Group #15777266 — subgroups

Subgroups — #15777266

Order 3328.

206 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 71 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	#13117111	C₂	1	1	yes	maximal, characteristic, complemented
2	#13114235	C₂	1	2	yes	maximal
3	#13112910	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112893	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	—	2	1	no	—
7	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	#11718669	—	2	2	no	—
9	#11718343	C₂²	1	2	yes	—
10	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	#11718096	—	2	1	no	—
14	#11718096	—	2	1	no	—
15	#11718073	—	2	2	no	—
16	#11718050	C₂²	1	2	yes	—
17	C₂²×C₈×C₁₃	—	2	1	no	—
18	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
19	#88123	—	2	2	no	—
20	C₁₃×M₃₂	—	4	2	no	—
21	C₂×C₁₃×C₁₆	—	2	1	no	—
22	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
24	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
25	C₁₃×C₂²⋊C₈	—	2	1	no	—
26	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	#88072	—	4	2	no	—
28	#88058	—	2	2	no	—
29	#88058	—	2	2	no	—
30	#88048	—	4	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	#12455	—	13	1	no	maximal, Sylow
33	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	—	2	1	no	—
35	C₂×Dic₂₆	—	4	2	no	—
36	C₂×C₈×C₁₃	—	2	1	no	—
37	C₂×C₈×C₁₃	—	4	1	no	—
38	C₂×C₈×C₁₃	—	4	1	no	—
39	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
41	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
42	C₁₃×C₂²⋊C₄	—	2	1	no	—
43	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₄⋊Dic₁₃	—	4	2	no	—
45	C₄⋊Dic₁₃	—	4	2	no	—
46	C₂×#1055	—	4	2	no	—
47	C₁₃×C₁₆	—	2	1	no	—
48	C₁₃×C₁₆	—	4	1	no	—
49	C₁₃×C₁₆	SD₁₆	1	2	yes	—
50	#2252	—	13	1	no	—
51	#2169	D₁₃	1	1	yes	characteristic
52	#1675	—	13	2	no	—
53	#1372	—	13	1	no	—
54	#1365	—	13	2	no	—
55	C₂³×C₁₃	—	2	1	no	—
56	D₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	4	1	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
63	C₂×Dic₁₃	—	8	2	no	—
64	Dic₂₆	—	16	2	no	—
65	C₈×C₁₃	—	2	1	no	—
66	C₈×C₁₃	—	4	1	no	—
67	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₈×C₁₃	#137	1	1	yes	characteristic
69	#1055	—	8	2	no	—
70	C₂×M₃₂	—	2	1	no	—
71	#500	—	13	2	no	—
72	#476	—	26	2	no	—
73	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
74	#366	—	26	1	no	—
75	#366	—	26	1	no	—
76	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
77	C₂²⋊C₁₆	—	2	1	no	—
78	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
79	#335	—	13	2	no	—
80	#333	—	26	2	no	—
81	C₂²×C₁₃	—	4	1	no	—
82	C₂²×C₁₃	—	4	1	no	—
83	C₂²×C₁₃	—	4	1	no	—
84	C₂²×C₁₃	#466	1	1	yes	characteristic
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	—	4	1	no	—
87	C₄×C₁₃	#461	1	1	yes	characteristic
88	C₄×C₁₃	#473	1	1	yes	characteristic
89	C₄×C₁₃	#510	1	2	yes	—
90	Dic₁₃	—	16	2	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₄⋊Q₈	—	26	2	no	—
93	C₄×D₄	D₅₂	1	1	yes	characteristic
94	M₃₂	—	4	2	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
98	#107	—	26	2	no	—
99	#107	—	26	2	no	—
100	#106	—	52	2	no	—
101	C₄⋊C₈	—	26	2	no	—
102	C₄⋊C₈	D₅₂	1	1	yes	characteristic
103	#102	—	52	2	no	—
104	C₂²⋊C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
106	C₂×C₁₃	—	8	1	no	—
107	C₂×C₁₃	#1673	1	1	yes	characteristic
108	C₂×C₁₃	#2219	1	1	yes	characteristic
109	C₂×C₁₃	#2221	1	1	yes	characteristic
110	C₂×Q₈	—	52	2	no	—
111	C₂×D₄	D₁₀₄	1	1	yes	characteristic
112	C₂²×C₄	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	52	2	no	—
117	C₂×C₈	#6127	1	1	yes	characteristic
118	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
119	C₄⋊C₄	—	52	2	no	—
120	C₄⋊C₄	—	52	2	no	—
121	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂×D₅₂	1	1	yes	characteristic
124	C₁₆	—	2	1	no	—
125	C₁₆	—	4	1	no	—
126	C₁₆	#6105	1	2	yes	—
127	C₁₃	#12455	1	1	yes	characteristic, complemented, Sylow
128	C₂³	—	2	1	no	—
129	Q₈	—	208	2	no	—
130	D₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	104	2	no	—
135	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
136	C₂×C₄	#88158	1	1	yes	characteristic
137	C₂×C₄	#88184	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	104	2	no	—
141	C₈	C₂×#6105	1	1	yes	characteristic
142	C₈	#88205	1	1	yes	characteristic
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#11718776	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	208	2	no	—
150	C₄	#11718570	1	2	yes	—
151	C₄	#11718680	1	1	yes	characteristic
152	C₄	#11718804	1	1	yes	characteristic
153	C₂	—	8	1	no	—
154	C₂	#13114225	1	1	yes	characteristic
155	C₂	#13116944	1	1	yes	characteristic
156	C₂	#13116956	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117111

C₂

yes

maximal, characteristic, complemented

#13114235

C₂

yes

maximal

#13112910

C₂

yes

maximal, characteristic, complemented

#13112893

C₂

yes

maximal, complemented

C₁₃×#2169

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

—

C₈×D₄×C₁₃

C₂²

yes

characteristic

#11718669

—

#11718343

C₂²

yes

—

C₁₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₁₃×C₂²⋊C₁₆

—

C₄×C₁₃×C₁₆

C₂²

yes

—

—

—

C₂²

yes

—

C₂²×C₈×C₁₃

—

C₄×D₄×C₁₃

D₄

yes

characteristic

#88123

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

D₄

yes

characteristic

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

—

C₂×D₄×C₁₃

D₈

yes

characteristic

C₂²×C₄×C₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₁₃×C₄⋊C₄

D₈

yes

characteristic

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

SD₁₆

yes

—

#2252

—

#2169

D₁₃

yes

—

—

—

C₂³×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×SD₁₆

yes

C₈×C₁₃

yes

—

C₂×M₃₂

—

#500

—

#476

—

C₈×D₄

C₂×D₁₃

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

C₂×D₁₃

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

—

#333

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#466

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×D₄

D₅₂

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

#107

—

#107

—

100

#106

—

101

C₄⋊C₈

—

102

C₄⋊C₈

D₅₂

yes

characteristic

103

#102

—

104

C₂²⋊C₈

—

105

C₄×C₈

C₂²×D₁₃

yes

characteristic

106

C₂×C₁₃

—

107

C₂×C₁₃

#1673

yes

characteristic

108

C₂×C₁₃

#2219

yes

characteristic

109

C₂×C₁₃

#2221

yes

characteristic

110

C₂×Q₈

—

111

C₂×D₄

D₁₀₄

yes

characteristic

112

C₂²×C₄

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

#6127

yes

characteristic

118

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

D₁₀₄

yes

characteristic

122

C₂²⋊C₄

—

123

C₄²

C₂×D₅₂

yes

characteristic

124

C₁₆

—

125

C₁₆

—

126

C₁₆

#6105

yes

—

127

C₁₃

#12455

yes

characteristic, complemented, Sylow

128

C₂³

—

129

Q₈

—

208

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

104

—

135

C₂×C₄

C₂×D₁₀₄

yes

characteristic

136

C₂×C₄

#88158

yes

characteristic

137

C₂×C₄

#88184

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

104

—

141

C₈

C₂×#6105

yes

characteristic

142

C₈

#88205

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#11718776

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

208

—

150

C₄

#11718570

yes

—

151

C₄

#11718680

yes

characteristic

152

C₄

#11718804

yes

characteristic

153

C₂

—

154

C₂

#13114225

yes

characteristic

155

C₂

#13116944

yes

characteristic

156

C₂

#13116956

yes

characteristic

	subgroup	G/H
center	C₂²	#11718776
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2169	C₂
socle	C₂²×C₁₃	#466
prime radical	C₂×D₄×C₁₃	D₈
hypercenter	#2169	D₁₃
hypocenter	C₁₃	#12455
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

subgroup

G/H

center

C₂²

#11718776

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2169

C₂

socle

C₂²×C₁₃

#466

prime radical

C₂×D₄×C₁₃

D₈

D₁₃

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169

N	H ≅ G/N	type
1. #13117111	C₂	semidirect
3. #13112910	C₂	semidirect
4. #13112893	C₂	semidirect
127. C₁₃	#12455	semidirect

H ≅ G/N

type

1. #13117111

C₂

semidirect

3. #13112910

C₂

semidirect

4. #13112893

C₂

semidirect

127. C₁₃

#12455

semidirect

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

subgroup	index
#13117111	2
#13114235	2
#13112910	2
#13112893	2
C₁₃×#2169	2
#12455	13

subgroup

index

C₁₃×#2169

Subgroups — #15777266

Conjugacy classes of subgroups

Complements

Maximal subgroups