Group #13120209 — subgroups

Subgroups — #13120209

Order 1664.

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

normal subgroups only

This group has 484 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	C₂×C₄×Dic₂₆	C₂	1	2	yes	maximal
2	C₁₃×#384	C₂	1	1	yes	maximal, characteristic
3	#11718646	C₂	1	4	yes	maximal
4	#11718644	C₂	1	4	yes	maximal
5	#11718635	C₂	1	4	yes	maximal
6	C₂²×Dic₂₆	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
12	C₂×#6101	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
14	C₄×Dic₂₆	—	2	8	no	—
15	#88071	C₂²	1	4	yes	—
16	#88071	C₂²	1	8	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
20	C₂²×Dic₁₃	C₂³	1	4	yes	—
21	C₂²×Dic₁₃	C₂³	1	4	yes	—
22	C₂×Dic₂₆	—	2	4	no	—
23	C₂×Dic₂₆	—	2	4	no	—
24	C₂×Dic₂₆	Q₈	1	8	yes	complemented
25	C₁₃×C₄⋊C₄	—	2	8	no	—
26	C₁₃×C₄⋊C₄	—	4	2	no	—
27	C₄²×C₁₃	D₄	1	4	yes	—
28	C₄⋊Dic₁₃	—	2	4	no	—
29	C₄⋊Dic₁₃	—	4	4	no	—
30	#6101	—	2	8	no	—
31	C₄×Dic₁₃	—	2	8	no	—
32	#2582	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
34	C₂×C₄×C₁₃	—	2	1	no	—
35	C₂×C₄×C₁₃	—	2	1	no	—
36	C₂×C₄×C₁₃	—	2	8	no	—
37	C₂×C₄×C₁₃	—	2	4	no	—
38	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₃	#41	1	4	yes	—
42	C₂×Dic₁₃	—	2	4	no	—
43	C₂×Dic₁₃	—	2	4	no	—
44	C₂×Dic₁₃	—	2	4	no	—
45	C₂×Dic₁₃	—	2	4	no	—
46	C₂×Dic₁₃	—	2	4	no	—
47	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
48	Dic₂₆	—	2	16	no	—
49	C₂×C₄×Q₈	—	13	2	no	—
50	#398	—	13	4	no	—
51	#391	—	13	4	no	—
52	#389	—	13	4	no	—
53	#384	D₁₃	1	1	yes	characteristic
54	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
55	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
56	C₂²×C₁₃	C₂×#41	1	2	yes	—
57	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
58	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
59	C₄×C₁₃	—	2	2	no	—
60	C₄×C₁₃	—	4	4	no	—
61	C₄×C₁₃	C₂²⋊Q₈	1	8	yes	—
62	Dic₁₃	—	2	8	no	—
63	Dic₁₃	—	4	4	no	—
64	C₂²×Q₈	—	13	2	no	—
65	C₄×Q₈	—	26	8	no	—
66	C₂×C₄⋊C₄	—	13	4	no	—
67	C₂×C₄⋊C₄	—	13	2	no	—
68	C₂×C₄⋊C₄	—	13	4	no	—
69	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
70	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
71	C₂×C₄²	—	13	4	no	—
72	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
73	#95	—	13	8	no	—
74	#95	—	13	4	no	—
75	C₂×C₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
76	C₂×C₁₃	#537	1	1	yes	characteristic
77	C₂×C₁₃	#557	1	2	yes	—
78	C₂×C₁₃	Q₈²	1	1	yes	characteristic
79	C₂×C₁₃	#559	1	1	yes	characteristic
80	C₂×Q₈	—	13	8	no	—
81	C₂×Q₈	—	26	4	no	—
82	C₂×Q₈	—	26	4	no	—
83	C₂²×C₄	—	13	4	no	—
84	C₂²×C₄	—	13	4	no	—
85	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
86	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
88	C₄⋊C₄	—	2	8	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	26	8	no	—
91	C₄⋊C₄	—	26	4	no	—
92	C₄⋊C₄	—	52	4	no	—
93	C₄²	—	26	8	no	—
94	C₄²	D₅₂	1	4	yes	—
95	C₁₃	#2582	1	1	yes	characteristic, complemented, Sylow
96	C₂³	C₂³×D₁₃	1	1	yes	characteristic
97	Q₈	—	26	16	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	13	8	no	—
103	C₂×C₄	—	26	4	no	—
104	C₂×C₄	—	26	4	no	—
105	C₂×C₄	—	26	4	no	—
106	C₂×C₄	—	26	4	no	—
107	C₂×C₄	—	26	4	no	—
108	C₂×C₄	C₂×D₅₂	1	2	yes	—
109	C₂×C₄	C₂×D₅₂	1	4	yes	—
110	C₂×C₄	#6129	1	4	yes	—
111	C₂×C₄	Q₈×D₁₃	1	4	yes	—
112	C₂²	C₂²×D₅₂	1	1	yes	characteristic
113	C₂²	C₂×#6129	1	2	yes	—
114	C₂²	#88251	1	1	yes	characteristic
115	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
116	C₂²	#88254	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	4	4	no	—
119	C₄	—	26	8	no	—
120	C₄	—	52	4	no	—
121	C₄	#88151	1	8	yes	—
122	C₂	C₂×#88151	1	2	yes	—
123	C₂	#11719215	1	1	yes	characteristic
124	C₂	#11719418	1	1	yes	characteristic
125	C₂	#11719426	1	2	yes	—
126	C₂	#11719427	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₆

C₂

yes

maximal

C₁₃×#384

C₂

yes

maximal, characteristic

#11718646

C₂

yes

maximal

#11718644

C₂

yes

maximal

#11718635

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₂₆

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

Q₈

yes

complemented

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

#2582

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

C₂×C₄×Q₈

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

#95

—

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

—

C₂×C₁₃

#537

yes

characteristic

C₂×C₁₃

#557

yes

—

C₂×C₁₃

Q₈²

yes

characteristic

C₂×C₁₃

#559

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₅₂

yes

—

C₁₃

#2582

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×D₅₂

yes

—

109

C₂×C₄

C₂×D₅₂

yes

—

110

C₂×C₄

#6129

yes

—

111

C₂×C₄

Q₈×D₁₃

yes

—

112

C₂²

C₂²×D₅₂

yes

characteristic

113

C₂²

C₂×#6129

yes

—

114

C₂²

#88251

yes

characteristic

115

C₂²

C₂×Q₈×D₁₃

yes

—

116

C₂²

#88254

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#88151

yes

—

122

C₂

C₂×#88151

yes

—

123

C₂

#11719215

yes

characteristic

124

C₂

#11719418

yes

characteristic

125

C₂

#11719426

yes

—

126

C₂

#11719427

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#384	C₂
hypercenter	#384	D₁₃
hypocenter	C₁₃	#2582
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#384

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
24. C₂×Dic₂₆	Q₈	semidirect
95. C₁₃	#2582	semidirect

H ≅ G/N

type

24. C₂×Dic₂₆

Q₈

semidirect

95. C₁₃

#2582

semidirect

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

subgroup	index
C₂×C₄×Dic₂₆	2
C₁₃×#384	2
#11718646	2
#11718644	2
#11718635	2
#2582	13

subgroup

index

C₂×C₄×Dic₂₆

C₁₃×#384

Subgroups — #13120209

Conjugacy classes of subgroups

Complements

Maximal subgroups