Group #15777296 — subgroups

Subgroups — #15777296

Order 3328.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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
2	#13116744	C₂	1	1	yes	maximal, characteristic
3	#13114231	C₂	1	2	yes	maximal
4	#13112892	C₂	1	2	yes	maximal
5	C₁₃×#1372	C₂	1	1	yes	maximal, characteristic
6	#11718510	—	2	2	no	—
7	#11718343	C₂²	1	2	yes	—
8	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#335	C₂²	1	2	yes	—
11	#11718096	—	2	1	no	—
12	#11718096	—	2	1	no	—
13	#11718094	—	2	1	no	—
14	#11718093	—	2	1	no	—
15	#11718074	—	2	2	no	—
16	C₄⋊#6090	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₂	—	4	2	no	—
18	#88123	—	2	2	no	—
19	C₄×Dic₂₆	D₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	—	2	2	no	—
23	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	#88058	—	2	2	no	—
25	#88058	—	2	2	no	—
26	#88056	—	4	2	no	—
27	#88054	—	2	1	no	—
28	C₈×Dic₁₃	—	2	1	no	—
29	C₂×#6090	—	2	1	no	—
30	C₄⋊#1055	D₄	1	1	yes	characteristic
31	#12460	—	13	1	no	maximal, Sylow
32	C₂×Dic₂₆	—	4	2	no	—
33	C₂×Dic₂₆	D₈	1	1	yes	characteristic
34	C₂×C₈×C₁₃	—	4	2	no	—
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
37	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₁₃	—	4	2	no	—
39	C₄⋊Dic₁₃	—	4	2	no	—
40	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
41	#6101	—	2	1	no	—
42	C₄×Dic₁₃	—	2	1	no	—
43	C₂×#1055	—	2	1	no	—
44	Dic₅₂	—	8	4	no	—
45	C₁₃×C₁₆	—	2	1	no	—
46	C₁₃×C₁₆	Q₁₆	1	2	yes	complemented
47	#6090	—	4	1	no	—
48	#2252	—	13	1	no	—
49	#2183	—	13	1	no	—
50	#1674	—	13	2	no	—
51	#1372	D₁₃	1	1	yes	characteristic
52	#1364	—	13	2	no	—
53	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
56	C₂×Dic₁₃	—	2	1	no	—
57	C₂×Dic₁₃	—	2	1	no	—
58	C₂×Dic₁₃	—	8	2	no	—
59	Dic₂₆	—	2	2	no	—
60	Dic₂₆	—	8	2	no	—
61	Dic₂₆	—	8	2	no	—
62	C₈×C₁₃	—	2	1	no	—
63	C₈×C₁₃	—	8	2	no	—
64	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₁₃	#137	1	1	yes	characteristic
66	#1055	—	4	1	no	—
67	#500	—	13	2	no	—
68	#462	—	26	2	no	—
69	C₈×Q₈	—	13	1	no	—
70	#366	—	26	1	no	—
71	#366	—	26	1	no	—
72	C₄⋊C₁₆	—	13	1	no	—
73	C₄⋊C₁₆	—	26	1	no	—
74	#346	—	26	1	no	—
75	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
76	#335	C₂×D₁₃	1	2	yes	—
77	#332	—	26	2	no	—
78	C₂²×C₁₃	#466	1	1	yes	characteristic
79	C₄×C₁₃	—	2	1	no	—
80	C₄×C₁₃	#462	1	1	yes	characteristic
81	C₄×C₁₃	#473	1	1	yes	characteristic
82	C₄×C₁₃	#509	1	2	yes	—
83	Dic₁₃	—	4	1	no	—
84	Dic₁₃	—	4	1	no	—
85	Dic₁₃	—	16	2	no	—
86	C₂×Q₁₆	—	52	2	no	—
87	C₄⋊Q₈	—	26	2	no	—
88	C₄×Q₈	—	13	1	no	—
89	C₂×C₁₆	—	26	1	no	—
90	C₂×C₁₆	D₅₂	1	1	yes	characteristic
91	C₂×C₁₆	D₅₂	1	1	yes	characteristic
92	#107	—	26	2	no	—
93	#107	—	26	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	13	1	no	—
96	C₄⋊C₈	—	26	1	no	—
97	#103	—	52	2	no	—
98	C₄×C₈	—	26	1	no	—
99	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
100	C₂×C₁₃	#1674	1	1	yes	characteristic
101	C₂×C₁₃	#2218	1	1	yes	characteristic
102	C₂×C₁₃	#2222	1	1	yes	characteristic
103	C₂×Q₈	—	13	1	no	—
104	C₂×Q₈	—	52	2	no	—
105	Q₁₆	—	104	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	26	1	no	—
108	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
109	C₂×C₈	#6131	1	1	yes	characteristic
110	C₄⋊C₄	—	13	1	no	—
111	C₄⋊C₄	—	26	1	no	—
112	C₄⋊C₄	—	52	2	no	—
113	C₄⋊C₄	—	52	2	no	—
114	C₄²	—	26	1	no	—
115	C₄²	D₄×D₁₃	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	52	1	no	—
118	C₁₆	Dic₅₂	1	2	yes	complemented
119	C₁₃	#12460	1	1	yes	characteristic, complemented, Sylow
120	Q₈	—	26	2	no	—
121	Q₈	—	104	2	no	—
122	Q₈	—	104	2	no	—
123	C₂×C₄	—	26	1	no	—
124	C₂×C₄	—	26	1	no	—
125	C₂×C₄	—	104	2	no	—
126	C₂×C₄	#88149	1	1	yes	characteristic
127	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
128	C₂×C₄	#88173	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	8	2	no	—
131	C₈	—	52	1	no	—
132	C₈	C₂×Dic₅₂	1	1	yes	characteristic
133	C₈	#88163	1	1	yes	characteristic
134	C₂²	#11718544	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	52	1	no	—
137	C₄	—	52	1	no	—
138	C₄	—	208	2	no	—
139	C₄	#11718510	1	1	yes	characteristic
140	C₄	#11718557	1	1	yes	characteristic
141	C₄	#11718575	1	2	yes	—
142	C₂	#13114231	1	1	yes	characteristic
143	C₂	#13116940	1	1	yes	characteristic
144	C₂	#13116962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117111

C₂

yes

maximal, characteristic

#13116744

C₂

yes

maximal, characteristic

#13114231

C₂

yes

maximal

#13112892

C₂

yes

maximal

C₁₃×#1372

C₂

yes

maximal, characteristic

#11718510

—

#11718343

C₂²

yes

—

C₈×Dic₂₆

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#335

C₂²

yes

—

—

—

—

—

—

C₂²

yes

characteristic

C₂×Dic₅₂

—

#88123

—

C₄×Dic₂₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

C₈×Dic₁₃

—

C₂×#6090

—

C₄⋊#1055

D₄

yes

characteristic

#12460

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×Dic₂₆

D₈

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₈

yes

characteristic

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

C₁₃×C₁₆

—

C₁₃×C₁₆

Q₁₆

yes

—

—

—

—

D₁₃

yes

characteristic

#1364

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×Q₁₆

yes

C₈×C₁₃

yes

—

—

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

C₂×D₁₃

yes

—

#332

—

C₂²×C₁₃

#466

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

D₅₂

yes

characteristic

C₂×C₁₆

D₅₂

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

characteristic

100

C₂×C₁₃

#1674

yes

characteristic

101

C₂×C₁₃

#2218

yes

characteristic

102

C₂×C₁₃

#2222

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

Q₁₆

—

104

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×D₅₂

yes

characteristic

109

C₂×C₈

#6131

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

D₄×D₁₃

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

Dic₅₂

yes

complemented

119

C₁₃

#12460

yes

characteristic, complemented, Sylow

120

Q₈

—

121

Q₈

—

104

—

122

Q₈

—

104

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

104

—

126

C₂×C₄

#88149

yes

characteristic

127

C₂×C₄

D₈×D₁₃

yes

characteristic

128

C₂×C₄

#88173

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

C₂×Dic₅₂

yes

characteristic

133

C₈

#88163

yes

characteristic

134

C₂²

#11718544

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

208

—

139

C₄

#11718510

yes

characteristic

140

C₄

#11718557

yes

characteristic

141

C₄

#11718575

yes

—

142

C₂

#13114231

yes

characteristic

143

C₂

#13116940

yes

characteristic

144

C₂

#13116962

yes

characteristic

	subgroup	G/H
center	C₂²	#11718544
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1372	C₂
socle, prime radical	C₂²×C₁₃	#466
hypercenter	#1372	D₁₃
hypocenter	C₁₃	#12460
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#11718544

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1372

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
46. C₁₃×C₁₆	Q₁₆	semidirect
118. C₁₆	Dic₅₂	semidirect
119. C₁₃	#12460	semidirect

H ≅ G/N

type

46. C₁₃×C₁₆

Q₁₆

semidirect

118. C₁₆

Dic₅₂

semidirect

119. C₁₃

#12460

semidirect

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

subgroup	index
#13117111	2
#13116744	2
#13114231	2
#13112892	2
C₁₃×#1372	2
#12460	13

subgroup

index

C₁₃×#1372

Subgroups — #15777296

Conjugacy classes of subgroups

Complements

Maximal subgroups