Group #15699222 — subgroups

Subgroups — C₁₃×#12883 #15699222

Order 3328.

342 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 174 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×#2150	C₂	1	2	yes	maximal
2	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1376	C₂	1	4	yes	maximal
4	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄³×C₁₃	C₄	1	1	yes	characteristic
10	C₁₃×C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	—	2	2	no	—
15	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₁₃×M₃₂	—	2	8	no	—
18	C₂×C₁₃×C₁₆	—	2	4	no	—
19	C₄×C₈×C₁₃	—	2	4	no	—
20	C₄×C₈×C₁₃	—	2	4	no	—
21	C₄×C₈×C₁₃	C₈	1	2	yes	—
22	C₄×C₈×C₁₃	C₈	1	2	yes	—
23	C₄×C₈×C₁₃	D₄	1	2	yes	—
24	C₄×C₈×C₁₃	Q₈	1	2	yes	—
25	#12883	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₄×C₁₃	—	2	2	no	—
27	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₃	—	2	2	no	—
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	—	2	2	no	—
32	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	#41	1	4	yes	—
37	C₄²×C₁₃	—	2	8	no	—
38	C₄²×C₁₃	—	2	2	no	—
39	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
40	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₃	M₁₆	1	4	yes	—
43	C₁₃×C₁₆	—	4	4	no	—
44	#2150	C₂×C₁₃	1	2	yes	—
45	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic
46	#1376	C₂×C₁₃	1	4	yes	—
47	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₃	—	2	8	no	—
49	C₂×C₄×C₁₃	—	2	4	no	—
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#117	1	2	yes	—
55	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
57	C₈×C₁₃	—	2	2	no	—
58	C₈×C₁₃	—	2	4	no	—
59	C₈×C₁₃	#124	1	2	yes	—
60	C₈×C₁₃	#125	1	2	yes	—
61	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
62	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	—
63	C₂×C₄×C₈	C₄×C₁₃	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
65	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
66	C₄³	C₄×C₁₃	1	1	yes	characteristic
67	C₄⋊C₁₆	—	2	4	no	—
68	C₂²×C₁₃	—	2	2	no	—
69	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₂²×C₁₃	#432	1	2	yes	—
71	C₄×C₁₃	—	2	8	no	—
72	C₄×C₁₃	—	2	2	no	—
73	C₄×C₁₃	#364	1	4	yes	—
74	C₄×C₁₃	#382	1	1	yes	characteristic
75	C₄×C₁₃	#423	1	1	yes	characteristic
76	C₄×C₁₃	#431	1	2	yes	—
77	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
79	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
80	C₂×C₄²	—	2	2	no	—
81	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
82	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
83	M₃₂	—	2	8	no	—
84	C₂×C₁₆	—	2	4	no	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	—	2	4	no	—
87	C₄×C₈	C₈×C₁₃	1	2	yes	—
88	C₄×C₈	C₈×C₁₃	1	2	yes	—
89	C₄×C₈	D₄×C₁₃	1	2	yes	—
90	C₄×C₈	Q₈×C₁₃	1	2	yes	—
91	C₂×C₁₃	—	2	2	no	—
92	C₂×C₁₃	#1831	1	1	yes	characteristic
93	C₂×C₁₃	C₂×#364	1	2	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
96	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
101	C₂×C₈	C₂×C₈×C₁₃	1	4	yes	—
102	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
103	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
104	C₂×C₈	C₁₃×#41	1	4	yes	—
105	C₄²	—	2	8	no	—
106	C₄²	—	2	2	no	—
107	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
108	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
109	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
110	C₄²	C₁₃×M₁₆	1	4	yes	—
111	C₁₆	—	4	4	no	—
112	C₁₃	#12883	1	1	yes	characteristic, direct product factor, Sylow
113	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
118	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
119	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄	C₁₃×#117	1	2	yes	—
121	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
122	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	4	no	—
125	C₈	C₁₃×#124	1	2	yes	—
126	C₈	C₁₃×#125	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
129	C₂²	C₁₃×#432	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	C₁₃×#364	1	4	yes	—
133	C₄	C₁₃×#382	1	1	yes	characteristic
134	C₄	C₁₃×#423	1	1	yes	characteristic
135	C₄	C₁₃×#431	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	C₁₃×#1831	1	1	yes	characteristic
138	C₂	C₂×C₁₃×#364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#2150

C₂

yes

maximal

C₄²×C₈×C₁₃

C₂

yes

maximal, characteristic

C₁₃×#1376

C₂

yes

maximal

C₂×C₁₃×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₄

yes

—

C₂×C₄×C₈×C₁₃

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₄³×C₁₃

C₄

yes

characteristic

C₁₃×C₄⋊C₁₆

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₈

yes

—

C₄×C₈×C₁₃

C₈

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

#12883

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

—

C₁₃×C₁₆

—

#2150

C₂×C₁₃

yes

—

C₄²×C₈

C₂×C₁₃

yes

characteristic

#1376

C₂×C₁₃

yes

—

C₂³×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#124

yes

—

C₈×C₁₃

#125

yes

—

C₂×M₃₂

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₄×C₁₃

yes

—

C₂×C₄×C₈

C₄×C₁₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₂²×C₁₃

yes

characteristic

C₄³

C₄×C₁₃

yes

characteristic

C₄⋊C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₃

#432

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#364

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂²×C₈

C₂×C₄×C₁₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₃

yes

—

C₂²×C₈

C₂³×C₁₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₁₃

yes

—

C₂×C₄²

C₂×C₄×C₁₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈×C₁₃

yes

—

C₄×C₈

C₈×C₁₃

yes

—

C₄×C₈

D₄×C₁₃

yes

—

C₄×C₈

Q₈×C₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#1831

yes

characteristic

C₂×C₁₃

C₂×#364

yes

—

C₂²×C₄

—

C₂²×C₄

C₁₃×M₁₆

yes

—

C₂²×C₄

C₂²×C₄×C₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₁₃×C₄⋊C₄

yes

—

101

C₂×C₈

C₂×C₈×C₁₃

yes

—

102

C₂×C₈

C₂×D₄×C₁₃

yes

characteristic

103

C₂×C₈

C₂×Q₈×C₁₃

yes

characteristic

104

C₂×C₈

C₁₃×#41

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₁₃×C₄⋊C₄

yes

—

108

C₄²

C₂×C₈×C₁₃

yes

characteristic

109

C₄²

C₂×C₈×C₁₃

yes

characteristic

110

C₄²

C₁₃×M₁₆

yes

—

111

C₁₆

—

112

C₁₃

#12883

yes

characteristic, direct product factor, Sylow

113

C₂³

C₂×C₁₃×M₁₆

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₁₃×C₄⋊C₈

yes

—

118

C₂×C₄

C₁₃×C₄⋊C₈

yes

—

119

C₂×C₄

C₂×C₁₃×C₄⋊C₄

yes

characteristic

120

C₂×C₄

C₁₃×#117

yes

—

121

C₂×C₄

C₂²×C₈×C₁₃

yes

characteristic

122

C₂×C₄

C₂×C₁₃×M₁₆

yes

—

123

C₈

—

124

C₈

—

125

C₈

C₁₃×#124

yes

—

126

C₈

C₁₃×#125

yes

—

127

C₂²

—

128

C₂²

C₂×C₁₃×C₄⋊C₈

yes

characteristic

129

C₂²

C₁₃×#432

yes

—

130

C₄

—

131

C₄

—

132

C₄

C₁₃×#364

yes

—

133

C₄

C₁₃×#382

yes

characteristic

134

C₄

C₁₃×#423

yes

characteristic

135

C₄

C₁₃×#431

yes

—

136

C₂

—

137

C₂

C₁₃×#1831

yes

characteristic

138

C₂

C₂×C₁₃×#364

yes

—

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

subgroup

G/H

center

C₂×C₈×C₁₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₈×C₁₃

Frattini, cosocle

C₂²×C₈

C₂³×C₁₃

socle

C₂²×C₁₃

C₂×C₄⋊C₈

prime radical

C₂³×C₁₃

C₂×M₁₆

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

N	H ≅ G/N	type
25. #12883	C₁₃	direct (N × H)
112. C₁₃	#12883	direct (N × H)

H ≅ G/N

type

25. #12883

C₁₃

direct (N × H)

112. C₁₃

#12883

direct (N × H)

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

subgroup	index
C₁₃×#2150	2
C₄²×C₈×C₁₃	2
C₁₃×#1376	2
#12883	13

subgroup

index

C₁₃×#2150

C₄²×C₈×C₁₃

C₁₃×#1376

#12883

Subgroups — C₁₃×#12883 #15699222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#12883 #15699222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12883 #15699222