Group #15696178 — subgroups

Subgroups — C₁₃×#9839 #15696178

Order 3328.

542 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 214 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	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1990	C₂	1	4	yes	maximal
3	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×Q₁₆	—	2	2	no	—
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₃×#494	—	2	2	no	—
8	C₁₃×#494	C₄	1	4	yes	complemented
9	C₂×C₁₃×#106	—	2	2	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#384	C₂²	1	2	yes	—
13	C₁₃×#340	—	2	8	no	—
14	C₁₃×#322	—	2	2	no	—
15	C₂²×Q₈×C₁₃	—	2	2	no	—
16	C₂×C₁₃×Q₁₆	—	2	8	no	—
17	C₂×C₁₃×Q₁₆	—	4	4	no	—
18	C₂×C₁₃×Q₁₆	—	4	4	no	—
19	C₂²×C₈×C₁₃	D₄	1	4	yes	—
20	C₂²×C₈×C₁₃	D₄	1	2	yes	—
21	C₁₃×C₄⋊Q₈	—	2	2	no	—
22	C₁₃×C₄⋊Q₈	—	2	4	no	—
23	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#106	—	4	4	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃	D₄	1	2	yes	—
33	#9839	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×Q₈×C₁₃	—	2	4	no	—
35	C₂×Q₈×C₁₃	—	4	2	no	—
36	C₂×Q₈×C₁₃	—	4	4	no	—
37	C₂×Q₈×C₁₃	—	4	4	no	—
38	C₂²×C₄×C₁₃	—	4	2	no	—
39	C₂²×C₄×C₁₃	—	4	2	no	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
42	C₁₃×Q₁₆	—	4	16	no	—
43	C₂×C₈×C₁₃	—	2	4	no	—
44	C₂×C₈×C₁₃	—	2	8	no	—
45	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
46	C₂×C₈×C₁₃	D₈	1	4	yes	—
47	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
48	C₁₃×C₄⋊C₄	—	2	8	no	—
49	C₁₃×C₄⋊C₄	—	4	4	no	—
50	C₁₃×C₄⋊C₄	—	4	4	no	—
51	C₁₃×C₄⋊C₄	—	4	8	no	—
52	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₃	#41	1	2	yes	—
55	C₂×#494	C₂×C₁₃	1	1	yes	characteristic
56	#1990	C₂×C₁₃	1	4	yes	—
57	#1844	C₂×C₁₃	1	1	yes	characteristic
58	C₂×#322	C₂×C₁₃	1	1	yes	characteristic
59	C₂³×C₁₃	#127	1	1	yes	characteristic
60	Q₈×C₁₃	—	4	8	no	—
61	Q₈×C₁₃	—	8	4	no	—
62	C₂×C₄×C₁₃	—	4	4	no	—
63	C₂×C₄×C₁₃	—	8	2	no	—
64	C₂×C₄×C₁₃	—	8	2	no	—
65	C₂×C₄×C₁₃	—	8	2	no	—
66	C₂×C₄×C₁₃	—	8	4	no	—
67	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₁₃	#121	1	2	yes	—
70	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₁₃	#137	1	8	yes	—
74	C₈×C₁₃	—	4	4	no	—
75	C₈×C₁₃	#102	1	8	yes	—
76	C₂²×Q₁₆	—	2	2	no	—
77	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
78	#494	—	2	2	no	—
79	#494	C₄×C₁₃	1	4	yes	complemented
80	C₂×#106	—	2	2	no	—
81	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
82	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
83	#384	C₂²×C₁₃	1	2	yes	—
84	#340	—	2	8	no	—
85	#322	—	2	2	no	—
86	C₂²×C₁₃	#390	1	1	yes	characteristic
87	C₂²×C₁₃	#492	1	1	yes	characteristic
88	C₂²×C₁₃	#493	1	1	yes	characteristic
89	C₂²×C₁₃	#497	1	4	yes	—
90	C₄×C₁₃	—	8	4	no	—
91	C₄×C₁₃	—	16	2	no	—
92	C₄×C₁₃	C₂×#102	1	2	yes	—
93	C₄×C₁₃	#441	1	4	yes	—
94	C₄×C₁₃	#470	1	4	yes	—
95	C₄×C₁₃	#486	1	2	yes	—
96	C₂²×Q₈	—	2	2	no	—
97	C₂×Q₁₆	—	2	8	no	—
98	C₂×Q₁₆	—	4	4	no	—
99	C₂×Q₁₆	—	4	4	no	—
100	C₂²×C₈	D₄×C₁₃	1	4	yes	—
101	C₂²×C₈	D₄×C₁₃	1	2	yes	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	—	2	4	no	—
104	C₄⋊Q₈	C₂×C₄×C₁₃	1	4	yes	—
105	C₂×C₄⋊C₄	—	2	4	no	—
106	C₂×C₄⋊C₄	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
109	#106	—	4	4	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
113	C₄×C₈	D₄×C₁₃	1	2	yes	—
114	C₂×C₁₃	#1716	1	2	yes	—
115	C₂×C₁₃	#1722	1	2	yes	—
116	C₂×C₁₃	#1957	1	1	yes	characteristic
117	C₂×C₁₃	#1971	1	2	yes	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂×Q₈	—	4	4	no	—
121	C₂×Q₈	—	4	4	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
125	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
126	Q₁₆	—	4	16	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
130	C₂×C₈	C₁₃×D₈	1	4	yes	—
131	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
132	C₄⋊C₄	—	2	8	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	8	no	—
136	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
137	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
138	C₄²	C₁₃×#41	1	2	yes	—
139	C₁₃	#9839	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³	C₁₃×#127	1	1	yes	characteristic
141	Q₈	—	4	8	no	—
142	Q₈	—	8	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	8	4	no	—
148	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
150	C₂×C₄	C₁₃×#121	1	2	yes	—
151	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
152	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
153	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
154	C₂×C₄	C₁₃×#137	1	8	yes	—
155	C₈	—	4	4	no	—
156	C₈	C₁₃×#102	1	8	yes	—
157	C₂²	C₁₃×#390	1	1	yes	characteristic
158	C₂²	C₁₃×#492	1	1	yes	characteristic
159	C₂²	C₁₃×#493	1	1	yes	characteristic
160	C₂²	C₁₃×#497	1	4	yes	—
161	C₄	—	8	4	no	—
162	C₄	—	16	2	no	—
163	C₄	C₂×C₁₃×#102	1	2	yes	—
164	C₄	C₁₃×#441	1	4	yes	—
165	C₄	C₁₃×#470	1	4	yes	—
166	C₄	C₁₃×#486	1	2	yes	—
167	C₂	C₁₃×#1716	1	2	yes	—
168	C₂	C₁₃×#1722	1	2	yes	—
169	C₂	C₁₃×#1957	1	1	yes	characteristic
170	C₂	C₁₃×#1971	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#494

C₂

yes

maximal, characteristic

C₁₃×#1990

C₂

yes

maximal

C₁₃×#1844

C₂

yes

maximal, characteristic

C₂×C₁₃×#322

C₂

yes

maximal, characteristic

C₂²×C₁₃×Q₁₆

—

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×#494

—

C₁₃×#494

C₄

yes

complemented

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

C₁₃×#340

—

C₁₃×#322

—

C₂²×Q₈×C₁₃

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×#106

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

#9839

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₂×#494

C₂×C₁₃

yes

characteristic

#1990

C₂×C₁₃

yes

—

#1844

C₂×C₁₃

yes

characteristic

C₂×#322

C₂×C₁₃

yes

characteristic

C₂³×C₁₃

#127

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×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

#124

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₈×C₁₃

—

C₈×C₁₃

#102

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²×C₁₃

yes

—

#494

—

#494

C₄×C₁₃

yes

complemented

C₂×#106

—

C₂×C₄×C₈

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₂²×C₁₃

yes

characteristic

#384

C₂²×C₁₃

yes

—

#340

—

#322

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#102

yes

—

C₄×C₁₃

#441

yes

—

C₄×C₁₃

#470

yes

—

C₄×C₁₃

#486

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₂²×C₈

D₄×C₁₃

yes

—

101

C₂²×C₈

D₄×C₁₃

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

C₂×C₄×C₁₃

yes

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

C₂³×C₁₃

yes

characteristic

109

#106

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₂×C₄×C₁₃

yes

—

113

C₄×C₈

D₄×C₁₃

yes

—

114

C₂×C₁₃

#1716

yes

—

115

C₂×C₁₃

#1722

yes

—

116

C₂×C₁₃

#1957

yes

characteristic

117

C₂×C₁₃

#1971

yes

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂×D₄×C₁₃

yes

characteristic

125

C₂²×C₄

C₂×D₄×C₁₃

yes

—

126

Q₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₁₃×C₂²⋊C₄

yes

—

130

C₂×C₈

C₁₃×D₈

yes

—

131

C₂×C₈

C₁₃×SD₁₆

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

C₂²×C₄×C₁₃

yes

characteristic

137

C₄²

C₂×D₄×C₁₃

yes

characteristic

138

C₄²

C₁₃×#41

yes

—

139

C₁₃

#9839

yes

characteristic, direct product factor, Sylow

140

C₂³

C₁₃×#127

yes

characteristic

141

Q₈

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₂×C₁₃×C₂²⋊C₄

yes

characteristic

149

C₂×C₄

C₄×D₄×C₁₃

yes

—

150

C₂×C₄

C₁₃×#121

yes

—

151

C₂×C₄

C₁₃×#124

yes

characteristic

152

C₂×C₄

C₂×C₁₃×D₈

yes

—

153

C₂×C₄

C₂×C₁₃×SD₁₆

yes

—

154

C₂×C₄

C₁₃×#137

yes

—

155

C₈

—

156

C₈

C₁₃×#102

yes

—

157

C₂²

C₁₃×#390

yes

characteristic

158

C₂²

C₁₃×#492

yes

characteristic

159

C₂²

C₁₃×#493

yes

characteristic

160

C₂²

C₁₃×#497

yes

—

161

C₄

—

162

C₄

—

163

C₄

C₂×C₁₃×#102

yes

—

164

C₄

C₁₃×#441

yes

—

165

C₄

C₁₃×#470

yes

—

166

C₄

C₁₃×#486

yes

—

167

C₂

C₁₃×#1716

yes

—

168

C₂

C₁₃×#1722

yes

—

169

C₂

C₁₃×#1957

yes

characteristic

170

C₂

C₁₃×#1971

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃

#127

commutator

C₄²

C₂²×C₄×C₁₃

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
8. C₁₃×#494	C₄	semidirect
33. #9839	C₁₃	direct (N × H)
79. #494	C₄×C₁₃	semidirect
139. C₁₃	#9839	direct (N × H)

H ≅ G/N

type

8. C₁₃×#494

C₄

semidirect

33. #9839

C₁₃

direct (N × H)

79. #494

C₄×C₁₃

semidirect

139. C₁₃

#9839

direct (N × H)

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

subgroup	index
C₂×C₁₃×#494	2
C₁₃×#1990	2
C₁₃×#1844	2
C₂×C₁₃×#322	2
#9839	13

subgroup

index

C₂×C₁₃×#494

C₁₃×#1990

C₁₃×#1844

C₂×C₁₃×#322

#9839

Subgroups — C₁₃×#9839 #15696178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#9839 #15696178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9839 #15696178