Group #15698843 — subgroups

Subgroups — C₁₃×#12504 #15698843

Order 3328.

262 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2252	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2241	C₂	1	2	yes	maximal
4	C₁₃×#1707	C₂	1	2	yes	maximal
5	C₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×Q₃₂	—	2	2	no	—
7	C₁₃×#500	C₂²	1	2	yes	—
8	C₁₃×#498	—	2	2	no	—
9	C₁₃×#494	C₂²	1	2	yes	—
10	C₁₃×#487	—	2	2	no	—
11	C₁₃×#366	—	2	2	no	—
12	C₁₃×#358	—	2	4	no	—
13	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₈²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₁₆	—	2	2	no	—
17	C₂×C₁₃×Q₁₆	—	2	2	no	—
18	C₁₃×C₄⋊Q₈	—	2	2	no	—
19	C₁₃×C₄⋊Q₈	—	2	2	no	—
20	C₁₃×Q₃₂	—	4	4	no	—
21	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
22	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
23	C₁₃×#107	—	2	2	no	—
24	C₁₃×#107	—	2	2	no	—
25	C₁₃×#106	—	4	4	no	—
26	C₁₃×#103	—	4	4	no	—
27	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	#12504	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×Q₈×C₁₃	—	4	2	no	—
32	C₂×Q₈×C₁₃	—	4	2	no	—
33	C₁₃×Q₁₆	—	4	4	no	—
34	C₁₃×Q₁₆	—	4	4	no	—
35	C₂×C₈×C₁₃	D₈	1	2	yes	—
36	C₂×C₈×C₁₃	D₈	1	2	yes	—
37	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₁₃×C₄⋊C₄	—	4	2	no	—
40	C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₁₃×C₄⋊C₄	—	4	2	no	—
42	C₁₃×C₄⋊C₄	—	4	2	no	—
43	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₁₃×C₁₆	—	2	2	no	—
45	C₁₃×C₁₆	SD₁₆	1	4	yes	—
46	#2252	C₂×C₁₃	1	1	yes	characteristic
47	#2247	C₂×C₁₃	1	1	yes	characteristic
48	#2241	C₂×C₁₃	1	2	yes	—
49	#1707	C₂×C₁₃	1	2	yes	—
50	C₈×C₁₆	C₂×C₁₃	1	1	yes	characteristic
51	Q₈×C₁₃	—	8	2	no	—
52	Q₈×C₁₃	—	8	2	no	—
53	Q₈×C₁₃	—	8	4	no	—
54	C₂×C₄×C₁₃	—	8	2	no	—
55	C₂×C₄×C₁₃	—	8	2	no	—
56	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
59	C₈×C₁₃	—	2	2	no	—
60	C₈×C₁₃	Q₃₂	1	4	yes	complemented
61	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
62	C₈×C₁₃	#135	1	2	yes	—
63	C₂×Q₃₂	—	2	2	no	—
64	#500	C₂²×C₁₃	1	2	yes	—
65	#498	—	2	2	no	—
66	#494	C₂²×C₁₃	1	2	yes	—
67	#487	—	2	2	no	—
68	#366	—	2	2	no	—
69	#358	—	2	4	no	—
70	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
71	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
72	C₈²	C₂²×C₁₃	1	1	yes	characteristic
73	C₂²×C₁₃	#493	1	1	yes	characteristic
74	C₄×C₁₃	—	16	2	no	—
75	C₄×C₁₃	—	16	2	no	—
76	C₄×C₁₃	#492	1	1	yes	characteristic
77	C₄×C₁₃	#495	1	1	yes	characteristic
78	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
79	C₄×C₁₃	#508	1	2	yes	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	2	2	no	—
84	Q₃₂	—	4	4	no	—
85	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
86	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
87	#107	—	2	2	no	—
88	#107	—	2	2	no	—
89	#106	—	4	4	no	—
90	#103	—	4	4	no	—
91	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
92	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
93	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
94	C₂×C₁₃	#1706	1	1	yes	characteristic
95	C₂×C₁₃	#2247	1	1	yes	characteristic
96	C₂×C₁₃	#2249	1	1	yes	characteristic
97	C₂×Q₈	—	4	2	no	—
98	C₂×Q₈	—	4	2	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	4	4	no	—
101	C₂×C₈	C₁₃×D₈	1	2	yes	—
102	C₂×C₈	C₁₃×D₈	1	2	yes	—
103	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	C₁₃×SD₁₆	1	4	yes	—
112	C₁₃	#12504	1	1	yes	characteristic, direct product factor, Sylow
113	Q₈	—	8	2	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	8	4	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
119	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
120	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	C₁₃×Q₃₂	1	4	yes	complemented
123	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
124	C₈	C₁₃×#135	1	2	yes	—
125	C₂²	C₁₃×#493	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	16	2	no	—
128	C₄	C₁₃×#492	1	1	yes	characteristic
129	C₄	C₁₃×#495	1	1	yes	characteristic
130	C₄	C₂×C₁₃×Q₃₂	1	2	yes	—
131	C₄	C₁₃×#508	1	2	yes	—
132	C₂	C₁₃×#1706	1	1	yes	characteristic
133	C₂	C₁₃×#2247	1	1	yes	characteristic
134	C₂	C₁₃×#2249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#2252

C₂

yes

maximal, characteristic

C₁₃×#2247

C₂

yes

maximal, characteristic

C₁₃×#2241

C₂

yes

maximal

C₁₃×#1707

C₂

yes

maximal

C₈×C₁₃×C₁₆

C₂

yes

maximal, characteristic

C₂×C₁₃×Q₃₂

—

C₁₃×#500

C₂²

yes

—

C₁₃×#498

—

C₁₃×#494

C₂²

yes

—

C₁₃×#487

—

C₁₃×#366

—

C₁₃×#358

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₈²×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×Q₃₂

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₁₃×#107

—

C₁₃×#107

—

C₁₃×#106

—

C₁₃×#103

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

#12504

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₁₆

—

C₁₃×C₁₆

SD₁₆

yes

—

#2252

C₂×C₁₃

yes

characteristic

#2247

C₂×C₁₃

yes

characteristic

#2241

C₂×C₁₃

yes

—

#1707

C₂×C₁₃

yes

—

C₈×C₁₆

C₂×C₁₃

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

Q₃₂

yes

complemented

C₈×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

#135

yes

—

C₂×Q₃₂

—

#500

C₂²×C₁₃

yes

—

#498

—

#494

C₂²×C₁₃

yes

—

#487

—

#366

—

#358

—

C₄×C₁₆

C₂²×C₁₃

yes

characteristic

C₄×C₁₆

C₂²×C₁₃

yes

characteristic

C₈²

C₂²×C₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×Q₃₂

yes

—

C₄×C₁₃

#508

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₄×C₁₃

yes

—

C₂×C₁₆

D₄×C₁₃

yes

—

—

—

—

—

C₄×C₈

D₄×C₁₃

yes

characteristic

C₄×C₈

D₄×C₁₃

yes

characteristic

C₄×C₈

C₂³×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

C₁₃×D₈

yes

—

102

C₂×C₈

C₁₃×D₈

yes

—

103

C₂×C₈

C₂×D₄×C₁₃

yes

characteristic

104

C₂×C₈

C₂×D₄×C₁₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

C₂×D₄×C₁₃

yes

characteristic

110

C₁₆

—

111

C₁₆

C₁₃×SD₁₆

yes

—

112

C₁₃

#12504

yes

characteristic, direct product factor, Sylow

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₁₃×#127

yes

characteristic

119

C₂×C₄

C₂×C₁₃×D₈

yes

characteristic

120

C₂×C₄

C₂×C₁₃×D₈

yes

characteristic

121

C₈

—

122

C₈

C₁₃×Q₃₂

yes

complemented

123

C₈

C₂×C₁₃×SD₁₆

yes

—

124

C₈

C₁₃×#135

yes

—

125

C₂²

C₁₃×#493

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

C₁₃×#492

yes

characteristic

129

C₄

C₁₃×#495

yes

characteristic

130

C₄

C₂×C₁₃×Q₃₂

yes

—

131

C₄

C₁₃×#508

yes

—

132

C₂

C₁₃×#1706

yes

characteristic

133

C₂

C₁₃×#2247

yes

characteristic

134

C₂

C₁₃×#2249

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄²×C₁₃ ◁ C₈²×C₁₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₃

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃ ◁ C₄²×C₁₃ ◁ C₈²×C₁₃ ◁ G

N	H ≅ G/N	type
30. #12504	C₁₃	direct (N × H)
60. C₈×C₁₃	Q₃₂	semidirect
112. C₁₃	#12504	direct (N × H)
122. C₈	C₁₃×Q₃₂	semidirect

H ≅ G/N

type

30. #12504

C₁₃

direct (N × H)

60. C₈×C₁₃

Q₃₂

semidirect

112. C₁₃

#12504

direct (N × H)

122. C₈

C₁₃×Q₃₂

semidirect

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

subgroup	index
C₁₃×#2252	2
C₁₃×#2247	2
C₁₃×#2241	2
C₁₃×#1707	2
C₈×C₁₃×C₁₆	2
#12504	13

subgroup

index

C₁₃×#2252

C₁₃×#2247

C₁₃×#2241

C₁₃×#1707

C₈×C₁₃×C₁₆

#12504

Subgroups — C₁₃×#12504 #15698843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#12504 #15698843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12504 #15698843