Group #13125210 — subgroups

Subgroups — #13125210

Order 1664.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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×#127	C₂	1	1	yes	maximal, characteristic
2	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718876	C₂	1	4	yes	maximal, complemented
4	#11718732	C₂	1	4	yes	maximal, complemented
5	#11718731	C₂	1	4	yes	maximal, complemented
6	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄×C₁₃	—	2	2	no	—
8	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
9	C₁₃×#127	—	2	2	no	—
10	C₁₃×#127	C₂²	1	2	yes	—
11	C₁₃×#127	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	#88180	C₂²	1	2	yes	—
14	#88178	—	2	2	no	—
15	C₂×#6099	C₂²	1	2	yes	—
16	#88124	C₂²	1	2	yes	complemented
17	#88123	C₂²	1	2	yes	complemented
18	C₄×Dic₂₆	—	2	2	no	—
19	#88072	C₂²	1	16	yes	complemented
20	#88043	C₂²	1	2	yes	complemented
21	C₄×#1055	C₂²	1	2	yes	complemented
22	C₂⁴×C₁₃	—	2	2	no	—
23	C₂×D₄×C₁₃	—	2	2	no	—
24	C₂×D₄×C₁₃	—	2	4	no	—
25	C₂×D₄×C₁₃	—	2	4	no	—
26	C₂×D₄×C₁₃	—	2	8	no	—
27	C₂×D₄×C₁₃	—	4	8	no	—
28	C₂×D₄×C₁₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₃	D₄	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₂₆	—	2	2	no	—
32	C₄²×C₁₃	D₄	1	2	yes	—
33	C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₁₃	—	2	2	no	—
36	C₄⋊Dic₁₃	—	2	2	no	—
37	C₄⋊Dic₁₃	C₂³	1	4	yes	—
38	#6101	—	2	4	no	—
39	C₄×Dic₁₃	—	2	2	no	—
40	#6099	—	2	4	no	—
41	C₂×#1055	C₂³	1	4	yes	—
42	#3138	—	13	1	no	maximal, Sylow
43	C₂³×C₁₃	—	2	4	no	—
44	C₂³×C₁₃	—	4	8	no	—
45	C₂³×C₁₃	—	4	4	no	—
46	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
47	D₄×C₁₃	—	2	8	no	—
48	D₄×C₁₃	—	4	8	no	—
49	D₄×C₁₃	—	4	4	no	—
50	D₄×C₁₃	—	4	8	no	—
51	C₂×C₄×C₁₃	—	2	4	no	—
52	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃	#41	1	4	yes	—
56	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₁₃	—	2	4	no	—
58	Dic₂₆	—	4	4	no	—
59	#1055	—	2	4	no	—
60	#533	—	13	1	no	—
61	C₂×#127	D₁₃	1	1	yes	characteristic
62	#490	—	13	4	no	—
63	#486	—	13	4	no	—
64	#418	—	13	4	no	—
65	C₄×M₁₆	—	13	1	no	—
66	C₂²×C₁₃	—	2	2	no	—
67	C₂²×C₁₃	—	4	4	no	—
68	C₂²×C₁₃	—	4	8	no	—
69	C₂²×C₁₃	—	8	8	no	—
70	C₂²×C₁₃	#124	1	2	yes	—
71	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
72	C₄×C₁₃	—	2	2	no	—
73	C₄×C₁₃	#124	1	2	yes	—
74	C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
75	C₄×C₁₃	C₂×#41	1	2	yes	—
76	Dic₁₃	—	4	4	no	—
77	C₂²×D₄	—	2	2	no	—
78	C₂²×D₄	C₂×D₁₃	1	2	yes	—
79	C₂×M₁₆	—	13	2	no	—
80	C₄⋊Q₈	—	13	2	no	—
81	#127	—	2	2	no	—
82	#127	C₂×D₁₃	1	2	yes	—
83	#127	C₂×D₁₃	1	2	yes	—
84	#125	—	13	2	no	—
85	C₂²⋊Q₈	—	26	2	no	—
86	C₄×Q₈	—	26	2	no	—
87	#117	—	13	2	no	—
88	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
89	#102	—	13	16	no	—
90	#97	—	13	2	no	—
91	C₄×C₈	—	13	2	no	—
92	C₂×C₁₃	—	2	2	no	—
93	C₂×C₁₃	—	8	4	no	—
94	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
95	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
96	C₂⁴	—	2	2	no	—
97	C₂×Q₈	—	26	2	no	—
98	C₂×D₄	—	2	8	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	4	8	no	—
103	C₂×D₄	C₂²×D₁₃	1	4	yes	—
104	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
105	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
106	M₁₆	—	26	4	no	—
107	C₂×C₈	—	13	4	no	—
108	C₄⋊C₄	—	13	4	no	—
109	C₄⋊C₄	—	26	4	no	—
110	C₄⋊C₄	—	26	2	no	—
111	C₂²⋊C₄	—	26	2	no	—
112	C₄²	—	26	2	no	—
113	C₄²	C₂²⋊D₁₃	1	2	yes	—
114	C₄²	C₂²×D₁₃	1	1	yes	characteristic
115	C₄²	C₂²×D₁₃	1	1	yes	characteristic
116	C₁₃	#3138	1	1	yes	characteristic, complemented, Sylow
117	C₂³	—	2	4	no	—
118	C₂³	—	4	8	no	—
119	C₂³	—	4	4	no	—
120	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
121	Q₈	—	52	4	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	8	no	—
124	D₄	—	4	8	no	—
125	D₄	—	4	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	26	4	no	—
128	C₂×C₄	#6129	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
132	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
133	C₈	—	26	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	8	no	—
136	C₂²	—	4	4	no	—
137	C₂²	—	8	8	no	—
138	C₂²	#88190	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	52	4	no	—
142	C₄	#88186	1	4	yes	—
143	C₄	#88190	1	2	yes	—
144	C₄	C₂×#6129	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	8	4	no	—
147	C₂	C₂×#88186	1	2	yes	—
148	C₂	C₂×#88190	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#127

C₂

yes

maximal, characteristic

#11719175

C₂

yes

maximal, characteristic, complemented

#11718876

C₂

yes

maximal, complemented

#11718732

C₂

yes

maximal, complemented

#11718731

C₂

yes

maximal, complemented

C₄×#6099

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

C₂²

yes

—

C₁₃×#127

—

C₁₃×#127

C₂²

yes

—

C₁₃×#127

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88180

C₂²

yes

—

#88178

—

C₂×#6099

C₂²

yes

—

#88124

C₂²

yes

complemented

#88123

C₂²

yes

complemented

C₄×Dic₂₆

—

#88072

C₂²

yes

complemented

#88043

C₂²

yes

complemented

C₄×#1055

C₂²

yes

complemented

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

C₄×Dic₁₃

—

#6099

—

C₂×#1055

C₂³

yes

—

#3138

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

#1055

—

#533

—

C₂×#127

D₁₃

yes

—

—

—

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#124

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

C₂×#41

yes

—

Dic₁₃

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#127

C₂×D₁₃

yes

—

#127

C₂×D₁₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#124

yes

characteristic

C₂×C₁₃

C₂×AGL(1,ℤ/8)

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

C₂²×D₁₃

yes

—

104

C₂²×C₄

C₂²⋊D₁₃

yes

—

105

C₂²×C₄

C₂²×D₁₃

yes

characteristic

106

M₁₆

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

C₂²⋊D₁₃

yes

—

114

C₄²

C₂²×D₁₃

yes

characteristic

115

C₄²

C₂²×D₁₃

yes

characteristic

116

C₁₃

#3138

yes

characteristic, complemented, Sylow

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

121

Q₈

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#6129

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

132

C₂×C₄

C₂³×D₁₃

yes

characteristic

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#88190

yes

—

139

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#88186

yes

—

143

C₄

#88190

yes

—

144

C₄

C₂×#6129

yes

—

145

C₂

—

146

C₂

—

147

C₂

C₂×#88186

yes

—

148

C₂

C₂×#88190

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×#127	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	C₂×#127	D₁₃
hypocenter	C₁₃	#3138
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting, prime radical

C₂×C₁₃×#127

C₂

socle

C₂²×C₁₃

C₂²×D₄

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
2. #11719175	C₂	semidirect
3. #11718876	C₂	semidirect
4. #11718732	C₂	semidirect
5. #11718731	C₂	semidirect
6. C₄×#6099	C₂	semidirect
16. #88124	C₂²	semidirect
17. #88123	C₂²	semidirect
19. #88072	C₂²	semidirect
20. #88043	C₂²	semidirect
21. C₄×#1055	C₂²	semidirect
116. C₁₃	#3138	semidirect

H ≅ G/N

type

2. #11719175

C₂

semidirect

3. #11718876

C₂

semidirect

4. #11718732

C₂

semidirect

5. #11718731

C₂

semidirect

6. C₄×#6099

C₂

semidirect

16. #88124

C₂²

semidirect

17. #88123

C₂²

semidirect

19. #88072

C₂²

semidirect

20. #88043

C₂²

semidirect

21. C₄×#1055

C₂²

semidirect

116. C₁₃

#3138

semidirect

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

subgroup	index
C₂×C₁₃×#127	2
#11719175	2
#11718876	2
#11718732	2
#11718731	2
C₄×#6099	2
#3138	13

subgroup

index

C₂×C₁₃×#127

Subgroups — #13125210

Conjugacy classes of subgroups

Complements

Maximal subgroups