Group #15777300 — subgroups

Subgroups — #15777300

Order 3328.

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

normal subgroups only

This group has 188 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
2	#13114240	C₂	1	2	yes	maximal
3	#13112910	C₂	1	1	yes	maximal, characteristic
4	#13112893	C₂	1	2	yes	maximal
5	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	#11718682	—	2	2	no	—
8	#11718343	C₂²	1	2	yes	—
9	C₁₃×C₄⋊C₁₆	—	2	1	no	—
10	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×#346	—	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	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
18	#88123	—	2	2	no	—
19	C₂×C₁₃×C₁₆	—	2	1	no	—
20	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	—	2	1	no	—
23	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	—	2	1	no	—
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	#88075	—	4	2	no	—
27	#88058	—	2	2	no	—
28	#88058	—	2	2	no	—
29	#88047	—	4	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	#12461	—	13	1	no	maximal, Sylow
32	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
33	C₂×Dic₂₆	—	4	2	no	—
34	C₂×C₈×C₁₃	—	2	1	no	—
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
37	C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
39	C₄²×C₁₃	—	2	1	no	—
40	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₁₃	—	4	2	no	—
42	C₄⋊Dic₁₃	—	4	2	no	—
43	C₂×#1055	—	4	2	no	—
44	C₁₃×C₁₆	—	2	1	no	—
45	C₁₃×C₁₆	—	4	1	no	—
46	C₁₃×C₁₆	Q₁₆	1	2	yes	—
47	#2252	—	13	1	no	—
48	#2183	D₁₃	1	1	yes	characteristic
49	#1676	—	13	2	no	—
50	#1372	—	13	1	no	—
51	#1365	—	13	2	no	—
52	Q₈×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
58	C₂×Dic₁₃	—	8	2	no	—
59	Dic₂₆	—	16	2	no	—
60	C₈×C₁₃	—	2	1	no	—
61	C₈×C₁₃	—	4	1	no	—
62	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₈×C₁₃	#137	1	1	yes	characteristic
64	#1055	—	8	2	no	—
65	#500	—	13	2	no	—
66	#477	—	26	2	no	—
67	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
68	#366	—	26	1	no	—
69	#366	—	26	1	no	—
70	C₄⋊C₁₆	—	2	1	no	—
71	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
72	#346	—	2	1	no	—
73	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
74	#335	—	13	2	no	—
75	#333	—	26	2	no	—
76	C₂²×C₁₃	#466	1	1	yes	characteristic
77	C₄×C₁₃	—	2	1	no	—
78	C₄×C₁₃	—	4	1	no	—
79	C₄×C₁₃	—	4	1	no	—
80	C₄×C₁₃	#462	1	1	yes	characteristic
81	C₄×C₁₃	#473	1	1	yes	characteristic
82	C₄×C₁₃	#510	1	2	yes	—
83	Dic₁₃	—	16	2	no	—
84	C₄⋊Q₈	—	26	2	no	—
85	C₄×Q₈	D₅₂	1	1	yes	characteristic
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
88	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
89	#107	—	26	2	no	—
90	#107	—	26	2	no	—
91	#107	—	52	2	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	—	26	2	no	—
94	C₄⋊C₈	D₅₂	1	1	yes	characteristic
95	#103	—	52	2	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
98	C₂×C₁₃	#1674	1	1	yes	characteristic
99	C₂×C₁₃	#2219	1	1	yes	characteristic
100	C₂×C₁₃	#2221	1	1	yes	characteristic
101	C₂×Q₈	—	52	2	no	—
102	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	52	2	no	—
105	C₂×C₈	#6127	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	52	2	no	—
109	C₄⋊C₄	—	52	2	no	—
110	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
111	C₄²	—	2	1	no	—
112	C₄²	C₂×D₅₂	1	1	yes	characteristic
113	C₁₆	—	2	1	no	—
114	C₁₆	—	4	1	no	—
115	C₁₆	#6107	1	2	yes	—
116	C₁₃	#12461	1	1	yes	characteristic, complemented, Sylow
117	Q₈	—	2	2	no	—
118	Q₈	—	208	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	104	2	no	—
122	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
123	C₂×C₄	#88158	1	1	yes	characteristic
124	C₂×C₄	#88184	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	104	2	no	—
128	C₈	C₂×#6107	1	1	yes	characteristic
129	C₈	#88205	1	1	yes	characteristic
130	C₂²	#11718776	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	208	2	no	—
135	C₄	#11718570	1	2	yes	—
136	C₄	#11718692	1	1	yes	characteristic
137	C₄	#11718804	1	1	yes	characteristic
138	C₂	#13114230	1	1	yes	characteristic
139	C₂	#13116944	1	1	yes	characteristic
140	C₂	#13116956	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117111

C₂

yes

maximal, characteristic

#13114240

C₂

yes

maximal

#13112910

C₂

yes

maximal, characteristic

#13112893

C₂

yes

maximal

C₁₃×#2183

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₃

C₂²

yes

characteristic

#11718682

—

#11718343

C₂²

yes

—

C₁₃×C₄⋊C₁₆

—

C₁₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₁₃×#346

—

C₄×C₁₃×C₁₆

C₂²

yes

—

—

—

C₂²

yes

—

C₄×Q₈×C₁₃

D₄

yes

characteristic

#88123

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

D₄

yes

characteristic

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

—

C₂×Q₈×C₁₃

D₈

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

D₈

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

Q₁₆

yes

—

#2252

—

#2183

D₁₃

yes

—

—

—

Q₈×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×Q₁₆

yes

C₈×C₁₃

yes

—

—

—

C₈×Q₈

C₂×D₁₃

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₃

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

—

#333

—

C₂²×C₁₃

#466

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

C₄⋊Q₈

—

C₄×Q₈

D₅₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

C₂×C₁₆

C₂²⋊D₁₃

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₅₂

yes

characteristic

#103

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

100

C₂×C₁₃

#2221

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

D₁₀₄

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

#6127

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

D₁₀₄

yes

characteristic

111

C₄²

—

112

C₄²

C₂×D₅₂

yes

characteristic

113

C₁₆

—

114

C₁₆

—

115

C₁₆

#6107

yes

—

116

C₁₃

#12461

yes

characteristic, complemented, Sylow

117

Q₈

—

118

Q₈

—

208

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

104

—

122

C₂×C₄

C₂×D₁₀₄

yes

characteristic

123

C₂×C₄

#88158

yes

characteristic

124

C₂×C₄

#88184

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

104

—

128

C₈

C₂×#6107

yes

characteristic

129

C₈

#88205

yes

characteristic

130

C₂²

#11718776

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

208

—

135

C₄

#11718570

yes

—

136

C₄

#11718692

yes

characteristic

137

C₄

#11718804

yes

characteristic

138

C₂

#13114230

yes

characteristic

139

C₂

#13116944

yes

characteristic

140

C₂

#13116956

yes

characteristic

	subgroup	G/H
center	C₂²	#11718776
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2183	C₂
socle, prime radical	C₂²×C₁₃	#466
hypercenter	#2183	D₁₃
hypocenter	C₁₃	#12461
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#11718776

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2183

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
116. C₁₃	#12461	semidirect

H ≅ G/N

type

116. C₁₃

#12461

semidirect

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

subgroup	index
#13117111	2
#13114240	2
#13112910	2
#13112893	2
C₁₃×#2183	2
#12461	13

subgroup

index

C₁₃×#2183

Subgroups — #15777300

Conjugacy classes of subgroups

Complements

Maximal subgroups