Group #15783804 — subgroups

Subgroups — #15783804

Order 3328.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 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	#13117110	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13113038	C₂	1	1	yes	maximal, characteristic
4	#13113037	C₂	1	1	yes	maximal, characteristic
5	#13113031	C₂	1	1	yes	maximal, characteristic
6	#13113017	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₁₃	—	2	1	no	—
10	#11718535	—	2	1	no	—
11	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
12	C₁₃×#366	C₂²	1	1	yes	characteristic
13	#11718156	—	2	1	no	—
14	#11718096	C₂²	1	1	yes	characteristic
15	#11718096	C₂²	1	1	yes	characteristic
16	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88034	C₂²	1	1	yes	characteristic
18	#11718081	—	2	1	no	—
19	C₂×C₁₃×Q₁₆	—	2	1	no	—
20	Q₈×Dic₁₃	—	4	1	no	—
21	#88142	—	4	1	no	—
22	C₁₃×Q₃₂	—	2	1	no	—
23	C₁₃×Q₃₂	Q₈	1	2	yes	complemented
24	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
25	C₁₃×#107	—	2	1	no	—
26	#88075	—	4	1	no	—
27	#88058	—	2	1	no	—
28	#88058	—	2	1	no	—
29	C₈×Dic₁₃	D₄	1	1	yes	characteristic
30	C₂×#6090	D₄	1	1	yes	characteristic
31	#88047	—	4	1	no	—
32	C₁₃×C₃₂	—	2	1	no	—
33	#88034	—	2	1	no	—
34	#13704	—	13	1	no	maximal, Sylow
35	C₂×Q₈×C₁₃	—	4	1	no	—
36	C₂×Dic₂₆	—	8	1	no	—
37	C₁₃×Q₁₆	—	2	2	no	—
38	C₁₃×Q₁₆	—	4	1	no	—
39	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₁₃×C₄⋊C₄	—	4	1	no	—
41	C₄⋊Dic₁₃	—	4	1	no	—
42	C₄⋊Dic₁₃	—	4	1	no	—
43	C₄⋊Dic₁₃	—	8	1	no	—
44	#6101	—	8	1	no	—
45	C₄×Dic₁₃	—	8	1	no	—
46	C₄×Dic₁₃	D₈	1	1	yes	characteristic
47	C₂×#1055	D₈	1	1	yes	characteristic
48	C₁₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₁₃×C₁₆	#41	1	1	yes	characteristic
50	#6090	—	2	1	no	—
51	#2252	—	13	1	no	—
52	C₄×Q₃₂	—	13	1	no	—
53	#1424	—	13	1	no	—
54	#1424	—	13	1	no	—
55	C₄⋊C₃₂	—	13	1	no	—
56	#1416	—	13	1	no	—
57	#1416	D₁₃	1	1	yes	characteristic
58	Q₈×C₁₃	—	4	2	no	—
59	Q₈×C₁₃	—	8	1	no	—
60	C₂×C₄×C₁₃	—	8	1	no	—
61	C₂×C₄×C₁₃	—	8	1	no	—
62	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
63	C₂×Dic₁₃	—	8	1	no	—
64	C₂×Dic₁₃	—	8	1	no	—
65	C₂×Dic₁₃	D₁₆	1	1	yes	characteristic
66	C₂×Dic₁₃	D₁₆	1	1	yes	characteristic
67	Dic₂₆	—	16	2	no	—
68	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₈×C₁₃	#137	1	1	yes	characteristic
70	#1055	—	2	1	no	—
71	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
72	#500	—	26	1	no	—
73	C₄×Q₁₆	—	26	1	no	—
74	C₂×C₃₂	—	13	1	no	—
75	C₂×C₃₂	C₂×D₁₃	1	1	yes	characteristic
76	#366	—	13	1	no	—
77	#366	—	13	1	no	—
78	#366	—	26	1	no	—
79	#366	C₂×D₁₃	1	1	yes	characteristic
80	#358	—	26	1	no	—
81	C₄×C₁₆	—	13	1	no	—
82	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
83	C₄×C₁₃	—	8	2	no	—
84	C₄×C₁₃	—	16	1	no	—
85	C₄×C₁₃	#474	1	1	yes	characteristic
86	C₄×C₁₃	#510	1	1	yes	characteristic
87	Dic₁₃	—	2	1	no	—
88	Dic₁₃	—	16	1	no	—
89	Dic₁₃	—	16	1	no	—
90	Dic₁₃	SD₆₄	1	2	yes	—
91	C₂×Q₁₆	—	2	1	no	—
92	C₄⋊Q₈	—	52	1	no	—
93	C₄×Q₈	—	52	1	no	—
94	Q₃₂	—	2	1	no	—
95	Q₃₂	Dic₂₆	1	2	yes	complemented
96	C₂×C₁₆	—	13	1	no	—
97	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
98	#107	—	2	1	no	—
99	#107	—	26	1	no	—
100	#107	—	26	1	no	—
101	#107	—	52	1	no	—
102	#103	—	52	1	no	—
103	C₄×C₈	—	13	1	no	—
104	C₃₂	—	2	1	no	—
105	C₃₂	—	26	1	no	—
106	C₂×C₁₃	#2224	1	1	yes	characteristic
107	C₂×C₁₃	C₂×SD₆₄	1	1	yes	characteristic
108	C₂×C₁₃	#2264	1	1	yes	characteristic
109	C₂×Q₈	—	4	1	no	—
110	C₂×Q₈	—	104	1	no	—
111	Q₁₆	—	2	2	no	—
112	Q₁₆	—	4	1	no	—
113	C₂×C₈	—	13	1	no	—
114	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	52	1	no	—
117	C₄⋊C₄	—	52	1	no	—
118	C₄⋊C₄	—	104	1	no	—
119	C₄⋊C₄	—	104	1	no	—
120	C₄²	—	13	1	no	—
121	C₄²	—	104	1	no	—
122	C₁₆	—	26	1	no	—
123	C₁₆	C₂×Dic₂₆	1	1	yes	characteristic
124	C₁₆	#6129	1	1	yes	characteristic
125	C₁₃	#13704	1	1	yes	characteristic, complemented, Sylow
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	208	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	13	1	no	—
132	C₂×C₄	—	13	1	no	—
133	C₂×C₄	—	104	1	no	—
134	C₂×C₄	—	104	1	no	—
135	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
136	C₈	—	26	1	no	—
137	C₈	#88132	1	1	yes	characteristic
138	C₈	#88169	1	1	yes	characteristic
139	C₂²	D₁₃×D₁₆	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	13	2	no	—
142	C₄	—	16	1	no	—
143	C₄	—	26	1	no	—
144	C₄	—	208	1	no	—
145	C₄	—	208	1	no	—
146	C₄	#11718420	1	1	yes	characteristic
147	C₄	#11718576	1	1	yes	characteristic
148	C₂	#13116971	1	1	yes	characteristic
149	C₂	D₁₃×SD₆₄	1	1	yes	characteristic
150	C₂	#13117155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117110

C₂

yes

maximal, characteristic

Q₃₂×Dic₁₃

C₂

yes

maximal, characteristic

#13113038

C₂

yes

maximal, characteristic

#13113037

C₂

yes

maximal, characteristic

#13113031

C₂

yes

maximal, characteristic

#13113017

C₂

yes

maximal, characteristic

C₁₃×#1416

C₂

yes

maximal, characteristic

C₂×C₁₃×Q₃₂

C₂²

yes

characteristic

Q₁₆×Dic₁₃

—

#11718535

—

C₂×C₁₃×C₃₂

C₂²

yes

characteristic

C₁₃×#366

C₂²

yes

characteristic

#11718156

—

#11718096

C₂²

yes

characteristic

#11718096

C₂²

yes

characteristic

C₁₆×Dic₁₃

C₂²

yes

characteristic

C₂×#88034

C₂²

yes

characteristic

#11718081

—

C₂×C₁₃×Q₁₆

—

Q₈×Dic₁₃

—

#88142

—

C₁₃×Q₃₂

—

C₁₃×Q₃₂

Q₈

yes

complemented

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₁₃×#107

—

#88075

—

#88058

—

#88058

—

C₈×Dic₁₃

D₄

yes

characteristic

C₂×#6090

D₄

yes

characteristic

#88047

—

C₁₃×C₃₂

—

#88034

—

#13704

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₈

yes

characteristic

C₂×#1055

D₈

yes

characteristic

C₁₃×C₁₆

C₂×Q₈

yes

C₁₃×C₁₆

yes

—

—

C₄×Q₃₂

—

#1424

—

#1424

—

C₄⋊C₃₂

—

#1416

—

#1416

D₁₃

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

D₁₆

yes

characteristic

C₂×Dic₁₃

D₁₆

yes

characteristic

Dic₂₆

—

C₈×C₁₃

C₂²⋊Q₈

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

C₂×D₁₃

yes

characteristic

#500

—

C₄×Q₁₆

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₁₃

yes

—

—

—

C₂×D₁₃

yes

characteristic

#358

—

C₄×C₁₆

—

C₂²×C₁₃

C₂×D₁₆

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

SD₆₄

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

Dic₂₆

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₁₃

yes

characteristic

#107

—

#107

—

100

#107

—

101

#107

—

102

#103

—

103

C₄×C₈

—

104

C₃₂

—

105

C₃₂

—

106

C₂×C₁₃

#2224

yes

characteristic

107

C₂×C₁₃

C₂×SD₆₄

yes

characteristic

108

C₂×C₁₃

#2264

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

104

—

111

Q₁₆

—

112

Q₁₆

—

113

C₂×C₈

—

114

C₂×C₈

D₄×D₁₃

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

104

—

119

C₄⋊C₄

—

104

—

120

C₄²

—

121

C₄²

—

104

—

122

C₁₆

—

123

C₁₆

C₂×Dic₂₆

yes

characteristic

124

C₁₆

#6129

yes

characteristic

125

C₁₃

#13704

yes

characteristic, complemented, Sylow

126

Q₈

—

127

Q₈

—

128

Q₈

—

208

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

104

—

134

C₂×C₄

—

104

—

135

C₂×C₄

D₈×D₁₃

yes

characteristic

136

C₈

—

137

C₈

#88132

yes

characteristic

138

C₈

#88169

yes

characteristic

139

C₂²

D₁₃×D₁₆

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

208

—

145

C₄

—

208

—

146

C₄

#11718420

yes

characteristic

147

C₄

#11718576

yes

characteristic

148

C₂

#13116971

yes

characteristic

149

C₂

D₁₃×SD₆₄

yes

characteristic

150

C₂

#13117155

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₃×D₁₆
commutator, cosocle	C₂×C₁₃×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₁₃×#1416	C₂
socle, prime radical	C₂²×C₁₃	C₂×D₁₆
hypercenter	#1416	D₁₃
hypocenter	C₁₃	#13704
LCS	G ▷ C₂×C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₃×D₁₆

commutator, cosocle

C₂×C₁₃×C₁₆

C₂³

Frattini

C₂×C₁₆

C₂²×D₁₃

Fitting

C₁₃×#1416

C₂

socle, prime radical

C₂²×C₁₃

C₂×D₁₆

D₁₃

C₁₃

G ▷ C₂×C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
23. C₁₃×Q₃₂	Q₈	semidirect
95. Q₃₂	Dic₂₆	semidirect
125. C₁₃	#13704	semidirect

H ≅ G/N

type

23. C₁₃×Q₃₂

Q₈

semidirect

95. Q₃₂

Dic₂₆

semidirect

125. C₁₃

#13704

semidirect

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

subgroup	index
#13117110	2
Q₃₂×Dic₁₃	2
#13113038	2
#13113037	2
#13113031	2
#13113017	2
C₁₃×#1416	2
#13704	13

subgroup

index

Q₃₂×Dic₁₃

C₁₃×#1416

Subgroups — #15783804

Conjugacy classes of subgroups

Complements

Maximal subgroups