Group #13127514 — subgroups

Subgroups — #13127514

Order 1664.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
2	#11719275	C₂	1	1	yes	maximal, characteristic
3	#11718754	C₂	1	3	yes	maximal
4	#11718752	C₂	1	3	yes	maximal
5	#11718692	C₂	1	3	yes	maximal
6	C₄×#6107	C₂	1	3	yes	maximal
7	Q₈×#1055	C₂	1	1	yes	maximal, characteristic
8	C₁₃×C₄⋊Q₈	—	2	3	no	—
9	C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
10	C₄×Q₈×C₁₃	—	2	1	no	—
11	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
12	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	Q₈×Dic₁₃	—	2	1	no	—
14	C₂×#6107	C₂²	1	3	yes	—
15	#88144	—	2	3	no	—
16	#88123	C₂²	1	3	yes	—
17	C₄×Dic₂₆	C₂²	1	3	yes	—
18	#88075	C₂²	1	1	yes	characteristic
19	#88050	C₂²	1	3	yes	—
20	#88050	C₂²	1	6	yes	—
21	#88047	C₂²	1	3	yes	—
22	C₄⋊#1055	C₂²	1	3	yes	—
23	C₄×#1055	C₂²	1	3	yes	—
24	C₂×Q₈×C₁₃	—	2	3	no	—
25	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₂₆	C₂³	1	3	yes	—
28	C₁₃×C₄⋊C₄	—	2	3	no	—
29	C₁₃×C₄⋊C₄	—	2	3	no	—
30	C₁₃×C₄⋊C₄	D₄	1	3	yes	—
31	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
32	C₄²×C₁₃	—	2	3	no	—
33	C₄²×C₁₃	C₂³	1	3	yes	—
34	#6107	—	2	6	no	—
35	C₄⋊Dic₁₃	—	2	3	no	—
36	C₄⋊Dic₁₃	—	2	3	no	—
37	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
38	#6101	—	2	3	no	—
39	C₄×Dic₁₃	—	2	3	no	—
40	C₂×#1055	C₂³	1	1	yes	characteristic
41	C₂×#1055	C₂³	1	3	yes	—
42	#3363	—	13	1	no	maximal, Sylow
43	Q₈×C₁₃	—	2	1	no	—
44	Q₈×C₁₃	—	8	3	no	—
45	Q₈×C₁₃	Q₁₆	1	4	yes	complemented
46	Q₈×C₁₃	#41	1	2	yes	—
47	C₂×C₄×C₁₃	—	2	1	no	—
48	C₂×C₄×C₁₃	—	2	3	no	—
49	C₂×C₄×C₁₃	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₁₃	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₁₃	—	2	1	no	—
53	C₂×Dic₁₃	—	2	3	no	—
54	Dic₂₆	—	2	3	no	—
55	Dic₂₆	—	2	3	no	—
56	#1055	—	2	3	no	—
57	#1055	—	2	1	no	—
58	Q₈²	D₁₃	1	1	yes	characteristic
59	#557	—	13	1	no	—
60	#487	—	13	3	no	—
61	#477	—	13	3	no	—
62	#462	—	13	3	no	—
63	C₈×Q₈	—	13	1	no	—
64	C₄×Q₁₆	—	13	3	no	—
65	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	—	2	3	no	—
68	C₄×C₁₃	—	4	3	no	—
69	C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
70	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
71	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
72	Dic₁₃	—	4	3	no	—
73	Dic₁₃	—	4	1	no	—
74	C₂×Q₁₆	—	13	3	no	—
75	C₄⋊Q₈	—	2	3	no	—
76	C₄⋊Q₈	—	13	3	no	—
77	C₄⋊Q₈	C₂×D₁₃	1	3	yes	—
78	#125	—	26	3	no	—
79	C₄×Q₈	—	2	1	no	—
80	C₄×Q₈	—	13	3	no	—
81	C₄×Q₈	—	26	1	no	—
82	C₄×Q₈	C₂×D₁₃	1	3	yes	—
83	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
84	#107	—	13	3	no	—
85	C₄⋊C₈	—	13	3	no	—
86	#103	—	13	1	no	—
87	#103	—	13	3	no	—
88	#103	—	13	6	no	—
89	C₄×C₈	—	13	3	no	—
90	C₂×C₁₃	#548	1	1	yes	characteristic
91	C₂×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
92	C₂×C₁₃	#577	1	1	yes	characteristic
93	C₂×Q₈	—	2	3	no	—
94	C₂×Q₈	—	13	3	no	—
95	C₂×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
96	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
97	Q₁₆	—	26	6	no	—
98	C₂×C₈	—	13	1	no	—
99	C₂×C₈	—	13	3	no	—
100	C₄⋊C₄	—	2	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	13	1	no	—
103	C₄⋊C₄	—	26	3	no	—
104	C₄⋊C₄	—	26	3	no	—
105	C₄⋊C₄	—	26	3	no	—
106	C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
107	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
108	C₄²	—	2	3	no	—
109	C₄²	—	26	3	no	—
110	C₄²	C₂²×D₁₃	1	3	yes	—
111	C₁₃	#3363	1	1	yes	characteristic, complemented, Sylow
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	26	3	no	—
115	Q₈	—	26	3	no	—
116	Q₈	#6107	1	4	yes	complemented
117	Q₈	#6131	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	26	1	no	—
121	C₂×C₄	—	26	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
124	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
125	C₈	—	26	3	no	—
126	C₈	—	26	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	52	3	no	—
132	C₄	—	52	1	no	—
133	C₄	C₂×#6107	1	6	yes	—
134	C₄	C₂×#6131	1	1	yes	characteristic
135	C₄	#88254	1	1	yes	characteristic
136	C₂	C₂²×#6107	1	1	yes	characteristic
137	C₂	#11719522	1	1	yes	characteristic
138	C₂	#11719541	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₁₃

C₂

yes

maximal, characteristic

#11719275

C₂

yes

maximal, characteristic

#11718754

C₂

yes

maximal

#11718752

C₂

yes

maximal

#11718692

C₂

yes

maximal

C₄×#6107

C₂

yes

maximal

Q₈×#1055

C₂

yes

maximal, characteristic

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₄×Q₈×C₁₃

—

C₄×Q₈×C₁₃

C₂²

yes

—

C₄×Q₈×C₁₃

C₂²

yes

characteristic

Q₈×Dic₁₃

—

C₂×#6107

C₂²

yes

—

#88144

—

#88123

C₂²

yes

—

C₄×Dic₂₆

C₂²

yes

—

#88075

C₂²

yes

characteristic

#88050

C₂²

yes

—

#88050

C₂²

yes

—

#88047

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

D₄

yes

characteristic

C₂×Q₈×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

D₄

yes

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂³

yes

—

#6107

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

characteristic

#6101

—

C₄×Dic₁₃

—

C₂×#1055

C₂³

yes

characteristic

C₂×#1055

C₂³

yes

—

#3363

—

maximal, Sylow

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

Q₁₆

yes

complemented

Q₈×C₁₃

#41

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

#1055

—

#1055

—

Q₈²

D₁₃

yes

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×Q₁₆

yes

—

C₄×C₁₃

C₂×#41

yes

characteristic

C₄×C₁₃

2₋¹⁺⁴

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₃

yes

—

C₄×Q₈

C₂×D₁₃

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₂×C₁₃

#548

yes

characteristic

C₂×C₁₃

C₂²×Q₁₆

yes

characteristic

C₂×C₁₃

#577

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₃

yes

characteristic

C₂×Q₈

C₂²×D₁₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂²⋊D₁₃

yes

—

107

C₄⋊C₄

C₂²×D₁₃

yes

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂²×D₁₃

yes

—

111

C₁₃

#3363

yes

characteristic, complemented, Sylow

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

#6107

yes

complemented

117

Q₈

#6131

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

124

C₂×C₄

C₂³×D₁₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

C₂×#6107

yes

—

134

C₄

C₂×#6131

yes

characteristic

135

C₄

#88254

yes

characteristic

136

C₂

C₂²×#6107

yes

characteristic

137

C₂

#11719522

yes

characteristic

138

C₂

#11719541

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	Q₈²×C₁₃	C₂
socle, prime radical	C₂²×C₁₃	C₂²×D₄
hypercenter	Q₈²	D₁₃
hypocenter	C₁₃	#3363
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

Q₈²×C₁₃

C₂

socle, prime radical

C₂²×C₁₃

C₂²×D₄

hypercenter

Q₈²

D₁₃

hypocenter

C₁₃

#3363

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
45. Q₈×C₁₃	Q₁₆	semidirect
111. C₁₃	#3363	semidirect
116. Q₈	#6107	semidirect

H ≅ G/N

type

45. Q₈×C₁₃

Q₁₆

semidirect

111. C₁₃

#3363

semidirect

116. Q₈

#6107

semidirect

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

subgroup	index
Q₈²×C₁₃	2
#11719275	2
#11718754	2
#11718752	2
#11718692	2
C₄×#6107	2
Q₈×#1055	2
#3363	13

subgroup

index

Q₈²×C₁₃

Subgroups — #13127514

Conjugacy classes of subgroups

Complements

Maximal subgroups