Group #15774652 — subgroups

Subgroups — #15774652

Order 3328.

226 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#13113642	C₂	1	2	yes	maximal
3	#13112799	C₂	1	2	yes	maximal
4	C₁₆×#1055	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄	C₄	1	1	yes	characteristic
7	#11718491	—	2	2	no	—
8	C₄×Dic₅₂	C₂²	1	2	yes	—
9	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#335	C₂²	1	2	yes	—
11	#11718096	C₄	1	1	yes	characteristic
12	#11718095	C₄	1	2	yes	—
13	#11718076	—	2	2	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₂₆	—	2	2	no	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₁₃×C₄⋊C₈	—	2	2	no	—
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	#88058	C₂×C₄	1	2	yes	—
23	#88056	—	2	2	no	—
24	#88054	—	4	2	no	—
25	C₈×Dic₁₃	—	4	2	no	—
26	C₂×#6090	—	2	1	no	—
27	C₄×#1055	D₄	1	1	yes	characteristic
28	C₄×#1055	D₄	1	1	yes	characteristic
29	Dic₁₀₄	C₈	1	4	yes	complemented
30	C₈×Q₃₂	—	13	1	no	maximal, Sylow
31	C₂×Dic₂₆	—	2	2	no	—
32	C₂×C₈×C₁₃	—	4	2	no	—
33	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
35	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₃	—	2	2	no	—
37	#6101	—	4	2	no	—
38	C₄×Dic₁₃	—	4	2	no	—
39	C₂×#1055	—	2	1	no	—
40	C₂×#1055	D₈	1	1	yes	characteristic
41	C₂×#1055	D₈	1	1	yes	characteristic
42	Dic₅₂	—	2	2	no	—
43	Dic₅₂	C₂×C₈	1	4	yes	—
44	C₁₃×C₁₆	—	2	1	no	—
45	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
46	#6090	—	2	2	no	—
47	C₄×Q₃₂	—	13	1	no	—
48	C₈×Q₁₆	—	13	2	no	—
49	#1371	D₁₃	1	1	yes	characteristic
50	#1332	—	13	2	no	—
51	C₈×C₁₆	—	13	1	no	—
52	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
55	C₂×Dic₁₃	—	4	2	no	—
56	C₂×Dic₁₃	—	4	2	no	—
57	Dic₂₆	—	2	4	no	—
58	Dic₂₆	—	4	2	no	—
59	C₈×C₁₃	—	2	1	no	—
60	C₈×C₁₃	—	8	2	no	—
61	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₁₃	#131	1	1	yes	characteristic
63	#1055	—	2	2	no	—
64	#1055	—	2	1	no	—
65	#1055	Q₃₂	1	2	yes	complemented
66	C₂×Q₃₂	—	13	1	no	—
67	C₈×Q₈	—	26	2	no	—
68	C₄×Q₁₆	—	13	2	no	—
69	#366	—	13	1	no	—
70	#358	—	13	2	no	—
71	C₄×C₁₆	—	13	1	no	—
72	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
73	#335	C₂×D₁₃	1	2	yes	—
74	Q₈⋊C₈	—	26	2	no	—
75	C₈²	—	13	1	no	—
76	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
77	C₄×C₁₃	—	2	1	no	—
78	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
79	C₄×C₁₃	#443	1	1	yes	characteristic
80	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
81	C₄×C₁₃	#508	1	1	yes	characteristic
82	Dic₁₃	—	4	4	no	—
83	Dic₁₃	—	8	2	no	—
84	C₂×Q₁₆	—	13	2	no	—
85	C₄×Q₈	—	26	2	no	—
86	Q₃₂	—	13	4	no	—
87	C₂×C₁₆	—	26	1	no	—
88	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
89	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
90	#107	—	13	2	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	52	2	no	—
93	#103	—	26	2	no	—
94	C₄×C₈	—	13	1	no	—
95	C₄×C₈	—	13	1	no	—
96	C₄×C₈	—	52	2	no	—
97	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
98	C₂×C₁₃	C₈×D₈	1	1	yes	characteristic
99	C₂×C₁₃	C₄×Q₃₂	1	1	yes	characteristic
100	C₂×C₁₃	#2178	1	1	yes	characteristic
101	C₂×Q₈	—	26	2	no	—
102	Q₁₆	—	13	4	no	—
103	Q₁₆	—	26	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	13	1	no	—
106	C₂×C₈	—	13	1	no	—
107	C₂×C₈	—	26	1	no	—
108	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
109	C₂×C₈	#6131	1	1	yes	characteristic
110	C₄⋊C₄	—	26	2	no	—
111	C₄⋊C₄	—	52	2	no	—
112	C₄²	—	52	2	no	—
113	C₄²	D₄×D₁₃	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	26	2	no	—
116	C₁₆	C₈×D₁₃	1	2	yes	—
117	C₁₃	C₈×Q₃₂	1	1	yes	characteristic, complemented, Sylow
118	Q₈	—	26	4	no	—
119	Q₈	—	52	2	no	—
120	C₂×C₄	—	52	2	no	—
121	C₂×C₄	—	52	2	no	—
122	C₂×C₄	#88147	1	1	yes	characteristic
123	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
124	C₂×C₄	#88173	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	13	2	no	—
128	C₈	—	26	2	no	—
129	C₈	—	26	1	no	—
130	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
131	C₈	#88161	1	1	yes	characteristic
132	C₂²	#11718533	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	52	4	no	—
135	C₄	—	104	2	no	—
136	C₄	#11718495	1	1	yes	characteristic
137	C₄	#11718556	1	1	yes	characteristic
138	C₄	#11718573	1	1	yes	characteristic
139	C₄	D₁₃×Q₃₂	1	1	yes	characteristic
140	C₂	#13113630	1	1	yes	characteristic
141	C₂	#13116694	1	1	yes	characteristic
142	C₂	#13116716	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₁₀₄

C₂

yes

maximal, characteristic

#13113642

C₂

yes

maximal

#13112799

C₂

yes

maximal

C₁₆×#1055

C₂

yes

maximal, characteristic

C₁₃×#1371

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄

C₄

yes

characteristic

#11718491

—

C₄×Dic₅₂

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#335

C₂²

yes

—

#11718096

C₄

yes

characteristic

#11718095

C₄

yes

—

#11718076

—

C₄×#6090

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₂×Dic₅₂

C₂×C₄

yes

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88058

C₂×C₄

yes

—

#88056

—

#88054

—

C₈×Dic₁₃

—

C₂×#6090

—

C₄×#1055

D₄

yes

characteristic

C₄×#1055

D₄

yes

characteristic

Dic₁₀₄

C₈

yes

complemented

C₈×Q₃₂

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

D₈

yes

characteristic

C₂×#1055

D₈

yes

characteristic

Dic₅₂

—

Dic₅₂

C₂×C₈

yes

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

—

#6090

—

C₄×Q₃₂

—

C₈×Q₁₆

—

#1371

D₁₃

yes

characteristic

#1332

—

C₈×C₁₆

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂²×C₈

yes

C₈×C₁₃

yes

—

—

Q₃₂

yes

complemented

C₂×Q₃₂

—

C₈×Q₈

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

C₂×D₁₃

yes

—

Q₈⋊C₈

—

C₈²

—

C₂²×C₁₃

C₄×D₈

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

#443

yes

characteristic

C₄×C₁₃

C₂×Q₃₂

yes

characteristic

C₄×C₁₃

#508

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₃

yes

characteristic

C₂×C₁₆

C₄×D₁₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

C₈×D₈

yes

characteristic

C₂×C₁₃

C₄×Q₃₂

yes

characteristic

100

C₂×C₁₃

#2178

yes

characteristic

101

C₂×Q₈

—

102

Q₁₆

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

109

C₂×C₈

#6131

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

D₄×D₁₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

C₈×D₁₃

yes

—

117

C₁₃

C₈×Q₃₂

yes

characteristic, complemented, Sylow

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#88147

yes

characteristic

123

C₂×C₄

D₈×D₁₃

yes

characteristic

124

C₂×C₄

#88173

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

C₂×C₈×D₁₃

yes

characteristic

131

C₈

#88161

yes

characteristic

132

C₂²

#11718533

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

104

—

136

C₄

#11718495

yes

characteristic

137

C₄

#11718556

yes

characteristic

138

C₄

#11718573

yes

characteristic

139

C₄

D₁₃×Q₃₂

yes

characteristic

140

C₂

#13113630

yes

characteristic

141

C₂

#13116694

yes

characteristic

142

C₂

#13116716

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1371	C₂
socle, prime radical	C₂²×C₁₃	C₄×D₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1371	D₁₃
hypocenter	C₁₃	C₈×Q₃₂
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂×C₄

D₈×D₁₃

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1371

C₂

socle, prime radical

C₂²×C₁₃

C₄×D₈

cosocle

C₄×C₈×C₁₃

C₂³

hypercenter

#1371

D₁₃

hypocenter

C₁₃

C₈×Q₃₂

LCS

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
29. Dic₁₀₄	C₈	semidirect
65. #1055	Q₃₂	semidirect
117. C₁₃	C₈×Q₃₂	semidirect

H ≅ G/N

type

29. Dic₁₀₄

C₈

semidirect

65. #1055

Q₃₂

semidirect

117. C₁₃

C₈×Q₃₂

semidirect

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

subgroup	index
C₄×Dic₁₀₄	2
#13113642	2
#13112799	2
C₁₆×#1055	2
C₁₃×#1371	2
C₈×Q₃₂	13

subgroup

index

C₄×Dic₁₀₄

#13113642

#13112799

C₁₆×#1055

C₁₃×#1371

C₈×Q₃₂

Subgroups — #15774652

Conjugacy classes of subgroups

Complements

Maximal subgroups