Group #15782603 — subgroups

Subgroups — #15782603

Order 3328.

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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	#13115050	C₂	1	3	yes	maximal
2	#13112963	C₂	1	3	yes	maximal
3	C₁₃×#2811	C₂	1	1	yes	maximal, characteristic
4	#11719517	C₂²	1	1	yes	characteristic
5	C₁₃×#395	—	2	3	no	—
6	C₁₃×#391	—	2	3	no	—
7	C₁₃×#391	C₂²	1	3	yes	—
8	C₂×#88076	C₂²	1	3	yes	—
9	#11718625	—	2	3	no	—
10	#11718115	—	2	3	no	—
11	C₂²×Q₈×C₁₃	—	2	1	no	—
12	C₂²×Q₈×C₁₃	—	2	1	no	—
13	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
15	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
16	C₂×C₄²×C₁₃	D₄	1	3	yes	—
17	Q₈×Dic₁₃	—	4	1	no	—
18	Q₈×Dic₁₃	—	4	1	no	—
19	#88180	—	2	3	no	—
20	C₂×#6099	—	2	3	no	—
21	C₁₃×#95	—	2	3	no	—
22	C₁₃×#95	—	2	6	no	—
23	#88076	—	2	3	no	—
24	#88076	Q₈	1	6	yes	complemented
25	#88048	—	4	3	no	—
26	#88047	—	4	3	no	—
27	#13464	—	13	1	no	maximal, Sylow
28	C₂×Q₈×C₁₃	—	2	4	no	—
29	C₂×Q₈×C₁₃	—	4	1	no	—
30	C₂×Q₈×C₁₃	—	4	1	no	—
31	C₂×Q₈×C₁₃	—	8	1	no	—
32	C₂×Q₈×C₁₃	—	8	1	no	—
33	C₂×Q₈×C₁₃	—	8	2	no	—
34	C₂×Q₈×C₁₃	—	8	1	no	—
35	C₂×Q₈×C₁₃	C₂×Q₈	1	3	yes	—
36	C₂×Q₈×C₁₃	#41	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	—	2	3	no	—
38	C₂²×C₄×C₁₃	—	2	3	no	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
40	C₁₃×C₄⋊C₄	—	4	3	no	—
41	C₁₃×C₄⋊C₄	—	8	3	no	—
42	C₄²×C₁₃	—	4	3	no	—
43	C₂²⋊Dic₁₃	—	4	1	no	—
44	C₄⋊Dic₁₃	—	2	3	no	—
45	C₄⋊Dic₁₃	—	2	3	no	—
46	C₄×Dic₁₃	—	4	3	no	—
47	#6099	—	2	6	no	—
48	#6099	—	4	3	no	—
49	C₂×#1055	—	4	3	no	—
50	#2811	D₁₃	1	1	yes	characteristic
51	#1857	—	13	3	no	—
52	#1394	—	13	3	no	—
53	C₂³×C₁₃	#120	1	1	yes	characteristic
54	Q₈×C₁₃	—	4	3	no	—
55	Q₈×C₁₃	—	4	1	no	—
56	Q₈×C₁₃	—	8	4	no	—
57	Q₈×C₁₃	—	8	4	no	—
58	C₂×C₄×C₁₃	—	2	3	no	—
59	C₂×C₄×C₁₃	—	2	3	no	—
60	C₂×C₄×C₁₃	—	2	6	no	—
61	C₂×C₄×C₁₃	—	4	3	no	—
62	C₂×C₄×C₁₃	—	4	3	no	—
63	C₂×C₄×C₁₃	—	4	6	no	—
64	C₂×C₄×C₁₃	—	4	3	no	—
65	C₂×C₄×C₁₃	C₂²⋊Q₈	1	3	yes	—
66	C₂×C₄×C₁₃	C₄⋊Q₈	1	3	yes	—
67	C₂×Dic₁₃	—	4	1	no	—
68	C₂×Dic₁₃	—	4	1	no	—
69	#1055	—	4	6	no	—
70	#519	—	13	1	no	—
71	#428	—	26	3	no	—
72	C₂×#101	—	13	3	no	—
73	#395	—	2	3	no	—
74	#391	—	2	3	no	—
75	#391	C₂×D₁₃	1	3	yes	—
76	#339	—	26	3	no	—
77	C₂²×C₁₃	—	2	1	no	—
78	C₂²×C₁₃	—	2	1	no	—
79	C₂²×C₁₃	#395	1	1	yes	characteristic
80	C₂²×C₁₃	B(4,𝔽₂)	1	1	yes	characteristic
81	C₂²×C₁₃	#458	1	1	yes	characteristic
82	C₄×C₁₃	—	2	3	no	—
83	C₄×C₁₃	—	2	3	no	—
84	C₄×C₁₃	—	4	6	no	—
85	C₄×C₁₃	—	8	3	no	—
86	Dic₁₃	—	8	1	no	—
87	Dic₁₃	—	8	1	no	—
88	C₂²×Q₈	—	2	1	no	—
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×Q₈	C₂²×D₁₃	1	1	yes	characteristic
91	C₂×M₁₆	—	26	3	no	—
92	C₄×Q₈	—	52	1	no	—
93	C₄×Q₈	—	52	1	no	—
94	#117	—	26	3	no	—
95	C₂×C₄⋊C₄	—	2	3	no	—
96	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
97	C₂×C₄²	C₂²⋊D₁₃	1	3	yes	—
98	#107	—	52	3	no	—
99	#106	—	52	3	no	—
100	#101	—	13	6	no	—
101	#101	—	26	3	no	—
102	#95	—	2	3	no	—
103	#95	—	2	6	no	—
104	C₂×C₁₃	—	2	1	no	—
105	C₂×C₁₃	—	2	1	no	—
106	C₂×C₁₃	#2062	1	1	yes	characteristic
107	C₂×C₁₃	#2202	1	1	yes	characteristic
108	C₂×C₁₃	#2205	1	1	yes	characteristic
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	4	1	no	—
111	C₂×Q₈	—	4	1	no	—
112	C₂×Q₈	—	8	1	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	2	no	—
115	C₂×Q₈	—	8	1	no	—
116	C₂×Q₈	Q₈×D₁₃	1	3	yes	—
117	C₂×Q₈	#6131	1	1	yes	characteristic
118	C₂²×C₄	—	2	3	no	—
119	C₂²×C₄	—	2	3	no	—
120	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
121	M₁₆	—	26	6	no	—
122	M₁₆	—	52	3	no	—
123	C₂×C₈	—	52	3	no	—
124	C₄⋊C₄	—	4	3	no	—
125	C₄⋊C₄	—	8	3	no	—
126	C₄⋊C₄	—	26	3	no	—
127	C₄⋊C₄	—	26	3	no	—
128	C₂²⋊C₄	—	52	1	no	—
129	C₄²	—	4	3	no	—
130	C₄²	—	52	3	no	—
131	C₁₃	#13464	1	1	yes	characteristic, complemented, Sylow
132	C₂³	#88207	1	1	yes	characteristic
133	Q₈	—	4	3	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	8	4	no	—
136	Q₈	—	8	4	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	4	3	no	—
141	C₂×C₄	—	4	3	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	—	4	3	no	—
144	C₂×C₄	—	52	1	no	—
145	C₂×C₄	—	52	1	no	—
146	C₂×C₄	#88198	1	3	yes	—
147	C₂×C₄	#88200	1	3	yes	—
148	C₈	—	52	6	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#11718890	1	1	yes	characteristic
152	C₂²	#11718904	1	1	yes	characteristic
153	C₂²	#11718905	1	1	yes	characteristic
154	C₄	—	2	3	no	—
155	C₄	—	2	3	no	—
156	C₄	—	4	6	no	—
157	C₄	—	8	3	no	—
158	C₄	—	104	1	no	—
159	C₄	—	104	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	#13116188	1	1	yes	characteristic
163	C₂	#13116853	1	1	yes	characteristic
164	C₂	#13116866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115050

C₂

yes

maximal

#13112963

C₂

yes

maximal

C₁₃×#2811

C₂

yes

maximal, characteristic

#11719517

C₂²

yes

characteristic

C₁₃×#395

—

C₁₃×#391

—

C₁₃×#391

C₂²

yes

—

C₂×#88076

C₂²

yes

—

#11718625

—

#11718115

—

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

C₂³

yes

characteristic

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

Q₈×Dic₁₃

—

Q₈×Dic₁₃

—

#88180

—

C₂×#6099

—

C₁₃×#95

—

C₁₃×#95

—

#88076

—

#88076

Q₈

yes

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×Q₈

yes

—

C₂×Q₈×C₁₃

#41

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

D₁₃

yes

characteristic

#1857

—

#1394

—

C₂³×C₁₃

#120

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

—

—

—

—

—

—

C₂×D₁₃

yes

—

#339

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂²×D₁₃

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

C₂×C₄²

C₂²⋊D₁₃

yes

—

#107

—

#106

—

100

#101

—

101

#101

—

102

#95

—

103

#95

—

104

C₂×C₁₃

—

105

C₂×C₁₃

—

106

C₂×C₁₃

#2062

yes

characteristic

107

C₂×C₁₃

#2202

yes

characteristic

108

C₂×C₁₃

#2205

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

Q₈×D₁₃

yes

—

117

C₂×Q₈

#6131

yes

characteristic

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₁₃

#13464

yes

characteristic, complemented, Sylow

132

C₂³

#88207

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#88198

yes

—

147

C₂×C₄

#88200

yes

—

148

C₈

—

149

C₂²

—

150

C₂²

—

151

C₂²

#11718890

yes

characteristic

152

C₂²

#11718904

yes

characteristic

153

C₂²

#11718905

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

104

—

159

C₄

—

104

—

160

C₂

—

161

C₂

—

162

C₂

#13116188

yes

characteristic

163

C₂

#13116853

yes

characteristic

164

C₂

#13116866

yes

characteristic

	subgroup	G/H
center	C₂²	#11718905
commutator, cosocle	C₂²×Q₈×C₁₃	C₂³
Frattini	C₂²×Q₈	C₂²×D₁₃
Fitting	C₁₃×#2811	C₂
socle	C₂²×C₁₃	B(4,𝔽₂)
prime radical	C₂³×C₁₃	#120
hypercenter	#2811	D₁₃
hypocenter	C₁₃	#13464
LCS	G ▷ C₂²×Q₈×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #2811

subgroup

G/H

center

C₂²

#11718905

commutator, cosocle

C₂²×Q₈×C₁₃

C₂³

Frattini

C₂²×Q₈

C₂²×D₁₃

Fitting

C₁₃×#2811

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂²×Q₈×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #2811

N	H ≅ G/N	type
24. #88076	Q₈	semidirect
131. C₁₃	#13464	semidirect

H ≅ G/N

type

24. #88076

Q₈

semidirect

131. C₁₃

#13464

semidirect

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

subgroup	index
#13115050	2
#13112963	2
C₁₃×#2811	2
#13464	13

subgroup

index

#13115050

#13112963

C₁₃×#2811

#13464

Subgroups — #15782603

Conjugacy classes of subgroups

Complements

Maximal subgroups