Group #13125581 — subgroups

Subgroups — #13125581

Order 1664.

358 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 133 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	2	yes	maximal
2	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#11718798	C₂	1	4	yes	maximal
4	#11718377	C₂	1	4	yes	maximal, complemented
5	#11718360	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718342	C₂	1	2	yes	maximal, complemented
7	#11718341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	#88180	—	2	2	no	—
11	#88178	—	2	2	no	—
12	#88178	C₂²	1	4	yes	—
13	C₂×Dic₅₂	C₂²	1	4	yes	—
14	#88124	—	2	2	no	—
15	#88123	C₂²	1	2	yes	—
16	#88123	C₂²	1	2	yes	—
17	C₄×Dic₂₆	—	2	2	no	—
18	C₄×Dic₂₆	C₂²	1	4	yes	—
19	C₁₃×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	#88057	C₂²	1	4	yes	—
23	#88056	C₂²	1	8	yes	—
24	C₂²×C₄×C₁₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₂₆	—	2	2	no	—
27	C₂×Dic₂₆	C₂³	1	4	yes	—
28	C₁₃×M₁₆	—	2	4	no	—
29	C₂×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂×C₈×C₁₃	C₂³	1	2	yes	—
31	C₄²×C₁₃	D₄	1	2	yes	—
32	C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₁₃	—	2	4	no	—
35	C₄⋊Dic₁₃	—	2	2	no	—
36	C₄⋊Dic₁₃	—	2	2	no	—
37	C₄⋊Dic₁₃	—	2	2	no	—
38	C₄⋊Dic₁₃	C₂³	1	4	yes	—
39	#6101	—	2	4	no	—
40	#6101	—	2	4	no	—
41	C₄×Dic₁₃	—	2	4	no	—
42	Dic₅₂	—	2	8	no	—
43	#3185	—	13	1	no	maximal, Sylow
44	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	—	2	4	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₃	—	2	4	no	—
52	C₂×Dic₁₃	—	2	4	no	—
53	Dic₂₆	—	2	8	no	—
54	Dic₂₆	—	4	4	no	—
55	C₈×C₁₃	—	2	2	no	—
56	C₈×C₁₃	#41	1	4	yes	—
57	#533	—	13	2	no	—
58	#498	—	13	1	no	—
59	#494	—	13	1	no	—
60	#487	—	13	2	no	—
61	#470	—	13	4	no	—
62	#441	—	13	4	no	—
63	C₄×M₁₆	D₁₃	1	1	yes	characteristic
64	C₂²×C₁₃	—	2	1	no	—
65	C₂²×C₁₃	—	2	2	no	—
66	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
67	C₄×C₁₃	—	2	1	no	—
68	C₄×C₁₃	—	2	2	no	—
69	C₄×C₁₃	#137	1	4	yes	—
70	C₄×C₁₃	C₂×#41	1	2	yes	—
71	Dic₁₃	—	4	4	no	—
72	Dic₁₃	—	4	4	no	—
73	C₂×Q₁₆	—	13	4	no	—
74	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
75	C₄⋊Q₈	—	13	2	no	—
76	C₄⋊Q₈	—	13	2	no	—
77	#125	—	26	2	no	—
78	C₂²⋊Q₈	—	13	4	no	—
79	C₂²⋊Q₈	—	26	2	no	—
80	C₄×Q₈	—	13	4	no	—
81	C₄×Q₈	—	26	2	no	—
82	#117	—	26	2	no	—
83	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
84	#106	—	13	4	no	—
85	#103	—	13	8	no	—
86	#97	C₂×D₁₃	1	2	yes	—
87	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
88	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
89	C₂×C₁₃	—	4	1	no	—
90	C₂×C₁₃	#532	1	1	yes	characteristic
91	C₂×C₁₃	C₂×#137	1	2	yes	—
92	C₂×Q₈	—	13	4	no	—
93	C₂×Q₈	—	26	2	no	—
94	C₂²×C₄	D₅₂	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
96	Q₁₆	—	26	8	no	—
97	M₁₆	—	2	4	no	—
98	C₂×C₈	C₂²×D₁₃	1	2	yes	—
99	C₂×C₈	C₂²×D₁₃	1	2	yes	—
100	C₄⋊C₄	—	13	4	no	—
101	C₄⋊C₄	—	26	2	no	—
102	C₄⋊C₄	—	26	2	no	—
103	C₄⋊C₄	—	26	4	no	—
104	C₄⋊C₄	—	26	4	no	—
105	C₄⋊C₄	—	26	2	no	—
106	C₂²⋊C₄	—	26	4	no	—
107	C₄²	—	26	4	no	—
108	C₄²	D₅₂	1	2	yes	—
109	C₄²	C₂²×D₁₃	1	1	yes	characteristic
110	C₄²	C₂²×D₁₃	1	1	yes	characteristic
111	C₁₃	#3185	1	1	yes	characteristic, complemented, Sylow
112	C₂³	C₂×D₅₂	1	1	yes	characteristic
113	Q₈	—	26	8	no	—
114	Q₈	—	52	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	26	4	no	—
118	C₂×C₄	—	26	4	no	—
119	C₂×C₄	C₂×D₅₂	1	2	yes	—
120	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
121	C₂×C₄	C₂×D₅₂	1	2	yes	—
122	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#6127	1	4	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂²×D₅₂	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	52	4	no	—
131	C₄	—	52	4	no	—
132	C₄	#88163	1	4	yes	—
133	C₄	C₂×#6127	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#11719182	1	1	yes	characteristic
136	C₂	C₂×#88163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719175

C₂

yes

maximal

C₄×C₁₃×M₁₆

C₂

yes

maximal, characteristic

#11718798

C₂

yes

maximal

#11718377

C₂

yes

maximal, complemented

#11718360

C₂

yes

maximal, characteristic, complemented

#11718342

C₂

yes

maximal, complemented

#11718341

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₂×Dic₅₂

C₂²

yes

—

#88124

—

#88123

C₂²

yes

—

#88123

C₂²

yes

—

C₄×Dic₂₆

—

C₄×Dic₂₆

C₂²

yes

—

C₁₃×#97

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₃

C₂²

yes

characteristic

#88057

C₂²

yes

—

#88056

C₂²

yes

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂³

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂³

yes

—

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

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

#3185

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

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

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

#41

yes

—

—

—

—

—

—

—

C₄×M₁₆

D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#137

yes

—

C₄×C₁₃

C₂×#41

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

C₄×C₈

C₂×D₁₃

yes

characteristic

C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#532

yes

characteristic

C₂×C₁₃

C₂×#137

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

D₅₂

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

C₂²×D₁₃

yes

—

C₂×C₈

C₂²×D₁₃

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

D₅₂

yes

—

109

C₄²

C₂²×D₁₃

yes

characteristic

110

C₄²

C₂²×D₁₃

yes

characteristic

111

C₁₃

#3185

yes

characteristic, complemented, Sylow

112

C₂³

C₂×D₅₂

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×D₅₂

yes

—

120

C₂×C₄

C₂×D₅₂

yes

characteristic

121

C₂×C₄

C₂×D₅₂

yes

—

122

C₂×C₄

C₂³×D₁₃

yes

characteristic

123

C₈

—

124

C₈

#6127

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₅₂

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#88163

yes

—

133

C₄

C₂×#6127

yes

—

134

C₂

—

135

C₂

#11719182

yes

characteristic

136

C₂

C₂×#88163

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	#3185
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

#3185

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
4. #11718377	C₂	semidirect
5. #11718360	C₂	semidirect
6. #11718342	C₂	semidirect
7. #11718341	C₂	semidirect
111. C₁₃	#3185	semidirect

H ≅ G/N

type

4. #11718377

C₂

semidirect

5. #11718360

C₂

semidirect

6. #11718342

C₂

semidirect

7. #11718341

C₂

semidirect

111. C₁₃

#3185

semidirect

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

subgroup	index
#11719175	2
C₄×C₁₃×M₁₆	2
#11718798	2
#11718377	2
#11718360	2
#11718342	2
#11718341	2
#3185	13

subgroup

index

C₄×C₁₃×M₁₆

Subgroups — #13125581

Conjugacy classes of subgroups

Complements

Maximal subgroups