Group #13123488 — subgroups

Subgroups — #13123488

Order 1664.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#6127	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718790	C₂	1	2	yes	maximal
4	#11718780	C₂	1	2	yes	maximal, complemented
5	C₂×#88043	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718366	C₂	1	4	yes	maximal, complemented
7	#11718351	C₂	1	4	yes	maximal, complemented
8	C₂×#6127	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₂²⋊D₁₃	C₄	1	4	yes	—
12	#88180	C₂²	1	1	yes	characteristic
13	C₂²×#1055	C₂²	1	2	yes	—
14	C₄×D₅₂	C₄	1	2	yes	—
15	#88126	C₂²	1	2	yes	—
16	C₄²×D₁₃	C₂²	1	2	yes	—
17	C₄×Dic₂₆	C₄	1	2	yes	—
18	C₁₃×#97	C₂²	1	2	yes	—
19	C₄×C₈×C₁₃	C₂²	1	2	yes	—
20	#88060	C₂²	1	8	yes	—
21	#88054	C₂²	1	8	yes	—
22	#88043	C₂²	1	2	yes	—
23	#88043	C₂²	1	2	yes	—
24	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
27	#6127	—	2	4	no	—
28	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₁₃	C₂³	1	2	yes	—
31	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
32	C₁₃×M₁₆	—	2	4	no	—
33	C₂×C₈×C₁₃	C₂³	1	4	yes	—
34	C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
38	#6103	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
40	#6101	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₃	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₃	C₂³	1	2	yes	—
43	C₂×#1055	—	2	4	no	—
44	C₂×#1055	C₂³	1	4	yes	—
45	#2923	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₃	—	4	2	no	—
49	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#41	1	4	yes	—
53	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₁₃	—	2	4	no	—
55	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
56	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
57	D₅₂	—	2	2	no	—
58	C₄×D₁₃	—	2	4	no	—
59	C₄×D₁₃	—	4	4	no	—
60	Dic₂₆	—	2	2	no	—
61	C₈×C₁₃	—	2	4	no	—
62	#1055	—	2	4	no	—
63	C₄×#41	—	13	1	no	—
64	#433	—	13	4	no	—
65	#432	—	13	4	no	—
66	#424	—	13	2	no	—
67	#408	—	13	2	no	—
68	C₄×M₁₆	D₁₃	1	1	yes	characteristic, complemented
69	C₂×#97	—	13	1	no	—
70	C₂²×C₁₃	—	2	2	no	—
71	C₂²×C₁₃	#117	1	2	yes	—
72	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×D₁₃	—	2	2	no	—
74	C₂×D₁₃	—	2	4	no	—
75	C₄×C₁₃	—	2	2	no	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	#117	1	2	yes	—
78	C₄×C₁₃	C₂×#41	1	2	yes	—
79	Dic₁₃	—	2	2	no	—
80	Dic₁₃	—	2	4	no	—
81	C₂×#41	—	13	1	no	—
82	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
83	C₂²×C₈	—	13	2	no	—
84	C₄×Q₈	—	13	2	no	—
85	C₄×D₄	—	13	4	no	—
86	C₄×D₄	—	13	2	no	—
87	#117	—	13	1	no	—
88	#117	—	13	2	no	—
89	C₂×C₄²	—	13	2	no	—
90	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
91	C₄⋊C₈	—	13	8	no	—
92	C₂²⋊C₈	—	13	8	no	—
93	#97	—	13	2	no	—
94	#97	—	13	2	no	—
95	#97	C₂×D₁₃	1	2	yes	—
96	C₄×C₈	C₂×D₁₃	1	2	yes	—
97	C₂×C₁₃	—	2	2	no	—
98	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
99	C₂×C₁₃	#568	1	2	yes	—
100	D₁₃	—	4	2	no	—
101	#41	—	26	4	no	—
102	C₂×Q₈	—	13	1	no	—
103	C₂×D₄	—	13	2	no	—
104	C₂×D₄	—	13	1	no	—
105	C₂²×C₄	—	13	2	no	—
106	C₂²×C₄	—	13	4	no	—
107	C₂²×C₄	C₄×D₁₃	1	2	yes	—
108	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
109	M₁₆	—	2	4	no	—
110	C₂×C₈	—	13	4	no	—
111	C₂×C₈	—	26	4	no	—
112	C₂×C₈	C₂²×D₁₃	1	4	yes	—
113	C₄⋊C₄	—	13	4	no	—
114	C₄⋊C₄	—	13	2	no	—
115	C₂²⋊C₄	—	13	2	no	—
116	C₂²⋊C₄	—	13	4	no	—
117	C₄²	—	13	4	no	—
118	C₄²	—	13	2	no	—
119	C₄²	C₄×D₁₃	1	2	yes	—
120	C₄²	C₂²×D₁₃	1	1	yes	characteristic
121	C₄²	C₂²×D₁₃	1	1	yes	characteristic
122	C₁₃	#2923	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	13	2	no	—
124	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
125	Q₈	—	26	2	no	—
126	D₄	—	26	4	no	—
127	D₄	—	26	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	13	4	no	—
130	C₂×C₄	—	13	2	no	—
131	C₂×C₄	—	26	4	no	—
132	C₂×C₄	—	52	4	no	—
133	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
136	C₂×C₄	#6127	1	4	yes	—
137	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	26	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	26	2	no	—
142	C₂²	—	26	4	no	—
143	C₂²	#88126	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	26	2	no	—
148	C₄	—	26	4	no	—
149	C₄	#88126	1	2	yes	—
150	C₄	C₂×#6127	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	52	2	no	—
153	C₂	C₂×#88126	1	1	yes	characteristic
154	C₂	#11719450	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#6127

C₂

yes

maximal, characteristic

C₄×C₁₃×M₁₆

C₂

yes

maximal, characteristic, complemented

#11718790

C₂

yes

maximal

#11718780

C₂

yes

maximal, complemented

C₂×#88043

C₂

yes

maximal, characteristic, complemented

#11718366

C₂

yes

maximal, complemented

#11718351

C₂

yes

maximal, complemented

C₂×#6127

C₂²

yes

characteristic

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₄×C₂²⋊D₁₃

C₄

yes

—

#88180

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

—

C₄×D₅₂

C₄

yes

—

#88126

C₂²

yes

—

C₄²×D₁₃

C₂²

yes

—

C₄×Dic₂₆

C₄

yes

—

C₁₃×#97

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

—

#88060

C₂²

yes

—

#88054

C₂²

yes

—

#88043

C₂²

yes

—

#88043

C₂²

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₂²⋊D₁₃

C₂×C₄

yes

—

#6127

—

C₂×D₅₂

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

C₂×Dic₂₆

C₂×C₄

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

C₂×C₄

yes

—

#6103

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

C₂×C₄

yes

—

C₄×Dic₁₃

C₂×C₄

yes

—

C₄×Dic₁₃

C₂³

yes

—

C₂×#1055

—

C₂×#1055

C₂³

yes

—

#2923

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂²×D₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂²×C₄

yes

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₄×#41

—

—

—

—

—

C₄×M₁₆

D₁₃

yes

characteristic, complemented

C₂×#97

—

C₂²×C₁₃

—

C₂²×C₁₃

#117

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#117

yes

—

C₄×C₁₃

C₂×#41

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×#41

—

C₂×M₁₆

C₂×D₁₃

yes

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

C₂×D₁₃

yes

—

C₄×C₈

C₂×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#117

yes

characteristic

C₂×C₁₃

#568

yes

—

100

D₁₃

—

101

#41

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₁₃

yes

—

108

C₂²×C₄

C₂²×D₁₃

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₁₃

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄×D₁₃

yes

—

120

C₄²

C₂²×D₁₃

yes

characteristic

121

C₄²

C₂²×D₁₃

yes

characteristic

122

C₁₃

#2923

yes

characteristic, complemented, Sylow

123

C₂³

—

124

C₂³

C₂×C₄×D₁₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₄×D₁₃

yes

—

134

C₂×C₄

C₂×C₄×D₁₃

yes

—

135

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

136

C₂×C₄

#6127

yes

—

137

C₂×C₄

C₂³×D₁₃

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#88126

yes

—

144

C₂²

C₂²×C₄×D₁₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#88126

yes

—

150

C₄

C₂×#6127

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#88126

yes

characteristic

154

C₂

#11719450

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×M₁₆	C₂
prime radical	C₂×#6127	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×M₁₆	D₁₃
hypocenter	C₁₃	#2923
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₄×C₁₃×M₁₆

C₂

prime radical

C₂×#6127

C₂²

cosocle

C₂×C₄×C₁₃

C₂⁴

hypercenter

C₄×M₁₆

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. C₄×C₁₃×M₁₆	C₂	semidirect
4. #11718780	C₂	semidirect
5. C₂×#88043	C₂	semidirect
6. #11718366	C₂	semidirect
7. #11718351	C₂	semidirect
68. C₄×M₁₆	D₁₃	semidirect
122. C₁₃	#2923	semidirect

H ≅ G/N

type

2. C₄×C₁₃×M₁₆

C₂

semidirect

4. #11718780

C₂

semidirect

5. C₂×#88043

C₂

semidirect

6. #11718366

C₂

semidirect

7. #11718351

C₂

semidirect

68. C₄×M₁₆

D₁₃

semidirect

122. C₁₃

#2923

semidirect

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

subgroup	index
C₄×#6127	2
C₄×C₁₃×M₁₆	2
#11718790	2
#11718780	2
C₂×#88043	2
#11718366	2
#11718351	2
#2923	13

subgroup

index

C₄×C₁₃×M₁₆

Subgroups — #13123488

Conjugacy classes of subgroups

Complements

Maximal subgroups