Group #13125216 — subgroups

Subgroups — #13125216

Order 1664.

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

normal subgroups only

This group has 380 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	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11719175	C₂	1	1	yes	maximal, characteristic
3	#11718886	C₂	1	4	yes	maximal
4	#11718743	C₂	1	4	yes	maximal, complemented
5	#11718742	C₂	1	4	yes	maximal, complemented
6	C₄×#6099	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₁₃×C₄⋊Q₈	—	2	2	no	—
9	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	#88180	C₂²	1	2	yes	—
14	#88178	—	2	2	no	—
15	C₂×#6099	C₂²	1	2	yes	—
16	#88124	C₂²	1	2	yes	—
17	#88123	C₂²	1	2	yes	—
18	C₄×Dic₂₆	—	2	2	no	—
19	#88075	C₂²	1	16	yes	—
20	#88043	C₂²	1	2	yes	—
21	C₄×#1055	C₂²	1	2	yes	—
22	C₂×Q₈×C₁₃	—	2	2	no	—
23	C₂×Q₈×C₁₃	—	2	8	no	—
24	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₁₃	—	2	2	no	—
26	C₂²×C₄×C₁₃	D₄	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₂₆	—	2	2	no	—
29	C₁₃×C₄⋊C₄	—	2	4	no	—
30	C₁₃×C₄⋊C₄	—	2	4	no	—
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	2	no	—
35	C₄⋊Dic₁₃	—	2	2	no	—
36	C₄⋊Dic₁₃	C₂³	1	4	yes	—
37	#6101	—	2	4	no	—
38	C₄×Dic₁₃	—	2	2	no	—
39	#6099	—	2	4	no	—
40	C₂×#1055	C₂³	1	4	yes	—
41	#3139	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₃	—	2	8	no	—
44	Q₈×C₁₃	—	4	4	no	—
45	C₂×C₄×C₁₃	—	2	4	no	—
46	C₂×C₄×C₁₃	—	2	4	no	—
47	C₂×C₄×C₁₃	—	4	4	no	—
48	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃	#41	1	4	yes	—
52	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₁₃	—	2	4	no	—
54	Dic₂₆	—	4	4	no	—
55	#1055	—	2	4	no	—
56	#533	—	13	1	no	—
57	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
58	#491	—	13	4	no	—
59	#487	—	13	4	no	—
60	#419	—	13	4	no	—
61	C₄×M₁₆	—	13	1	no	—
62	C₂²×C₁₃	—	2	2	no	—
63	C₂²×C₁₃	#124	1	2	yes	—
64	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₃	—	2	2	no	—
66	C₄×C₁₃	—	4	4	no	—
67	C₄×C₁₃	#124	1	2	yes	—
68	C₄×C₁₃	#137	1	4	yes	—
69	C₄×C₁₃	C₂×#41	1	2	yes	—
70	Dic₁₃	—	4	4	no	—
71	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
72	C₂×M₁₆	—	13	2	no	—
73	C₄⋊Q₈	—	2	2	no	—
74	C₄⋊Q₈	—	13	2	no	—
75	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
76	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
77	#125	—	13	2	no	—
78	C₂²⋊Q₈	—	26	2	no	—
79	C₄×Q₈	—	26	2	no	—
80	#117	—	13	2	no	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
83	#103	—	13	16	no	—
84	#97	—	13	2	no	—
85	C₄×C₈	—	13	2	no	—
86	C₂×C₁₃	—	2	2	no	—
87	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
88	C₂×C₁₃	C₂×#137	1	2	yes	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	8	no	—
91	C₂×Q₈	—	26	2	no	—
92	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
96	M₁₆	—	26	4	no	—
97	C₂×C₈	—	13	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	13	4	no	—
101	C₄⋊C₄	—	26	4	no	—
102	C₄⋊C₄	—	26	2	no	—
103	C₂²⋊C₄	—	26	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	#3139	1	1	yes	characteristic, complemented, Sylow
109	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	52	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	26	4	no	—
117	C₂×C₄	#6129	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
121	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
122	C₈	—	26	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#88190	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	52	4	no	—
129	C₄	#88190	1	2	yes	—
130	C₄	#88196	1	4	yes	—
131	C₄	C₂×#6129	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#88190	1	1	yes	characteristic
134	C₂	C₂×#88196	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

#11719175

C₂

yes

maximal, characteristic

#11718886

C₂

yes

maximal

#11718743

C₂

yes

maximal, complemented

#11718742

C₂

yes

maximal, complemented

C₄×#6099

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88180

C₂²

yes

—

#88178

—

C₂×#6099

C₂²

yes

—

#88124

C₂²

yes

—

#88123

C₂²

yes

—

C₄×Dic₂₆

—

#88075

C₂²

yes

—

#88043

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

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

—

#6099

—

C₂×#1055

C₂³

yes

—

#3139

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

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

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

#1055

—

#533

—

C₂×C₄⋊Q₈

D₁₃

yes

—

—

—

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

#124

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

C₄×C₁₃

#137

yes

—

C₄×C₁₃

C₂×#41

yes

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃

yes

—

C₄⋊Q₈

C₂×D₁₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#124

yes

characteristic

C₂×C₁₃

C₂×#137

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

C₂²⋊D₁₃

yes

—

106

C₄²

C₂²×D₁₃

yes

characteristic

107

C₄²

C₂²×D₁₃

yes

characteristic

108

C₁₃

#3139

yes

characteristic, complemented, Sylow

109

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#6129

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

121

C₂×C₄

C₂³×D₁₃

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

#88190

yes

—

125

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#88190

yes

—

130

C₄

#88196

yes

—

131

C₄

C₂×#6129

yes

—

132

C₂

—

133

C₂

C₂×#88190

yes

characteristic

134

C₂

C₂×#88196

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#3139
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

socle

C₂²×C₁₃

C₂²×D₄

prime radical

C₂³×C₁₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₃

hypocenter

C₁₃

#3139

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
4. #11718743	C₂	semidirect
5. #11718742	C₂	semidirect
108. C₁₃	#3139	semidirect

H ≅ G/N

type

4. #11718743

C₂

semidirect

5. #11718742

C₂

semidirect

108. C₁₃

#3139

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
#11719175	2
#11718886	2
#11718743	2
#11718742	2
C₄×#6099	2
#3139	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

Subgroups — #13125216

Conjugacy classes of subgroups

Complements

Maximal subgroups