Group #15774725 — subgroups

Subgroups — #15774725

Order 3328.

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

normal subgroups only

This group has 206 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	#13116745	C₂	1	2	yes	maximal
2	#13112906	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#6090	C₂	1	2	yes	maximal
4	C₈×#6090	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
8	C₁₃×#335	C₄	1	1	yes	characteristic
9	C₁₃×Q₈⋊C₈	C₄	1	2	yes	—
10	C₈²×C₁₃	C₂²	1	1	yes	characteristic
11	C₄⋊#6090	—	2	2	no	—
12	C₄⋊#6090	C₂²	1	2	yes	—
13	#11718053	—	2	2	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₄×#6090	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₁₆	C₈	1	1	yes	characteristic
17	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₁₃×#107	C₈	1	1	yes	characteristic
19	C₁₃×C₄⋊C₈	—	2	2	no	—
20	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
21	C₁₃×#103	C₈	1	2	yes	—
22	C₄×C₈×C₁₃	—	2	2	no	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×#6090	—	2	1	no	—
27	C₂×#6090	—	2	2	no	—
28	C₂×#6090	D₄	1	1	yes	characteristic
29	C₂×#6090	D₄	1	1	yes	characteristic
30	#12054	—	13	1	no	maximal, Sylow
31	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
32	C₁₃×Q₁₆	—	2	2	no	—
33	C₂×C₈×C₁₃	—	2	1	no	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
39	C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
41	C₄²×C₁₃	—	2	2	no	—
42	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	#6090	—	2	2	no	—
44	#6090	—	2	1	no	—
45	#6090	—	4	2	no	—
46	#6090	Q₁₆	1	2	yes	complemented
47	#2183	—	13	2	no	—
48	C₈×Q₁₆	D₁₃	1	1	yes	characteristic
49	#1370	—	13	1	no	—
50	Q₈⋊C₁₆	—	13	2	no	—
51	C₈×C₁₆	—	13	1	no	—
52	Q₈×C₁₃	—	2	2	no	—
53	Q₈×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
59	C₈×C₁₃	—	2	2	no	—
60	C₈×C₁₃	—	2	1	no	—
61	C₈×C₁₃	—	2	1	no	—
62	C₈×C₁₃	—	4	2	no	—
63	C₈×C₁₃	M₃₂	1	2	yes	—
64	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₁₃	#135	1	1	yes	characteristic
66	C₈×Q₈	C₂×D₁₃	1	2	yes	—
67	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
68	C₄⋊C₁₆	—	13	2	no	—
69	C₄⋊C₁₆	—	26	2	no	—
70	#346	—	26	2	no	—
71	C₄×C₁₆	—	13	1	no	—
72	C₄×C₁₆	—	13	1	no	—
73	#335	Dic₁₃	1	1	yes	characteristic
74	Q₈⋊C₈	Dic₁₃	1	2	yes	—
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	—	4	2	no	—
79	C₄×C₁₃	—	4	2	no	—
80	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
81	C₄×C₁₃	#443	1	1	yes	characteristic
82	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
83	C₄×C₁₃	#504	1	1	yes	characteristic
84	C₂×Q₁₆	#1055	1	1	yes	characteristic
85	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
86	C₂×C₁₆	—	13	1	no	—
87	C₂×C₁₆	—	13	1	no	—
88	C₂×C₁₆	—	26	1	no	—
89	C₂×C₁₆	—	26	2	no	—
90	#107	#1055	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
93	#103	#1055	1	2	yes	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₁₃	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	C₂×#1055	1	2	yes	—
102	Q₁₆	—	2	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₂×#1055	1	1	yes	characteristic
106	C₂×C₈	C₂×#1055	1	1	yes	characteristic
107	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
108	C₂×C₈	#6129	1	1	yes	characteristic
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	C₂×#1055	1	2	yes	—
111	C₄²	—	2	2	no	—
112	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
113	C₁₆	—	13	2	no	—
114	C₁₆	—	26	2	no	—
115	C₁₆	—	26	1	no	—
116	C₁₆	—	52	2	no	—
117	C₁₃	#12054	1	1	yes	characteristic, complemented, Sylow
118	Q₈	—	2	2	no	—
119	Q₈	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
123	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
124	C₂×C₄	#88202	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	4	2	no	—
129	C₈	#88052	1	2	yes	—
130	C₈	Q₁₆×D₁₃	1	1	yes	characteristic
131	C₈	#88173	1	1	yes	characteristic
132	C₂²	D₄×#1055	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	2	no	—
136	C₄	C₂×#88052	1	1	yes	characteristic
137	C₄	#11718801	1	1	yes	characteristic
138	C₄	Q₁₆×Dic₁₃	1	1	yes	characteristic
139	C₄	#11718857	1	1	yes	characteristic
140	C₂	Q₁₆×#1055	1	1	yes	characteristic
141	C₂	#13116668	1	1	yes	characteristic
142	C₂	#13116673	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116745

C₂

yes

maximal

#13112906

C₂

yes

maximal, characteristic

Q₈⋊#6090

C₂

yes

maximal

C₈×#6090

C₂

yes

maximal, characteristic

C₈×C₁₃×Q₁₆

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₃

C₂²

yes

—

C₄×C₁₃×Q₁₆

C₄

yes

characteristic

C₁₃×#335

C₄

yes

characteristic

C₁₃×Q₈⋊C₈

C₄

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

C₄⋊#6090

—

C₄⋊#6090

C₂²

yes

—

#11718053

—

C₄×#6090

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

C₈

yes

characteristic

C₄×Q₈×C₁₃

C₂×C₄

yes

—

C₁₃×#107

C₈

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₁₃×#103

C₈

yes

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

D₄

yes

characteristic

C₂×#6090

D₄

yes

characteristic

#12054

—

maximal, Sylow

C₂×Q₈×C₁₃

C₂×C₈

yes

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

—

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

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₈

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²×C₄

yes

—

—

—

Q₁₆

yes

complemented

#2183

—

C₈×Q₁₆

D₁₃

yes

characteristic

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

M₃₂

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

characteristic

C₈×C₁₃

#135

yes

characteristic

C₈×Q₈

C₂×D₁₃

yes

—

C₄×Q₁₆

Dic₁₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₁₃

yes

characteristic

Q₈⋊C₈

Dic₁₃

yes

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₁₆

yes

characteristic

C₄×C₁₃

#443

yes

characteristic

C₄×C₁₃

C₂×M₃₂

yes

C₄×C₁₃

yes

C₂×Q₁₆

yes

C₄×Q₈

C₂×Dic₁₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

#1055

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×Dic₁₃

yes

—

#103

#1055

yes

—

C₄×C₈

—

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

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

C₂×#1055

yes

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×#1055

yes

characteristic

106

C₂×C₈

C₂×#1055

yes

characteristic

107

C₂×C₈

D₄×D₁₃

yes

characteristic

108

C₂×C₈

#6129

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×#1055

yes

—

111

C₄²

—

112

C₄²

C₂²×Dic₁₃

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

C₂²×#1055

yes

characteristic

123

C₂×C₄

D₄×Dic₁₃

yes

characteristic

124

C₂×C₄

#88202

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#88052

yes

—

130

C₈

Q₁₆×D₁₃

yes

characteristic

131

C₈

#88173

yes

characteristic

132

C₂²

D₄×#1055

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×#88052

yes

characteristic

137

C₄

#11718801

yes

characteristic

138

C₄

Q₁₆×Dic₁₃

yes

characteristic

139

C₄

#11718857

yes

characteristic

140

C₂

Q₁₆×#1055

yes

characteristic

141

C₂

#13116668

yes

characteristic

142

C₂

#13116673

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×Q₁₆	C₂
socle, prime radical	C₂²×C₁₃	C₈×D₄
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₈×Q₁₆	D₁₃
hypocenter	C₁₃	#12054
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₈×C₁₃×Q₁₆

C₂

socle, prime radical

C₂²×C₁₃

C₈×D₄

cosocle

C₄×C₈×C₁₃

C₂³

hypercenter

C₈×Q₁₆

D₁₃

hypocenter

C₁₃

#12054

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

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
#13116745	2
#13112906	2
Q₈⋊#6090	2
C₈×#6090	2
C₈×C₁₃×Q₁₆	2
#12054	13

subgroup

index

C₈×C₁₃×Q₁₆

Subgroups — #15774725

Conjugacy classes of subgroups

Complements

Maximal subgroups