Group #13116166 — subgroups

Subgroups — #13116166

Order 1664.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#88048	C₂	1	2	yes	maximal
3	#11718589	C₂	1	1	yes	maximal, characteristic
4	#11718121	C₂	1	2	yes	maximal
5	#11718116	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃	—	2	1	no	—
7	C₁₃×C₄⋊Q₈	—	2	4	no	—
8	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6101	—	2	1	no	—
14	C₂²×#1055	C₂²	1	2	yes	—
15	#88071	—	2	1	no	—
16	#88048	—	2	4	no	—
17	C₂×Q₈×C₁₃	—	2	4	no	—
18	C₂×Q₈×C₁₃	—	4	2	no	—
19	C₂×Q₈×C₁₃	—	4	2	no	—
20	C₂²×C₄×C₁₃	—	2	2	no	—
21	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₃	—	2	1	no	—
25	C₂²×Dic₁₃	—	2	1	no	—
26	C₁₃×C₄⋊C₄	—	2	4	no	—
27	C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
29	C₄²×C₁₃	—	2	2	no	—
30	C₄⋊Dic₁₃	—	2	1	no	—
31	C₄⋊Dic₁₃	—	2	2	no	—
32	C₄⋊Dic₁₃	Q₈	1	2	yes	complemented
33	#6101	—	4	2	no	—
34	C₂×#1055	—	2	2	no	—
35	C₂×#1055	—	2	2	no	—
36	C₂×#1055	Q₈	1	4	yes	complemented
37	#2058	—	13	1	no	maximal, Sylow
38	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₃	—	4	8	no	—
40	C₂×C₄×C₁₃	—	2	1	no	—
41	C₂×C₄×C₁₃	—	2	1	no	—
42	C₂×C₄×C₁₃	—	2	1	no	—
43	C₂×C₄×C₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	—	2	4	no	—
45	C₂×C₄×C₁₃	—	4	2	no	—
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
49	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
53	C₂×Dic₁₃	—	2	2	no	—
54	C₂×Dic₁₃	—	4	1	no	—
55	C₂×Dic₁₃	—	4	1	no	—
56	C₂×Dic₁₃	—	4	1	no	—
57	C₂×Dic₁₃	—	4	1	no	—
58	C₂×Dic₁₃	—	4	1	no	—
59	#1055	—	2	4	no	—
60	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
61	C₂×#106	—	13	2	no	—
62	#389	—	13	1	no	—
63	#340	—	13	2	no	—
64	#336	—	13	1	no	—
65	C₂²×C₁₃	#120	1	1	yes	characteristic
66	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₂²×C₁₃	#137	1	1	yes	characteristic
72	C₄×C₁₃	—	2	2	no	—
73	C₄×C₁₃	—	2	2	no	—
74	C₄×C₁₃	—	4	4	no	—
75	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
76	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
77	Dic₁₃	—	4	2	no	—
78	Dic₁₃	—	8	1	no	—
79	C₂²×Q₈	—	2	1	no	—
80	C₂²×C₈	—	13	2	no	—
81	C₄⋊Q₈	—	2	4	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	13	1	no	—
84	C₂×C₄⋊C₄	—	13	1	no	—
85	C₂×C₄⋊C₄	—	26	1	no	—
86	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
87	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
88	#106	—	26	4	no	—
89	#95	—	26	1	no	—
90	C₂×C₁₃	#395	1	1	yes	characteristic
91	C₂×C₁₃	#448	1	1	yes	characteristic
92	C₂×C₁₃	#450	1	1	yes	characteristic
93	C₂×C₁₃	#475	1	1	yes	characteristic
94	C₂×C₁₃	#476	1	1	yes	characteristic
95	C₂×C₁₃	#498	1	1	yes	characteristic
96	C₂×C₁₃	#501	1	1	yes	characteristic
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	26	1	no	—
102	C₂²×C₄	—	26	1	no	—
103	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
106	C₂×C₈	—	13	4	no	—
107	C₂×C₈	—	26	2	no	—
108	C₂×C₈	—	26	2	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	13	2	no	—
112	C₄⋊C₄	—	26	1	no	—
113	C₄⋊C₄	—	26	2	no	—
114	C₄⋊C₄	—	52	2	no	—
115	C₄⋊C₄	C₂²⋊D₁₃	1	4	yes	—
116	C₄²	—	2	2	no	—
117	C₁₃	#2058	1	1	yes	characteristic, complemented, Sylow
118	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	26	2	no	—
128	C₂×C₄	—	52	1	no	—
129	C₂×C₄	—	52	1	no	—
130	C₂×C₄	—	52	1	no	—
131	C₂×C₄	—	52	1	no	—
132	C₂×C₄	—	52	1	no	—
133	C₂×C₄	#6105	1	2	yes	—
134	C₂×C₄	#6106	1	2	yes	—
135	C₂×C₄	Q₈×D₁₃	1	1	yes	characteristic
136	C₂×C₄	Q₈×D₁₃	1	2	yes	—
137	C₂×C₄	#6131	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
139	C₈	—	26	4	no	—
140	C₂²	#88186	1	1	yes	characteristic
141	C₂²	C₂×#6105	1	1	yes	characteristic
142	C₂²	C₂×#6106	1	1	yes	characteristic
143	C₂²	#88196	1	1	yes	characteristic
144	C₂²	#88198	1	1	yes	characteristic
145	C₂²	#88200	1	1	yes	characteristic
146	C₂²	#88207	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	52	2	no	—
151	C₄	—	104	1	no	—
152	C₄	#88198	1	2	yes	—
153	C₄	#88200	1	2	yes	—
154	C₂	#11718740	1	1	yes	characteristic
155	C₂	#11718741	1	1	yes	characteristic
156	C₂	#11718745	1	1	yes	characteristic
157	C₂	#11718755	1	1	yes	characteristic
158	C₂	#11718879	1	1	yes	characteristic
159	C₂	#11718888	1	1	yes	characteristic
160	C₂	#11718890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#88048

C₂

yes

maximal

#11718589

C₂

yes

maximal, characteristic

#11718121

C₂

yes

maximal

#11718116

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×#6101

—

C₂²×#1055

C₂²

yes

—

#88071

—

#88048

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

D₄

yes

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

Q₈

yes

—

—

—

Q₈

yes

complemented

#2058

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

C₂×C₄⋊Q₈

D₁₃

yes

—

—

—

—

C₂²×C₁₃

#120

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃

C₂×SD₁₆

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#106

—

#95

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

104

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

105

C₂²×C₄

C₂²×D₁₃

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²⋊D₁₃

yes

—

116

C₄²

—

117

C₁₃

#2058

yes

characteristic, complemented, Sylow

118

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#6105

yes

—

134

C₂×C₄

#6106

yes

—

135

C₂×C₄

Q₈×D₁₃

yes

characteristic

136

C₂×C₄

Q₈×D₁₃

yes

—

137

C₂×C₄

#6131

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

139

C₈

—

140

C₂²

#88186

yes

characteristic

141

C₂²

C₂×#6105

yes

characteristic

142

C₂²

C₂×#6106

yes

characteristic

143

C₂²

#88196

yes

characteristic

144

C₂²

#88198

yes

characteristic

145

C₂²

#88200

yes

characteristic

146

C₂²

#88207

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

104

—

152

C₄

#88198

yes

—

153

C₄

#88200

yes

—

154

C₂

#11718740

yes

characteristic

155

C₂

#11718741

yes

characteristic

156

C₂

#11718745

yes

characteristic

157

C₂

#11718755

yes

characteristic

158

C₂

#11718879

yes

characteristic

159

C₂

#11718888

yes

characteristic

160

C₂

#11718890

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2058
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₃

hypocenter

C₁₃

#2058

LCS

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
32. C₄⋊Dic₁₃	Q₈	semidirect
36. C₂×#1055	Q₈	semidirect
117. C₁₃	#2058	semidirect

H ≅ G/N

type

32. C₄⋊Dic₁₃

Q₈

semidirect

36. C₂×#1055

Q₈

semidirect

117. C₁₃

#2058

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88048	2
#11718589	2
#11718121	2
#11718116	2
#2058	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

Subgroups — #13116166

Conjugacy classes of subgroups

Complements

Maximal subgroups