Group #13114496 — subgroups

Subgroups — #13114496

Order 1664.

186 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 61 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, complemented
2	#11718355	C₂	1	1	yes	maximal, characteristic
3	#11718355	C₂	1	1	yes	maximal, characteristic
4	#11718355	C₂	1	1	yes	maximal, characteristic
5	#11718342	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₂	—	2	2	no	—
9	#88128	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	#88061	—	2	2	no	—
17	#88061	—	2	2	no	—
18	#88061	—	2	2	no	—
19	#88057	—	2	2	no	—
20	#88056	—	2	2	no	—
21	C₂×D₅₂	—	2	1	no	—
22	C₂×D₅₂	—	2	1	no	—
23	C₂×D₅₂	—	2	1	no	—
24	C₂×Dic₂₆	—	2	1	no	—
25	C₂×Dic₂₆	—	2	1	no	—
26	C₂×Dic₂₆	—	2	1	no	—
27	C₂×C₈×C₁₃	D₄	1	2	yes	—
28	C₂×C₈×C₁₃	D₄	1	2	yes	—
29	C₂×C₈×C₁₃	D₄	1	2	yes	—
30	C₄²×C₁₃	C₂³	1	1	yes	characteristic
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	#1720	—	13	1	no	maximal, Sylow
39	C₂²×D₁₃	—	4	1	no	—
40	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₁₃	—	4	1	no	—
44	C₂×Dic₁₃	—	4	1	no	—
45	C₂×Dic₁₃	—	4	1	no	—
46	D₅₂	—	4	2	no	—
47	D₅₂	—	4	2	no	—
48	D₅₂	—	4	2	no	—
49	Dic₂₆	—	4	2	no	—
50	Dic₂₆	—	4	2	no	—
51	Dic₂₆	—	4	2	no	—
52	C₈×C₁₃	—	2	2	no	—
53	C₈×C₁₃	—	2	2	no	—
54	C₈×C₁₃	—	2	2	no	—
55	#498	—	13	1	no	—
56	#494	—	13	1	no	—
57	#487	—	13	1	no	—
58	#486	—	13	1	no	—
59	#486	—	13	1	no	—
60	#486	—	13	1	no	—
61	#322	D₁₃	1	1	yes	characteristic, complemented
62	C₂²×C₁₃	#127	1	1	yes	characteristic
63	C₂×D₁₃	—	8	1	no	—
64	C₂×D₁₃	—	8	1	no	—
65	C₂×D₁₃	—	8	1	no	—
66	C₄×C₁₃	#135	1	2	yes	—
67	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₁₃	#137	1	2	yes	—
69	Dic₁₃	—	8	1	no	—
70	Dic₁₃	—	8	1	no	—
71	Dic₁₃	—	8	1	no	—
72	C₂×Q₁₆	—	26	2	no	—
73	C₄⋊Q₈	—	13	1	no	—
74	C₄⋊Q₈	—	13	1	no	—
75	C₄⋊Q₈	—	13	1	no	—
76	#127	—	13	1	no	—
77	#106	—	26	2	no	—
78	#103	—	26	2	no	—
79	#102	—	26	2	no	—
80	#102	—	26	2	no	—
81	#102	—	26	2	no	—
82	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
84	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
85	C₂×C₁₃	#495	1	1	yes	characteristic
86	C₂×C₁₃	#496	1	1	yes	characteristic
87	C₂×C₁₃	#497	1	1	yes	characteristic
88	D₁₃	—	16	1	no	—
89	C₂×Q₈	—	26	1	no	—
90	C₂×Q₈	—	26	1	no	—
91	C₂×Q₈	—	26	1	no	—
92	C₂×D₄	—	26	1	no	—
93	C₂×D₄	—	26	1	no	—
94	C₂×D₄	—	26	1	no	—
95	Q₁₆	—	52	4	no	—
96	C₂×C₈	D₅₂	1	2	yes	—
97	C₂×C₈	D₅₂	1	2	yes	—
98	C₂×C₈	D₅₂	1	2	yes	—
99	C₄⋊C₄	—	26	1	no	—
100	C₄⋊C₄	—	26	1	no	—
101	C₄⋊C₄	—	26	1	no	—
102	C₄⋊C₄	—	26	1	no	—
103	C₄⋊C₄	—	26	1	no	—
104	C₄⋊C₄	—	26	1	no	—
105	C₄²	C₂²×D₁₃	1	1	yes	characteristic
106	C₁₃	#1720	1	1	yes	characteristic, complemented, Sylow
107	C₂³	—	52	1	no	—
108	Q₈	—	52	2	no	—
109	Q₈	—	52	2	no	—
110	Q₈	—	52	2	no	—
111	D₄	—	52	2	no	—
112	D₄	—	52	2	no	—
113	D₄	—	52	2	no	—
114	C₂×C₄	—	52	1	no	—
115	C₂×C₄	—	52	1	no	—
116	C₂×C₄	—	52	1	no	—
117	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
118	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	2	2	no	—
122	C₈	—	2	2	no	—
123	C₂²	—	104	1	no	—
124	C₂²	—	104	1	no	—
125	C₂²	—	104	1	no	—
126	C₂²	#88128	1	1	yes	characteristic
127	C₄	—	104	1	no	—
128	C₄	—	104	1	no	—
129	C₄	—	104	1	no	—
130	C₄	#88158	1	2	yes	—
131	C₄	#88162	1	2	yes	—
132	C₄	#88163	1	2	yes	—
133	C₂	—	208	1	no	—
134	C₂	#11718357	1	1	yes	characteristic
135	C₂	#11718373	1	1	yes	characteristic
136	C₂	#11718376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11718360

C₂

yes

maximal, characteristic, complemented

#11718355

C₂

yes

maximal, characteristic

#11718355

C₂

yes

maximal, characteristic

#11718355

C₂

yes

maximal, characteristic

#11718342

C₂

yes

maximal, characteristic, complemented

#11718341

C₂

yes

maximal, characteristic, complemented

C₁₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₂

—

#88128

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×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

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

—

Dic₅₂

—

#1720

—

maximal, Sylow

C₂²×D₁₃

—

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

—

D₅₂

—

D₅₂

—

D₅₂

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

#127

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#135

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

#137

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

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

D₁₃

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

C₂×C₈

D₅₂

yes

—

C₂×C₈

D₅₂

yes

—

C₂×C₈

D₅₂

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

C₂²×D₁₃

yes

characteristic

106

C₁₃

#1720

yes

characteristic, complemented, Sylow

107

C₂³

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₅₂

yes

characteristic

118

C₂×C₄

C₂×D₅₂

yes

characteristic

119

C₂×C₄

C₂×D₅₂

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

104

—

124

C₂²

—

104

—

125

C₂²

—

104

—

126

C₂²

#88128

yes

characteristic

127

C₄

—

104

—

128

C₄

—

104

—

129

C₄

—

104

—

130

C₄

#88158

yes

—

131

C₄

#88162

yes

—

132

C₄

#88163

yes

—

133

C₂

—

208

—

134

C₂

#11718357

yes

characteristic

135

C₂

#11718373

yes

characteristic

136

C₂

#11718376

yes

characteristic

	subgroup	G/H
center	C₂²	#88128
commutator, cosocle	C₄²×C₁₃	C₂³
Frattini	C₄²	C₂²×D₁₃
Fitting	C₁₃×#322	C₂
socle	C₂²×C₁₃	#127
prime radical	#88128	C₂²
hypercenter	#322	D₁₃
hypocenter	C₁₃	#1720
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

C₂²

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #11718360	C₂	semidirect
5. #11718342	C₂	semidirect
6. #11718341	C₂	semidirect
7. C₁₃×#322	C₂	semidirect
61. #322	D₁₃	semidirect
106. C₁₃	#1720	semidirect

H ≅ G/N

type

1. #11718360

C₂

semidirect

5. #11718342

C₂

semidirect

6. #11718341

C₂

semidirect

7. C₁₃×#322

C₂

semidirect

61. #322

D₁₃

semidirect

106. C₁₃

#1720

semidirect

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

subgroup	index
#11718360	2
#11718355	2
#11718342	2
#11718341	2
C₁₃×#322	2
#1720	13

subgroup

index

C₁₃×#322

Subgroups — #13114496

Conjugacy classes of subgroups

Complements

Maximal subgroups