×#6101

yes

characteristic

123

C₂×C₄

C₂×#6105

yes

—

124

C₂×C₄

#88194

yes

characteristic

125

C₂×C₄

C₂×#6106

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

104

—

130

C₈

#88048

yes

—

131

C₈

#88048

yes

—

132

C₂²

—

133

C₂²

#11718637

yes

characteristic

134

C₂²

#11718740

yes

—

135

C₄

—

136

C₄

D₁₃×#108

yes

—

137

C₄

C₂×#88048

yes

—

138

C₄

#11718737

yes

characteristic

139

C₄

#11718746

yes

characteristic

140

C₂

—

141

C₂

#13114992

yes

characteristic

142

C₂

#13115006

yes

—

	subgroup	G/H
center	C₂×C₄	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#8822
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1848

C₂

socle

C₂²×C₁₃

#384

prime radical

C₂³×C₁₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
64. #1055	#108	semidirect
117. C₁₃	#8822	semidirect

H ≅ G/N

type

64. #1055

#108

semidirect

117. C₁₃

#8822

semidirect

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

subgroup	index
#13115008	2
#13113591	2
C₂×C₈×#1055	2
C₁₃×#1848	2
#8822	13

subgroup

index

#13115008

#13113591

C₂×C₈×#1055

C₁₃×#1848

#8822

Subgroups — #15756718

Conjugacy classes of subgroups

Complements

Maximal subgroups