Group #13115004 — subgroups

Subgroups — #13115004

Order 1664.

256 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 141 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×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#88058	C₂	1	2	yes	maximal
3	C₂×#88057	C₂	1	2	yes	maximal
4	#11718597	C₂	1	1	yes	maximal, characteristic
5	#11718596	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃	—	2	2	no	—
9	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#97	C₄	1	4	yes	complemented
13	#88071	—	2	2	no	—
14	#88058	—	2	4	no	—
15	#88057	—	2	4	no	—
16	C₂²×C₄×C₁₃	D₄	1	2	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₃	—	2	1	no	—
19	C₂²×Dic₁₃	—	2	1	no	—
20	C₂²×Dic₁₃	—	2	2	no	—
21	C₂×C₈×C₁₃	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₁₃	D₄	1	4	yes	—
23	C₂×C₈×C₁₃	Q₈	1	4	yes	—
24	C₄²×C₁₃	C₂×C₄	1	2	yes	—
25	C₄²×C₁₃	Q₈	1	2	yes	—
26	C₄⋊Dic₁₃	—	2	2	no	—
27	C₄⋊Dic₁₃	—	2	2	no	—
28	C₄⋊Dic₁₃	—	2	2	no	—
29	C₄⋊Dic₁₃	—	2	2	no	—
30	C₄⋊Dic₁₃	—	4	4	no	—
31	#1847	—	13	1	no	maximal, Sylow
32	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃	—	2	2	no	—
34	C₂×C₄×C₁₃	—	2	2	no	—
35	C₂×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×Dic₁₃	—	4	2	no	—
41	C₂×Dic₁₃	—	4	1	no	—
42	C₂×Dic₁₃	—	4	1	no	—
43	C₂×Dic₁₃	—	4	4	no	—
44	C₂×Dic₁₃	—	4	2	no	—
45	C₂×Dic₁₃	—	4	2	no	—
46	C₈×C₁₃	C₄⋊C₄	1	8	yes	complemented
47	C₂×#107	—	13	2	no	—
48	C₂×#106	—	13	2	no	—
49	C₂×#97	D₁₃	1	1	yes	characteristic
50	#384	—	13	1	no	—
51	#382	—	13	1	no	—
52	C₂²×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₃	#127	1	1	yes	characteristic
54	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
56	C₂²×C₁₃	#137	1	2	yes	—
57	C₄×C₁₃	—	2	4	no	—
58	C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
59	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
60	Dic₁₃	—	8	2	no	—
61	Dic₁₃	—	8	1	no	—
62	Dic₁₃	—	8	1	no	—
63	C₂²×C₈	C₂×D₁₃	1	2	yes	—
64	C₂×C₄⋊C₄	—	13	1	no	—
65	C₂×C₄⋊C₄	—	13	1	no	—
66	C₂×C₄⋊C₄	—	13	2	no	—
67	C₂×C₄⋊C₄	—	26	2	no	—
68	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
69	#107	—	26	4	no	—
70	#106	—	26	4	no	—
71	#97	Dic₁₃	1	4	yes	complemented
72	#95	—	26	2	no	—
73	C₂×C₁₃	#384	1	1	yes	characteristic
74	C₂×C₁₃	#428	1	2	yes	—
75	C₂×C₁₃	#496	1	1	yes	characteristic
76	C₂×C₁₃	#497	1	1	yes	characteristic
77	C₂×C₁₃	#501	1	2	yes	—
78	C₂²×C₄	—	26	1	no	—
79	C₂²×C₄	—	26	1	no	—
80	C₂²×C₄	—	26	2	no	—
81	C₂²×C₄	D₅₂	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
83	C₂×C₈	Dic₂₆	1	4	yes	—
84	C₂×C₈	D₅₂	1	4	yes	—
85	C₂×C₈	C₂×Dic₁₃	1	4	yes	—
86	C₄⋊C₄	—	26	2	no	—
87	C₄⋊C₄	—	26	2	no	—
88	C₄⋊C₄	—	26	2	no	—
89	C₄⋊C₄	—	26	2	no	—
90	C₄⋊C₄	—	52	4	no	—
91	C₄²	Dic₂₆	1	2	yes	—
92	C₄²	C₂×Dic₁₃	1	2	yes	—
93	C₁₃	#1847	1	1	yes	characteristic, complemented, Sylow
94	C₂³	C₂×D₅₂	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	52	2	no	—
98	C₂×C₄	—	52	1	no	—
99	C₂×C₄	—	52	1	no	—
100	C₂×C₄	—	52	4	no	—
101	C₂×C₄	—	52	2	no	—
102	C₂×C₄	—	52	2	no	—
103	C₂×C₄	C₄⋊Dic₁₃	1	4	yes	—
104	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
105	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
106	C₂×C₄	C₂×D₅₂	1	2	yes	—
107	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
108	C₈	C₄⋊Dic₁₃	1	8	yes	complemented
109	C₂²	#88123	1	1	yes	characteristic
110	C₂²	#88128	1	1	yes	characteristic
111	C₂²	#88162	1	2	yes	—
112	C₂²	#88163	1	2	yes	—
113	C₂²	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
114	C₄	—	2	4	no	—
115	C₄	—	104	2	no	—
116	C₄	—	104	1	no	—
117	C₄	—	104	1	no	—
118	C₄	#88123	1	2	yes	—
119	C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
120	C₂	#11718363	1	2	yes	—
121	C₂	#11718373	1	1	yes	characteristic
122	C₂	#11718376	1	1	yes	characteristic
123	C₂	#11718596	1	1	yes	characteristic
124	C₂	#11718782	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#97

C₂

yes

maximal, characteristic

C₂×#88058

C₂

yes

maximal

C₂×#88057

C₂

yes

maximal

#11718597

C₂

yes

maximal, characteristic

#11718596

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₁₃×#97

C₄

yes

—

—

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

Q₈

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#1847

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

C₄⋊C₄

yes

complemented

C₂×#107

—

C₂×#106

—

C₂×#97

D₁₃

yes

characteristic

#384

—

#382

—

C₂²×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃

#127

yes

characteristic

C₂²×C₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

—

Dic₁₃

yes

complemented

#95

—

C₂×C₁₃

#384

yes

characteristic

C₂×C₁₃

#428

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₅₂

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

Dic₂₆

yes

—

C₂×C₈

D₅₂

yes

—

C₂×C₈

C₂×Dic₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₂₆

yes

—

C₄²

C₂×Dic₁₃

yes

—

C₁₃

#1847

yes

characteristic, complemented, Sylow

C₂³

C₂×D₅₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₄⋊Dic₁₃

yes

—

104

C₂×C₄

C₂×Dic₂₆

yes

characteristic

105

C₂×C₄

C₂×Dic₂₆

yes

—

106

C₂×C₄

C₂×D₅₂

yes

—

107

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

108

C₈

C₄⋊Dic₁₃

yes

complemented

109

C₂²

#88123

yes

characteristic

110

C₂²

#88128

yes

characteristic

111

C₂²

#88162

yes

—

112

C₂²

#88163

yes

—

113

C₂²

C₂×C₄⋊Dic₁₃

yes

characteristic

114

C₄

—

115

C₄

—

104

—

116

C₄

—

104

—

117

C₄

—

104

—

118

C₄

#88123

yes

—

119

C₄

C₂×C₄⋊Dic₁₃

yes

—

120

C₂

#11718363

yes

—

121

C₂

#11718373

yes

characteristic

122

C₂

#11718376

yes

characteristic

123

C₂

#11718596

yes

characteristic

124

C₂

#11718782

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#97	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×#97	D₁₃
hypocenter	C₁₃	#1847
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×#97

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂×#97

D₁₃

hypocenter

C₁₃

#1847

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. C₁₃×#97	C₄	semidirect
46. C₈×C₁₃	C₄⋊C₄	semidirect
71. #97	Dic₁₃	semidirect
93. C₁₃	#1847	semidirect
108. C₈	C₄⋊Dic₁₃	semidirect

H ≅ G/N

type

12. C₁₃×#97

C₄

semidirect

46. C₈×C₁₃

C₄⋊C₄

semidirect

71. #97

Dic₁₃

semidirect

93. C₁₃

#1847

semidirect

108. C₈

C₄⋊Dic₁₃

semidirect

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

subgroup	index
C₂×C₁₃×#97	2
C₂×#88058	2
C₂×#88057	2
#11718597	2
#11718596	2
#1847	13

subgroup

index

C₂×C₁₃×#97

Subgroups — #13115004

Conjugacy classes of subgroups

Complements

Maximal subgroups