Group #15699224 — subgroups

Subgroups — C₁₃×#12885 #15699224

Order 3328.

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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	C₁₃×#2153	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1747	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1379	C₂	1	2	yes	maximal
5	C₁₃×#1378	C₂	1	2	yes	maximal
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₁₃×#405	C₂²	1	2	yes	—
10	C₁₃×#405	C₂²	1	1	yes	characteristic
11	C₁₃×#364	—	2	2	no	—
12	C₁₃×C₄⋊C₁₆	—	2	2	no	—
13	C₁₃×#338	C₄	1	1	yes	characteristic
14	C₁₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
15	C₁₃×H(3,ℤ/4)	C₄	1	2	yes	—
16	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₁₃×#117	C₂×C₄	1	2	yes	—
20	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
21	C₁₃×M₃₂	—	2	2	no	—
22	C₁₃×M₃₂	—	2	2	no	—
23	C₁₃×M₃₂	—	2	2	no	—
24	C₂×C₁₃×C₁₆	—	2	1	no	—
25	C₂×C₁₃×C₁₆	—	2	1	no	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₂×C₁₃×C₁₆	—	2	2	no	—
28	C₁₃×#97	—	2	2	no	—
29	C₁₃×#97	Q₈	1	2	yes	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₄×C₈×C₁₃	D₄	1	2	yes	—
32	#12885	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₁₃×M₁₆	—	2	2	no	—
35	C₁₃×M₁₆	—	2	2	no	—
36	C₁₃×M₁₆	—	2	2	no	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	C₄⋊C₄	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	#41	1	2	yes	—
42	C₂×C₈×C₁₃	#41	1	2	yes	—
43	C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₁₃×C₂²⋊C₄	—	2	1	no	—
46	C₁₃×C₂²⋊C₄	—	2	2	no	—
47	C₄²×C₁₃	—	2	2	no	—
48	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₁₃×C₁₆	—	4	1	no	—
50	C₁₃×C₁₆	—	4	1	no	—
51	C₁₃×C₁₆	—	4	2	no	—
52	#2153	C₂×C₁₃	1	1	yes	characteristic
53	#2151	C₂×C₁₃	1	1	yes	characteristic
54	#1747	C₂×C₁₃	1	1	yes	characteristic
55	#1379	C₂×C₁₃	1	2	yes	—
56	#1378	C₂×C₁₃	1	2	yes	—
57	C₂³×C₁₃	#131	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	4	2	no	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#117	1	2	yes	—
62	C₂×C₄×C₁₃	#131	1	2	yes	—
63	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
64	C₈×C₁₃	—	2	2	no	—
65	C₈×C₁₃	—	4	2	no	—
66	C₈×C₁₃	#124	1	1	yes	characteristic
67	C₈×C₁₃	#124	1	1	yes	characteristic
68	C₈×C₁₃	#125	1	2	yes	—
69	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
70	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
71	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
72	#405	C₂²×C₁₃	1	2	yes	—
73	#405	C₂²×C₁₃	1	1	yes	characteristic
74	#364	—	2	2	no	—
75	C₄⋊C₁₆	—	2	2	no	—
76	#338	C₄×C₁₃	1	1	yes	characteristic
77	H(3,ℤ/4)	C₄×C₁₃	1	1	yes	characteristic
78	H(3,ℤ/4)	C₄×C₁₃	1	2	yes	—
79	C₂²×C₁₃	—	4	1	no	—
80	C₂²×C₁₃	#424	1	1	yes	characteristic
81	C₂²×C₁₃	#433	1	2	yes	—
82	C₄×C₁₃	—	4	2	no	—
83	C₄×C₁₃	—	8	2	no	—
84	C₄×C₁₃	#382	1	1	yes	characteristic
85	C₄×C₁₃	#424	1	1	yes	characteristic
86	C₄×C₁₃	#433	1	2	yes	—
87	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
88	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
89	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
90	#117	C₂×C₄×C₁₃	1	2	yes	—
91	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
92	M₃₂	—	2	2	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	2	2	no	—
99	#97	—	2	2	no	—
100	#97	Q₈×C₁₃	1	2	yes	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	D₄×C₁₃	1	2	yes	—
103	C₂×C₁₃	—	2	1	no	—
104	C₂×C₁₃	—	2	2	no	—
105	C₂×C₁₃	#1832	1	1	yes	characteristic
106	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
112	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
113	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
114	C₂×C₈	C₁₃×#41	1	2	yes	—
115	C₂×C₈	C₁₃×#41	1	2	yes	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
122	C₁₆	—	4	1	no	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	2	no	—
125	C₁₃	#12885	1	1	yes	characteristic, direct product factor, Sylow
126	C₂³	C₁₃×#131	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄	C₁₃×#117	1	2	yes	—
131	C₂×C₄	C₁₃×#131	1	2	yes	—
132	C₂×C₄	C₁₃×#131	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	C₁₃×#124	1	1	yes	characteristic
136	C₈	C₁₃×#124	1	1	yes	characteristic
137	C₈	C₁₃×#125	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	C₁₃×#424	1	1	yes	characteristic
140	C₂²	C₁₃×#433	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	C₁₃×#382	1	1	yes	characteristic
144	C₄	C₁₃×#424	1	1	yes	characteristic
145	C₄	C₁₃×#433	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	C₁₃×#1832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#2153

C₂

yes

maximal, characteristic

C₁₃×#2151

C₂

yes

maximal, characteristic

C₁₃×#1747

C₂

yes

maximal, characteristic

C₁₃×#1379

C₂

yes

maximal

C₁₃×#1378

C₂

yes

maximal

C₂×C₁₃×M₃₂

C₂²

yes

characteristic

C₂×C₁₃×M₃₂

C₂²

yes

—

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#405

C₂²

yes

—

C₁₃×#405

C₂²

yes

characteristic

C₁₃×#364

—

C₁₃×C₄⋊C₁₆

—

C₁₃×#338

C₄

yes

characteristic

C₁₃×H(3,ℤ/4)

C₄

yes

characteristic

C₁₃×H(3,ℤ/4)

C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

C₂×C₄

yes

—

C₁₃×#117

C₂×C₄

yes

characteristic

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

C₁₃×#97

—

C₁₃×#97

Q₈

yes

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

D₄

yes

—

#12885

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

#2153

C₂×C₁₃

yes

characteristic

#2151

C₂×C₁₃

yes

characteristic

#1747

C₂×C₁₃

yes

characteristic

#1379

C₂×C₁₃

yes

—

#1378

C₂×C₁₃

yes

—

C₂³×C₁₃

#131

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

C₈×C₁₃

yes

—

C₂×M₃₂

C₂²×C₁₃

yes

characteristic

C₂×M₃₂

C₂²×C₁₃

yes

—

C₂²×C₁₆

C₂²×C₁₃

yes

characteristic

#405

C₂²×C₁₃

yes

—

#405

C₂²×C₁₃

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

C₄×C₁₃

yes

characteristic

H(3,ℤ/4)

C₄×C₁₃

yes

characteristic

H(3,ℤ/4)

C₄×C₁₃

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

#424

yes

characteristic

C₂²×C₁₃

#433

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂×M₁₆

C₂×C₄×C₁₃

yes

—

C₂×M₁₆

C₂×C₄×C₁₃

yes

characteristic

C₂²×C₈

C₂³×C₁₃

yes

characteristic

#117

C₂×C₄×C₁₃

yes

—

#117

C₂×C₄×C₁₃

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

100

#97

Q₈×C₁₃

yes

—

101

C₄×C₈

—

102

C₄×C₈

D₄×C₁₃

yes

—

103

C₂×C₁₃

—

104

C₂×C₁₃

—

105

C₂×C₁₃

#1832

yes

characteristic

106

C₂²×C₄

C₂²×C₄×C₁₃

yes

characteristic

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

C₁₃×C₄⋊C₄

yes

—

112

C₂×C₈

C₂×D₄×C₁₃

yes

characteristic

113

C₂×C₈

C₂×Q₈×C₁₃

yes

characteristic

114

C₂×C₈

C₁₃×#41

yes

—

115

C₂×C₈

C₁₃×#41

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₁₃×C₄⋊C₄

yes

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₁₃

#12885

yes

characteristic, direct product factor, Sylow

126

C₂³

C₁₃×#131

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₁₃×C₄⋊C₄

yes

characteristic

130

C₂×C₄

C₁₃×#117

yes

—

131

C₂×C₄

C₁₃×#131

yes

—

132

C₂×C₄

C₁₃×#131

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

C₁₃×#124

yes

characteristic

136

C₈

C₁₃×#124

yes

characteristic

137

C₈

C₁₃×#125

yes

—

138

C₂²

—

139

C₂²

C₁₃×#424

yes

characteristic

140

C₂²

C₁₃×#433

yes

—

141

C₄

—

142

C₄

—

143

C₄

C₁₃×#382

yes

characteristic

144

C₄

C₁₃×#424

yes

characteristic

145

C₄

C₁₃×#433

yes

—

146

C₂

—

147

C₂

—

148

C₂

C₁₃×#1832

yes

characteristic

	subgroup	G/H
center	C₈×C₁₃	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1832
prime radical	C₂³×C₁₃	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

subgroup

G/H

center

C₈×C₁₃

#124

commutator

C₂²×C₄

C₂²×C₄×C₁₃

Frattini, cosocle

C₂²×C₈

C₂³×C₁₃

C₂×C₁₃

C₂³×C₁₃

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

N	H ≅ G/N	type
32. #12885	C₁₃	direct (N × H)
125. C₁₃	#12885	direct (N × H)

H ≅ G/N

type

32. #12885

C₁₃

direct (N × H)

125. C₁₃

#12885

direct (N × H)

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

subgroup	index
C₁₃×#2153	2
C₁₃×#2151	2
C₁₃×#1747	2
C₁₃×#1379	2
C₁₃×#1378	2
#12885	13

subgroup

index

C₁₃×#2153

C₁₃×#2151

C₁₃×#1747

C₁₃×#1379

C₁₃×#1378

#12885

Subgroups — C₁₃×#12885 #15699224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#12885 #15699224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12885 #15699224