Group #13112197 — subgroups

Subgroups — C₁₃×#3139 #13112197

Order 1664.

380 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 192 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#491	C₂	1	4	yes	maximal, complemented
4	C₁₃×#487	C₂	1	4	yes	maximal, complemented
5	C₁₃×#419	C₂	1	4	yes	maximal
6	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₁₃×C₄⋊Q₈	—	2	2	no	—
10	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₁₃×#125	C₂²	1	2	yes	—
14	C₁₃×C₂²⋊Q₈	—	2	2	no	—
15	C₄×Q₈×C₁₃	—	2	2	no	—
16	C₁₃×#117	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#103	C₂²	1	16	yes	—
20	C₁₃×#97	C₂²	1	2	yes	—
21	C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₂×Q₈×C₁₃	—	2	2	no	—
23	C₂×Q₈×C₁₃	—	2	2	no	—
24	C₂×Q₈×C₁₃	—	2	8	no	—
25	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₃	—	2	2	no	—
27	C₂²×C₄×C₁₃	D₄	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×M₁₆	—	2	4	no	—
30	C₂×C₈×C₁₃	C₂³	1	4	yes	—
31	C₁₃×C₄⋊C₄	—	2	4	no	—
32	C₁₃×C₄⋊C₄	—	2	4	no	—
33	C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₄²×C₁₃	—	2	2	no	—
38	C₄²×C₁₃	D₄	1	2	yes	—
39	C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	#3139	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₃	—	2	8	no	—
44	Q₈×C₁₃	—	4	4	no	—
45	Q₈×C₁₃	—	4	4	no	—
46	C₂×C₄×C₁₃	—	2	4	no	—
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	—	4	4	no	—
50	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃	#41	1	4	yes	—
54	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₈×C₁₃	—	2	4	no	—
56	#533	C₂×C₁₃	1	1	yes	characteristic
57	C₂×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
58	#491	C₂×C₁₃	1	4	yes	complemented
59	#487	C₂×C₁₃	1	4	yes	complemented
60	#419	C₂×C₁₃	1	4	yes	—
61	C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic
62	C₂²×C₁₃	—	2	2	no	—
63	C₂²×C₁₃	#124	1	2	yes	—
64	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₃	—	2	2	no	—
66	C₄×C₁₃	—	4	4	no	—
67	C₄×C₁₃	—	4	4	no	—
68	C₄×C₁₃	#124	1	2	yes	—
69	C₄×C₁₃	#137	1	4	yes	—
70	C₄×C₁₃	C₂×#41	1	2	yes	—
71	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
72	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
73	C₄⋊Q₈	—	2	2	no	—
74	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
75	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
76	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
77	#125	C₂²×C₁₃	1	2	yes	—
78	C₂²⋊Q₈	—	2	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	#117	C₂²×C₁₃	1	2	yes	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
83	#103	C₂²×C₁₃	1	16	yes	—
84	#97	C₂²×C₁₃	1	2	yes	—
85	C₄×C₈	C₂²×C₁₃	1	2	yes	—
86	C₂×C₁₃	—	2	2	no	—
87	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
88	C₂×C₁₃	C₂×#137	1	2	yes	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	C₂³×C₁₃	1	4	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	D₄×C₁₃	1	2	yes	—
95	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
96	M₁₆	—	2	4	no	—
97	C₂×C₈	C₂³×C₁₃	1	4	yes	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
103	C₂²⋊C₄	—	2	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	D₄×C₁₃	1	2	yes	—
106	C₄²	C₂³×C₁₃	1	1	yes	characteristic
107	C₄²	C₂³×C₁₃	1	1	yes	characteristic
108	C₁₃	#3139	1	1	yes	characteristic, direct product factor, Sylow
109	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	4	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
118	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
120	C₂×C₄	C₁₃×#41	1	4	yes	—
121	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₁₃×#124	1	2	yes	—
125	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	4	4	no	—
129	C₄	C₁₃×#124	1	2	yes	—
130	C₄	C₁₃×#137	1	4	yes	—
131	C₄	C₂×C₁₃×#41	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×C₁₃×#124	1	1	yes	characteristic
134	C₂	C₂×C₁₃×#137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#533

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₁₃×#491

C₂

yes

maximal, complemented

C₁₃×#487

C₂

yes

maximal, complemented

C₁₃×#419

C₂

yes

maximal

C₄×C₁₃×M₁₆

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂×C₁₃×M₁₆

C₂²

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×#125

C₂²

yes

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₁₃×#103

C₂²

yes

—

C₁₃×#97

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

#3139

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₈×C₁₃

—

#533

C₂×C₁₃

yes

characteristic

C₂×C₄⋊Q₈

C₂×C₁₃

yes

characteristic

#491

C₂×C₁₃

yes

complemented

#487

C₂×C₁₃

yes

complemented

#419

C₂×C₁₃

yes

—

C₄×M₁₆

C₂×C₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

#124

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

C₄×C₁₃

#137

yes

—

C₄×C₁₃

C₂×#41

yes

—

C₂²×Q₈

C₂²×C₁₃

yes

—

C₂×M₁₆

C₂²×C₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₃

yes

—

C₄⋊Q₈

C₂²×C₁₃

yes

—

C₄⋊Q₈

C₂²×C₁₃

yes

—

#125

C₂²×C₁₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂²×C₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×C₁₃

yes

characteristic

#103

C₂²×C₁₃

yes

—

#97

C₂²×C₁₃

yes

—

C₄×C₈

C₂²×C₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#124

yes

characteristic

C₂×C₁₃

C₂×#137

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₃

yes

—

C₂²×C₄

C₂³×C₁₃

yes

characteristic

M₁₆

—

C₂×C₈

C₂³×C₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂³×C₁₃

yes

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

D₄×C₁₃

yes

—

106

C₄²

C₂³×C₁₃

yes

characteristic

107

C₄²

C₂³×C₁₃

yes

characteristic

108

C₁₃

#3139

yes

characteristic, direct product factor, Sylow

109

C₂³

C₂×D₄×C₁₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₄×C₁₃

yes

—

118

C₂×C₄

C₂×D₄×C₁₃

yes

characteristic

119

C₂×C₄

C₂×D₄×C₁₃

yes

—

120

C₂×C₄

C₁₃×#41

yes

—

121

C₂×C₄

C₂⁴×C₁₃

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

C₁₃×#124

yes

—

125

C₂²

C₂²×D₄×C₁₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₁₃×#124

yes

—

130

C₄

C₁₃×#137

yes

—

131

C₄

C₂×C₁₃×#41

yes

—

132

C₂

—

133

C₂

C₂×C₁₃×#124

yes

characteristic

134

C₂

C₂×C₁₃×#137

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₃

prime radical

C₂³×C₁₃

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
3. C₁₃×#491	C₂	semidirect
4. C₁₃×#487	C₂	semidirect
41. #3139	C₁₃	direct (N × H)
58. #491	C₂×C₁₃	semidirect
59. #487	C₂×C₁₃	semidirect
108. C₁₃	#3139	direct (N × H)

H ≅ G/N

type

3. C₁₃×#491

C₂

semidirect

4. C₁₃×#487

C₂

semidirect

41. #3139

C₁₃

direct (N × H)

58. #491

C₂×C₁₃

semidirect

59. #487

C₂×C₁₃

semidirect

108. C₁₃

#3139

direct (N × H)

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

subgroup	index
C₁₃×#533	2
C₂×C₁₃×C₄⋊Q₈	2
C₁₃×#491	2
C₁₃×#487	2
C₁₃×#419	2
C₄×C₁₃×M₁₆	2
#3139	13

subgroup

index

C₁₃×#533

C₂×C₁₃×C₄⋊Q₈

C₁₃×#491

C₁₃×#487

C₁₃×#419

C₄×C₁₃×M₁₆

#3139

Subgroups — C₁₃×#3139 #13112197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#3139 #13112197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3139 #13112197