Group #13130185 — subgroups

Subgroups — #13130185

Order 1664.

232 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 74 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	#11719433	C₂	1	1	yes	maximal, characteristic
2	#11719052	C₂	1	1	yes	maximal, characteristic
3	C₄×D₁₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#1057	C₂	1	4	yes	maximal, complemented
5	C₂×#6118	—	2	1	no	—
6	C₂×D₁₃⋊C₈	—	2	1	no	—
7	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	#88200	C₄	1	2	yes	—
9	Q₈×Dic₁₃	C₂²	1	2	yes	—
10	#88151	—	2	1	no	—
11	#88146	—	2	1	no	—
12	#88144	—	2	1	no	—
13	#88141	—	2	1	no	—
14	C₄²×D₁₃	C₂²	1	1	yes	characteristic
15	#88123	C₄	1	1	yes	characteristic
16	C₄⋊#1057	—	2	1	no	—
17	C₄⋊#1057	C₂²	1	2	yes	—
18	C₄×#1057	—	2	1	no	—
19	C₄×#1057	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
24	C₂×Dic₂₆	—	2	1	no	—
25	C₂×#1057	—	2	2	no	—
26	C₂×#1057	—	2	2	no	—
27	#6118	—	4	2	no	—
28	D₁₃⋊C₈	—	2	4	no	—
29	C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	#6103	—	2	2	no	—
33	C₄⋊Dic₁₃	—	2	1	no	—
34	C₄⋊Dic₁₃	—	2	2	no	—
35	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
36	#6101	—	2	2	no	—
37	C₄×Dic₁₃	—	2	2	no	—
38	C₄×Dic₁₃	D₄	1	1	yes	characteristic
39	C₄×Dic₁₃	D₄	1	1	yes	characteristic
40	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
41	#1493	—	13	1	no	maximal, Sylow
42	C₂²×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₁₃	—	2	4	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₃	—	2	2	no	—
49	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
52	C₄×D₁₃	—	2	2	no	—
53	C₄×D₁₃	—	2	2	no	—
54	C₄×D₁₃	SD₁₆	1	2	yes	—
55	C₄×D₁₃	Q₁₆	1	2	yes	—
56	Dic₂₆	—	4	2	no	—
57	#1057	—	2	4	no	—
58	#1057	—	4	2	no	—
59	#533	—	13	1	no	—
60	#423	—	13	1	no	—
61	C₂×C₄×C₈	—	13	1	no	—
62	Q₈⋊C₈	—	13	4	no	—
63	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₁₃	—	2	1	no	—
65	C₂×D₁₃	—	2	1	no	—
66	C₂×D₁₃	#103	1	2	yes	—
67	C₄×C₁₃	—	4	2	no	—
68	C₄×C₁₃	#103	1	2	yes	—
69	C₄×C₁₃	#104	1	2	yes	—
70	C₄×C₁₃	#131	1	2	yes	—
71	Dic₁₃	—	2	1	no	—
72	Dic₁₃	—	2	1	no	—
73	Dic₁₃	—	4	2	no	—
74	Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
75	Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×M₁₆	—	26	1	no	—
77	C₂²×C₈	—	26	1	no	—
78	C₄⋊Q₈	—	13	1	no	—
79	C₄⋊Q₈	#261	1	1	yes	characteristic
80	#125	—	26	1	no	—
81	C₂²⋊Q₈	—	13	2	no	—
82	C₂²⋊Q₈	—	26	1	no	—
83	C₄×Q₈	—	13	2	no	—
84	C₄×Q₈	—	26	1	no	—
85	#117	—	26	1	no	—
86	C₂×C₄²	—	13	1	no	—
87	C₄⋊C₈	—	13	2	no	—
88	C₄⋊C₈	—	26	1	no	—
89	C₄×C₈	—	13	2	no	—
90	C₄×C₈	—	26	1	no	—
91	C₂×C₁₃	#408	1	1	yes	characteristic
92	C₂×C₁₃	C₂×#103	1	1	yes	characteristic
93	C₂×C₁₃	C₂×#104	1	1	yes	characteristic
94	D₁₃	—	2	2	no	—
95	C₂×Q₈	—	26	1	no	—
96	C₂×Q₈	C₂×#261	1	2	yes	—
97	C₂²×C₄	—	13	1	no	—
98	C₂²×C₄	—	13	1	no	—
99	C₂²×C₄	—	13	1	no	—
100	M₁₆	—	52	2	no	—
101	C₂×C₈	—	26	2	no	—
102	C₂×C₈	—	26	4	no	—
103	C₂×C₈	—	26	2	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	13	2	no	—
107	C₄⋊C₄	—	26	2	no	—
108	C₄⋊C₄	—	26	2	no	—
109	C₄⋊C₄	—	26	1	no	—
110	C₂²⋊C₄	—	26	2	no	—
111	C₄²	—	13	1	no	—
112	C₄²	—	13	1	no	—
113	C₄²	—	13	1	no	—
114	C₄²	—	26	2	no	—
115	C₄²	C₂×#261	1	1	yes	characteristic
116	C₁₃	#1493	1	1	yes	characteristic, complemented, Sylow
117	C₂³	—	13	1	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	52	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	13	1	no	—
122	C₂×C₄	—	13	2	no	—
123	C₂×C₄	—	13	2	no	—
124	C₂×C₄	—	13	1	no	—
125	C₂×C₄	—	13	1	no	—
126	C₂×C₄	—	26	2	no	—
127	C₂×C₄	—	26	2	no	—
128	C₂×C₄	—	26	2	no	—
129	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
131	C₂×C₄	C₂²×#261	1	1	yes	characteristic
132	C₈	—	26	4	no	—
133	C₈	—	52	2	no	—
134	C₂²	—	13	2	no	—
135	C₂²	—	26	1	no	—
136	C₂²	—	26	1	no	—
137	C₂²	C₂×C₂²⋊#261	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	—	13	1	no	—
140	C₄	—	13	1	no	—
141	C₄	—	26	1	no	—
142	C₄	—	26	1	no	—
143	C₄	—	52	2	no	—
144	C₄	#88117	1	2	yes	—
145	C₄	#88118	1	2	yes	—
146	C₄	#88238	1	2	yes	—
147	C₂	—	26	2	no	—
148	C₂	#11719147	1	1	yes	characteristic
149	C₂	C₂×#88117	1	1	yes	characteristic
150	C₂	C₂×#88118	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719433

C₂

yes

maximal, characteristic

#11719052

C₂

yes

maximal, characteristic

C₄×D₁₃⋊C₈

C₂

yes

maximal, characteristic

Q₈⋊#1057

C₂

yes

maximal, complemented

C₂×#6118

—

C₂×D₁₃⋊C₈

—

C₁₃×C₄⋊Q₈

C₄

yes

characteristic

#88200

C₄

yes

—

Q₈×Dic₁₃

C₂²

yes

—

—

—

—

—

C₄²×D₁₃

C₂²

yes

characteristic

#88123

C₄

yes

characteristic

C₄⋊#1057

—

C₄⋊#1057

C₂²

yes

—

C₄×#1057

—

C₄×#1057

C₂²

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃

D₄

yes

characteristic

C₂×C₄×D₁₃

D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×#1057

—

C₂×#1057

—

#6118

—

D₁₃⋊C₈

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

D₄

yes

characteristic

C₄×Dic₁₃

C₂³

yes

characteristic

#1493

—

maximal, Sylow

C₂²×D₁₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

SD₁₆

yes

—

C₄×D₁₃

Q₁₆

yes

—

Dic₂₆

—

—

—

—

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

#103

yes

—

C₄×C₁₃

—

C₄×C₁₃

#103

yes

—

C₄×C₁₃

#104

yes

—

C₄×C₁₃

#131

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

C₂×SD₁₆

yes

characteristic

Dic₁₃

C₂×Q₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#261

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

D₁₃

—

C₂×Q₈

—

C₂×Q₈

C₂×#261

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂×#261

yes

characteristic

116

C₁₃

#1493

yes

characteristic, complemented, Sylow

117

C₂³

—

118

Q₈

—

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

C₂²⋊#261

yes

characteristic

130

C₂×C₄

C₂²⋊#261

yes

characteristic

131

C₂×C₄

C₂²×#261

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×C₂²⋊#261

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#88117

yes

—

145

C₄

#88118

yes

—

146

C₄

#88238

yes

—

147

C₂

—

148

C₂

#11719147

yes

characteristic

149

C₂

C₂×#88117

yes

characteristic

150

C₂

C₂×#88118

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#261
Fitting	C₁₃×C₄⋊Q₈	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃	C₂²⋊C₄
cosocle	C₄×Dic₁₃	C₂³
hypercenter	C₄⋊Q₈	#261
hypocenter	C₁₃	#1493
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#261

Fitting

C₁₃×C₄⋊Q₈

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃

C₂²⋊C₄

cosocle

C₄×Dic₁₃

C₂³

C₄⋊Q₈

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
4. Q₈⋊#1057	C₂	semidirect
116. C₁₃	#1493	semidirect

H ≅ G/N

type

4. Q₈⋊#1057

C₂

semidirect

116. C₁₃

#1493

semidirect

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

subgroup	index
#11719433	2
#11719052	2
C₄×D₁₃⋊C₈	2
Q₈⋊#1057	2
#1493	13

subgroup

index

C₄×D₁₃⋊C₈

Subgroups — #13130185

Conjugacy classes of subgroups

Complements

Maximal subgroups