Group #13117555 — subgroups

Subgroups — #13117555

Order 1664.

516 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 205 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×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆	C₂	1	2	yes	maximal
3	C₁₃×#376	C₂	1	1	yes	maximal, characteristic
4	#11718597	C₂	1	1	yes	maximal, characteristic
5	#11718590	C₂	1	2	yes	maximal
6	#11718314	C₂	1	8	yes	maximal
7	C₂²×Dic₂₆	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
13	C₂×#6101	C₂²	1	4	yes	—
14	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
15	#88123	C₄	1	8	yes	complemented
16	C₄×Dic₂₆	—	2	8	no	—
17	C₁₃×#95	C₂²	1	4	yes	—
18	#88071	C₂²	1	8	yes	—
19	#88071	C₂²	1	4	yes	—
20	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₃	C₂³	1	4	yes	—
25	C₂²×Dic₁₃	C₂³	1	4	yes	—
26	C₂×Dic₂₆	—	2	4	no	—
27	C₂×Dic₂₆	—	2	4	no	—
28	C₂×Dic₂₆	C₂×C₄	1	8	yes	—
29	C₄²×C₁₃	—	2	2	no	—
30	C₄²×C₁₃	C₂×C₄	1	4	yes	—
31	C₄²×C₁₃	D₄	1	4	yes	—
32	C₄⋊Dic₁₃	—	2	4	no	—
33	C₄⋊Dic₁₃	C₂×C₄	1	16	yes	—
34	#6101	—	2	8	no	—
35	C₄×Dic₁₃	—	2	8	no	—
36	#2327	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
38	C₂×C₄×C₁₃	—	2	4	no	—
39	C₂×C₄×C₁₃	—	2	4	no	—
40	C₂×C₄×C₁₃	—	2	2	no	—
41	C₂×C₄×C₁₃	—	2	4	no	—
42	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
43	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃	#41	1	2	yes	—
47	C₂×Dic₁₃	—	2	4	no	—
48	C₂×Dic₁₃	—	2	4	no	—
49	C₂×Dic₁₃	—	2	4	no	—
50	C₂×Dic₁₃	—	2	8	no	—
51	C₂×Dic₁₃	C₂²×C₄	1	8	yes	—
52	Dic₂₆	—	2	16	no	—
53	C₂×C₄⋊Q₈	—	13	1	no	—
54	C₂×C₄×Q₈	—	13	2	no	—
55	#391	—	13	2	no	—
56	#387	—	13	8	no	—
57	#382	—	13	1	no	—
58	#376	D₁₃	1	1	yes	characteristic
59	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
62	C₂²×C₁₃	2₋¹⁺⁴	1	4	yes	—
63	C₄×C₁₃	—	2	4	no	—
64	C₄×C₁₃	—	2	4	no	—
65	C₄×C₁₃	—	4	2	no	—
66	C₄×C₁₃	C₄×D₄	1	4	yes	—
67	Dic₁₃	—	2	8	no	—
68	Dic₁₃	—	4	4	no	—
69	C₂²×Q₈	—	13	2	no	—
70	C₄⋊Q₈	—	13	8	no	—
71	C₄×Q₈	—	26	8	no	—
72	C₂×C₄⋊C₄	—	13	4	no	—
73	C₂×C₄⋊C₄	—	13	4	no	—
74	C₂×C₄⋊C₄	—	13	2	no	—
75	C₂×C₄²	—	13	4	no	—
76	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
79	#95	—	13	4	no	—
80	#95	—	13	8	no	—
81	#95	C₂×D₁₃	1	4	yes	—
82	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
83	C₂×C₁₃	#519	1	2	yes	—
84	C₂×C₁₃	#536	1	2	yes	—
85	C₂×C₁₃	#541	1	2	yes	—
86	C₂×Q₈	—	13	8	no	—
87	C₂×Q₈	—	26	4	no	—
88	C₂×Q₈	—	26	4	no	—
89	C₂²×C₄	—	13	4	no	—
90	C₂²×C₄	—	13	4	no	—
91	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
94	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
95	C₄⋊C₄	—	13	16	no	—
96	C₄⋊C₄	—	26	8	no	—
97	C₄⋊C₄	—	26	4	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	26	8	no	—
100	C₄²	C₄×D₁₃	1	4	yes	—
101	C₄²	D₅₂	1	4	yes	—
102	C₁₃	#2327	1	1	yes	characteristic, complemented, Sylow
103	C₂³	C₂³×D₁₃	1	1	yes	characteristic
104	Q₈	—	26	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	13	8	no	—
110	C₂×C₄	—	26	4	no	—
111	C₂×C₄	—	26	4	no	—
112	C₂×C₄	—	26	4	no	—
113	C₂×C₄	—	26	8	no	—
114	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
116	C₂×C₄	C₂×D₅₂	1	4	yes	—
117	C₂×C₄	C₂×D₅₂	1	2	yes	—
118	C₂×C₄	#6127	1	2	yes	—
119	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
120	C₂²	C₂²×D₅₂	1	1	yes	characteristic
121	C₂²	C₂×#6127	1	1	yes	characteristic
122	C₂²	#88257	1	4	yes	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	—	4	2	no	—
126	C₄	—	26	8	no	—
127	C₄	—	52	4	no	—
128	C₄	C₄×D₅₂	1	4	yes	—
129	C₂	C₂×C₄×D₅₂	1	1	yes	characteristic
130	C₂	#11719221	1	2	yes	—
131	C₂	#11719223	1	2	yes	—
132	C₂	#11719231	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₂

yes

maximal

C₁₃×#376

C₂

yes

maximal, characteristic

#11718597

C₂

yes

maximal, characteristic

#11718590

C₂

yes

maximal

#11718314

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

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

—

#88123

C₄

yes

complemented

C₄×Dic₂₆

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

—

C₄×Dic₁₃

—

#2327

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²×C₄

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

C₂×C₄×D₄

yes

characteristic

C₂×C₁₃

#519

yes

—

C₂×C₁₃

#536

yes

—

C₂×C₁₃

#541

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

C₄×D₁₃

yes

—

101

C₄²

D₅₂

yes

—

102

C₁₃

#2327

yes

characteristic, complemented, Sylow

103

C₂³

C₂³×D₁₃

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄×D₁₃

yes

—

115

C₂×C₄

C₂×C₄×D₁₃

yes

—

116

C₂×C₄

C₂×D₅₂

yes

—

117

C₂×C₄

C₂×D₅₂

yes

—

118

C₂×C₄

#6127

yes

—

119

C₂²

C₂²×C₄×D₁₃

yes

characteristic

120

C₂²

C₂²×D₅₂

yes

characteristic

121

C₂²

C₂×#6127

yes

characteristic

122

C₂²

#88257

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₄×D₅₂

yes

—

129

C₂

C₂×C₄×D₅₂

yes

characteristic

130

C₂

#11719221

yes

—

131

C₂

#11719223

yes

—

132

C₂

#11719231

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#376	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	#376	D₁₃
hypocenter	C₁₃	#2327
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×#376

C₂

socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
15. #88123	C₄	semidirect
102. C₁₃	#2327	semidirect

H ≅ G/N

type

15. #88123

C₄

semidirect

102. C₁₃

#2327

semidirect

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

subgroup	index
C₂×#88123	2
C₂×C₄×Dic₂₆	2
C₁₃×#376	2
#11718597	2
#11718590	2
#11718314	2
#2327	13

subgroup

index

C₂×#88123

C₂×C₄×Dic₂₆

C₁₃×#376

Subgroups — #13117555

Conjugacy classes of subgroups

Complements

Maximal subgroups