Group #15696194 — subgroups

Subgroups — C₁₃×#9855 #15696194

Order 3328.

422 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1493	C₂	1	4	yes	maximal
4	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#533	C₂²	1	2	yes	—
6	C₁₃×#500	C₄	1	2	yes	—
7	C₁₃×#499	—	2	2	no	—
8	C₁₃×#498	C₄	1	2	yes	—
9	C₂×C₁₃×#108	—	2	2	no	—
10	C₁₃×#427	—	2	2	no	—
11	C₁₃×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×Q₈⋊C₈	—	2	8	no	—
15	C₈²×C₁₃	—	2	2	no	—
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	D₄	1	2	yes	—
19	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
20	C₁₃×#125	—	2	2	no	—
21	C₁₃×C₂²⋊Q₈	—	2	4	no	—
22	C₄×Q₈×C₁₃	—	2	4	no	—
23	C₁₃×#117	—	2	2	no	—
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₁₃×#108	—	4	4	no	—
26	C₁₃×#107	—	2	4	no	—
27	C₁₃×#106	—	2	4	no	—
28	C₁₃×C₄⋊C₈	—	2	4	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	D₄	1	4	yes	—
33	C₄×C₈×C₁₃	D₄	1	2	yes	—
34	#9855	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×Q₈×C₁₃	—	2	4	no	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	#41	1	2	yes	—
38	C₁₃×M₁₆	—	4	4	no	—
39	C₂×C₈×C₁₃	—	2	8	no	—
40	C₂×C₈×C₁₃	—	2	4	no	—
41	C₂×C₈×C₁₃	—	4	2	no	—
42	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
45	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
46	C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₄⋊C₄	—	4	4	no	—
49	C₁₃×C₂²⋊C₄	—	4	2	no	—
50	C₄²×C₁₃	—	4	2	no	—
51	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×D₄	1	2	yes	—
54	#3160	C₂×C₁₃	1	1	yes	characteristic
55	#1848	C₂×C₁₃	1	1	yes	characteristic
56	#1493	C₂×C₁₃	1	4	yes	—
57	C₂×C₈²	C₂×C₁₃	1	1	yes	characteristic
58	C₂³×C₁₃	#124	1	1	yes	characteristic
59	Q₈×C₁₃	—	4	8	no	—
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	—	4	4	no	—
62	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₁₃	#121	1	2	yes	—
65	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
68	C₈×C₁₃	—	2	8	no	—
69	C₈×C₁₃	—	8	2	no	—
70	C₈×C₁₃	#103	1	4	yes	—
71	C₈×C₁₃	#103	1	4	yes	—
72	#533	C₂²×C₁₃	1	2	yes	—
73	#500	C₄×C₁₃	1	2	yes	—
74	#499	—	2	2	no	—
75	#498	C₄×C₁₃	1	2	yes	—
76	C₂×#108	—	2	2	no	—
77	#427	—	2	2	no	—
78	#423	C₂²×C₁₃	1	2	yes	—
79	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
81	Q₈⋊C₈	—	2	8	no	—
82	C₈²	—	2	2	no	—
83	C₂²×C₁₃	—	2	2	no	—
84	C₂²×C₁₃	#390	1	1	yes	characteristic
85	C₂²×C₁₃	#487	1	2	yes	—
86	C₄×C₁₃	—	2	2	no	—
87	C₄×C₁₃	—	8	4	no	—
88	C₄×C₁₃	C₂×#103	1	2	yes	—
89	C₄×C₁₃	#443	1	4	yes	—
90	C₄×C₁₃	#492	1	1	yes	characteristic
91	C₄×C₁₃	#494	1	1	yes	characteristic
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	D₄×C₁₃	1	2	yes	—
95	C₄⋊Q₈	C₂×C₄×C₁₃	1	4	yes	—
96	#125	—	2	2	no	—
97	C₂²⋊Q₈	—	2	4	no	—
98	C₄×Q₈	—	2	4	no	—
99	#117	—	2	2	no	—
100	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
101	#108	—	4	4	no	—
102	#107	—	2	4	no	—
103	#106	—	2	4	no	—
104	C₄⋊C₈	—	2	4	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
107	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
108	C₄×C₈	D₄×C₁₃	1	4	yes	—
109	C₄×C₈	D₄×C₁₃	1	2	yes	—
110	C₂×C₁₃	—	2	2	no	—
111	C₂×C₁₃	#1956	1	1	yes	characteristic
112	C₂×C₁₃	#1974	1	2	yes	—
113	C₂×Q₈	—	2	4	no	—
114	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
115	C₂²×C₄	C₁₃×#41	1	2	yes	—
116	M₁₆	—	4	4	no	—
117	C₂×C₈	—	2	8	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
121	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
122	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
123	C₂×C₈	C₁₃×Q₁₆	1	4	yes	—
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₂²⋊C₄	—	4	2	no	—
128	C₄²	—	4	2	no	—
129	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
130	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
131	C₄²	C₂×D₄×C₁₃	1	2	yes	—
132	C₁₃	#9855	1	1	yes	characteristic, direct product factor, Sylow
133	C₂³	C₁₃×#124	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
139	C₂×C₄	C₁₃×#121	1	2	yes	—
140	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
141	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
142	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
143	C₈	—	2	8	no	—
144	C₈	—	8	2	no	—
145	C₈	C₁₃×#103	1	4	yes	—
146	C₈	C₁₃×#103	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₁₃×#390	1	1	yes	characteristic
149	C₂²	C₁₃×#487	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×C₁₃×#103	1	2	yes	—
153	C₄	C₁₃×#443	1	4	yes	—
154	C₄	C₁₃×#492	1	1	yes	characteristic
155	C₄	C₁₃×#494	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₁₃×#1956	1	1	yes	characteristic
158	C₂	C₁₃×#1974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#3160

C₂

yes

maximal, characteristic

C₁₃×#1848

C₂

yes

maximal, characteristic

C₁₃×#1493

C₂

yes

maximal

C₂×C₈²×C₁₃

C₂

yes

maximal, characteristic

C₁₃×#533

C₂²

yes

—

C₁₃×#500

C₄

yes

—

C₁₃×#499

—

C₁₃×#498

C₄

yes

—

C₂×C₁₃×#108

—

C₁₃×#427

—

C₁₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×Q₈⋊C₈

—

C₈²×C₁₃

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₁₃×#125

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×#108

—

C₁₃×#107

—

C₁₃×#106

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

#9855

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

#3160

C₂×C₁₃

yes

characteristic

#1848

C₂×C₁₃

yes

characteristic

#1493

C₂×C₁₃

yes

—

C₂×C₈²

C₂×C₁₃

yes

characteristic

C₂³×C₁₃

#124

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#103

yes

—

C₈×C₁₃

#103

yes

—

#533

C₂²×C₁₃

yes

—

#500

C₄×C₁₃

yes

—

#499

—

#498

C₄×C₁₃

yes

—

C₂×#108

—

#427

—

#423

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₂²×C₁₃

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂²×C₁₃

—

C₂²×C₁₃

#390

yes

characteristic

C₂²×C₁₃

#487

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#103

yes

—

C₄×C₁₃

#443

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄×C₁₃

yes

—

C₄⋊Q₈

C₂×C₄×C₁₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

100

C₂×C₄²

C₂³×C₁₃

yes

characteristic

101

#108

—

102

#107

—

103

#106

—

104

C₄⋊C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×C₄×C₁₃

yes

characteristic

107

C₄×C₈

C₂×C₄×C₁₃

yes

characteristic

108

C₄×C₈

D₄×C₁₃

yes

—

109

C₄×C₈

D₄×C₁₃

yes

—

110

C₂×C₁₃

—

111

C₂×C₁₃

#1956

yes

characteristic

112

C₂×C₁₃

#1974

yes

—

113

C₂×Q₈

—

114

C₂²×C₄

C₂×D₄×C₁₃

yes

characteristic

115

C₂²×C₄

C₁₃×#41

yes

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₁₃×C₂²⋊C₄

yes

—

121

C₂×C₈

C₁₃×C₂²⋊C₄

yes

—

122

C₂×C₈

C₁₃×SD₁₆

yes

—

123

C₂×C₈

C₁₃×Q₁₆

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

C₂²×C₄×C₁₃

yes

characteristic

130

C₄²

C₂×D₄×C₁₃

yes

characteristic

131

C₄²

C₂×D₄×C₁₃

yes

—

132

C₁₃

#9855

yes

characteristic, direct product factor, Sylow

133

C₂³

C₁₃×#124

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×C₁₃×C₂²⋊C₄

yes

characteristic

138

C₂×C₄

C₄×D₄×C₁₃

yes

—

139

C₂×C₄

C₁₃×#121

yes

—

140

C₂×C₄

C₁₃×#127

yes

characteristic

141

C₂×C₄

C₂×C₁₃×SD₁₆

yes

—

142

C₂×C₄

C₂×C₁₃×Q₁₆

yes

—

143

C₈

—

144

C₈

—

145

C₈

C₁₃×#103

yes

—

146

C₈

C₁₃×#103

yes

—

147

C₂²

—

148

C₂²

C₁₃×#390

yes

characteristic

149

C₂²

C₁₃×#487

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×C₁₃×#103

yes

—

153

C₄

C₁₃×#443

yes

—

154

C₄

C₁₃×#492

yes

characteristic

155

C₄

C₁₃×#494

yes

characteristic

156

C₂

—

157

C₂

C₁₃×#1956

yes

characteristic

158

C₂

C₁₃×#1974

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₃	#127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#390
prime radical	C₂³×C₁₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃

#127

commutator

C₄²

C₂²×C₄×C₁₃

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃

C₂²×C₁₃

C₂³×C₁₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
34. #9855	C₁₃	direct (N × H)
132. C₁₃	#9855	direct (N × H)

H ≅ G/N

type

34. #9855

C₁₃

direct (N × H)

132. C₁₃

#9855

direct (N × H)

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

subgroup	index
C₁₃×#3160	2
C₁₃×#1848	2
C₁₃×#1493	2
C₂×C₈²×C₁₃	2
#9855	13

subgroup

index

C₁₃×#3160

C₁₃×#1848

C₁₃×#1493

C₂×C₈²×C₁₃

#9855

Subgroups — C₁₃×#9855 #15696194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#9855 #15696194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9855 #15696194