Group #13129714 — subgroups

Subgroups — #13129714

Order 1664.

188 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 76 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×#88112	C₂	1	2	yes	maximal
2	#11718637	C₂	1	1	yes	maximal, characteristic
3	C₂²×#1057	—	2	2	no	—
4	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
5	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
6	C₂×#6101	—	2	2	no	—
7	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
8	#88112	—	2	2	no	—
9	#88112	C₄	1	4	yes	complemented
10	C₂²×C₄×C₁₃	—	2	1	no	—
11	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×Dic₁₃	—	2	1	no	—
13	C₂²×Dic₁₃	D₄	1	2	yes	—
14	C₂×#1057	—	2	4	no	—
15	C₂×#1057	—	4	2	no	—
16	C₂×#1057	—	4	2	no	—
17	C₁₃×C₄⋊C₄	—	4	1	no	—
18	C₄⋊Dic₁₃	—	4	1	no	—
19	#6101	—	4	2	no	—
20	#6101	—	4	2	no	—
21	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₃	D₄	1	1	yes	characteristic
23	C₄×Dic₁₃	Q₈	1	1	yes	characteristic
24	#1296	—	13	1	no	maximal, Sylow
25	C₂³×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃	—	2	1	no	—
27	C₂×C₄×C₁₃	—	2	1	no	—
28	C₂×C₄×C₁₃	—	4	1	no	—
29	C₂×C₄×C₁₃	—	4	2	no	—
30	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₃	—	4	1	no	—
33	C₂×Dic₁₃	—	4	2	no	—
34	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
35	C₂×Dic₁₃	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₃	D₈	1	2	yes	—
37	C₂×Dic₁₃	SD₁₆	1	2	yes	—
38	C₂×Dic₁₃	SD₁₆	1	2	yes	—
39	C₂×Dic₁₃	Q₁₆	1	2	yes	—
40	#1057	—	4	4	no	—
41	C₂×C₄⋊C₈	—	13	2	no	—
42	#384	—	13	1	no	—
43	C₂²×C₁₃	#95	1	1	yes	characteristic
44	C₂²×C₁₃	#100	1	1	yes	characteristic
45	C₂²×C₁₃	#101	1	1	yes	characteristic
46	C₂²×C₁₃	#102	1	2	yes	—
47	C₂²×C₁₃	#103	1	2	yes	—
48	C₄×C₁₃	—	2	1	no	—
49	C₄×C₁₃	—	2	1	no	—
50	C₄×C₁₃	—	8	1	no	—
51	Dic₁₃	—	8	1	no	—
52	Dic₁₃	#102	1	2	yes	—
53	Dic₁₃	#103	1	2	yes	—
54	Dic₁₃	#106	1	2	yes	—
55	Dic₁₃	#107	1	2	yes	—
56	C₂²×C₈	—	26	2	no	—
57	C₂×C₄⋊C₄	—	13	1	no	—
58	C₂×C₄⋊C₄	—	26	2	no	—
59	C₂×C₄⋊C₄	#261	1	1	yes	characteristic
60	C₂×C₄²	—	13	1	no	—
61	C₄⋊C₈	—	13	4	no	—
62	C₄⋊C₈	—	26	2	no	—
63	C₂×C₁₃	#331	1	1	yes	characteristic
64	C₂×C₁₃	#332	1	2	yes	—
65	C₂×C₁₃	#333	1	1	yes	characteristic
66	C₂×C₁₃	#340	1	2	yes	—
67	C₂×C₁₃	#343	1	1	yes	characteristic
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	13	2	no	—
70	C₂²×C₄	—	26	1	no	—
71	C₂²×C₄	C₂×#261	1	1	yes	characteristic
72	C₂×C₈	—	26	4	no	—
73	C₂×C₈	—	52	2	no	—
74	C₂×C₈	—	52	2	no	—
75	C₄⋊C₄	—	4	1	no	—
76	C₄⋊C₄	—	52	1	no	—
77	C₄⋊C₄	—	52	2	no	—
78	C₄⋊C₄	—	52	2	no	—
79	C₄²	—	13	2	no	—
80	C₄²	—	13	1	no	—
81	C₄²	—	13	1	no	—
82	C₁₃	#1296	1	1	yes	characteristic, complemented, Sylow
83	C₂³	C₂²⋊#261	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	13	2	no	—
89	C₂×C₄	—	13	2	no	—
90	C₂×C₄	—	13	2	no	—
91	C₂×C₄	—	13	2	no	—
92	C₂×C₄	—	13	2	no	—
93	C₂×C₄	—	13	2	no	—
94	C₂×C₄	—	52	2	no	—
95	C₂×C₄	—	52	1	no	—
96	C₂×C₄	C₄×#261	1	1	yes	characteristic
97	C₂×C₄	C₄⋊#261	1	1	yes	characteristic
98	C₈	—	52	4	no	—
99	C₂²	#88113	1	1	yes	characteristic
100	C₂²	(C₂×#261)⋊C₄	1	1	yes	characteristic
101	C₂²	#88115	1	2	yes	—
102	C₂²	#88117	1	2	yes	—
103	C₂²	#88121	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	8	1	no	—
107	C₄	—	13	2	no	—
108	C₄	—	13	2	no	—
109	C₄	—	13	2	no	—
110	C₄	—	13	2	no	—
111	C₄	—	104	1	no	—
112	C₂	#11718245	1	2	yes	—
113	C₂	#11718246	1	2	yes	—
114	C₂	#11718291	1	1	yes	characteristic
115	C₂	#11718297	1	1	yes	characteristic
116	C₂	#11718303	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88112

C₂

yes

maximal

#11718637

C₂

yes

maximal, characteristic

C₂²×#1057

—

C₂×C₁₃×C₄⋊C₄

C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

C₄

yes

characteristic

C₂×#6101

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88112

—

#88112

C₄

yes

complemented

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

D₄

yes

—

C₂×#1057

—

C₂×#1057

—

C₂×#1057

—

C₁₃×C₄⋊C₄

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

C₂×C₄

yes

—

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

Q₈

yes

characteristic

#1296

—

maximal, Sylow

C₂³×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄²

yes

characteristic

C₂×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₃

C₄⋊C₄

yes

—

C₂×Dic₁₃

D₈

yes

—

C₂×Dic₁₃

SD₁₆

yes

—

C₂×Dic₁₃

SD₁₆

yes

—

C₂×Dic₁₃

Q₁₆

yes

—

#1057

—

C₂×C₄⋊C₈

—

#384

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#103

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

#102

yes

—

Dic₁₃

#103

yes

—

Dic₁₃

#106

yes

—

Dic₁₃

#107

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#261

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂×C₁₃

#331

yes

characteristic

C₂×C₁₃

#332

yes

—

C₂×C₁₃

#333

yes

characteristic

C₂×C₁₃

#340

yes

—

C₂×C₁₃

#343

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#261

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₁₃

#1296

yes

characteristic, complemented, Sylow

C₂³

C₂²⋊#261

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₂²

#88113

yes

characteristic

100

C₂²

(C₂×#261)⋊C₄

yes

characteristic

101

C₂²

#88115

yes

—

102

C₂²

#88117

yes

—

103

C₂²

#88121

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

104

—

112

C₂

#11718245

yes

—

113

C₂

#11718246

yes

—

114

C₂

#11718291

yes

characteristic

115

C₂

#11718297

yes

characteristic

116

C₂

#11718303

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₄²
Frattini	C₂²×C₄	C₂×#261
Fitting	C₂×C₁₃×C₄⋊C₄	C₄
socle, prime radical	C₂³×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂²
hypercenter	C₂×C₄⋊C₄	#261
hypocenter	C₁₃	#1296
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₄²

Frattini

C₂²×C₄

C₂×#261

Fitting

C₂×C₁₃×C₄⋊C₄

C₄

socle, prime radical

C₂³×C₁₃

C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₃

C₂²

C₂×C₄⋊C₄

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
9. #88112	C₄	semidirect
82. C₁₃	#1296	semidirect

H ≅ G/N

type

9. #88112

C₄

semidirect

82. C₁₃

#1296

semidirect

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

subgroup	index
C₂×#88112	2
#11718637	2
#1296	13

subgroup

index

C₂×#88112

#11718637

#1296

Subgroups — #13129714

Conjugacy classes of subgroups

Complements

Maximal subgroups