Group #15779342 — subgroups

Subgroups — #15779342

Order 3328.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#13116601	C₂	1	2	yes	maximal
2	#13116596	C₂	1	2	yes	maximal
3	#13116595	C₂	1	1	yes	maximal, characteristic
4	#13114591	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₁₃×#405	C₂²	1	1	yes	characteristic
8	#11718762	C₂²	1	2	yes	—
9	C₂×#88052	C₂²	1	1	yes	characteristic
10	C₂²×#6090	C₂²	1	1	yes	characteristic
11	C₁₃×#364	—	2	2	no	—
12	#11718152	C₄	1	1	yes	characteristic
13	#11718149	C₄	1	1	yes	characteristic
14	#11718123	C₄	1	2	yes	—
15	#11718103	—	2	4	no	—
16	#11718093	—	2	4	no	—
17	C₄⋊#6090	—	2	2	no	—
18	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
21	#88180	C₂×C₄	1	2	yes	—
22	C₂×#6099	C₂×C₄	1	2	yes	—
23	C₁₃×M₃₂	—	2	2	no	—
24	C₁₃×M₃₂	—	2	2	no	—
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₁₃×#97	Q₈	1	2	yes	—
27	C₄×C₈×C₁₃	D₄	1	2	yes	—
28	#88055	Q₈	1	4	yes	—
29	C₈×Dic₁₃	D₄	1	4	yes	—
30	#88052	—	2	2	no	—
31	C₂×#6090	—	2	1	no	—
32	C₂×#6090	—	2	2	no	—
33	C₂×#6090	—	2	1	no	—
34	#12902	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₁₃×M₁₆	—	2	2	no	—
37	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
42	C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₁₃×C₂²⋊C₄	—	2	1	no	—
44	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂²⋊Dic₁₃	—	2	2	no	—
46	C₄⋊Dic₁₃	—	2	2	no	—
47	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
48	#6099	—	2	2	no	—
49	#6099	—	2	2	no	—
50	C₂×#1055	C₄⋊C₄	1	4	yes	—
51	C₁₃×C₁₆	—	4	2	no	—
52	#6090	—	4	1	no	—
53	#6090	—	4	1	no	—
54	#2153	—	13	2	no	—
55	#2153	D₁₃	1	1	yes	characteristic
56	#2151	—	13	2	no	—
57	#2151	—	13	1	no	—
58	#1747	—	13	1	no	—
59	C₂³×C₁₃	#131	1	1	yes	characteristic
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#131	1	2	yes	—
64	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
65	C₂×Dic₁₃	—	2	4	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	#127	1	1	yes	characteristic
68	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
69	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
70	#1055	—	2	4	no	—
71	C₂×M₃₂	—	13	1	no	—
72	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
73	C₂²×C₁₆	—	13	1	no	—
74	#405	—	13	2	no	—
75	#405	C₂×D₁₃	1	1	yes	characteristic
76	#364	—	2	2	no	—
77	#364	—	26	4	no	—
78	C₄⋊C₁₆	—	26	4	no	—
79	C₄⋊C₁₆	—	26	2	no	—
80	#338	—	13	1	no	—
81	H(3,ℤ/4)	—	13	1	no	—
82	H(3,ℤ/4)	—	13	2	no	—
83	C₂²×C₁₃	—	4	1	no	—
84	C₂²×C₁₃	#424	1	2	yes	—
85	C₂²×C₁₃	#424	1	1	yes	characteristic
86	C₄×C₁₃	—	4	2	no	—
87	C₄×C₁₃	#384	1	1	yes	characteristic
88	C₄×C₁₃	#424	1	2	yes	—
89	C₄×C₁₃	#424	1	1	yes	characteristic
90	Dic₁₃	—	4	4	no	—
91	C₂×M₁₆	—	13	2	no	—
92	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
94	#117	—	13	2	no	—
95	#117	C₄×D₁₃	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	2	no	—
98	M₃₂	—	26	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	26	2	no	—
101	C₂×C₁₆	—	26	1	no	—
102	C₂×C₁₆	—	26	1	no	—
103	#97	—	13	4	no	—
104	#97	Dic₂₆	1	2	yes	—
105	C₄×C₈	—	13	4	no	—
106	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
107	C₂×C₁₃	—	2	2	no	—
108	C₂×C₁₃	—	2	1	no	—
109	C₂×C₁₃	#1843	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
111	M₁₆	—	2	2	no	—
112	M₁₆	—	26	2	no	—
113	M₁₆	—	26	2	no	—
114	C₂×C₈	—	13	4	no	—
115	C₂×C₈	#6101	1	2	yes	—
116	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
117	C₂×C₈	D₄×D₁₃	1	2	yes	—
118	C₂×C₈	Q₈×D₁₃	1	2	yes	—
119	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	26	2	no	—
122	C₂²⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	26	2	no	—
124	C₄²	—	13	4	no	—
125	C₄²	#6101	1	2	yes	—
126	C₁₆	—	4	2	no	—
127	C₁₆	—	52	1	no	—
128	C₁₆	—	52	1	no	—
129	C₁₃	#12902	1	1	yes	characteristic, complemented, Sylow
130	C₂³	#88161	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	26	4	no	—
133	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
134	C₂×C₄	#88155	1	2	yes	—
135	C₂×C₄	#88161	1	1	yes	characteristic
136	C₂×C₄	C₂×#6101	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	26	4	no	—
139	C₈	#88142	1	2	yes	—
140	C₈	#88194	1	1	yes	characteristic
141	C₈	#88198	1	1	yes	characteristic
142	C₂²	—	4	1	no	—
143	C₂²	#11718500	1	2	yes	—
144	C₂²	#11718780	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	52	4	no	—
147	C₄	#11718500	1	2	yes	—
148	C₄	#11718637	1	1	yes	characteristic
149	C₄	#11718763	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#13114989	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116601

C₂

yes

maximal

#13116596

C₂

yes

maximal

#13116595

C₂

yes

maximal, characteristic

#13114591

C₂

yes

maximal, characteristic

C₁₃×#2153

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₂²

yes

—

C₁₃×#405

C₂²

yes

characteristic

#11718762

C₂²

yes

—

C₂×#88052

C₂²

yes

characteristic

C₂²×#6090

C₂²

yes

characteristic

C₁₃×#364

—

#11718152

C₄

yes

characteristic

#11718149

C₄

yes

characteristic

#11718123

C₄

yes

—

#11718103

—

#11718093

—

C₄⋊#6090

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

C₂×C₄

yes

characteristic

#88180

C₂×C₄

yes

—

C₂×#6099

C₂×C₄

yes

—

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₁₃×#97

Q₈

yes

—

C₄×C₈×C₁₃

D₄

yes

—

#88055

Q₈

yes

—

C₈×Dic₁₃

D₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

C₄⋊C₄

yes

—

#6099

—

#6099

—

C₂×#1055

C₄⋊C₄

yes

—

C₁₃×C₁₆

—

—

—

—

D₁₃

yes

—

—

—

C₂³×C₁₃

#131

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

characteristic

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#127

yes

characteristic

C₈×C₁₃

C₄⋊Q₈

yes

—

C₈×C₁₃

C₄⋊Q₈

yes

characteristic

#1055

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₁₃

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₁₃

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂²×C₁₃

—

C₂²×C₁₃

#424

yes

—

C₂²×C₁₃

#424

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#384

yes

characteristic

C₄×C₁₃

#424

yes

—

C₄×C₁₃

#424

yes

characteristic

Dic₁₃

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₁₃

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

#117

—

#117

C₄×D₁₃

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

#97

—

104

#97

Dic₂₆

yes

—

105

C₄×C₈

—

106

C₄×C₈

C₂²⋊D₁₃

yes

—

107

C₂×C₁₃

—

108

C₂×C₁₃

—

109

C₂×C₁₃

#1843

yes

characteristic

110

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

#6101

yes

—

116

C₂×C₈

C₂×Dic₂₆

yes

characteristic

117

C₂×C₈

D₄×D₁₃

yes

—

118

C₂×C₈

Q₈×D₁₃

yes

—

119

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

#6101

yes

—

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

129

C₁₃

#12902

yes

characteristic, complemented, Sylow

130

C₂³

#88161

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊C₄×D₁₃

yes

—

134

C₂×C₄

#88155

yes

—

135

C₂×C₄

#88161

yes

characteristic

136

C₂×C₄

C₂×#6101

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#88142

yes

—

140

C₈

#88194

yes

characteristic

141

C₈

#88198

yes

characteristic

142

C₂²

—

143

C₂²

#11718500

yes

—

144

C₂²

#11718780

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

#11718500

yes

—

148

C₄

#11718637

yes

characteristic

149

C₄

#11718763

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#13114989

yes

characteristic

	subgroup	G/H
center	C₈	#88194
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2153	C₂
socle	C₂×C₁₃	#1843
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2153	D₁₃
hypocenter	C₁₃	#12902
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#88194

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#2153

C₂

C₂×C₁₃

C₂³×C₁₃

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
129. C₁₃	#12902	semidirect

H ≅ G/N

type

129. C₁₃

#12902

semidirect

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

subgroup	index
#13116601	2
#13116596	2
#13116595	2
#13114591	2
C₁₃×#2153	2
#12902	13

subgroup

index

C₁₃×#2153

Subgroups — #15779342

Conjugacy classes of subgroups

Complements

Maximal subgroups