Group #13114475 — subgroups

Subgroups — #13114475

Order 1664.

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

normal subgroups only

This group has 208 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	#11718356	C₂	1	1	yes	maximal, characteristic
2	#11718355	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718355	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718355	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718354	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718342	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₀₄	—	2	2	no	—
9	C₂×#6095	—	2	2	no	—
10	#88128	C₂²	1	1	yes	characteristic
11	#88128	C₂²	1	1	yes	characteristic
12	#88123	C₂²	1	1	yes	characteristic
13	#88123	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#88061	—	2	2	no	—
18	#88061	—	2	2	no	—
19	#88061	—	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×D₅₂	—	2	1	no	—
25	C₂×D₅₂	—	2	1	no	—
26	C₂×D₅₂	—	2	1	no	—
27	C₂×Dic₂₆	—	2	1	no	—
28	C₂×Dic₂₆	—	2	1	no	—
29	C₂×C₈×C₁₃	D₄	1	2	yes	—
30	C₂×C₈×C₁₃	D₄	1	2	yes	—
31	C₂×C₈×C₁₃	D₄	1	2	yes	—
32	C₄²×C₁₃	C₂³	1	1	yes	characteristic
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	D₁₀₄	—	4	4	no	—
38	#6095	—	4	4	no	—
39	#1717	—	13	1	no	maximal, Sylow
40	C₂²×D₁₃	—	4	1	no	—
41	C₂²×D₁₃	—	4	1	no	—
42	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₃	—	4	1	no	—
46	C₂×Dic₁₃	—	4	1	no	—
47	D₅₂	—	4	2	no	—
48	D₅₂	—	4	2	no	—
49	D₅₂	—	4	2	no	—
50	D₅₂	—	4	2	no	—
51	D₅₂	—	4	2	no	—
52	D₅₂	—	4	2	no	—
53	Dic₂₆	—	4	2	no	—
54	Dic₂₆	—	4	2	no	—
55	C₈×C₁₃	—	2	2	no	—
56	C₈×C₁₃	—	2	2	no	—
57	C₈×C₁₃	—	2	2	no	—
58	#493	—	13	1	no	—
59	#492	—	13	1	no	—
60	#487	—	13	1	no	—
61	#486	—	13	1	no	—
62	#486	—	13	1	no	—
63	#486	—	13	1	no	—
64	#322	D₁₃	1	1	yes	characteristic, complemented
65	C₂²×C₁₃	#127	1	1	yes	characteristic
66	C₂×D₁₃	—	8	1	no	—
67	C₂×D₁₃	—	8	1	no	—
68	C₂×D₁₃	—	8	1	no	—
69	C₂×D₁₃	—	8	1	no	—
70	C₂×D₁₃	—	8	1	no	—
71	C₂×D₁₃	—	8	1	no	—
72	C₄×C₁₃	#135	1	2	yes	—
73	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
75	Dic₁₃	—	8	1	no	—
76	Dic₁₃	—	8	1	no	—
77	C₂×SD₁₆	—	26	2	no	—
78	C₂×D₈	—	26	2	no	—
79	C₄⋊Q₈	—	13	1	no	—
80	C₄⋊Q₈	—	13	1	no	—
81	#127	—	13	1	no	—
82	#127	—	13	1	no	—
83	#103	—	26	2	no	—
84	#102	—	26	2	no	—
85	#102	—	26	2	no	—
86	#102	—	26	2	no	—
87	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
88	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
89	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
90	C₂×C₁₃	#495	1	1	yes	characteristic
91	C₂×C₁₃	#496	1	1	yes	characteristic
92	C₂×C₁₃	#496	1	1	yes	characteristic
93	D₁₃	—	16	1	no	—
94	D₁₃	—	16	1	no	—
95	C₂×Q₈	—	26	1	no	—
96	C₂×Q₈	—	26	1	no	—
97	C₂×D₄	—	26	1	no	—
98	C₂×D₄	—	26	1	no	—
99	C₂×D₄	—	26	1	no	—
100	C₂×D₄	—	26	1	no	—
101	C₂×D₄	—	26	1	no	—
102	C₂×D₄	—	26	1	no	—
103	SD₁₆	—	52	4	no	—
104	D₈	—	52	4	no	—
105	C₂×C₈	D₅₂	1	2	yes	—
106	C₂×C₈	D₅₂	1	2	yes	—
107	C₂×C₈	D₅₂	1	2	yes	—
108	C₄⋊C₄	—	26	1	no	—
109	C₄⋊C₄	—	26	1	no	—
110	C₄⋊C₄	—	26	1	no	—
111	C₄⋊C₄	—	26	1	no	—
112	C₄²	C₂²×D₁₃	1	1	yes	characteristic
113	C₁₃	#1717	1	1	yes	characteristic, complemented, Sylow
114	C₂³	—	52	1	no	—
115	C₂³	—	52	1	no	—
116	Q₈	—	52	2	no	—
117	Q₈	—	52	2	no	—
118	D₄	—	52	2	no	—
119	D₄	—	52	2	no	—
120	D₄	—	52	2	no	—
121	D₄	—	52	2	no	—
122	D₄	—	52	2	no	—
123	D₄	—	52	2	no	—
124	C₂×C₄	—	52	1	no	—
125	C₂×C₄	—	52	1	no	—
126	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
127	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
128	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	2	2	no	—
131	C₈	—	2	2	no	—
132	C₂²	—	104	1	no	—
133	C₂²	—	104	1	no	—
134	C₂²	—	104	1	no	—
135	C₂²	—	104	1	no	—
136	C₂²	—	104	1	no	—
137	C₂²	—	104	1	no	—
138	C₂²	#88128	1	1	yes	characteristic
139	C₄	—	104	1	no	—
140	C₄	—	104	1	no	—
141	C₄	#88158	1	2	yes	—
142	C₄	#88162	1	2	yes	—
143	C₄	#88162	1	2	yes	—
144	C₂	—	208	1	no	—
145	C₂	—	208	1	no	—
146	C₂	#11718357	1	1	yes	characteristic
147	C₂	#11718373	1	1	yes	characteristic
148	C₂	#11718373	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11718356

C₂

yes

maximal, characteristic

#11718355

C₂

yes

maximal, characteristic, complemented

#11718355

C₂

yes

maximal, characteristic, complemented

#11718355

C₂

yes

maximal, characteristic, complemented

#11718354

C₂

yes

maximal, characteristic, complemented

#11718342

C₂

yes

maximal, characteristic, complemented

C₁₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₄

—

C₂×#6095

—

#88128

C₂²

yes

characteristic

#88128

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×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

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

—

D₁₀₄

—

#6095

—

#1717

—

maximal, Sylow

C₂²×D₁₃

—

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

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

#127

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#135

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×SD₁₆

—

C₂×D₈

—

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

—

D₁₃

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

SD₁₆

—

104

D₈

—

105

C₂×C₈

D₅₂

yes

—

106

C₂×C₈

D₅₂

yes

—

107

C₂×C₈

D₅₂

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

C₂²×D₁₃

yes

characteristic

113

C₁₃

#1717

yes

characteristic, complemented, Sylow

114

C₂³

—

115

C₂³

—

116

Q₈

—

117

Q₈

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×D₅₂

yes

characteristic

127

C₂×C₄

C₂×D₅₂

yes

characteristic

128

C₂×C₄

C₂×D₅₂

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

104

—

133

C₂²

—

104

—

134

C₂²

—

104

—

135

C₂²

—

104

—

136

C₂²

—

104

—

137

C₂²

—

104

—

138

C₂²

#88128

yes

characteristic

139

C₄

—

104

—

140

C₄

—

104

—

141

C₄

#88158

yes

—

142

C₄

#88162

yes

—

143

C₄

#88162

yes

—

144

C₂

—

208

—

145

C₂

—

208

—

146

C₂

#11718357

yes

characteristic

147

C₂

#11718373

yes

characteristic

148

C₂

#11718373

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	#11718356	C₂
hypercenter	#322	D₁₃
hypocenter	C₁₃	#1717
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

C₂

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
2. #11718355	C₂	semidirect
3. #11718355	C₂	semidirect
4. #11718355	C₂	semidirect
5. #11718354	C₂	semidirect
6. #11718342	C₂	semidirect
7. C₁₃×#322	C₂	semidirect
64. #322	D₁₃	semidirect
113. C₁₃	#1717	semidirect

H ≅ G/N

type

2. #11718355

C₂

semidirect

3. #11718355

C₂

semidirect

4. #11718355

C₂

semidirect

5. #11718354

C₂

semidirect

6. #11718342

C₂

semidirect

7. C₁₃×#322

C₂

semidirect

64. #322

D₁₃

semidirect

113. C₁₃

#1717

semidirect

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

subgroup	index
#11718356	2
#11718355	2
#11718354	2
#11718342	2
C₁₃×#322	2
#1717	13

subgroup

index

C₁₃×#322

Subgroups — #13114475

Conjugacy classes of subgroups

Complements

Maximal subgroups