Group #13117397 — subgroups

Subgroups — #125×Dic₁₃ #13117397

Order 1664.

484 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 249 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×#125	C₂	1	1	yes	maximal, characteristic
2	#11718646	C₂	1	2	yes	maximal
3	#11718642	C₂	1	4	yes	maximal
4	C₄⋊C₄×Dic₁₃	C₂	1	4	yes	maximal
5	C₄⋊C₄×Dic₁₃	C₂	1	2	yes	maximal
6	#11718597	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#125	C₄	1	8	yes	complemented
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
14	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
18	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
19	#88071	C₂²	1	4	yes	—
20	#88071	C₂²	1	4	yes	—
21	#88071	C₂²	1	4	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₃	C₂³	1	4	yes	—
26	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₃	C₂³	1	2	yes	—
28	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
30	C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
31	C₄²×C₁₃	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₁₃	—	2	4	no	—
33	C₄⋊Dic₁₃	—	2	8	no	—
34	C₄×Dic₁₃	—	2	2	no	—
35	C₄×Dic₁₃	—	2	4	no	—
36	C₄×Dic₁₃	—	2	4	no	—
37	C₄×Dic₁₃	—	2	8	no	—
38	C₄×Dic₁₃	Q₈	1	4	yes	—
39	C₄×#125	—	13	1	no	maximal, Sylow
40	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	2	4	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	4	no	—
45	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₁₃	C₂²×C₄	1	8	yes	—
47	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₃	#41	1	2	yes	—
50	C₂×Dic₁₃	—	2	2	no	—
51	C₂×Dic₁₃	—	2	2	no	—
52	C₂×Dic₁₃	—	2	2	no	—
53	C₂×Dic₁₃	—	2	4	no	—
54	C₂×Dic₁₃	—	2	4	no	—
55	C₂×Dic₁₃	—	2	4	no	—
56	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
57	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
58	C₂×Dic₁₃	#41	1	4	yes	—
59	C₂×Dic₁₃	#41	1	4	yes	—
60	C₂×#125	D₁₃	1	1	yes	characteristic
61	#389	—	13	2	no	—
62	#387	—	13	4	no	—
63	#382	—	13	1	no	—
64	C₄×C₄⋊C₄	—	13	4	no	—
65	C₄×C₄⋊C₄	—	13	2	no	—
66	C₄³	—	13	1	no	—
67	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
68	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
70	C₂²×C₁₃	C₂×#41	1	2	yes	—
71	C₂²×C₁₃	C₂×#41	1	2	yes	—
72	C₄×C₁₃	—	2	4	no	—
73	C₄×C₁₃	—	2	8	no	—
74	C₄×C₁₃	C₄×Q₈	1	4	yes	—
75	Dic₁₃	—	2	4	no	—
76	Dic₁₃	—	2	2	no	—
77	Dic₁₃	—	4	4	no	—
78	Dic₁₃	#125	1	4	yes	direct product factor
79	#125	Dic₁₃	1	8	yes	direct product factor
80	C₂×C₄⋊C₄	—	13	2	no	—
81	C₂×C₄⋊C₄	—	13	4	no	—
82	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
83	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
84	C₂×C₄²	—	13	1	no	—
85	C₂×C₄²	—	13	2	no	—
86	C₂×C₄²	—	13	2	no	—
87	C₂×C₄²	—	13	1	no	—
88	C₂×C₄²	—	13	4	no	—
89	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
90	#95	—	13	4	no	—
91	#95	—	13	4	no	—
92	#95	—	13	4	no	—
93	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₂×C₁₃	C₄×#41	1	2	yes	—
95	C₂×C₁₃	C₂×#125	1	1	yes	characteristic
96	C₂×C₁₃	#529	1	2	yes	—
97	C₂×C₁₃	#533	1	1	yes	characteristic
98	C₂²×C₄	—	13	1	no	—
99	C₂²×C₄	—	13	1	no	—
100	C₂²×C₄	—	13	2	no	—
101	C₂²×C₄	—	13	4	no	—
102	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
103	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
104	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
105	C₄⋊C₄	—	26	8	no	—
106	C₄⋊C₄	—	26	4	no	—
107	C₄⋊C₄	C₂×Dic₁₃	1	16	yes	—
108	C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
109	C₄²	—	13	4	no	—
110	C₄²	—	26	4	no	—
111	C₄²	—	26	8	no	—
112	C₄²	—	26	4	no	—
113	C₄²	—	26	2	no	—
114	C₄²	C₂×Dic₁₃	1	4	yes	—
115	C₁₃	C₄×#125	1	1	yes	characteristic, complemented, Sylow
116	C₂³	C₂³×D₁₃	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	13	2	no	—
122	C₂×C₄	—	13	4	no	—
123	C₂×C₄	—	13	2	no	—
124	C₂×C₄	—	13	4	no	—
125	C₂×C₄	—	26	2	no	—
126	C₂×C₄	—	26	2	no	—
127	C₂×C₄	—	26	2	no	—
128	C₂×C₄	—	26	4	no	—
129	C₂×C₄	—	26	4	no	—
130	C₂×C₄	—	26	4	no	—
131	C₂×C₄	Q₈×D₁₃	1	2	yes	—
132	C₂×C₄	#6131	1	2	yes	—
133	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
134	C₂×C₄	C₂²×Dic₁₃	1	8	yes	—
135	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
136	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
137	C₂²	C₂×#6131	1	1	yes	characteristic
138	C₂²	#41×D₁₃	1	2	yes	—
139	C₂²	#41×D₁₃	1	2	yes	—
140	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	8	no	—
143	C₄	—	13	4	no	—
144	C₄	—	26	2	no	—
145	C₄	—	26	4	no	—
146	C₄	—	52	4	no	—
147	C₄	Q₈×Dic₁₃	1	4	yes	—
148	C₂	D₁₃×#125	1	1	yes	characteristic
149	C₂	#11719393	1	1	yes	characteristic
150	C₂	#11719396	1	2	yes	—
151	C₂	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
152	C₂	#41×Dic₁₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#125

C₂

yes

maximal, characteristic

#11718646

C₂

yes

maximal

#11718642

C₂

yes

maximal

C₄⋊C₄×Dic₁₃

C₂

yes

maximal

C₄⋊C₄×Dic₁₃

C₂

yes

maximal

#11718597

C₂

yes

maximal, characteristic

C₄²×Dic₁₃

C₂

yes

maximal, characteristic

C₁₃×#125

C₄

yes

complemented

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

Q₈

yes

—

C₄×#125

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

#41

yes

—

C₂×Dic₁₃

#41

yes

—

C₂×#125

D₁₃

yes

—

—

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#125

yes

direct product factor

#125

Dic₁₃

yes

direct product factor

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₃

C₄×#41

yes

—

C₂×C₁₃

C₂×#125

yes

characteristic

C₂×C₁₃

#529

yes

—

C₂×C₁₃

#533

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₁₃

yes

—

103

C₂²×C₄

C₂²×D₁₃

yes

—

104

C₂²×C₄

C₂²×D₁₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×Dic₁₃

yes

—

108

C₄⋊C₄

C₂×Dic₁₃

yes

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×Dic₁₃

yes

—

115

C₁₃

C₄×#125

yes

characteristic, complemented, Sylow

116

C₂³

C₂³×D₁₃

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

Q₈×D₁₃

yes

—

132

C₂×C₄

#6131

yes

—

133

C₂×C₄

C₂²×Dic₁₃

yes

—

134

C₂×C₄

C₂²×Dic₁₃

yes

—

135

C₂×C₄

C₂²×Dic₁₃

yes

—

136

C₂²

C₂×Q₈×D₁₃

yes

characteristic

137

C₂²

C₂×#6131

yes

characteristic

138

C₂²

#41×D₁₃

yes

—

139

C₂²

#41×D₁₃

yes

—

140

C₂²

C₂³×Dic₁₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

Q₈×Dic₁₃

yes

—

148

C₂

D₁₃×#125

yes

characteristic

149

C₂

#11719393

yes

characteristic

150

C₂

#11719396

yes

—

151

C₂

C₂×Q₈×Dic₁₃

yes

characteristic

152

C₂

#41×Dic₁₃

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×#125	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	C₄×#125
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator

C₂²×C₁₃

C₂³×C₄

Fitting

C₂×C₁₃×#125

C₂

socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
8. C₁₃×#125	C₄	semidirect
78. Dic₁₃	#125	direct (N × H)
79. #125	Dic₁₃	direct (N × H)
115. C₁₃	C₄×#125	semidirect

H ≅ G/N

type

8. C₁₃×#125

C₄

semidirect

78. Dic₁₃

#125

direct (N × H)

79. #125

Dic₁₃

direct (N × H)

115. C₁₃

C₄×#125

semidirect

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

subgroup	index
C₂×C₁₃×#125	2
#11718646	2
#11718642	2
C₄⋊C₄×Dic₁₃	2
#11718597	2
C₄²×Dic₁₃	2
C₄×#125	13

subgroup

index

C₂×C₁₃×#125

C₄⋊C₄×Dic₁₃

C₄²×Dic₁₃

Subgroups — #125×Dic₁₃ #13117397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #125×Dic13 #13117397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #125×Dic₁₃ #13117397