Group #13125358 — subgroups

Subgroups — #13125358

Order 1664.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	#11719175	C₂	1	1	yes	maximal, characteristic
3	#11718741	C₂	1	4	yes	maximal, complemented
4	#11718726	C₂	1	4	yes	maximal, complemented
5	#11718659	C₂	1	4	yes	maximal
6	C₂×#88043	C₂	1	1	yes	maximal, characteristic
7	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₁₃×#125	C₂²	1	2	yes	—
9	C₁₃×C₂²⋊Q₈	—	2	2	no	—
10	C₄×Q₈×C₁₃	—	2	2	no	—
11	C₁₃×#117	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	#88180	C₂²	1	2	yes	—
14	#88178	—	2	2	no	—
15	C₂²×#1055	C₂²	1	2	yes	—
16	#88124	C₂²	1	2	yes	—
17	#88123	C₂²	1	2	yes	—
18	C₄×Dic₂₆	—	2	2	no	—
19	#88048	C₂²	1	8	yes	—
20	#88047	C₂²	1	8	yes	—
21	#88043	C₂²	1	2	yes	—
22	#88043	C₂²	1	2	yes	—
23	C₂×Q₈×C₁₃	—	2	2	no	—
24	C₂²×C₄×C₁₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₂₆	—	2	2	no	—
27	C₁₃×C₄⋊C₄	—	2	4	no	—
28	C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
30	C₁₃×C₂²⋊C₄	—	2	2	no	—
31	C₄²×C₁₃	—	2	2	no	—
32	C₄²×C₁₃	D₄	1	2	yes	—
33	C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₁₃	—	2	2	no	—
36	C₄⋊Dic₁₃	—	2	2	no	—
37	C₄⋊Dic₁₃	C₂³	1	4	yes	—
38	#6101	—	2	4	no	—
39	C₄×Dic₁₃	—	2	2	no	—
40	C₂×#1055	Q₈	1	8	yes	complemented
41	C₂×#1055	C₂³	1	4	yes	—
42	#3162	—	13	1	no	maximal, Sylow
43	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₃	—	4	4	no	—
45	C₂×C₄×C₁₃	—	2	4	no	—
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
51	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₁₃	—	2	4	no	—
53	Dic₂₆	—	4	4	no	—
54	#1055	C₂×Q₈	1	8	yes	—
55	#533	—	13	1	no	—
56	#533	D₁₃	1	1	yes	characteristic
57	#501	—	13	4	no	—
58	#501	—	13	4	no	—
59	#427	—	13	4	no	—
60	C₂×#97	—	13	1	no	—
61	C₂²×C₁₃	—	2	2	no	—
62	C₂²×C₁₃	C₄⋊Q₈	1	2	yes	—
63	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₃	—	2	2	no	—
65	C₄×C₁₃	—	2	2	no	—
66	C₄×C₁₃	—	4	4	no	—
67	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
68	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
69	Dic₁₃	—	4	4	no	—
70	C₂²×C₈	—	13	2	no	—
71	C₄⋊Q₈	—	13	2	no	—
72	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
73	#125	—	13	2	no	—
74	#125	C₂×D₁₃	1	2	yes	—
75	C₂²⋊Q₈	—	2	2	no	—
76	C₂²⋊Q₈	—	26	2	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	26	2	no	—
79	#117	—	13	2	no	—
80	#117	C₂×D₁₃	1	2	yes	—
81	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
82	#107	—	13	8	no	—
83	#106	—	13	8	no	—
84	#97	—	13	2	no	—
85	#97	—	13	2	no	—
86	C₂×C₁₃	—	2	2	no	—
87	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₁₃	#575	1	2	yes	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	26	2	no	—
91	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
93	C₂×C₈	—	13	8	no	—
94	C₂×C₈	—	13	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	13	4	no	—
98	C₄⋊C₄	—	26	4	no	—
99	C₄⋊C₄	—	26	2	no	—
100	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
101	C₂²⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	26	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	26	2	no	—
105	C₄²	C₂²⋊D₁₃	1	2	yes	—
106	C₄²	C₂²×D₁₃	1	1	yes	characteristic
107	C₄²	C₂²×D₁₃	1	1	yes	characteristic
108	C₁₃	#3162	1	1	yes	characteristic, complemented, Sylow
109	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	52	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	26	4	no	—
115	C₂×C₄	Q₈×D₁₃	1	4	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
119	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
120	C₈	—	13	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#88198	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	52	4	no	—
128	C₄	#88198	1	2	yes	—
129	C₄	C₂×Q₈×D₁₃	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#88198	1	1	yes	characteristic
132	C₂	#11719527	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#533

C₂

yes

maximal, characteristic

#11719175

C₂

yes

maximal, characteristic

#11718741

C₂

yes

maximal, complemented

#11718726

C₂

yes

maximal, complemented

#11718659

C₂

yes

maximal

C₂×#88043

C₂

yes

maximal, characteristic

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×#125

C₂²

yes

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88180

C₂²

yes

—

#88178

—

C₂²×#1055

C₂²

yes

—

#88124

C₂²

yes

—

#88123

C₂²

yes

—

C₄×Dic₂₆

—

#88048

C₂²

yes

—

#88047

C₂²

yes

—

#88043

C₂²

yes

—

#88043

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

Q₈

yes

complemented

C₂×#1055

C₂³

yes

—

#3162

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

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

—

Dic₂₆

—

#1055

C₂×Q₈

yes

—

#533

—

#533

D₁₃

yes

—

—

—

C₂×#97

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄⋊Q₈

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄⋊Q₈

yes

—

C₄×C₁₃

C₂²×Q₈

yes

—

Dic₁₃

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃

yes

—

#125

—

#125

C₂×D₁₃

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃

#575

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

C₂²×D₁₃

yes

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²⋊D₁₃

yes

—

106

C₄²

C₂²×D₁₃

yes

characteristic

107

C₄²

C₂²×D₁₃

yes

characteristic

108

C₁₃

#3162

yes

characteristic, complemented, Sylow

109

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

Q₈×D₁₃

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

119

C₂×C₄

C₂³×D₁₃

yes

characteristic

120

C₈

—

121

C₂²

—

122

C₂²

#88198

yes

—

123

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#88198

yes

—

129

C₄

C₂×Q₈×D₁₃

yes

—

130

C₂

—

131

C₂

C₂×#88198

yes

characteristic

132

C₂

#11719527

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#533	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3162
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₁₃×#533

C₂

socle

C₂²×C₁₃

C₂²×D₄

prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #11718741	C₂	semidirect
4. #11718726	C₂	semidirect
40. C₂×#1055	Q₈	semidirect
108. C₁₃	#3162	semidirect

H ≅ G/N

type

3. #11718741

C₂

semidirect

4. #11718726

C₂

semidirect

40. C₂×#1055

Q₈

semidirect

108. C₁₃

#3162

semidirect

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

subgroup	index
C₁₃×#533	2
#11719175	2
#11718741	2
#11718726	2
#11718659	2
C₂×#88043	2
#3162	13

subgroup

index

C₁₃×#533

Subgroups — #13125358

Conjugacy classes of subgroups

Complements

Maximal subgroups