Group #15778299 — subgroups

Subgroups — #15778299

Order 3328.

308 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 131 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	#13116610	C₂	1	2	yes	maximal
2	M₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13114718	C₂	1	2	yes	maximal
4	#13112938	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#108	C₂²	1	2	yes	—
8	#11718783	—	2	2	no	—
9	#11718767	C₂²	1	2	yes	—
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88052	C₂²	1	1	yes	characteristic
12	#11718147	—	2	2	no	—
13	#11718094	C₄	1	2	yes	complemented
14	C₁₆×Dic₁₃	C₄	1	2	yes	complemented
15	#11718086	C₄	1	8	yes	complemented
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	#88180	—	2	2	no	—
19	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
20	C₂²×#1055	D₄	1	1	yes	characteristic
21	C₁₃×M₃₂	—	2	2	no	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₁₃×#108	—	2	2	no	—
24	C₁₃×#108	C₂×C₄	1	4	yes	—
25	#88058	—	2	2	no	—
26	#88057	C₂×C₄	1	4	yes	—
27	#88055	—	4	2	no	—
28	C₈×Dic₁₃	—	4	2	no	—
29	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
32	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
33	#88052	C₂×C₄	1	4	yes	—
34	C₂×#6090	C₂×C₄	1	2	yes	—
35	#12668	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₃	—	2	1	no	—
38	C₁₃×M₁₆	—	2	4	no	—
39	C₁₃×M₁₆	—	4	2	no	—
40	C₂×C₈×C₁₃	—	4	2	no	—
41	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
43	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₃	#41	1	2	yes	—
45	C₂²⋊Dic₁₃	—	4	2	no	—
46	C₄⋊Dic₁₃	—	2	4	no	—
47	C₄×Dic₁₃	—	2	1	no	—
48	C₄×Dic₁₃	—	4	2	no	—
49	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
53	C₂×#1055	SD₁₆	1	2	yes	—
54	C₂×#1055	SD₁₆	1	2	yes	—
55	C₁₃×C₁₆	—	2	2	no	—
56	#6090	C₄²	1	4	yes	—
57	#2155	—	13	2	no	—
58	C₄×M₃₂	—	13	1	no	—
59	#1772	—	13	2	no	—
60	#1385	D₁₃	1	1	yes	characteristic
61	#1384	—	13	1	no	—
62	C₂³×C₁₃	#135	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
67	C₂×Dic₁₃	—	2	1	no	—
68	C₂×Dic₁₃	—	2	1	no	—
69	C₂×Dic₁₃	—	4	2	no	—
70	C₂×Dic₁₃	—	4	4	no	—
71	C₈×C₁₃	—	4	4	no	—
72	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
73	C₈×C₁₃	#117	1	1	yes	characteristic
74	C₈×C₁₃	C₄×Q₈	1	2	yes	—
75	#1055	#106	1	1	yes	characteristic
76	#1055	#106	1	2	yes	—
77	#1055	#106	1	1	yes	characteristic
78	C₂×M₃₂	—	13	1	no	—
79	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
80	C₂×#108	C₂×D₁₃	1	2	yes	—
81	#427	—	13	2	no	—
82	#405	—	26	2	no	—
83	C₂×C₄×C₈	—	13	1	no	—
84	#365	—	13	8	no	—
85	#346	—	13	2	no	—
86	C₄×C₁₆	—	13	2	no	—
87	#339	—	26	2	no	—
88	C₂²×C₁₃	—	4	1	no	—
89	C₂²×C₁₃	#427	1	1	yes	characteristic
90	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
91	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
93	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
94	Dic₁₃	—	2	2	no	—
95	Dic₁₃	—	4	1	no	—
96	Dic₁₃	—	8	4	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂²×C₈	—	13	1	no	—
99	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
100	#117	—	26	2	no	—
101	C₂×C₄²	—	13	1	no	—
102	M₃₂	—	2	2	no	—
103	M₃₂	—	13	4	no	—
104	C₂×C₁₆	—	13	2	no	—
105	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
106	#108	—	2	2	no	—
107	#108	C₄×D₁₃	1	4	yes	—
108	#107	—	26	2	no	—
109	#106	—	13	4	no	—
110	#97	—	52	2	no	—
111	C₄×C₈	—	13	1	no	—
112	C₄×C₈	—	13	1	no	—
113	C₄×C₈	—	13	1	no	—
114	C₄×C₈	—	13	1	no	—
115	C₄×C₈	—	52	2	no	—
116	C₂×C₁₃	—	2	1	no	—
117	C₂×C₁₃	—	2	2	no	—
118	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
119	C₂²×C₄	—	26	1	no	—
120	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
121	M₁₆	—	2	4	no	—
122	M₁₆	—	4	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	13	1	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	—	13	2	no	—
128	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
130	C₂×C₈	#6129	1	2	yes	—
131	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
132	C₄⋊C₄	—	26	4	no	—
133	C₂²⋊C₄	—	52	2	no	—
134	C₄²	—	13	1	no	—
135	C₄²	—	13	1	no	—
136	C₄²	—	26	1	no	—
137	C₄²	—	52	2	no	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	13	4	no	—
140	C₁₃	#12668	1	1	yes	characteristic, complemented, Sylow
141	C₂³	#88173	1	1	yes	characteristic
142	C₂×C₄	—	26	1	no	—
143	C₂×C₄	—	26	1	no	—
144	C₂×C₄	—	52	2	no	—
145	C₂×C₄	—	52	4	no	—
146	C₂×C₄	#88135	1	2	yes	—
147	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
148	C₂×C₄	#88166	1	1	yes	characteristic
149	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
150	C₈	—	4	4	no	—
151	C₈	—	13	1	no	—
152	C₈	—	13	2	no	—
153	C₈	—	13	1	no	—
154	C₈	C₄²×D₁₃	1	1	yes	characteristic
155	C₈	#88131	1	1	yes	characteristic
156	C₈	#88141	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#11718449	1	2	yes	—
159	C₂²	#11718541	1	1	yes	characteristic
160	C₄	—	26	2	no	—
161	C₄	—	52	1	no	—
162	C₄	—	104	4	no	—
163	C₄	#11718310	1	1	yes	characteristic
164	C₄	#11718419	1	2	yes	—
165	C₄	D₁₃×#106	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	#13114724	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116610

C₂

yes

maximal

M₃₂×Dic₁₃

C₂

yes

maximal, characteristic

#13114718

C₂

yes

maximal

#13112938

C₂

yes

maximal, characteristic

C₁₃×#1385

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₂²

yes

characteristic

C₂×C₁₃×#108

C₂²

yes

—

#11718783

—

#11718767

C₂²

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

characteristic

C₂×#88052

C₂²

yes

characteristic

#11718147

—

#11718094

C₄

yes

complemented

C₁₆×Dic₁₃

C₄

yes

complemented

#11718086

C₄

yes

complemented

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

#88180

—

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂²×#1055

D₄

yes

characteristic

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₁₃×#108

—

C₁₃×#108

C₂×C₄

yes

—

#88058

—

#88057

C₂×C₄

yes

—

#88055

—

C₈×Dic₁₃

—

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

#88052

C₂×C₄

yes

—

C₂×#6090

C₂×C₄

yes

—

#12668

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

C₄⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

SD₁₆

yes

—

C₂×#1055

SD₁₆

yes

—

C₁₃×C₁₆

—

#6090

C₄²

yes

—

#2155

—

C₄×M₃₂

—

#1772

—

#1385

D₁₃

yes

characteristic

#1384

—

C₂³×C₁₃

#135

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×C₄²

yes

characteristic

C₈×C₁₃

#117

yes

characteristic

C₈×C₁₃

C₄×Q₈

yes

—

yes

yes

—

#1055

#106

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₁₃

yes

characteristic

C₂×#108

C₂×D₁₃

yes

—

#427

—

#405

—

C₂×C₄×C₈

—

#365

—

#346

—

C₄×C₁₆

—

#339

—

C₂²×C₁₃

—

C₂²×C₁₃

#427

yes

characteristic

C₂²×C₁₃

C₄×SD₁₆

yes

—

C₄×C₁₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₁₃

C₂×#106

yes

characteristic

C₄×C₁₃

C₄×SD₁₆

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₁₃

yes

characteristic

100

#117

—

101

C₂×C₄²

—

102

M₃₂

—

103

M₃₂

—

104

C₂×C₁₆

—

105

C₂×C₁₆

C₄×D₁₃

yes

—

106

#108

—

107

#108

C₄×D₁₃

yes

—

108

#107

—

109

#106

—

110

#97

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₂×C₁₃

—

117

C₂×C₁₃

—

118

C₂×C₁₃

C₄×#106

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

D₄×D₁₃

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

129

C₂×C₈

C₂×C₄×D₁₃

yes

—

130

C₂×C₈

#6129

yes

—

131

C₂×C₈

Q₈×D₁₃

yes

characteristic

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₁₆

—

139

C₁₆

—

140

C₁₃

#12668

yes

characteristic, complemented, Sylow

141

C₂³

#88173

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#88135

yes

—

147

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

148

C₂×C₄

#88166

yes

characteristic

149

C₂×C₄

SD₁₆×D₁₃

yes

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₄²×D₁₃

yes

characteristic

155

C₈

#88131

yes

characteristic

156

C₈

#88141

yes

—

157

C₂²

—

158

C₂²

#11718449

yes

—

159

C₂²

#11718541

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

104

—

163

C₄

#11718310

yes

characteristic

164

C₄

#11718419

yes

—

165

C₄

D₁₃×#106

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#13114724

yes

characteristic

	subgroup	G/H
center	C₄	D₁₃×#106
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1385	C₂
socle	C₂×C₁₃	C₄×#106
prime radical	C₂³×C₁₃	#135
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1385	D₁₃
hypocenter	C₁₃	#12668
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

D₁₃×#106

commutator

C₈×C₁₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1385

C₂

C₂×C₁₃

C₂³×C₁₃

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
13. #11718094	C₄	semidirect
14. C₁₆×Dic₁₃	C₄	semidirect
15. #11718086	C₄	semidirect
140. C₁₃	#12668	semidirect

H ≅ G/N

type

13. #11718094

C₄

semidirect

14. C₁₆×Dic₁₃

C₄

semidirect

15. #11718086

C₄

semidirect

140. C₁₃

#12668

semidirect

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

subgroup	index
#13116610	2
M₃₂×Dic₁₃	2
#13114718	2
#13112938	2
C₁₃×#1385	2
#12668	13

subgroup

index

M₃₂×Dic₁₃

C₁₃×#1385

Subgroups — #15778299

Conjugacy classes of subgroups

Complements

Maximal subgroups