Group #13123766 — subgroups

Subgroups — #13123766

Order 1664.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#11719175	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#423	C₂	1	1	yes	maximal, characteristic
3	#11718783	C₂	1	2	yes	maximal
4	#11718780	C₂	1	2	yes	maximal
5	C₂×C₄×#1055	C₂	1	1	yes	maximal, characteristic
6	#11718492	C₂	1	4	yes	maximal, complemented
7	#11718491	C₂	1	4	yes	maximal, complemented
8	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	#88180	C₂²	1	2	yes	—
11	#88178	C₄	1	4	yes	—
12	C₂²×#1055	C₂²	1	1	yes	characteristic
13	C₂²×#1055	C₂²	1	1	yes	characteristic
14	#88124	C₄	1	2	yes	—
15	#88123	C₄	1	2	yes	—
16	C₄×Dic₂₆	C₂²	1	4	yes	—
17	C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
18	#88055	C₂²	1	4	yes	—
19	#88054	C₂²	1	8	yes	—
20	C₈×Dic₁₃	C₂²	1	4	yes	—
21	C₄×#1055	C₂²	1	2	yes	—
22	C₄×#1055	C₂²	1	2	yes	—
23	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
26	C₁₃×M₁₆	—	2	4	no	—
27	C₂×C₈×C₁₃	C₂³	1	4	yes	—
28	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₃	C₂³	1	2	yes	—
31	C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
33	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
34	#6101	C₂×C₄	1	8	yes	—
35	C₄×Dic₁₃	C₂³	1	4	yes	—
36	C₂×#1055	—	2	2	no	—
37	C₂×#1055	Q₈	1	4	yes	complemented
38	C₂×#1055	C₂³	1	2	yes	—
39	C₂×#1055	C₂³	1	1	yes	characteristic
40	C₂×#1055	C₂³	1	1	yes	characteristic
41	#2960	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	—	2	4	no	—
44	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₃	#41	1	2	yes	—
49	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₃	C₂²×C₄	1	8	yes	—
51	Dic₂₆	—	2	8	no	—
52	C₈×C₁₃	—	2	4	no	—
53	#1055	—	2	2	no	—
54	#1055	C₂×Q₈	1	2	yes	—
55	#1055	C₂×Q₈	1	2	yes	—
56	#533	—	13	1	no	—
57	#446	—	13	4	no	—
58	C₈×Q₈	—	13	4	no	—
59	#424	—	13	2	no	—
60	#423	D₁₃	1	1	yes	characteristic
61	#405	—	13	2	no	—
62	C₂×C₄×C₈	—	13	1	no	—
63	C₂²×C₁₃	—	2	2	no	—
64	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
65	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	C₄×Q₈	1	2	yes	—
68	C₄×C₁₃	#131	1	4	yes	—
69	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
71	Dic₁₃	—	2	8	no	—
72	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
73	C₂²×C₈	—	13	1	no	—
74	C₂²×C₈	—	13	1	no	—
75	C₄⋊Q₈	—	13	2	no	—
76	#125	—	13	2	no	—
77	C₂²⋊Q₈	—	13	4	no	—
78	C₄×Q₈	—	13	4	no	—
79	#117	—	13	2	no	—
80	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
81	C₄⋊C₈	—	13	8	no	—
82	C₄⋊C₈	C₂×D₁₃	1	4	yes	—
83	#97	—	13	4	no	—
84	C₄×C₈	—	13	4	no	—
85	C₄×C₈	—	13	2	no	—
86	C₄×C₈	—	13	2	no	—
87	C₂×C₁₃	—	2	2	no	—
88	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₁₃	C₂×#131	1	2	yes	—
90	C₂×Q₈	—	13	4	no	—
91	C₂²×C₄	C₄×D₁₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
93	M₁₆	—	2	4	no	—
94	C₂×C₈	—	13	4	no	—
95	C₂×C₈	—	13	1	no	—
96	C₂×C₈	—	13	1	no	—
97	C₂×C₈	—	13	2	no	—
98	C₂×C₈	—	26	2	no	—
99	C₂×C₈	C₂²×D₁₃	1	4	yes	—
100	C₄⋊C₄	—	13	4	no	—
101	C₄⋊C₄	—	13	8	no	—
102	C₄⋊C₄	—	13	4	no	—
103	C₂²⋊C₄	—	13	4	no	—
104	C₄²	—	13	4	no	—
105	C₄²	C₄×D₁₃	1	1	yes	characteristic
106	C₄²	C₄×D₁₃	1	1	yes	characteristic
107	C₄²	C₂²×D₁₃	1	2	yes	—
108	C₁₃	#2960	1	1	yes	characteristic, complemented, Sylow
109	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
110	Q₈	—	26	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	13	8	no	—
113	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
116	C₂×C₄	#6129	1	2	yes	—
117	C₂×C₄	Q₈×D₁₃	1	2	yes	—
118	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	13	2	no	—
121	C₈	—	13	2	no	—
122	C₈	—	26	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#88141	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	26	8	no	—
128	C₄	#88141	1	2	yes	—
129	C₄	#88161	1	4	yes	—
130	C₄	C₂×#6129	1	1	yes	characteristic
131	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#88141	1	1	yes	characteristic
134	C₂	C₂×#88161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719175

C₂

yes

maximal, characteristic

C₁₃×#423

C₂

yes

maximal, characteristic

#11718783

C₂

yes

maximal

#11718780

C₂

yes

maximal

C₂×C₄×#1055

C₂

yes

maximal, characteristic

#11718492

C₂

yes

maximal, complemented

#11718491

C₂

yes

maximal, complemented

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88180

C₂²

yes

—

#88178

C₄

yes

—

C₂²×#1055

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

characteristic

#88124

C₄

yes

—

#88123

C₄

yes

—

C₄×Dic₂₆

C₂²

yes

—

C₁₃×C₄⋊C₈

C₂²

yes

—

#88055

C₂²

yes

—

#88054

C₂²

yes

—

C₈×Dic₁₃

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

C₂×C₄

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

—

C₂²⋊Dic₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

C₂×C₄

yes

—

C₄×Dic₁₃

C₂³

yes

—

C₂×#1055

—

C₂×#1055

Q₈

yes

complemented

C₂×#1055

C₂³

yes

—

C₂×#1055

C₂³

yes

characteristic

C₂×#1055

C₂³

yes

characteristic

#2960

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

C₂²×C₄

yes

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

#1055

C₂×Q₈

yes

—

#1055

C₂×Q₈

yes

—

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

D₁₃

yes

characteristic

#405

—

C₂×C₄×C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄×Q₈

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₈

yes

—

C₄×C₁₃

#131

yes

—

C₄×C₁₃

C₂²×Q₈

yes

characteristic

C₄×C₁₃

C₂×#41

yes

characteristic

Dic₁₃

—

C₂×M₁₆

C₂×D₁₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₃

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₃

C₂×#131

yes

—

C₂×Q₈

—

C₂²×C₄

C₄×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₃

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

C₄×D₁₃

yes

characteristic

106

C₄²

C₄×D₁₃

yes

characteristic

107

C₄²

C₂²×D₁₃

yes

—

108

C₁₃

#2960

yes

characteristic, complemented, Sylow

109

C₂³

C₂×C₄×D₁₃

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₁₃

yes

—

114

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

115

C₂×C₄

C₂×C₄×D₁₃

yes

—

116

C₂×C₄

#6129

yes

—

117

C₂×C₄

Q₈×D₁₃

yes

—

118

C₂×C₄

C₂³×D₁₃

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

#88141

yes

—

125

C₂²

C₂²×C₄×D₁₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#88141

yes

—

129

C₄

#88161

yes

—

130

C₄

C₂×#6129

yes

characteristic

131

C₄

C₂×Q₈×D₁₃

yes

characteristic

132

C₂

—

133

C₂

C₂×#88141

yes

characteristic

134

C₂

C₂×#88161

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#423	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#423	D₁₃
hypocenter	C₁₃	#2960
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×#423

C₂

prime radical

C₂³×C₁₃

C₂²×C₄

cosocle

C₂×C₄×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
6. #11718492	C₂	semidirect
7. #11718491	C₂	semidirect
37. C₂×#1055	Q₈	semidirect
108. C₁₃	#2960	semidirect

H ≅ G/N

type

6. #11718492

C₂

semidirect

7. #11718491

C₂

semidirect

37. C₂×#1055

Q₈

semidirect

108. C₁₃

#2960

semidirect

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

subgroup	index
#11719175	2
C₁₃×#423	2
#11718783	2
#11718780	2
C₂×C₄×#1055	2
#11718492	2
#11718491	2
#2960	13

subgroup

index

C₁₃×#423

C₂×C₄×#1055

Subgroups — #13123766

Conjugacy classes of subgroups

Complements

Maximal subgroups