Group #13114513 — subgroups

Subgroups — #13114513

Order 1664.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#11718360	C₂	1	1	yes	maximal, characteristic
2	#11718343	C₂	1	1	yes	maximal, characteristic
3	#11718343	C₂	1	1	yes	maximal, characteristic
4	#11718342	C₂	1	1	yes	maximal, characteristic
5	#11718342	C₂	1	1	yes	maximal, characteristic
6	#11718341	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₂	—	2	2	no	—
9	#88123	C₂²	1	1	yes	characteristic
10	#88123	C₂²	1	1	yes	characteristic
11	#88123	C₂²	1	1	yes	characteristic
12	#88123	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#88058	—	2	2	no	—
17	#88058	—	2	2	no	—
18	#88057	—	2	2	no	—
19	#88056	—	2	2	no	—
20	#88056	—	2	2	no	—
21	C₂×Dic₂₆	—	2	1	no	—
22	C₂×Dic₂₆	—	2	1	no	—
23	C₂×Dic₂₆	—	2	1	no	—
24	C₂×Dic₂₆	—	2	1	no	—
25	C₂×C₈×C₁₃	D₄	1	2	yes	—
26	C₂×C₈×C₁₃	D₄	1	2	yes	—
27	C₂×C₈×C₁₃	D₄	1	2	yes	—
28	C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₃	—	2	1	no	—
30	C₄⋊Dic₁₃	—	2	1	no	—
31	C₄⋊Dic₁₃	—	2	1	no	—
32	C₄⋊Dic₁₃	—	2	1	no	—
33	C₄⋊Dic₁₃	—	2	1	no	—
34	C₄⋊Dic₁₃	—	2	1	no	—
35	C₄⋊Dic₁₃	—	2	1	no	—
36	C₄⋊Dic₁₃	—	2	1	no	—
37	Dic₅₂	—	4	4	no	—
38	#1723	—	13	1	no	maximal, Sylow
39	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₁₃	—	4	1	no	—
43	C₂×Dic₁₃	—	4	1	no	—
44	C₂×Dic₁₃	—	4	1	no	—
45	C₂×Dic₁₃	—	4	1	no	—
46	Dic₂₆	—	4	2	no	—
47	Dic₂₆	—	4	2	no	—
48	Dic₂₆	—	4	2	no	—
49	Dic₂₆	—	4	2	no	—
50	C₈×C₁₃	—	2	2	no	—
51	C₈×C₁₃	—	2	2	no	—
52	C₈×C₁₃	Q₁₆	1	4	yes	complemented
53	#500	—	13	1	no	—
54	#500	—	13	1	no	—
55	#498	—	13	1	no	—
56	#494	—	13	1	no	—
57	#487	—	13	1	no	—
58	#487	—	13	1	no	—
59	#322	D₁₃	1	1	yes	characteristic
60	C₂²×C₁₃	#127	1	1	yes	characteristic
61	C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
62	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₃	#137	1	2	yes	—
64	Dic₁₃	—	8	1	no	—
65	Dic₁₃	—	8	1	no	—
66	Dic₁₃	—	8	1	no	—
67	Dic₁₃	—	8	1	no	—
68	C₂×Q₁₆	—	26	2	no	—
69	C₄⋊Q₈	—	13	1	no	—
70	C₄⋊Q₈	—	13	1	no	—
71	C₄⋊Q₈	—	13	1	no	—
72	C₄⋊Q₈	—	13	1	no	—
73	#107	—	26	2	no	—
74	#107	—	26	2	no	—
75	#106	—	26	2	no	—
76	#103	—	26	2	no	—
77	#103	—	26	2	no	—
78	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
79	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
80	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
81	C₂×C₁₃	#494	1	1	yes	characteristic
82	C₂×C₁₃	#496	1	1	yes	characteristic
83	C₂×C₁₃	#497	1	1	yes	characteristic
84	C₂×Q₈	—	26	1	no	—
85	C₂×Q₈	—	26	1	no	—
86	C₂×Q₈	—	26	1	no	—
87	C₂×Q₈	—	26	1	no	—
88	Q₁₆	—	52	4	no	—
89	C₂×C₈	D₅₂	1	2	yes	—
90	C₂×C₈	D₅₂	1	2	yes	—
91	C₂×C₈	D₅₂	1	2	yes	—
92	C₄⋊C₄	—	26	1	no	—
93	C₄⋊C₄	—	26	1	no	—
94	C₄⋊C₄	—	26	1	no	—
95	C₄⋊C₄	—	26	1	no	—
96	C₄⋊C₄	—	26	1	no	—
97	C₄⋊C₄	—	26	1	no	—
98	C₄⋊C₄	—	26	1	no	—
99	C₄⋊C₄	—	26	1	no	—
100	C₄²	C₂²×D₁₃	1	1	yes	characteristic
101	C₁₃	#1723	1	1	yes	characteristic, complemented, Sylow
102	Q₈	—	52	2	no	—
103	Q₈	—	52	2	no	—
104	Q₈	—	52	2	no	—
105	Q₈	—	52	2	no	—
106	C₂×C₄	—	52	1	no	—
107	C₂×C₄	—	52	1	no	—
108	C₂×C₄	—	52	1	no	—
109	C₂×C₄	—	52	1	no	—
110	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
111	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
112	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₅₂	1	4	yes	complemented
116	C₂²	#88128	1	1	yes	characteristic
117	C₄	—	104	1	no	—
118	C₄	—	104	1	no	—
119	C₄	—	104	1	no	—
120	C₄	—	104	1	no	—
121	C₄	C₂×Dic₅₂	1	2	yes	—
122	C₄	#88162	1	2	yes	—
123	C₄	#88163	1	2	yes	—
124	C₂	#11718360	1	1	yes	characteristic
125	C₂	#11718373	1	1	yes	characteristic
126	C₂	#11718376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11718360

C₂

yes

maximal, characteristic

#11718343

C₂

yes

maximal, characteristic

#11718343

C₂

yes

maximal, characteristic

#11718342

C₂

yes

maximal, characteristic

#11718342

C₂

yes

maximal, characteristic

#11718341

C₂

yes

maximal, characteristic

C₁₃×#322

C₂

yes

maximal, characteristic

C₂×Dic₅₂

—

#88123

C₂²

yes

characteristic

#88123

C₂²

yes

characteristic

#88123

C₂²

yes

characteristic

#88123

C₂²

yes

characteristic

C₄×C₈×C₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₃

C₂²

yes

—

—

—

—

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

#1723

—

maximal, Sylow

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

Q₁₆

yes

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

#127

yes

characteristic

C₄×C₁₃

C₂×Q₁₆

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

#137

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

C₄×C₈

C₂×D₁₃

yes

characteristic

C₄×C₈

C₂×D₁₃

yes

characteristic

C₄×C₈

C₂×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

D₅₂

yes

—

C₂×C₈

D₅₂

yes

—

C₂×C₈

D₅₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

C₂²×D₁₃

yes

characteristic

101

C₁₃

#1723

yes

characteristic, complemented, Sylow

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×D₅₂

yes

characteristic

111

C₂×C₄

C₂×D₅₂

yes

characteristic

112

C₂×C₄

C₂×D₅₂

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₅₂

yes

complemented

116

C₂²

#88128

yes

characteristic

117

C₄

—

104

—

118

C₄

—

104

—

119

C₄

—

104

—

120

C₄

—

104

—

121

C₄

C₂×Dic₅₂

yes

—

122

C₄

#88162

yes

—

123

C₄

#88163

yes

—

124

C₂

#11718360

yes

characteristic

125

C₂

#11718373

yes

characteristic

126

C₂

#11718376

yes

characteristic

	subgroup	G/H
center	C₂²	#88128
commutator, cosocle	C₄²×C₁₃	C₂³
Frattini	C₄²	C₂²×D₁₃
Fitting	C₁₃×#322	C₂
socle, prime radical	C₂²×C₁₃	#127
hypercenter	#322	D₁₃
hypocenter	C₁₃	#1723
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#88128

commutator, cosocle

C₄²×C₁₃

C₂³

Frattini

C₄²

C₂²×D₁₃

Fitting

C₁₃×#322

C₂

C₂²×C₁₃

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
52. C₈×C₁₃	Q₁₆	semidirect
101. C₁₃	#1723	semidirect
115. C₈	Dic₅₂	semidirect

H ≅ G/N

type

52. C₈×C₁₃

Q₁₆

semidirect

101. C₁₃

#1723

semidirect

115. C₈

Dic₅₂

semidirect

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

subgroup	index
#11718360	2
#11718343	2
#11718342	2
#11718341	2
C₁₃×#322	2
#1723	13

subgroup

index

C₁₃×#322

Subgroups — #13114513

Conjugacy classes of subgroups

Complements

Maximal subgroups