Group #13123771 — subgroups

Subgroups — #13123771

Order 1664.

364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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	C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	#11718685	C₂	1	4	yes	maximal, complemented
3	Q₈×#1055	C₂	1	4	yes	maximal, complemented
4	#11718660	C₂	1	2	yes	maximal
5	#11718658	C₂	1	2	yes	maximal
6	#11718585	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×#1055	C₂	1	1	yes	maximal, characteristic
8	C₁₃×C₄⋊Q₈	C₄	1	2	yes	—
9	C₁₃×#125	C₄	1	2	yes	—
10	C₁₃×C₂²⋊Q₈	C₄	1	4	yes	—
11	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
12	C₁₃×#117	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#6099	C₂²	1	2	yes	—
15	C₂²×#1055	C₂²	1	1	yes	characteristic
16	C₂²×#1055	C₂²	1	1	yes	characteristic
17	C₄⋊#1055	C₂²	1	8	yes	—
18	C₄⋊#1055	C₂²	1	4	yes	—
19	#88043	C₂²	1	4	yes	—
20	C₄×#1055	C₂²	1	4	yes	—
21	C₄×#1055	C₂²	1	2	yes	—
22	C₄×#1055	C₂²	1	2	yes	—
23	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
27	C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
28	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
30	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₁₃	C₂³	1	2	yes	—
33	C₄²×C₁₃	C₂³	1	4	yes	—
34	#6099	—	2	4	no	—
35	C₂×#1055	—	2	2	no	—
36	C₂×#1055	Q₈	1	4	yes	complemented
37	C₂×#1055	C₂³	1	2	yes	—
38	C₂×#1055	C₂³	1	4	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	Q₈×C₁₃	—	2	8	no	—
44	C₂×C₄×C₁₃	—	2	4	no	—
45	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₃	C₂²×C₄	1	8	yes	—
47	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₃	#41	1	2	yes	—
51	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	#1055	—	2	2	no	—
53	#1055	—	2	4	no	—
54	#1055	C₂×Q₈	1	2	yes	—
55	#1055	C₂×Q₈	1	2	yes	—
56	#533	D₁₃	1	1	yes	characteristic
57	#446	—	13	4	no	—
58	C₈×Q₈	—	13	4	no	—
59	#424	—	13	2	no	—
60	#423	—	13	1	no	—
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	—	2	8	no	—
68	C₄×C₁₃	C₄×Q₈	1	2	yes	—
69	C₄×C₁₃	#131	1	4	yes	—
70	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
71	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
72	C₂×M₁₆	—	13	2	no	—
73	C₂²×C₈	—	13	1	no	—
74	C₂²×C₈	—	13	1	no	—
75	C₄⋊Q₈	Dic₁₃	1	2	yes	—
76	#125	Dic₁₃	1	2	yes	—
77	C₂²⋊Q₈	Dic₁₃	1	4	yes	—
78	C₄×Q₈	C₂×D₁₃	1	4	yes	—
79	#117	C₂×D₁₃	1	2	yes	—
80	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
81	C₄⋊C₈	—	13	4	no	—
82	C₄⋊C₈	—	13	8	no	—
83	#97	—	13	4	no	—
84	C₄×C₈	—	13	2	no	—
85	C₄×C₈	—	13	2	no	—
86	C₄×C₈	—	13	4	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	C₂×Dic₁₃	1	4	yes	—
91	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
93	M₁₆	—	26	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	—	13	4	no	—
99	C₂×C₈	—	26	2	no	—
100	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
101	C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
102	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
103	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
104	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
105	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
106	C₄²	C₂²×D₁₃	1	2	yes	—
107	C₄²	C₂²×D₁₃	1	4	yes	—
108	C₁₃	#2960	1	1	yes	characteristic, complemented, Sylow
109	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	Q₈×D₁₃	1	2	yes	—
113	C₂×C₄	#6131	1	2	yes	—
114	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
115	C₂×C₄	C₂²×Dic₁₃	1	8	yes	—
116	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
118	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
119	C₈	—	13	2	no	—
120	C₈	—	13	2	no	—
121	C₈	—	26	2	no	—
122	C₈	—	26	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₁₃	1	2	yes	—
125	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₁₃	1	2	yes	—
129	C₄	#88202	1	4	yes	—
130	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
131	C₄	C₂×#6131	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
134	C₂	C₂×#88202	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#533

C₂

yes

maximal, characteristic

#11718685

C₂

yes

maximal, complemented

Q₈×#1055

C₂

yes

maximal, complemented

#11718660

C₂

yes

maximal

#11718658

C₂

yes

maximal

#11718585

C₂

yes

maximal, characteristic

C₂×C₄×#1055

C₂

yes

maximal, characteristic

C₁₃×C₄⋊Q₈

C₄

yes

—

C₁₃×#125

C₄

yes

—

C₁₃×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₁₃

C₂²

yes

—

C₁₃×#117

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

—

C₂²×#1055

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

characteristic

C₄⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

#88043

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂³

yes

—

#6099

—

C₂×#1055

—

C₂×#1055

Q₈

yes

complemented

C₂×#1055

C₂³

yes

—

C₂×#1055

C₂³

yes

—

C₂×#1055

C₂³

yes

characteristic

C₂×#1055

C₂³

yes

characteristic

#2960

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

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

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

—

—

C₂×Q₈

yes

—

#1055

C₂×Q₈

yes

—

#533

D₁₃

yes

characteristic

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄×Q₈

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₈

yes

—

C₄×C₁₃

#131

yes

—

C₄×C₁₃

C₂²×Q₈

yes

characteristic

C₄×C₁₃

C₂×#41

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₁₃

yes

—

#125

Dic₁₃

yes

—

C₂²⋊Q₈

Dic₁₃

yes

—

C₄×Q₈

C₂×D₁₃

yes

—

#117

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₃

C₂×#131

yes

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

C₂×Dic₁₃

yes

—

101

C₄⋊C₄

C₂×Dic₁₃

yes

—

102

C₄⋊C₄

C₂×Dic₁₃

yes

—

103

C₂²⋊C₄

C₂×Dic₁₃

yes

—

104

C₄²

C₂×Dic₁₃

yes

characteristic

105

C₄²

C₂×Dic₁₃

yes

characteristic

106

C₄²

C₂²×D₁₃

yes

—

107

C₄²

C₂²×D₁₃

yes

—

108

C₁₃

#2960

yes

characteristic, complemented, Sylow

109

C₂³

C₂²×Dic₁₃

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

Q₈×D₁₃

yes

—

113

C₂×C₄

#6131

yes

—

114

C₂×C₄

C₂²×Dic₁₃

yes

—

115

C₂×C₄

C₂²×Dic₁₃

yes

—

116

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

117

C₂×C₄

C₂²×Dic₁₃

yes

—

118

C₂×C₄

C₂³×D₁₃

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

Q₈×Dic₁₃

yes

—

125

C₂²

C₂³×Dic₁₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₁₃

yes

—

129

C₄

#88202

yes

—

130

C₄

C₂×Q₈×D₁₃

yes

characteristic

131

C₄

C₂×#6131

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₁₃

yes

characteristic

134

C₂

C₂×#88202

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#533	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#533	D₁₃
hypocenter	C₁₃	#2960
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×#533

C₂

prime radical

C₂³×C₁₃

C₂²×C₄

cosocle

C₂×C₄×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
2. #11718685	C₂	semidirect
3. Q₈×#1055	C₂	semidirect
36. C₂×#1055	Q₈	semidirect
108. C₁₃	#2960	semidirect

H ≅ G/N

type

2. #11718685

C₂

semidirect

3. Q₈×#1055

C₂

semidirect

36. 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
C₁₃×#533	2
#11718685	2
Q₈×#1055	2
#11718660	2
#11718658	2
#11718585	2
C₂×C₄×#1055	2
#2960	13

subgroup

index

C₁₃×#533

C₂×C₄×#1055

Subgroups — #13123771

Conjugacy classes of subgroups

Complements

Maximal subgroups