Group #13120560 — subgroups

Subgroups — #13120560

Order 1664.

932 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂²×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88184	C₂	1	3	yes	maximal, complemented
3	#11718892	C₂	1	2	yes	maximal, complemented
4	#11718884	C₂	1	6	yes	maximal, complemented
5	#11718873	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₃×#41	—	2	12	no	—
7	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₃	C₂²	1	3	yes	—
9	C₂³×C₄×C₁₃	C₂²	1	3	yes	—
10	C₂²×C₂²⋊D₁₃	C₂²	1	6	yes	—
11	C₂²×D₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Dic₁₃	C₂²	1	6	yes	—
13	#88184	—	2	12	no	—
14	C₂×#6103	C₂²	1	6	yes	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
16	#88071	C₂²	1	6	yes	—
17	C₂⁴×C₁₃	C₂³	1	3	yes	—
18	C₂³×D₁₃	C₂³	1	2	yes	—
19	C₁₃×#41	—	2	32	no	—
20	C₂×Q₈×C₁₃	—	2	6	no	—
21	C₂×D₄×C₁₃	—	2	12	no	—
22	C₂×D₄×C₁₃	—	2	6	no	—
23	C₂²×C₄×C₁₃	—	2	12	no	—
24	C₂²×C₄×C₁₃	D₄	1	12	yes	complemented
25	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₃	—	2	12	no	—
28	C₂×C₂²⋊D₁₃	—	4	12	no	—
29	C₂²×Dic₁₃	C₂³	1	6	yes	—
30	C₂×D₅₂	—	2	6	no	—
31	C₂²⋊Dic₁₃	—	4	6	no	—
32	#6103	—	2	12	no	—
33	C₄⋊Dic₁₃	—	2	6	no	—
34	#2615	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	—	2	6	no	—
36	C₂³×C₁₃	—	2	3	no	—
37	C₂³×C₁₃	—	2	6	no	—
38	C₂³×C₁₃	—	2	3	no	—
39	C₂³×C₁₃	C₂×D₄	1	6	yes	—
40	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂²×D₁₃	—	2	6	no	—
42	C₂²×D₁₃	—	2	6	no	—
43	C₂²×D₁₃	—	2	2	no	—
44	Q₈×C₁₃	—	2	8	no	—
45	D₄×C₁₃	—	2	24	no	—
46	C₂×C₄×C₁₃	—	2	6	no	—
47	C₂×C₄×C₁₃	—	2	24	no	—
48	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
49	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
50	C₂²⋊D₁₃	—	4	24	no	—
51	C₂×Dic₁₃	—	2	6	no	—
52	C₂×Dic₁₃	—	2	12	no	—
53	D₅₂	—	4	4	no	—
54	C₂²×#41	D₁₃	1	1	yes	characteristic, complemented
55	C₂×#121	—	13	3	no	—
56	#393	—	13	2	no	—
57	#392	—	13	6	no	—
58	#380	—	13	3	no	—
59	C₂²×C₁₃	—	2	6	no	—
60	C₂²×C₁₃	—	2	6	no	—
61	C₂²×C₁₃	—	2	6	no	—
62	C₂²×C₁₃	—	4	6	no	—
63	C₂²×C₁₃	—	4	6	no	—
64	C₂²×C₁₃	C₂²×D₄	1	3	yes	—
65	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
66	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
67	C₂×D₁₃	—	4	2	no	—
68	C₂×D₁₃	—	4	6	no	—
69	C₂×D₁₃	—	4	6	no	—
70	C₄×C₁₃	—	2	6	no	—
71	C₄×C₁₃	#120	1	4	yes	—
72	Dic₁₃	—	4	6	no	—
73	C₂×#41	—	2	12	no	—
74	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
75	C₂²×D₄	—	13	6	no	—
76	C₂²×D₄	—	13	1	no	—
77	C₂²×D₄	C₂×D₁₃	1	3	yes	—
78	C₂³×C₄	C₂×D₁₃	1	3	yes	—
79	#121	—	26	12	no	—
80	C₂×C₄⋊C₄	—	13	3	no	—
81	C₂×C₂²⋊C₄	—	13	6	no	—
82	C₂×C₂²⋊C₄	—	13	6	no	—
83	#95	—	13	6	no	—
84	C₂×C₁₃	—	4	6	no	—
85	C₂×C₁₃	C₂×#120	1	1	yes	characteristic
86	C₂×C₁₃	#535	1	3	yes	—
87	C₂×C₁₃	#537	1	3	yes	—
88	D₁₃	—	8	2	no	—
89	C₂⁴	—	13	2	no	—
90	C₂⁴	C₂²×D₁₃	1	3	yes	—
91	#41	—	2	32	no	—
92	C₂×Q₈	—	2	6	no	—
93	C₂×D₄	—	2	12	no	—
94	C₂×D₄	—	2	6	no	—
95	C₂×D₄	—	26	12	no	—
96	C₂×D₄	—	26	6	no	—
97	C₂×D₄	—	52	12	no	—
98	C₂²×C₄	—	2	12	no	—
99	C₂²×C₄	—	13	6	no	—
100	C₂²×C₄	C₂²⋊D₁₃	1	12	yes	complemented
101	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
102	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
103	C₄⋊C₄	—	26	6	no	—
104	C₂²⋊C₄	—	26	12	no	—
105	C₂²⋊C₄	—	52	6	no	—
106	C₁₃	#2615	1	1	yes	characteristic, complemented, Sylow
107	C₂³	—	2	3	no	—
108	C₂³	—	2	6	no	—
109	C₂³	—	2	3	no	—
110	C₂³	—	2	6	no	—
111	C₂³	—	26	6	no	—
112	C₂³	—	26	2	no	—
113	C₂³	—	26	6	no	—
114	C₂³	C₂×C₂²⋊D₁₃	1	6	yes	—
115	C₂³	C₂³×D₁₃	1	1	yes	characteristic
116	Q₈	—	2	8	no	—
117	D₄	—	2	24	no	—
118	D₄	—	52	24	no	—
119	D₄	—	52	4	no	—
120	C₂×C₄	—	2	6	no	—
121	C₂×C₄	—	2	24	no	—
122	C₂×C₄	—	26	12	no	—
123	C₂×C₄	—	26	6	no	—
124	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
125	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
126	C₂²	—	2	6	no	—
127	C₂²	—	2	6	no	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	6	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	52	6	no	—
132	C₂²	—	52	6	no	—
133	C₂²	—	52	2	no	—
134	C₂²	#88256	1	3	yes	—
135	C₂²	#88257	1	1	yes	characteristic
136	C₂²	C₂²×C₂²⋊D₁₃	1	3	yes	—
137	C₄	—	2	6	no	—
138	C₄	—	52	6	no	—
139	C₄	#88207	1	4	yes	—
140	C₂	—	4	6	no	—
141	C₂	—	104	2	no	—
142	C₂	#11719535	1	3	yes	—
143	C₂	#11719538	1	3	yes	—
144	C₂	C₂×#88207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃×#41

C₂

yes

maximal, characteristic, complemented

C₂×#88184

C₂

yes

maximal, complemented

#11718892

C₂

yes

maximal, complemented

#11718884

C₂

yes

maximal, complemented

#11718873

C₂

yes

maximal, complemented

C₂×C₁₃×#41

—

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×D₄×C₁₃

C₂²

yes

—

C₂³×C₄×C₁₃

C₂²

yes

—

C₂²×C₂²⋊D₁₃

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

#88184

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₂³×D₁₃

C₂³

yes

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

complemented

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₂²⋊D₁₃

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂²⋊Dic₁₃

—

#6103

—

C₄⋊Dic₁₃

—

#2615

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

C₂²×#41

D₁₃

yes

characteristic, complemented

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

—

Dic₁₃

—

C₂×#41

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₃

yes

—

C₂³×C₄

C₂×D₁₃

yes

—

#121

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#120

yes

characteristic

C₂×C₁₃

#535

yes

—

C₂×C₁₃

#537

yes

—

D₁₃

—

C₂⁴

—

C₂⁴

C₂²×D₁₃

yes

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂²⋊D₁₃

yes

complemented

101

C₂²×C₄

C₂²×D₁₃

yes

—

102

C₂²×C₄

C₂²×D₁₃

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₁₃

#2615

yes

characteristic, complemented, Sylow

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₁₃

yes

—

115

C₂³

C₂³×D₁₃

yes

characteristic

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

125

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#88256

yes

—

135

C₂²

#88257

yes

characteristic

136

C₂²

C₂²×C₂²⋊D₁₃

yes

—

137

C₄

—

138

C₄

—

139

C₄

#88207

yes

—

140

C₂

—

141

C₂

—

104

—

142

C₂

#11719535

yes

—

143

C₂

#11719538

yes

—

144

C₂

C₂×#88207

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂²×C₁₃×#41	C₂
hypercenter	C₂²×#41	D₁₃
hypocenter	C₁₃	#2615
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂²×C₁₃×#41

C₂

hypercenter

C₂²×#41

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂²×#41

N	H ≅ G/N	type
1. C₂²×C₁₃×#41	C₂	semidirect
2. C₂×#88184	C₂	semidirect
3. #11718892	C₂	semidirect
4. #11718884	C₂	semidirect
5. #11718873	C₂	semidirect
24. C₂²×C₄×C₁₃	D₄	semidirect
54. C₂²×#41	D₁₃	semidirect
100. C₂²×C₄	C₂²⋊D₁₃	semidirect
106. C₁₃	#2615	semidirect

H ≅ G/N

type

1. C₂²×C₁₃×#41

C₂

semidirect

2. C₂×#88184

C₂

semidirect

3. #11718892

C₂

semidirect

4. #11718884

C₂

semidirect

5. #11718873

C₂

semidirect

24. C₂²×C₄×C₁₃

D₄

semidirect

54. C₂²×#41

D₁₃

semidirect

100. C₂²×C₄

C₂²⋊D₁₃

semidirect

106. C₁₃

#2615

semidirect

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

subgroup	index
C₂²×C₁₃×#41	2
C₂×#88184	2
#11718892	2
#11718884	2
#11718873	2
#2615	13

subgroup

index

C₂²×C₁₃×#41

Subgroups — #13120560

Conjugacy classes of subgroups

Complements

Maximal subgroups