Group #13123197 — subgroups

Subgroups — #13123197

Order 1664.

472 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 197 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×#88144	C₂	1	2	yes	maximal
2	C₁₃×#384	C₂	1	1	yes	maximal, characteristic
3	#11718646	C₂	1	4	yes	maximal
4	#11718638	C₂	1	2	yes	maximal
5	#11718634	C₂	1	4	yes	maximal
6	#11718596	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
14	C₂×#6101	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
19	#88144	—	2	8	no	—
20	#88071	C₂²	1	8	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
24	C₂²×Dic₁₃	C₂³	1	4	yes	—
25	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₃	C₂³	1	2	yes	—
28	C₁₃×C₄⋊C₄	—	2	4	no	—
29	C₁₃×C₄⋊C₄	—	2	8	no	—
30	C₄²×C₁₃	Q₈	1	4	yes	—
31	C₄⋊Dic₁₃	—	2	4	no	—
32	C₄⋊Dic₁₃	—	2	8	no	—
33	C₄⋊Dic₁₃	—	2	8	no	—
34	#6101	—	2	8	no	—
35	C₄×Dic₁₃	—	2	4	no	—
36	C₄×Dic₁₃	—	2	4	no	—
37	C₄×Dic₁₃	D₄	1	4	yes	—
38	C₄×Dic₁₃	Q₈	1	4	yes	—
39	#2859	—	13	1	no	maximal, Sylow
40	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₁₃	—	2	4	no	—
42	C₂×C₄×C₁₃	—	2	8	no	—
43	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
45	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₃	#41	1	4	yes	—
48	C₂×C₄×C₁₃	#41	1	4	yes	—
49	C₂×Dic₁₃	—	2	2	no	—
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	8	no	—
55	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
56	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
57	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
58	C₂×#125	—	13	2	no	—
59	#389	—	13	4	no	—
60	#389	—	13	4	no	—
61	#384	—	13	1	no	—
62	#384	D₁₃	1	1	yes	characteristic
63	#382	—	13	2	no	—
64	C₄³	—	13	1	no	—
65	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
68	C₂²×C₁₃	C₂×#41	1	2	yes	—
69	C₂²×C₁₃	C₂×#41	1	2	yes	—
70	C₄×C₁₃	—	4	4	no	—
71	C₄×C₁₃	#125	1	8	yes	complemented
72	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
73	Dic₁₃	—	2	2	no	—
74	Dic₁₃	—	2	2	no	—
75	Dic₁₃	—	2	4	no	—
76	Dic₁₃	—	4	4	no	—
77	#125	—	26	8	no	—
78	C₂×C₄⋊C₄	—	13	4	no	—
79	C₂×C₄⋊C₄	—	13	2	no	—
80	C₂×C₄⋊C₄	—	13	4	no	—
81	C₂×C₄⋊C₄	—	13	4	no	—
82	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
83	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
84	C₂×C₄²	—	13	1	no	—
85	C₂×C₄²	—	13	1	no	—
86	C₂×C₄²	—	13	2	no	—
87	C₂×C₄²	—	13	2	no	—
88	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
89	#95	—	13	8	no	—
90	C₂×C₁₃	C₂×#125	1	2	yes	—
91	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₁₃	#532	1	1	yes	characteristic
93	C₂×C₁₃	#532	1	1	yes	characteristic
94	C₂×C₁₃	#533	1	2	yes	—
95	C₂²×C₄	—	13	2	no	—
96	C₂²×C₄	—	13	1	no	—
97	C₂²×C₄	—	13	1	no	—
98	C₂²×C₄	—	13	4	no	—
99	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
100	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
101	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	26	8	no	—
105	C₄⋊C₄	—	26	8	no	—
106	C₄⋊C₄	—	26	4	no	—
107	C₄⋊C₄	—	26	8	no	—
108	C₄²	—	13	4	no	—
109	C₄²	—	13	4	no	—
110	C₄²	—	26	4	no	—
111	C₄²	—	26	4	no	—
112	C₄²	Dic₂₆	1	4	yes	—
113	C₁₃	#2859	1	1	yes	characteristic, complemented, Sylow
114	C₂³	C₂³×D₁₃	1	1	yes	characteristic
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	8	no	—
117	C₂×C₄	—	13	2	no	—
118	C₂×C₄	—	13	2	no	—
119	C₂×C₄	—	13	4	no	—
120	C₂×C₄	—	26	2	no	—
121	C₂×C₄	—	26	2	no	—
122	C₂×C₄	—	26	2	no	—
123	C₂×C₄	—	26	2	no	—
124	C₂×C₄	—	26	4	no	—
125	C₂×C₄	—	26	8	no	—
126	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
127	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
128	C₂×C₄	D₄×D₁₃	1	2	yes	—
129	C₂×C₄	#6129	1	4	yes	—
130	C₂×C₄	Q₈×D₁₃	1	2	yes	—
131	C₂×C₄	#6131	1	4	yes	—
132	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
133	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
134	C₂²	C₂×#6129	1	2	yes	—
135	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
136	C₂²	C₂×#6131	1	2	yes	—
137	C₄	—	4	4	no	—
138	C₄	—	26	2	no	—
139	C₄	—	26	4	no	—
140	C₄	—	26	2	no	—
141	C₄	—	52	4	no	—
142	C₄	#88142	1	4	yes	—
143	C₄	#88144	1	8	yes	complemented
144	C₂	C₂×#88142	1	1	yes	characteristic
145	C₂	C₂×#88144	1	2	yes	—
146	C₂	#11719421	1	1	yes	characteristic
147	C₂	#11719430	1	1	yes	characteristic
148	C₂	#11719433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88144

C₂

yes

maximal

C₁₃×#384

C₂

yes

maximal, characteristic

#11718646

C₂

yes

maximal

#11718638

C₂

yes

maximal

#11718634

C₂

yes

maximal

#11718596

C₂

yes

maximal, characteristic

C₄²×Dic₁₃

C₂

yes

maximal, characteristic

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×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88144

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

—

C₄×Dic₁₃

Q₈

yes

—

#2859

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

—

—

—

—

D₁₃

yes

characteristic

#382

—

C₄³

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₄×C₁₃

—

C₄×C₁₃

#125

yes

complemented

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

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×D₁₃

yes

characteristic

#95

—

C₂×C₁₃

C₂×#125

yes

—

C₂×C₁₃

C₂×C₄⋊Q₈

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

100

C₂²×C₄

C₂²×D₁₃

yes

characteristic

101

C₂²×C₄

C₂²×D₁₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

Dic₂₆

yes

—

113

C₁₃

#2859

yes

characteristic, complemented, Sylow

114

C₂³

C₂³×D₁₃

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

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

C₂×Dic₂₆

yes

—

127

C₂×C₄

C₂×Dic₂₆

yes

—

128

C₂×C₄

D₄×D₁₃

yes

—

129

C₂×C₄

#6129

yes

—

130

C₂×C₄

Q₈×D₁₃

yes

—

131

C₂×C₄

#6131

yes

—

132

C₂²

C₂²×Dic₂₆

yes

characteristic

133

C₂²

C₂×D₄×D₁₃

yes

characteristic

134

C₂²

C₂×#6129

yes

—

135

C₂²

C₂×Q₈×D₁₃

yes

characteristic

136

C₂²

C₂×#6131

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#88142

yes

—

143

C₄

#88144

yes

complemented

144

C₂

C₂×#88142

yes

characteristic

145

C₂

C₂×#88144

yes

—

146

C₂

#11719421

yes

characteristic

147

C₂

#11719430

yes

characteristic

148

C₂

#11719433

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#384	C₂
hypercenter	#384	D₁₃
hypocenter	C₁₃	#2859
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#384

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
71. C₄×C₁₃	#125	semidirect
113. C₁₃	#2859	semidirect
143. C₄	#88144	semidirect

H ≅ G/N

type

71. C₄×C₁₃

#125

semidirect

113. C₁₃

#2859

semidirect

143. C₄

#88144

semidirect

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

subgroup	index
C₂×#88144	2
C₁₃×#384	2
#11718646	2
#11718638	2
#11718634	2
#11718596	2
C₄²×Dic₁₃	2
#2859	13

subgroup

index

C₁₃×#384

C₄²×Dic₁₃

Subgroups — #13123197

Conjugacy classes of subgroups

Complements

Maximal subgroups