Group #13122829 — subgroups

Subgroups — #13122829

Order 1664.

432 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 157 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×#391	C₂	1	1	yes	maximal, characteristic
2	#11718891	C₂	1	2	yes	maximal
3	#11718640	C₂	1	2	yes	maximal
4	#11718590	C₂	1	2	yes	maximal
5	#11718317	C₂	1	4	yes	maximal
6	#11718307	C₂	1	4	yes	maximal
7	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₆	C₂²	1	2	yes	—
9	C₂²×Dic₂₆	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₂×#6101	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
16	C₁₃×#95	C₂²	1	4	yes	—
17	#88071	C₂²	1	2	yes	—
18	#88071	C₂²	1	4	yes	—
19	#88071	C₂²	1	4	yes	—
20	#88071	C₂²	1	4	yes	—
21	C₂×Q₈×C₁₃	—	4	1	no	—
22	C₂×Q₈×C₁₃	—	4	1	no	—
23	C₂×Q₈×C₁₃	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂²×Dic₁₃	C₂³	1	4	yes	—
29	C₂²×Dic₁₃	C₂³	1	4	yes	—
30	C₂×Dic₂₆	—	4	2	no	—
31	C₂×Dic₂₆	—	4	2	no	—
32	C₂×Dic₂₆	—	4	4	no	—
33	C₂×Dic₂₆	—	4	2	no	—
34	C₂×Dic₂₆	Q₈	1	8	yes	complemented
35	C₁₃×C₄⋊C₄	—	4	1	no	—
36	C₄²×C₁₃	—	4	1	no	—
37	C₄⋊Dic₁₃	—	4	2	no	—
38	C₄⋊Dic₁₃	—	4	2	no	—
39	#6101	—	4	4	no	—
40	C₄×Dic₁₃	—	4	2	no	—
41	#2811	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
43	Q₈×C₁₃	—	4	4	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	1	no	—
49	C₂×C₄×C₁₃	—	2	1	no	—
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂×Dic₁₃	—	2	4	no	—
56	C₂×Dic₁₃	—	2	4	no	—
57	C₂×Dic₁₃	—	2	4	no	—
58	C₂×Dic₁₃	—	2	4	no	—
59	C₂×Dic₁₃	—	2	4	no	—
60	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
61	Dic₂₆	—	4	8	no	—
62	Dic₂₆	—	4	8	no	—
63	#395	—	13	2	no	—
64	#395	—	13	4	no	—
65	#391	—	13	2	no	—
66	#391	—	13	2	no	—
67	#391	—	13	4	no	—
68	#391	D₁₃	1	1	yes	characteristic
69	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
70	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
71	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
74	C₄×C₁₃	—	2	4	no	—
75	C₄×C₁₃	—	2	4	no	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	4	2	no	—
78	Dic₁₃	—	2	8	no	—
79	Dic₁₃	—	4	4	no	—
80	C₂²×Q₈	—	13	2	no	—
81	C₂²×Q₈	—	13	2	no	—
82	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	13	4	no	—
84	C₂×C₄⋊C₄	—	13	2	no	—
85	C₂×C₄⋊C₄	—	13	2	no	—
86	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
87	C₂×C₄²	—	13	2	no	—
88	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
89	#95	—	13	2	no	—
90	#95	—	13	4	no	—
91	#95	—	13	4	no	—
92	#95	—	13	4	no	—
93	#95	C₂×D₁₃	1	4	yes	—
94	C₂×C₁₃	#543	1	1	yes	characteristic
95	C₂×C₁₃	#543	1	2	yes	—
96	C₂×C₁₃	Q₈²	1	1	yes	characteristic
97	C₂×C₁₃	Q₈²	1	2	yes	—
98	C₂×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
99	C₂×Q₈	—	4	1	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂×Q₈	—	13	8	no	—
102	C₂×Q₈	—	52	2	no	—
103	C₂×Q₈	—	52	2	no	—
104	C₂×Q₈	—	52	2	no	—
105	C₂×Q₈	—	52	4	no	—
106	C₂×Q₈	Dic₂₆	1	4	yes	complemented
107	C₂²×C₄	—	13	4	no	—
108	C₂²×C₄	—	13	4	no	—
109	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	52	4	no	—
115	C₄⋊C₄	—	52	2	no	—
116	C₄⋊C₄	—	52	2	no	—
117	C₄²	—	4	1	no	—
118	C₄²	—	52	2	no	—
119	C₁₃	#2811	1	1	yes	characteristic, complemented, Sylow
120	C₂³	C₂³×D₁₃	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	Q₈	—	52	8	no	—
123	Q₈	—	52	8	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	13	8	no	—
133	C₂×C₄	—	26	4	no	—
134	C₂×C₄	—	26	4	no	—
135	C₂×C₄	—	26	4	no	—
136	C₂×C₄	—	26	4	no	—
137	C₂×C₄	—	26	4	no	—
138	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
139	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
140	C₂×C₄	Q₈×D₁₃	1	4	yes	—
141	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
142	C₂²	#88251	1	1	yes	characteristic
143	C₂²	#88251	1	1	yes	characteristic
144	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
145	C₂²	#88256	1	2	yes	—
146	C₄	—	2	2	no	—
147	C₄	—	2	4	no	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	26	8	no	—
151	C₄	—	52	4	no	—
152	C₂	#11719188	1	1	yes	characteristic
153	C₂	Q₈×Dic₂₆	1	2	yes	—
154	C₂	#11719339	1	2	yes	—
155	C₂	#11719388	1	1	yes	characteristic
156	C₂	#11719427	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#391

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal

#11718640

C₂

yes

maximal

#11718590

C₂

yes

maximal

#11718317

C₂

yes

maximal

#11718307

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

Q₈

yes

complemented

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

Q₈

yes

complemented

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

#2811

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

Dic₂₆

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

#543

yes

characteristic

C₂×C₁₃

#543

yes

—

C₂×C₁₃

Q₈²

yes

characteristic

C₂×C₁₃

Q₈²

yes

—

C₂×C₁₃

H(3,𝔽₄)

yes

characteristic

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

Dic₂₆

yes

complemented

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₁₃

yes

—

110

C₂²×C₄

C₂²×D₁₃

yes

characteristic

111

C₂²×C₄

C₂²×D₁₃

yes

—

112

C₂²×C₄

C₂²×D₁₃

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₁₃

#2811

yes

characteristic, complemented, Sylow

120

C₂³

C₂³×D₁₃

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×Dic₂₆

yes

—

139

C₂×C₄

C₂×Dic₂₆

yes

—

140

C₂×C₄

Q₈×D₁₃

yes

—

141

C₂²

C₂²×Dic₂₆

yes

characteristic

142

C₂²

#88251

yes

characteristic

143

C₂²

#88251

yes

characteristic

144

C₂²

C₂×Q₈×D₁₃

yes

—

145

C₂²

#88256

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₂

#11719188

yes

characteristic

153

C₂

Q₈×Dic₂₆

yes

—

154

C₂

#11719339

yes

—

155

C₂

#11719388

yes

characteristic

156

C₂

#11719427

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#391	C₂
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2811
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#391

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
23. C₂×Q₈×C₁₃	Q₈	semidirect
34. C₂×Dic₂₆	Q₈	semidirect
106. C₂×Q₈	Dic₂₆	semidirect
119. C₁₃	#2811	semidirect

H ≅ G/N

type

23. C₂×Q₈×C₁₃

Q₈

semidirect

34. C₂×Dic₂₆

Q₈

semidirect

106. C₂×Q₈

Dic₂₆

semidirect

119. C₁₃

#2811

semidirect

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

subgroup	index
C₁₃×#391	2
#11718891	2
#11718640	2
#11718590	2
#11718317	2
#11718307	2
#2811	13

subgroup

index

C₁₃×#391

Subgroups — #13122829

Conjugacy classes of subgroups

Complements

Maximal subgroups