Group #13123925 — subgroups

Subgroups — #13123925

Order 1664.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 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×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	#11718685	C₂	1	6	yes	maximal, complemented
3	Q₈×#1055	C₂	1	2	yes	maximal, complemented
4	#11718587	C₂	1	3	yes	maximal
5	#11718586	C₂	1	3	yes	maximal
6	C₂²×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₃	C₄	1	3	yes	—
8	C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₄	1	3	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	3	yes	—
13	C₂²⋊#1055	C₂²	1	1	yes	characteristic
14	C₂²⋊#1055	C₂²	1	3	yes	—
15	C₄⋊#1055	C₂²	1	6	yes	—
16	C₄⋊#1055	C₂²	1	6	yes	—
17	#88043	C₂²	1	6	yes	—
18	C₄×#1055	C₂²	1	6	yes	—
19	C₂×Q₈×C₁₃	—	2	4	no	—
20	C₂×Q₈×C₁₃	C₂×C₄	1	3	yes	—
21	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₁₃	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
26	C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
27	C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
28	C₄²×C₁₃	C₂×C₄	1	3	yes	—
29	C₄²×C₁₃	C₂×C₄	1	3	yes	—
30	C₄²×C₁₃	C₂³	1	3	yes	—
31	C₄²×C₁₃	C₂³	1	3	yes	—
32	C₂×#1055	C₂³	1	2	yes	—
33	C₂×#1055	C₂³	1	6	yes	—
34	#2984	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₁₃	—	2	6	no	—
37	Q₈×C₁₃	M₁₆	1	4	yes	—
38	C₂×C₄×C₁₃	—	2	6	no	—
39	C₂×C₄×C₁₃	—	2	6	no	—
40	C₂×C₄×C₁₃	—	2	2	no	—
41	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
47	#1055	—	2	2	no	—
48	#1055	—	2	6	no	—
49	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic
50	#446	—	13	6	no	—
51	C₈×Q₈	—	13	2	no	—
52	#432	—	13	3	no	—
53	#431	—	13	3	no	—
54	C₂²×C₁₃	—	2	1	no	—
55	C₂²×C₁₃	—	2	1	no	—
56	C₂²×C₁₃	#131	1	2	yes	—
57	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₃	—	2	1	no	—
59	C₄×C₁₃	—	2	3	no	—
60	C₄×C₁₃	—	2	3	no	—
61	C₄×C₁₃	—	2	3	no	—
62	C₄×C₁₃	C₂×M₁₆	1	6	yes	—
63	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
64	C₂²×Q₈	Dic₁₃	1	1	yes	characteristic
65	C₄×Q₈	Dic₁₃	1	3	yes	—
66	C₄×Q₈	Dic₁₃	1	1	yes	characteristic
67	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
68	C₄×Q₈	C₂×D₁₃	1	3	yes	—
69	C₂×C₄⋊C₄	Dic₁₃	1	3	yes	—
70	C₂×C₄²	C₂×D₁₃	1	3	yes	—
71	C₄⋊C₈	—	13	6	no	—
72	C₄⋊C₈	—	13	6	no	—
73	C₂²⋊C₈	—	13	1	no	—
74	C₂²⋊C₈	—	13	3	no	—
75	#97	—	13	6	no	—
76	C₄×C₈	—	13	6	no	—
77	C₂×C₁₃	—	2	2	no	—
78	C₂×C₁₃	#519	1	1	yes	characteristic
79	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
80	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	C₂×Dic₁₃	1	3	yes	—
83	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
84	C₂²×C₄	C₂×Dic₁₃	1	3	yes	—
85	C₂²×C₄	C₂×Dic₁₃	1	3	yes	—
86	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
87	C₂×C₈	—	13	2	no	—
88	C₂×C₈	—	13	6	no	—
89	C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
90	C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
91	C₄⋊C₄	C₂×Dic₁₃	1	3	yes	—
92	C₄²	C₂×Dic₁₃	1	3	yes	—
93	C₄²	C₂×Dic₁₃	1	3	yes	—
94	C₄²	C₂²×D₁₃	1	3	yes	—
95	C₄²	C₂²×D₁₃	1	3	yes	—
96	C₁₃	#2984	1	1	yes	characteristic, complemented, Sylow
97	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#6099	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₁₃	1	3	yes	—
105	C₂×C₄	C₂²×Dic₁₃	1	3	yes	—
106	C₂×C₄	C₂²×Dic₁₃	1	3	yes	—
107	C₂×C₄	C₂²×Dic₁₃	1	3	yes	—
108	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
109	C₈	—	26	2	no	—
110	C₈	—	26	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#88202	1	2	yes	—
114	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂×#6099	1	6	yes	—
120	C₄	#88254	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	C₂²×#6099	1	1	yes	characteristic
123	C₂	#11719517	1	1	yes	characteristic
124	C₂	C₂×#88202	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₃

C₂

yes

maximal, characteristic

#11718685

C₂

yes

maximal, complemented

Q₈×#1055

C₂

yes

maximal, complemented

#11718587

C₂

yes

maximal

#11718586

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₄

yes

characteristic

C₄×Q₈×C₁₃

C₄

yes

—

C₄×Q₈×C₁₃

C₄

yes

characteristic

C₄×Q₈×C₁₃

C₂²

yes

characteristic

C₄×Q₈×C₁₃

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₄

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

C₂²⋊#1055

C₂²

yes

characteristic

C₂²⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

#88043

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×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

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂³

yes

—

C₂×#1055

C₂³

yes

—

C₂×#1055

C₂³

yes

—

#2984

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

M₁₆

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

#1055

—

#1055

—

C₂×C₄×Q₈

D₁₃

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#431

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#131

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×M₁₆

yes

—

C₄×C₁₃

2₋¹⁺⁴

yes

—

C₂²×Q₈

Dic₁₃

yes

characteristic

C₄×Q₈

Dic₁₃

yes

—

C₄×Q₈

Dic₁₃

yes

characteristic

C₄×Q₈

C₂×D₁₃

yes

characteristic

C₄×Q₈

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

Dic₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#519

yes

characteristic

C₂×C₁₃

C₂²×M₁₆

yes

characteristic

C₂×C₁₃

C₂×#131

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂×Q₈

C₂×Dic₁₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄²

C₂×Dic₁₃

yes

—

C₄²

C₂×Dic₁₃

yes

—

C₄²

C₂²×D₁₃

yes

—

C₄²

C₂²×D₁₃

yes

—

C₁₃

#2984

yes

characteristic, complemented, Sylow

C₂³

C₂²×Dic₁₃

yes

characteristic

Q₈

—

Q₈

#6099

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

104

C₂×C₄

C₂²×Dic₁₃

yes

—

105

C₂×C₄

C₂²×Dic₁₃

yes

—

106

C₂×C₄

C₂²×Dic₁₃

yes

—

107

C₂×C₄

C₂²×Dic₁₃

yes

—

108

C₂×C₄

C₂³×D₁₃

yes

characteristic

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

#88202

yes

—

114

C₂²

C₂³×Dic₁₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#6099

yes

—

120

C₄

#88254

yes

—

121

C₂

—

122

C₂

C₂²×#6099

yes

characteristic

123

C₂

#11719517

yes

characteristic

124

C₂

C₂×#88202

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₃	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	#2984
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₁₃

C₂

prime radical

C₂³×C₁₃

C₂²×C₄

cosocle

C₂×C₄×C₁₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
2. #11718685	C₂	semidirect
3. Q₈×#1055	C₂	semidirect
96. C₁₃	#2984	semidirect

H ≅ G/N

type

2. #11718685

C₂

semidirect

3. Q₈×#1055

C₂

semidirect

96. C₁₃

#2984

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
#11718685	2
Q₈×#1055	2
#11718587	2
#11718586	2
#2984	13

subgroup

index

C₂×C₄×Q₈×C₁₃

Subgroups — #13123925

Conjugacy classes of subgroups

Complements

Maximal subgroups