Group #15774724 — subgroups

Subgroups — #15774724

Order 3328.

206 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13116741	C₂	1	2	yes	maximal
2	C₈×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13112814	C₂	1	2	yes	maximal
4	C₈×#6090	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1370	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₅₂	C₄	1	1	yes	characteristic
7	C₈×Dic₂₆	C₂²	1	2	yes	—
8	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₈²×C₁₃	C₂²	1	1	yes	characteristic
10	#11718094	—	2	2	no	—
11	#11718093	—	2	2	no	—
12	C₄×#6090	C₂²	1	1	yes	characteristic
13	C₄×#6090	C₂²	1	1	yes	characteristic
14	#11718050	C₄	1	1	yes	characteristic
15	#11718048	C₄	1	2	yes	—
16	C₂×Dic₅₂	C₈	1	1	yes	characteristic
17	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₃×C₁₆	—	2	2	no	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	#88058	C₈	1	1	yes	characteristic
23	#88056	C₈	1	2	yes	—
24	#88054	—	2	2	no	—
25	C₈×Dic₁₃	—	2	2	no	—
26	C₂×#6090	—	2	1	no	—
27	C₂×#6090	D₄	1	1	yes	characteristic
28	C₂×#6090	D₄	1	1	yes	characteristic
29	C₄⋊#1055	C₂×C₄	1	2	yes	—
30	#12054	—	13	1	no	maximal, Sylow
31	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₁₃	—	2	1	no	—
33	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
37	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
39	#6101	—	2	2	no	—
40	C₄×Dic₁₃	—	2	2	no	—
41	C₂×#1055	—	2	2	no	—
42	Dic₅₂	—	2	2	no	—
43	C₁₃×C₁₆	—	4	2	no	—
44	#6090	—	2	2	no	—
45	#6090	—	2	1	no	—
46	#6090	Q₁₆	1	2	yes	complemented
47	#2183	—	13	2	no	—
48	C₈×Q₁₆	—	13	1	no	—
49	#1370	D₁₃	1	1	yes	characteristic
50	Q₈⋊C₁₆	—	13	2	no	—
51	C₈×C₁₆	—	13	1	no	—
52	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
55	C₂×Dic₁₃	—	2	2	no	—
56	C₂×Dic₁₃	—	2	2	no	—
57	Dic₂₆	—	2	2	no	—
58	Dic₂₆	—	2	2	no	—
59	C₈×C₁₃	—	2	2	no	—
60	C₈×C₁₃	—	2	1	no	—
61	C₈×C₁₃	—	2	1	no	—
62	C₈×C₁₃	M₃₂	1	2	yes	—
63	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₈×C₁₃	#135	1	1	yes	characteristic
65	#1055	—	4	2	no	—
66	C₈×Q₈	—	13	2	no	—
67	C₄×Q₁₆	—	13	1	no	—
68	C₄⋊C₁₆	—	26	2	no	—
69	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
70	#346	—	26	2	no	—
71	C₄×C₁₆	—	13	1	no	—
72	C₄×C₁₆	—	13	1	no	—
73	#335	—	13	1	no	—
74	Q₈⋊C₈	—	13	2	no	—
75	C₈²	C₂×D₁₃	1	1	yes	characteristic
76	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
77	C₄×C₁₃	—	2	1	no	—
78	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₁₃	#443	1	1	yes	characteristic
80	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
81	C₄×C₁₃	#504	1	1	yes	characteristic
82	Dic₁₃	—	4	2	no	—
83	Dic₁₃	—	4	2	no	—
84	C₂×Q₁₆	—	13	1	no	—
85	C₄×Q₈	—	13	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	—	13	1	no	—
88	C₂×C₁₆	—	13	1	no	—
89	C₂×C₁₆	—	26	1	no	—
90	#107	—	13	1	no	—
91	C₄⋊C₈	—	13	2	no	—
92	C₄⋊C₈	—	26	2	no	—
93	#103	—	13	2	no	—
94	C₄×C₈	—	26	2	no	—
95	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
96	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
98	C₂×C₁₃	C₈×Q₁₆	1	1	yes	characteristic
99	C₂×C₁₃	#2169	1	1	yes	characteristic
100	C₂×C₁₃	#2170	1	1	yes	characteristic
101	C₂×Q₈	—	13	2	no	—
102	Q₁₆	—	26	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	26	2	no	—
105	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
106	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
107	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
108	C₂×C₈	#6131	1	1	yes	characteristic
109	C₄⋊C₄	—	13	2	no	—
110	C₄⋊C₄	—	26	2	no	—
111	C₄²	—	26	2	no	—
112	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
113	C₁₆	—	4	2	no	—
114	C₁₆	—	13	2	no	—
115	C₁₆	—	26	2	no	—
116	C₁₆	—	26	1	no	—
117	C₁₃	#12054	1	1	yes	characteristic, complemented, Sylow
118	Q₈	—	26	2	no	—
119	Q₈	—	26	2	no	—
120	C₂×C₄	—	26	2	no	—
121	C₂×C₄	—	26	2	no	—
122	C₂×C₄	#88147	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
124	C₂×C₄	#88161	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	52	2	no	—
129	C₈	#88038	1	2	yes	—
130	C₈	#88166	1	1	yes	characteristic
131	C₈	Q₁₆×D₁₃	1	1	yes	characteristic
132	C₂²	#11718495	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	52	2	no	—
135	C₄	—	52	2	no	—
136	C₄	#11718534	1	1	yes	characteristic
137	C₄	#11718556	1	1	yes	characteristic
138	C₄	C₂×#88038	1	1	yes	characteristic
139	C₄	#11718568	1	1	yes	characteristic
140	C₂	#13113642	1	1	yes	characteristic
141	C₂	#13116667	1	1	yes	characteristic
142	C₂	#13116672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116741

C₂

yes

maximal

C₈×Dic₅₂

C₂

yes

maximal, characteristic

#13112814

C₂

yes

maximal

C₈×#6090

C₂

yes

maximal, characteristic

C₁₃×#1370

C₂

yes

maximal, characteristic

C₄×Dic₅₂

C₄

yes

characteristic

C₈×Dic₂₆

C₂²

yes

—

C₁₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

characteristic

#11718050

C₄

yes

characteristic

#11718048

C₄

yes

—

C₂×Dic₅₂

C₈

yes

characteristic

C₄×Dic₂₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88058

C₈

yes

characteristic

#88056

C₈

yes

—

#88054

—

C₈×Dic₁₃

—

C₂×#6090

—

C₂×#6090

D₄

yes

characteristic

C₂×#6090

D₄

yes

characteristic

C₄⋊#1055

C₂×C₄

yes

—

#12054

—

maximal, Sylow

C₂×Dic₂₆

C₂×C₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₁₃

C₂×C₈

yes

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

C₁₃×C₁₆

—

#6090

—

#6090

—

#6090

Q₁₆

yes

complemented

#2183

—

C₈×Q₁₆

—

#1370

D₁₃

yes

characteristic

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

M₃₂

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

C₈×C₁₃

yes

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₃

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₁₆

yes

characteristic

C₄×C₁₃

#443

yes

characteristic

C₄×C₁₃

C₂×M₃₂

yes

characteristic

C₄×C₁₃

#504

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₁₃

yes

characteristic

C₄×C₈

C₄×D₁₃

yes

characteristic

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

C₈×Q₁₆

yes

characteristic

C₂×C₁₃

#2169

yes

characteristic

100

C₂×C₁₃

#2170

yes

characteristic

101

C₂×Q₈

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₈×D₁₃

yes

characteristic

106

C₂×C₈

C₈×D₁₃

yes

characteristic

107

C₂×C₈

D₄×D₁₃

yes

characteristic

108

C₂×C₈

#6131

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

C₂×C₄×D₁₃

yes

characteristic

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₃

#12054

yes

characteristic, complemented, Sylow

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#88147

yes

characteristic

123

C₂×C₄

C₂×C₈×D₁₃

yes

characteristic

124

C₂×C₄

#88161

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#88038

yes

—

130

C₈

#88166

yes

characteristic

131

C₈

Q₁₆×D₁₃

yes

characteristic

132

C₂²

#11718495

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#11718534

yes

characteristic

137

C₄

#11718556

yes

characteristic

138

C₄

C₂×#88038

yes

characteristic

139

C₄

#11718568

yes

characteristic

140

C₂

#13113642

yes

characteristic

141

C₂

#13116667

yes

characteristic

142

C₂

#13116672

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1370	C₂
socle, prime radical	C₂²×C₁₃	C₈×D₄
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1370	D₁₃
hypocenter	C₁₃	#12054
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1370

C₂

socle, prime radical

C₂²×C₁₃

C₈×D₄

cosocle

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
46. #6090	Q₁₆	semidirect
117. C₁₃	#12054	semidirect

H ≅ G/N

type

46. #6090

Q₁₆

semidirect

117. C₁₃

#12054

semidirect

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

subgroup	index
#13116741	2
C₈×Dic₅₂	2
#13112814	2
C₈×#6090	2
C₁₃×#1370	2
#12054	13

subgroup

index

C₈×Dic₅₂

C₁₃×#1370

Subgroups — #15774724

Conjugacy classes of subgroups

Complements

Maximal subgroups