Group #13123212 — subgroups

Subgroups — #13123212

Order 1664.

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

normal subgroups only

This group has 504 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	C₂×#88142	C₂	1	2	yes	maximal
2	C₁₃×#382	C₂	1	1	yes	maximal, characteristic
3	#11718638	C₂	1	2	yes	maximal
4	#11718597	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	#11718307	C₂	1	8	yes	maximal
7	C₂²×Dic₂₆	C₂²	1	4	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×#6101	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
15	#88142	—	2	8	no	—
16	C₁₃×#95	C₂²	1	4	yes	—
17	#88071	C₂²	1	4	yes	—
18	#88071	C₂²	1	8	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
20	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₃	C₂³	1	4	yes	—
23	C₂²×Dic₁₃	C₂³	1	2	yes	—
24	C₂²×Dic₁₃	C₂³	1	2	yes	—
25	C₂×Dic₂₆	—	2	8	no	—
26	C₂×Dic₂₆	—	2	16	no	—
27	C₁₃×C₄⋊C₄	—	2	4	no	—
28	C₄²×C₁₃	Q₈	1	4	yes	complemented
29	C₄⋊Dic₁₃	—	2	4	no	—
30	#6101	—	2	8	no	—
31	C₄×Dic₁₃	—	2	8	no	—
32	C₄×Dic₁₃	D₄	1	4	yes	—
33	C₄×Dic₁₃	Q₈	1	4	yes	complemented
34	#2862	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	—	2	4	no	—
38	C₂×C₄×C₁₃	—	2	8	no	—
39	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
43	C₂×Dic₁₃	—	2	4	no	—
44	C₂×Dic₁₃	—	2	4	no	—
45	C₂×Dic₁₃	—	2	8	no	—
46	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
47	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
48	C₂×Dic₁₃	#41	1	8	yes	—
49	Dic₂₆	—	4	16	no	—
50	C₂×C₄⋊Q₈	—	13	2	no	—
51	#391	—	13	8	no	—
52	#382	—	13	2	no	—
53	#382	—	13	1	no	—
54	#382	D₁₃	1	1	yes	characteristic
55	C₄³	—	13	1	no	—
56	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
57	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
58	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
59	C₂²×C₁₃	C₂×#41	1	4	yes	—
60	C₄×C₁₃	—	2	4	no	—
61	C₄×C₁₃	—	4	4	no	—
62	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
63	Dic₁₃	—	2	4	no	—
64	Dic₁₃	—	4	4	no	—
65	Dic₁₃	#124	1	8	yes	—
66	C₂²×Q₈	—	13	4	no	—
67	C₄⋊Q₈	—	26	8	no	—
68	C₂×C₄⋊C₄	—	13	4	no	—
69	C₂×C₄⋊C₄	—	13	2	no	—
70	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
71	C₂×C₄²	—	13	1	no	—
72	C₂×C₄²	—	13	1	no	—
73	C₂×C₄²	—	13	4	no	—
74	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
75	#95	—	13	8	no	—
76	#95	—	13	4	no	—
77	#95	C₂×D₁₃	1	4	yes	—
78	C₂×C₁₃	C₂×#124	1	2	yes	—
79	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₁₃	#533	1	2	yes	—
81	C₂×C₁₃	#533	1	2	yes	—
82	C₂×Q₈	—	26	8	no	—
83	C₂×Q₈	—	26	16	no	—
84	C₂²×C₄	—	13	2	no	—
85	C₂²×C₄	—	13	2	no	—
86	C₂²×C₄	—	13	4	no	—
87	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
88	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	26	8	no	—
92	C₄⋊C₄	—	26	4	no	—
93	C₄²	—	13	4	no	—
94	C₄²	—	13	4	no	—
95	C₄²	—	26	8	no	—
96	C₄²	Dic₂₆	1	4	yes	complemented
97	C₁₃	#2862	1	1	yes	characteristic, complemented, Sylow
98	C₂³	C₂³×D₁₃	1	1	yes	characteristic
99	Q₈	—	52	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	13	4	no	—
104	C₂×C₄	—	13	8	no	—
105	C₂×C₄	—	13	4	no	—
106	C₂×C₄	—	26	4	no	—
107	C₂×C₄	—	26	4	no	—
108	C₂×C₄	—	26	8	no	—
109	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
110	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
111	C₂×C₄	D₄×D₁₃	1	2	yes	—
112	C₂×C₄	Q₈×D₁₃	1	2	yes	—
113	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
114	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
115	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
116	C₂²	#41×D₁₃	1	4	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	13	8	no	—
120	C₄	—	26	4	no	—
121	C₄	—	52	4	no	—
122	C₄	#88142	1	4	yes	—
123	C₂	C₂×#88142	1	1	yes	characteristic
124	C₂	#11719222	1	2	yes	—
125	C₂	D₁₃×#124	1	2	yes	—
126	C₂	#11719393	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88142

C₂

yes

maximal

C₁₃×#382

C₂

yes

maximal, characteristic

#11718638

C₂

yes

maximal

#11718597

C₂

yes

maximal, characteristic

C₄²×Dic₁₃

C₂

yes

maximal, characteristic

#11718307

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88142

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

Q₈

yes

complemented

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

—

C₄×Dic₁₃

Q₈

yes

complemented

#2862

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

#41

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

—

—

—

D₁₃

yes

characteristic

C₄³

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#124

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

C₂×#124

yes

—

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃

#533

yes

—

C₂×C₁₃

#533

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₂₆

yes

complemented

C₁₃

#2862

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×Dic₂₆

yes

—

110

C₂×C₄

C₂×Dic₂₆

yes

—

111

C₂×C₄

D₄×D₁₃

yes

—

112

C₂×C₄

Q₈×D₁₃

yes

—

113

C₂²

C₂²×Dic₂₆

yes

characteristic

114

C₂²

C₂×D₄×D₁₃

yes

characteristic

115

C₂²

C₂×Q₈×D₁₃

yes

characteristic

116

C₂²

#41×D₁₃

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#88142

yes

—

123

C₂

C₂×#88142

yes

characteristic

124

C₂

#11719222

yes

—

125

C₂

D₁₃×#124

yes

—

126

C₂

#11719393

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#382	C₂
hypercenter	#382	D₁₃
hypocenter	C₁₃	#2862
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#382

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
28. C₄²×C₁₃	Q₈	semidirect
33. C₄×Dic₁₃	Q₈	semidirect
96. C₄²	Dic₂₆	semidirect
97. C₁₃	#2862	semidirect

H ≅ G/N

type

28. C₄²×C₁₃

Q₈

semidirect

33. C₄×Dic₁₃

Q₈

semidirect

96. C₄²

Dic₂₆

semidirect

97. C₁₃

#2862

semidirect

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

subgroup	index
C₂×#88142	2
C₁₃×#382	2
#11718638	2
#11718597	2
C₄²×Dic₁₃	2
#11718307	2
#2862	13

subgroup

index

C₁₃×#382

C₄²×Dic₁₃

Subgroups — #13123212

Conjugacy classes of subgroups

Complements

Maximal subgroups