Group #13125388 — subgroups

Subgroups — #13125388

Order 1664.

348 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 165 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	#11719175	C₂	1	1	yes	maximal, characteristic
2	C₂×#88124	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#11718782	C₂	1	4	yes	maximal
5	#11718363	C₂	1	4	yes	maximal, complemented
6	#11718344	C₂	1	4	yes	maximal, complemented
7	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	#88180	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	#88178	—	2	2	no	—
12	C₂×#6101	—	2	2	no	—
13	#88124	—	2	2	no	—
14	#88124	C₂²	1	2	yes	—
15	#88124	C₂²	1	2	yes	—
16	#88124	C₂²	1	2	yes	—
17	#88123	C₂²	1	2	yes	—
18	C₄×Dic₂₆	—	2	2	no	—
19	C₁₃×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₁₃	C₂²	1	2	yes	—
21	#88058	C₂²	1	8	yes	—
22	#88057	C₂²	1	8	yes	—
23	C₂²×C₄×C₁₃	D₄	1	2	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₃	—	2	2	no	—
26	C₂×Dic₂₆	—	2	2	no	—
27	C₁₃×M₁₆	Q₈	1	8	yes	complemented
28	C₂×C₈×C₁₃	C₂³	1	4	yes	—
29	C₄²×C₁₃	D₄	1	2	yes	—
30	C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²⋊Dic₁₃	—	2	2	no	—
33	C₄⋊Dic₁₃	—	2	2	no	—
34	C₄⋊Dic₁₃	—	2	2	no	—
35	C₄⋊Dic₁₃	C₂³	1	4	yes	—
36	C₄⋊Dic₁₃	C₂³	1	4	yes	—
37	#6101	—	2	4	no	—
38	#6101	—	2	4	no	—
39	#6101	—	2	4	no	—
40	C₄×Dic₁₃	—	2	2	no	—
41	#3166	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	—	4	2	no	—
44	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
48	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₁₃	—	2	4	no	—
50	C₂×Dic₁₃	—	2	4	no	—
51	C₂×Dic₁₃	—	4	4	no	—
52	Dic₂₆	—	4	4	no	—
53	C₈×C₁₃	C₂×Q₈	1	8	yes	—
54	#533	—	13	1	no	—
55	C₂×#125	—	13	1	no	—
56	#501	—	13	4	no	—
57	#499	—	13	4	no	—
58	#428	—	13	4	no	—
59	C₄×M₁₆	D₁₃	1	1	yes	characteristic
60	C₂²×C₁₃	—	2	2	no	—
61	C₂²×C₁₃	C₄⋊Q₈	1	2	yes	—
62	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₃	—	2	2	no	—
64	C₄×C₁₃	—	2	2	no	—
65	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
66	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
67	Dic₁₃	—	4	4	no	—
68	Dic₁₃	—	4	4	no	—
69	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
70	C₄⋊Q₈	—	13	2	no	—
71	#125	—	13	2	no	—
72	#125	—	13	2	no	—
73	#125	—	13	2	no	—
74	#125	—	26	2	no	—
75	C₂²⋊Q₈	—	26	2	no	—
76	C₄×Q₈	—	26	2	no	—
77	#117	—	13	2	no	—
78	C₂×C₄⋊C₄	—	13	2	no	—
79	C₂×C₄⋊C₄	—	26	2	no	—
80	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
81	#107	—	13	8	no	—
82	#106	—	13	8	no	—
83	#97	C₂×D₁₃	1	2	yes	—
84	C₄×C₈	C₂×D₁₃	1	2	yes	—
85	C₂×C₁₃	—	2	2	no	—
86	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₁₃	#575	1	2	yes	—
88	C₂×Q₈	—	26	2	no	—
89	C₂²×C₄	—	26	2	no	—
90	C₂²×C₄	D₅₂	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
92	M₁₆	Dic₂₆	1	8	yes	complemented
93	C₂×C₈	C₂²×D₁₃	1	4	yes	—
94	C₄⋊C₄	—	13	4	no	—
95	C₄⋊C₄	—	13	4	no	—
96	C₄⋊C₄	—	26	2	no	—
97	C₄⋊C₄	—	26	4	no	—
98	C₄⋊C₄	—	26	4	no	—
99	C₄⋊C₄	—	26	4	no	—
100	C₄⋊C₄	—	26	2	no	—
101	C₂²⋊C₄	—	26	2	no	—
102	C₄²	—	26	2	no	—
103	C₄²	D₅₂	1	2	yes	—
104	C₄²	C₂²×D₁₃	1	1	yes	characteristic
105	C₄²	C₂²×D₁₃	1	1	yes	characteristic
106	C₁₃	#3166	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂×D₅₂	1	1	yes	characteristic
108	Q₈	—	52	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	26	4	no	—
111	C₂×C₄	—	26	4	no	—
112	C₂×C₄	—	52	4	no	—
113	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
114	C₂×C₄	C₂×D₅₂	1	2	yes	—
115	C₂×C₄	C₂×D₅₂	1	2	yes	—
116	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
118	C₈	C₂×Dic₂₆	1	8	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	#88123	1	2	yes	—
121	C₂²	C₂²×D₅₂	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	52	4	no	—
125	C₄	—	52	4	no	—
126	C₄	#88123	1	2	yes	—
127	C₄	C₂²×Dic₂₆	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#88123	1	1	yes	characteristic
130	C₂	#11719453	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719175

C₂

yes

maximal, characteristic

C₂×#88124

C₂

yes

maximal, characteristic

C₄×C₁₃×M₁₆

C₂

yes

maximal, characteristic

#11718782

C₂

yes

maximal

#11718363

C₂

yes

maximal, complemented

#11718344

C₂

yes

maximal, complemented

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88180

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

—

—

—

C₂²

yes

—

#88124

C₂²

yes

—

#88124

C₂²

yes

—

#88123

C₂²

yes

—

C₄×Dic₂₆

—

C₁₃×#97

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

—

#88058

C₂²

yes

—

#88057

C₂²

yes

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₁₃×M₁₆

Q₈

yes

complemented

C₂×C₈×C₁₃

C₂³

yes

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

#6101

—

#6101

—

C₄×Dic₁₃

—

#3166

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂×Q₈

yes

—

—

—

—

—

—

C₄×M₁₆

D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

C₄⋊Q₈

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄⋊Q₈

yes

—

C₄×C₁₃

C₂²×Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

C₂×D₁₃

yes

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

C₄×C₈

C₂×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃

#575

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₅₂

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

Dic₂₆

yes

complemented

C₂×C₈

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

D₅₂

yes

—

104

C₄²

C₂²×D₁₃

yes

characteristic

105

C₄²

C₂²×D₁₃

yes

characteristic

106

C₁₃

#3166

yes

characteristic, complemented, Sylow

107

C₂³

C₂×D₅₂

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×Dic₂₆

yes

—

114

C₂×C₄

C₂×D₅₂

yes

—

115

C₂×C₄

C₂×D₅₂

yes

—

116

C₂×C₄

C₂×D₅₂

yes

characteristic

117

C₂×C₄

C₂³×D₁₃

yes

characteristic

118

C₈

C₂×Dic₂₆

yes

—

119

C₂²

—

120

C₂²

#88123

yes

—

121

C₂²

C₂²×D₅₂

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#88123

yes

—

127

C₄

C₂²×Dic₂₆

yes

—

128

C₂

—

129

C₂

C₂×#88123

yes

characteristic

130

C₂

#11719453

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	C₄×M₁₆	D₁₃
hypocenter	C₁₃	#3166
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₅₂

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₄×C₁₃×M₁₆

C₂

socle

C₂²×C₁₃

C₂²×D₄

prime radical

C₂³×C₁₃

C₂×D₄

hypercenter

C₄×M₁₆

D₁₃

hypocenter

C₁₃

#3166

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
5. #11718363	C₂	semidirect
6. #11718344	C₂	semidirect
27. C₁₃×M₁₆	Q₈	semidirect
92. M₁₆	Dic₂₆	semidirect
106. C₁₃	#3166	semidirect

H ≅ G/N

type

5. #11718363

C₂

semidirect

6. #11718344

C₂

semidirect

27. C₁₃×M₁₆

Q₈

semidirect

92. M₁₆

Dic₂₆

semidirect

106. C₁₃

#3166

semidirect

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

subgroup	index
#11719175	2
C₂×#88124	2
C₄×C₁₃×M₁₆	2
#11718782	2
#11718363	2
#11718344	2
#3166	13

subgroup

index

C₄×C₁₃×M₁₆

Subgroups — #13125388

Conjugacy classes of subgroups

Complements

Maximal subgroups