Group #15749989 — subgroups

Subgroups — #15749989

Order 3328.

252 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 79 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	#13114882	C₂	1	2	yes	maximal
2	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
4	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic, complemented
6	#11718901	—	2	2	no	—
7	#11718767	—	2	2	no	—
8	C₁₃×C₂²⋊C₁₆	—	2	2	no	—
9	#11718149	C₄	1	4	yes	complemented
10	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
11	C₁₃×#131	—	4	1	no	—
12	C₁₃×#131	—	4	1	no	—
13	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
14	C₂×C₁₃×M₁₆	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	#88180	—	2	4	no	—
18	C₁₃×M₃₂	—	4	1	no	—
19	C₂×C₁₃×C₁₆	—	2	1	no	—
20	C₂×C₁₃×C₁₆	—	2	1	no	—
21	C₂×C₁₃×C₁₆	—	4	1	no	—
22	#88074	—	4	2	no	—
23	#88074	—	4	2	no	—
24	#88058	—	4	2	no	—
25	#88057	—	4	2	no	—
26	#7228	—	13	1	no	maximal, Sylow
27	C₁₃×#41	—	4	1	no	—
28	C₁₃×#41	—	4	1	no	—
29	C₂×Q₈×C₁₃	Q₁₆	1	1	yes	characteristic
30	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
31	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
34	C₁₃×M₁₆	—	4	2	no	—
35	C₁₃×M₁₆	—	4	1	no	—
36	C₂×C₈×C₁₃	—	2	1	no	—
37	C₂×C₈×C₁₃	—	2	1	no	—
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂²⋊Dic₁₃	—	4	4	no	—
44	C₄⋊Dic₁₃	—	4	4	no	—
45	C₄×Dic₁₃	—	4	4	no	—
46	C₁₃×C₁₆	—	4	1	no	—
47	C₁₃×C₁₆	—	4	1	no	—
48	#2113	D₁₃	1	1	yes	characteristic
49	#1812	—	13	2	no	—
50	C₂³×C₁₃	#102	1	2	yes	—
51	C₂³×C₁₃	#106	1	1	yes	characteristic
52	Q₈×C₁₃	—	4	1	no	—
53	D₄×C₁₃	—	4	1	no	—
54	D₄×C₁₃	—	4	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#103	1	2	yes	—
61	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
62	C₂×Dic₁₃	—	8	4	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	—	2	2	no	—
65	C₈×C₁₃	#99	1	1	yes	characteristic
66	C₈×C₁₃	#99	1	1	yes	characteristic
67	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
68	C₂×M₃₂	Dic₁₃	1	1	yes	characteristic, complemented
69	C₂²×C₁₆	Dic₁₃	1	1	yes	characteristic, complemented
70	#427	—	26	2	no	—
71	#410	—	26	2	no	—
72	C₂²⋊C₁₆	—	2	2	no	—
73	H(3,ℤ/4)	—	13	4	no	—
74	C₂²×C₁₃	—	2	2	no	—
75	C₂²×C₁₃	—	2	1	no	—
76	C₂²×C₁₃	—	4	2	no	—
77	C₂²×C₁₃	—	4	1	no	—
78	C₂²×C₁₃	#340	1	1	yes	characteristic
79	C₄×C₁₃	—	2	2	no	—
80	C₄×C₁₃	—	2	1	no	—
81	C₄×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
82	C₄×C₁₃	#342	1	1	yes	characteristic
83	Dic₁₃	—	16	4	no	—
84	C₂×#41	D₅₂	1	1	yes	characteristic
85	#131	—	4	1	no	—
86	#131	—	4	1	no	—
87	C₂×M₁₆	Dic₂₆	1	1	yes	characteristic
88	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
89	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
90	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
91	#117	—	26	4	no	—
92	M₃₂	—	4	1	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	4	1	no	—
96	#107	—	52	2	no	—
97	#106	—	52	2	no	—
98	#99	—	52	2	no	—
99	#99	—	52	2	no	—
100	C₂×C₁₃	—	2	1	no	—
101	C₂×C₁₃	—	4	2	no	—
102	C₂×C₁₃	—	4	1	no	—
103	C₂×C₁₃	#1289	1	1	yes	characteristic
104	#41	—	4	1	no	—
105	#41	—	4	1	no	—
106	C₂×Q₈	Dic₅₂	1	1	yes	characteristic
107	C₂×D₄	#6095	1	2	yes	—
108	C₂×D₄	D₁₀₄	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
110	C₂²×C₄	#6103	1	2	yes	—
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
118	C₂×C₈	#6101	1	2	yes	—
119	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
120	C₄⋊C₄	—	52	4	no	—
121	C₂²⋊C₄	—	52	4	no	—
122	C₄²	—	52	4	no	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₃	#7228	1	1	yes	characteristic, complemented, Sylow
126	C₂³	#88057	1	1	yes	characteristic
127	C₂³	#88061	1	2	yes	—
128	Q₈	—	4	1	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	104	4	no	—
136	C₂×C₄	#88056	1	2	yes	—
137	C₂×C₄	#88058	1	1	yes	characteristic
138	C₂×C₄	#88071	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	2	no	—
141	C₈	#88074	1	1	yes	characteristic
142	C₈	#88074	1	1	yes	characteristic
143	C₂²	—	2	2	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	2	no	—
147	C₂²	#11718143	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	1	no	—
150	C₄	—	208	4	no	—
151	C₄	#11718118	1	1	yes	characteristic
152	C₄	#11718149	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	4	2	no	—
155	C₂	—	4	1	no	—
156	C₂	#13112704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114882

C₂

yes

maximal

C₁₃×#2113

C₂

yes

maximal, characteristic

C₂×C₁₃×#131

C₂²

yes

characteristic

C₂×C₁₃×M₃₂

C₄

yes

characteristic, complemented

C₂²×C₁₃×C₁₆

C₄

yes

characteristic, complemented

#11718901

—

#11718767

—

C₁₃×C₂²⋊C₁₆

—

#11718149

C₄

yes

complemented

C₂×C₁₃×#41

D₄

yes

characteristic

C₁₃×#131

—

C₁₃×#131

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

Q₈

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

—

#88180

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

—

—

—

—

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

Q₁₆

yes

characteristic

C₂×D₄×C₁₃

D₈

yes

characteristic

C₂×D₄×C₁₃

SD₁₆

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

#2113

D₁₃

yes

characteristic

#1812

—

C₂³×C₁₃

#102

yes

—

C₂³×C₁₃

#106

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#95

yes

characteristic

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

#107

yes

characteristic

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

C₂×D₁₃

yes

characteristic

C₂×M₃₂

Dic₁₃

yes

characteristic, complemented

C₂²×C₁₆

Dic₁₃

yes

characteristic, complemented

#427

—

#410

—

C₂²⋊C₁₆

—

H(3,ℤ/4)

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#340

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×#41

D₅₂

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₂₆

yes

characteristic

C₂×M₁₆

C₄×D₁₃

yes

—

C₂²×C₈

C₂×Dic₁₃

yes

characteristic

C₂²×C₈

C₂²⋊D₁₃

yes

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

100

C₂×C₁₃

—

101

C₂×C₁₃

—

102

C₂×C₁₃

—

103

C₂×C₁₃

#1289

yes

characteristic

104

#41

—

105

#41

—

106

C₂×Q₈

Dic₅₂

yes

characteristic

107

C₂×D₄

#6095

yes

—

108

C₂×D₄

D₁₀₄

yes

characteristic

109

C₂²×C₄

C₄⋊Dic₁₃

yes

characteristic

110

C₂²×C₄

#6103

yes

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₄×Dic₁₃

yes

characteristic

118

C₂×C₈

#6101

yes

—

119

C₂×C₈

C₂²⋊Dic₁₃

yes

characteristic

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₁₆

—

124

C₁₆

—

125

C₁₃

#7228

yes

characteristic, complemented, Sylow

126

C₂³

#88057

yes

characteristic

127

C₂³

#88061

yes

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

104

—

136

C₂×C₄

#88056

yes

—

137

C₂×C₄

#88058

yes

characteristic

138

C₂×C₄

#88071

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

#88074

yes

characteristic

142

C₈

#88074

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11718143

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

208

—

151

C₄

#11718118

yes

characteristic

152

C₄

#11718149

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#13112704

yes

characteristic

	subgroup	G/H
center	C₄	#11718149
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂×#131	C₂×D₁₃
Fitting	C₁₃×#2113	C₂
socle	C₂×C₁₃	#1289
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂×C₁₃×#131	C₂²
hypercenter	#2113	D₁₃
hypocenter	C₁₃	#7228
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

subgroup

G/H

center

C₄

#11718149

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₂×#131

C₂×D₁₃

Fitting

C₁₃×#2113

C₂

socle

C₂×C₁₃

#1289

prime radical

C₂×C₁₃×#41

D₄

cosocle

C₂×C₁₃×#131

C₂²

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

N	H ≅ G/N	type
4. C₂×C₁₃×M₃₂	C₄	semidirect
5. C₂²×C₁₃×C₁₆	C₄	semidirect
9. #11718149	C₄	semidirect
68. C₂×M₃₂	Dic₁₃	semidirect
69. C₂²×C₁₆	Dic₁₃	semidirect
125. C₁₃	#7228	semidirect

H ≅ G/N

type

4. C₂×C₁₃×M₃₂

C₄

semidirect

5. C₂²×C₁₃×C₁₆

C₄

semidirect

9. #11718149

C₄

semidirect

68. C₂×M₃₂

Dic₁₃

semidirect

69. C₂²×C₁₆

Dic₁₃

semidirect

125. C₁₃

#7228

semidirect

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

subgroup	index
#13114882	2
C₁₃×#2113	2
#7228	13

subgroup

index

#13114882

C₁₃×#2113

#7228

Subgroups — #15749989

Conjugacy classes of subgroups

Complements

Maximal subgroups