Group #13120565 — subgroups

Subgroups — #13120565

Order 1664.

816 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂²×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#88178	C₂	1	3	yes	maximal, complemented
3	#11718890	C₂	1	2	yes	maximal, complemented
4	#11718883	C₂	1	6	yes	maximal, complemented
5	#11718873	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₃×#41	—	2	12	no	—
7	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₃	C₂²	1	3	yes	—
9	C₂³×C₄×C₁₃	C₂²	1	3	yes	—
10	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₁₃	C₂²	1	6	yes	—
12	C₂×C₂²⋊Dic₁₃	C₂²	1	6	yes	—
13	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
14	#88178	—	2	12	no	—
15	C₂×#6101	C₂²	1	6	yes	—
16	#88071	C₂²	1	6	yes	—
17	C₂⁴×C₁₃	C₂³	1	3	yes	—
18	C₁₃×#41	—	2	32	no	—
19	C₂×Q₈×C₁₃	—	2	6	no	—
20	C₂×D₄×C₁₃	—	2	12	no	—
21	C₂×D₄×C₁₃	—	2	6	no	—
22	C₂²×C₄×C₁₃	—	2	12	no	—
23	C₂²×C₄×C₁₃	D₄	1	12	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃	C₂³	1	6	yes	—
27	C₂²×Dic₁₃	C₂³	1	2	yes	—
28	C₂×Dic₂₆	—	2	6	no	—
29	C₂²⋊Dic₁₃	—	2	12	no	—
30	C₂²⋊Dic₁₃	—	4	6	no	—
31	C₄⋊Dic₁₃	—	2	6	no	—
32	#6101	—	2	12	no	—
33	#2616	—	13	1	no	maximal, Sylow
34	C₂³×C₁₃	—	2	6	no	—
35	C₂³×C₁₃	—	2	3	no	—
36	C₂³×C₁₃	—	2	6	no	—
37	C₂³×C₁₃	—	2	3	no	—
38	C₂³×C₁₃	C₂×D₄	1	6	yes	—
39	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
40	Q₈×C₁₃	—	2	8	no	—
41	D₄×C₁₃	—	2	24	no	—
42	C₂×C₄×C₁₃	—	2	6	no	—
43	C₂×C₄×C₁₃	—	2	24	no	—
44	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
45	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
46	C₂×Dic₁₃	—	2	6	no	—
47	C₂×Dic₁₃	—	2	6	no	—
48	C₂×Dic₁₃	—	2	6	no	—
49	C₂×Dic₁₃	—	2	6	no	—
50	Dic₂₆	—	4	4	no	—
51	C₂²×#41	D₁₃	1	1	yes	characteristic
52	C₂×C₂²⋊Q₈	—	13	3	no	—
53	#395	—	13	2	no	—
54	#394	—	13	6	no	—
55	#380	—	13	3	no	—
56	C₂²×C₁₃	—	2	6	no	—
57	C₂²×C₁₃	—	2	6	no	—
58	C₂²×C₁₃	—	2	6	no	—
59	C₂²×C₁₃	—	4	6	no	—
60	C₂²×C₁₃	—	4	6	no	—
61	C₂²×C₁₃	C₂²×D₄	1	3	yes	—
62	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
63	C₂²×C₁₃	2₋¹⁺⁴	1	3	yes	—
64	C₄×C₁₃	—	2	6	no	—
65	C₄×C₁₃	#120	1	4	yes	—
66	Dic₁₃	—	4	2	no	—
67	Dic₁₃	—	4	6	no	—
68	C₂×#41	—	2	12	no	—
69	C₂²×Q₈	—	13	1	no	—
70	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
71	C₂²×D₄	C₂×D₁₃	1	3	yes	—
72	C₂³×C₄	C₂×D₁₃	1	3	yes	—
73	C₂²⋊Q₈	—	26	12	no	—
74	C₂×C₄⋊C₄	—	13	6	no	—
75	C₂×C₄⋊C₄	—	13	3	no	—
76	C₂×C₂²⋊C₄	—	13	6	no	—
77	C₂×C₂²⋊C₄	—	13	6	no	—
78	#95	—	13	6	no	—
79	C₂×C₁₃	—	4	6	no	—
80	C₂×C₁₃	C₂×#120	1	1	yes	characteristic
81	C₂×C₁₃	#536	1	3	yes	—
82	C₂×C₁₃	#537	1	3	yes	—
83	C₂⁴	C₂²×D₁₃	1	3	yes	—
84	#41	—	2	32	no	—
85	C₂×Q₈	—	2	6	no	—
86	C₂×Q₈	—	26	6	no	—
87	C₂×D₄	—	2	12	no	—
88	C₂×D₄	—	2	6	no	—
89	C₂²×C₄	—	2	12	no	—
90	C₂²×C₄	—	13	6	no	—
91	C₂²×C₄	—	13	2	no	—
92	C₂²×C₄	C₂²⋊D₁₃	1	12	yes	—
93	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
94	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
95	C₄⋊C₄	—	26	12	no	—
96	C₄⋊C₄	—	26	6	no	—
97	C₂²⋊C₄	—	26	12	no	—
98	C₂²⋊C₄	—	52	6	no	—
99	C₁₃	#2616	1	1	yes	characteristic, complemented, Sylow
100	C₂³	—	2	3	no	—
101	C₂³	—	2	6	no	—
102	C₂³	—	2	3	no	—
103	C₂³	—	2	6	no	—
104	C₂³	C₂×C₂²⋊D₁₃	1	6	yes	—
105	C₂³	C₂³×D₁₃	1	1	yes	characteristic
106	Q₈	—	2	8	no	—
107	Q₈	—	52	4	no	—
108	D₄	—	2	24	no	—
109	C₂×C₄	—	2	6	no	—
110	C₂×C₄	—	2	24	no	—
111	C₂×C₄	—	26	6	no	—
112	C₂×C₄	—	26	6	no	—
113	C₂×C₄	—	26	6	no	—
114	C₂×C₄	—	26	6	no	—
115	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
117	C₂²	—	2	6	no	—
118	C₂²	—	2	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	6	no	—
121	C₂²	—	4	6	no	—
122	C₂²	#88256	1	1	yes	characteristic
123	C₂²	#88257	1	3	yes	—
124	C₂²	C₂²×C₂²⋊D₁₃	1	3	yes	—
125	C₄	—	2	6	no	—
126	C₄	—	52	6	no	—
127	C₄	—	52	2	no	—
128	C₄	#88207	1	4	yes	—
129	C₂	—	4	6	no	—
130	C₂	#11719530	1	3	yes	—
131	C₂	#11719538	1	3	yes	—
132	C₂	C₂×#88207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃×#41

C₂

yes

maximal, characteristic

C₂×#88178

C₂

yes

maximal, complemented

#11718890

C₂

yes

maximal, complemented

#11718883

C₂

yes

maximal, complemented

#11718873

C₂

yes

maximal, complemented

C₂×C₁₃×#41

—

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×D₄×C₁₃

C₂²

yes

—

C₂³×C₄×C₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

#88178

—

C₂×#6101

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#2616

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

C₂²×#41

D₁₃

yes

characteristic

C₂×C₂²⋊Q₈

—

#395

—

#394

—

#380

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×#41

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×D₄

C₂×D₁₃

yes

—

C₂³×C₄

C₂×D₁₃

yes

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#120

yes

characteristic

C₂×C₁₃

#536

yes

—

C₂×C₁₃

#537

yes

—

C₂⁴

C₂²×D₁₃

yes

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₁₃

#2616

yes

characteristic, complemented, Sylow

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

C₂×C₂²⋊D₁₃

yes

—

105

C₂³

C₂³×D₁₃

yes

characteristic

106

Q₈

—

107

Q₈

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#88256

yes

characteristic

123

C₂²

#88257

yes

—

124

C₂²

C₂²×C₂²⋊D₁₃

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#88207

yes

—

129

C₂

—

130

C₂

#11719530

yes

—

131

C₂

#11719538

yes

—

132

C₂

C₂×#88207

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting, prime radical	C₂²×C₁₃×#41	C₂
hypercenter	C₂²×#41	D₁₃
hypocenter	C₁₃	#2616
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting, prime radical

C₂²×C₁₃×#41

C₂

hypercenter

C₂²×#41

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂²×#41

N	H ≅ G/N	type
2. C₂×#88178	C₂	semidirect
3. #11718890	C₂	semidirect
4. #11718883	C₂	semidirect
5. #11718873	C₂	semidirect
99. C₁₃	#2616	semidirect

H ≅ G/N

type

2. C₂×#88178

C₂

semidirect

3. #11718890

C₂

semidirect

4. #11718883

C₂

semidirect

5. #11718873

C₂

semidirect

99. C₁₃

#2616

semidirect

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

subgroup	index
C₂²×C₁₃×#41	2
C₂×#88178	2
#11718890	2
#11718883	2
#11718873	2
#2616	13

subgroup

index

C₂²×C₁₃×#41

Subgroups — #13120565

Conjugacy classes of subgroups

Complements

Maximal subgroups