Group #13123196 — subgroups

Subgroups — #13123196

Order 1664.

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

normal subgroups only

This group has 472 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×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#88124	C₂	1	1	yes	maximal, characteristic
3	#11718638	C₂	1	2	yes	maximal
4	#11718637	C₂	1	2	yes	maximal
5	#11718633	C₂	1	8	yes	maximal
6	C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#125	—	2	4	no	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×#6101	C₂²	1	4	yes	—
13	C₂×#6101	C₂²	1	8	yes	—
14	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
17	#88124	—	2	4	no	—
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	C₁₃×C₄⋊C₄	—	2	8	no	—
26	C₁₃×C₄⋊C₄	—	2	4	no	—
27	C₄²×C₁₃	D₄	1	4	yes	—
28	C₄⋊Dic₁₃	—	2	4	no	—
29	#6101	—	2	8	no	—
30	#6101	—	2	16	no	—
31	C₄×Dic₁₃	—	2	8	no	—
32	C₄×Dic₁₃	Q₈	1	4	yes	—
33	C₄×Dic₁₃	Q₈	1	4	yes	—
34	#2859	—	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	4	yes	—
40	C₂×C₄×C₁₃	C₂×D₄	1	2	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×Q₈	1	4	yes	—
47	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
48	C₂×Dic₁₃	#41	1	8	yes	—
49	C₂×#125	—	13	1	no	—
50	C₂×#125	D₁₃	1	1	yes	characteristic
51	#389	—	13	8	no	—
52	#384	—	13	2	no	—
53	#382	—	13	2	no	—
54	C₄³	—	13	1	no	—
55	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
56	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
57	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
58	C₂²×C₁₃	C₂×#41	1	4	yes	—
59	C₄×C₁₃	—	2	4	no	—
60	C₄×C₁₃	—	4	4	no	—
61	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
62	Dic₁₃	—	2	4	no	—
63	Dic₁₃	—	4	4	no	—
64	Dic₁₃	#125	1	8	yes	complemented
65	#125	—	2	4	no	—
66	#125	—	26	4	no	—
67	C₂×C₄⋊C₄	—	13	4	no	—
68	C₂×C₄⋊C₄	—	13	2	no	—
69	C₂×C₄⋊C₄	—	13	8	no	—
70	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
71	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
72	C₂×C₄²	—	13	1	no	—
73	C₂×C₄²	—	13	1	no	—
74	C₂×C₄²	—	13	4	no	—
75	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
76	#95	—	13	8	no	—
77	C₂×C₁₃	C₂×#125	1	2	yes	—
78	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₁₃	#532	1	2	yes	—
80	C₂×C₁₃	#533	1	2	yes	—
81	C₂²×C₄	—	13	2	no	—
82	C₂²×C₄	—	13	2	no	—
83	C₂²×C₄	—	13	4	no	—
84	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
85	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
86	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	2	4	no	—
89	C₄⋊C₄	—	26	16	no	—
90	C₄⋊C₄	—	26	8	no	—
91	C₄⋊C₄	—	26	4	no	—
92	C₄²	—	13	4	no	—
93	C₄²	—	13	4	no	—
94	C₄²	—	26	8	no	—
95	C₄²	C₂²⋊D₁₃	1	4	yes	—
96	C₁₃	#2859	1	1	yes	characteristic, complemented, Sylow
97	C₂³	C₂³×D₁₃	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	13	4	no	—
102	C₂×C₄	—	13	8	no	—
103	C₂×C₄	—	13	4	no	—
104	C₂×C₄	—	26	4	no	—
105	C₂×C₄	—	26	8	no	—
106	C₂×C₄	—	26	4	no	—
107	C₂×C₄	Q₈×D₁₃	1	2	yes	—
108	C₂×C₄	Q₈×D₁₃	1	2	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
111	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
112	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
113	C₂²	#41×D₁₃	1	4	yes	—
114	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
115	C₄	—	2	4	no	—
116	C₄	—	4	4	no	—
117	C₄	—	13	8	no	—
118	C₄	—	26	4	no	—
119	C₄	—	52	4	no	—
120	C₄	#88198	1	4	yes	—
121	C₂	D₁₃×#125	1	2	yes	—
122	C₂	#11719393	1	2	yes	—
123	C₂	C₂×#88198	1	1	yes	characteristic
124	C₂	#11719537	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#125

C₂

yes

maximal, characteristic

C₂×#88124

C₂

yes

maximal, characteristic

#11718638

C₂

yes

maximal

#11718637

C₂

yes

maximal

#11718633

C₂

yes

maximal

C₄²×Dic₁₃

C₂

yes

maximal, characteristic

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88124

—

#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

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

Q₈

yes

—

C₄×Dic₁₃

Q₈

yes

—

#2859

—

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×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

#41

yes

—

C₂×#125

—

C₂×#125

D₁₃

yes

—

—

—

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

yes

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

C₂×C₁₃

C₂×#125

yes

—

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃

#532

yes

—

C₂×C₁₃

#533

yes

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₁₃

yes

—

C₁₃

#2859

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

Q₈×D₁₃

yes

—

108

C₂×C₄

Q₈×D₁₃

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

111

C₂²

C₂×Q₈×D₁₃

yes

characteristic

112

C₂²

C₂×Q₈×D₁₃

yes

characteristic

113

C₂²

#41×D₁₃

yes

—

114

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#88198

yes

—

121

C₂

D₁₃×#125

yes

—

122

C₂

#11719393

yes

—

123

C₂

C₂×#88198

yes

characteristic

124

C₂

#11719537

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×#125	C₂
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	#2859
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×#125

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
64. Dic₁₃	#125	semidirect
96. C₁₃	#2859	semidirect

H ≅ G/N

type

64. Dic₁₃

#125

semidirect

96. C₁₃

#2859

semidirect

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

subgroup	index
C₂×C₁₃×#125	2
C₂×#88124	2
#11718638	2
#11718637	2
#11718633	2
C₄²×Dic₁₃	2
#2859	13

subgroup

index

C₂×C₁₃×#125

C₄²×Dic₁₃

Subgroups — #13123196

Conjugacy classes of subgroups

Complements

Maximal subgroups