Group #13093231 — subgroups

Subgroups — #13093231

Order 1472.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	#10625161	C₂	1	2	yes	maximal, complemented
2	C₂₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625122	C₂	1	2	yes	maximal, complemented
4	Q₁₆×D₂₃	C₂	1	2	yes	maximal, complemented
5	#10625086	C₂	1	4	yes	maximal, complemented
6	#10625083	C₂	1	2	yes	maximal, complemented
7	C₂×Dic₉₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625072	C₂	1	1	yes	maximal, characteristic, complemented
9	#41×C₂₃	C₂²	1	2	yes	—
10	Q₈×D₂₃	C₂²	1	4	yes	—
11	#67290	—	2	4	no	—
12	#67290	C₂²	1	4	yes	—
13	#67288	—	2	2	no	—
14	#67288	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₆	—	2	2	no	—
16	C₂×Dic₄₆	C₂²	1	2	yes	—
17	Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	SD₁₆×C₂₃	C₂²	1	2	yes	—
19	D₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	#67275	C₂²	1	4	yes	—
22	#67273	C₂²	1	4	yes	—
23	#67267	C₂²	1	1	yes	characteristic
24	Dic₉₂	C₂²	1	2	yes	—
25	Dic₉₂	C₂²	1	2	yes	—
26	#67262	C₂²	1	2	yes	—
27	C₈×D₂₃	C₂²	1	2	yes	—
28	Q₈×C₂₃	C₂³	1	2	yes	—
29	D₄×C₂₃	—	2	2	no	—
30	D₄×C₂₃	C₂³	1	2	yes	—
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²⋊D₂₃	—	2	4	no	—
34	C₂²⋊D₂₃	D₄	1	2	yes	—
35	C₂×Dic₂₃	—	2	2	no	—
36	C₂×Dic₂₃	—	2	4	no	—
37	D₉₂	D₄	1	1	yes	characteristic
38	C₄×D₂₃	—	2	4	no	—
39	C₄×D₂₃	C₂³	1	2	yes	—
40	Dic₄₆	—	2	2	no	—
41	Dic₄₆	—	2	4	no	—
42	Dic₄₆	D₄	1	1	yes	characteristic
43	Dic₄₆	C₂³	1	4	yes	—
44	C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃⋊C₈	C₂³	1	2	yes	—
47	C₂²×C₂₃	—	2	2	no	—
48	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×D₂₃	C₂×D₄	1	2	yes	—
50	C₄×C₂₃	—	2	2	no	—
51	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₄×C₂₃	C₂⁴	1	1	yes	characteristic
53	Dic₂₃	—	2	4	no	—
54	Dic₂₃	C₂×D₄	1	2	yes	—
55	#578	—	23	1	no	maximal, Sylow
56	C₂×C₂₃	—	2	1	no	—
57	C₂×C₂₃	—	4	2	no	—
58	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
59	D₂₃	—	2	2	no	—
60	2₋¹⁺⁴	—	23	2	no	—
61	#137	—	23	2	no	—
62	#137	—	23	4	no	—
63	#135	—	23	2	no	—
64	#135	D₂₃	1	1	yes	characteristic, complemented
65	C₂×Q₁₆	—	23	2	no	—
66	C₂×Q₁₆	—	23	1	no	—
67	#131	—	23	1	no	—
68	C₂₃	#578	1	1	yes	characteristic, complemented, Sylow
69	#41	—	23	1	no	—
70	#41	—	23	4	no	—
71	#41	—	46	2	no	—
72	#41	—	46	4	no	—
73	#41	C₂×D₂₃	1	2	yes	—
74	C₂×Q₈	—	23	4	no	—
75	C₂×Q₈	—	23	2	no	—
76	C₂×Q₈	—	46	2	no	—
77	Q₁₆	—	23	4	no	—
78	Q₁₆	—	23	2	no	—
79	Q₁₆	—	23	2	no	—
80	Q₁₆	C₂×D₂₃	1	1	yes	characteristic
81	SD₁₆	—	23	4	no	—
82	SD₁₆	C₂×D₂₃	1	2	yes	—
83	D₈	C₂×D₂₃	1	1	yes	characteristic
84	M₁₆	—	23	1	no	—
85	M₁₆	—	23	2	no	—
86	C₂×C₈	—	23	2	no	—
87	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
88	Q₈	—	23	1	no	—
89	Q₈	—	23	4	no	—
90	Q₈	—	46	4	no	—
91	Q₈	—	46	2	no	—
92	Q₈	C₂²×D₂₃	1	2	yes	—
93	D₄	—	2	2	no	—
94	D₄	—	23	2	no	—
95	D₄	—	23	1	no	—
96	D₄	—	46	4	no	—
97	D₄	C₂²×D₂₃	1	2	yes	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	23	2	no	—
100	C₂×C₄	—	46	4	no	—
101	C₂×C₄	—	46	4	no	—
102	C₂×C₄	—	46	2	no	—
103	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
104	C₈	—	23	2	no	—
105	C₈	C₂²×D₂₃	1	1	yes	characteristic
106	C₈	C₂²×D₂₃	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	23	2	no	—
109	C₂²	D₄×D₂₃	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	23	2	no	—
112	C₄	—	46	4	no	—
113	C₄	D₄×D₂₃	1	1	yes	characteristic
114	C₄	C₂³×D₂₃	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	4	2	no	—
117	C₂	—	46	2	no	—
118	C₂	C₂×D₄×D₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10625161

C₂

yes

maximal, complemented

C₂₃×#135

C₂

yes

maximal, characteristic, complemented

#10625122

C₂

yes

maximal, complemented

Q₁₆×D₂₃

C₂

yes

maximal, complemented

#10625086

C₂

yes

maximal, complemented

#10625083

C₂

yes

maximal, complemented

C₂×Dic₉₂

C₂

yes

maximal, characteristic, complemented

#10625072

C₂

yes

maximal, characteristic, complemented

#41×C₂₃

C₂²

yes

—

Q₈×D₂₃

C₂²

yes

—

#67290

—

#67290

C₂²

yes

—

#67288

—

#67288

C₂²

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂²

yes

—

Q₁₆×C₂₃

C₂²

yes

characteristic

SD₁₆×C₂₃

C₂²

yes

—

D₈×C₂₃

C₂²

yes

characteristic

C₂×C₈×C₂₃

C₂²

yes

characteristic

#67275

C₂²

yes

—

#67273

C₂²

yes

—

#67267

C₂²

yes

characteristic

Dic₉₂

C₂²

yes

—

Dic₉₂

C₂²

yes

—

#67262

C₂²

yes

—

C₈×D₂₃

C₂²

yes

—

Q₈×C₂₃

C₂³

yes

—

D₄×C₂₃

—

D₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂²⋊D₂₃

D₄

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

D₉₂

D₄

yes

characteristic

C₄×D₂₃

—

C₄×D₂₃

C₂³

yes

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

D₄

yes

characteristic

Dic₄₆

C₂³

yes

—

C₈×C₂₃

C₂³

yes

characteristic

C₈×C₂₃

C₂³

yes

characteristic

C₂₃⋊C₈

C₂³

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×D₄

yes

characteristic

C₂×D₂₃

C₂×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×D₄

yes

characteristic

C₄×C₂₃

C₂⁴

yes

characteristic

Dic₂₃

—

Dic₂₃

C₂×D₄

yes

—

#578

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₄

yes

characteristic

D₂₃

—

2₋¹⁺⁴

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×Q₁₆

—

#131

—

C₂₃

#578

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×D₂₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×D₂₃

yes

characteristic

SD₁₆

—

SD₁₆

C₂×D₂₃

yes

—

D₈

C₂×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×D₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

C₂²×D₂₃

yes

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

C₂²×D₂₃

yes

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×D₂₃

yes

characteristic

104

C₈

—

105

C₈

C₂²×D₂₃

yes

characteristic

106

C₈

C₂²×D₂₃

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

D₄×D₂₃

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

D₄×D₂₃

yes

characteristic

114

C₄

C₂³×D₂₃

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

118

C₂

C₂×D₄×D₂₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₂₃
commutator, cosocle	C₄×C₂₃	C₂⁴
Frattini	C₄	C₂³×D₂₃
Fitting	C₂₃×#135	C₂
socle	C₂×C₂₃	C₂²×D₄
hypercenter	#135	D₂₃
hypocenter	C₂₃	#578
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₂

C₂×D₄×D₂₃

commutator, cosocle

C₄×C₂₃

C₂⁴

Frattini

C₄

C₂³×D₂₃

Fitting

C₂₃×#135

C₂

socle

C₂×C₂₃

C₂²×D₄

D₂₃

C₂₃

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #10625161	C₂	semidirect
2. C₂₃×#135	C₂	semidirect
3. #10625122	C₂	semidirect
4. Q₁₆×D₂₃	C₂	semidirect
5. #10625086	C₂	semidirect
6. #10625083	C₂	semidirect
7. C₂×Dic₉₂	C₂	semidirect
8. #10625072	C₂	semidirect
64. #135	D₂₃	semidirect
68. C₂₃	#578	semidirect

H ≅ G/N

type

1. #10625161

C₂

semidirect

2. C₂₃×#135

C₂

semidirect

3. #10625122

C₂

semidirect

4. Q₁₆×D₂₃

C₂

semidirect

5. #10625086

C₂

semidirect

6. #10625083

C₂

semidirect

7. C₂×Dic₉₂

C₂

semidirect

8. #10625072

C₂

semidirect

64. #135

D₂₃

semidirect

68. C₂₃

#578

semidirect

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

subgroup	index
#10625161	2
C₂₃×#135	2
#10625122	2
Q₁₆×D₂₃	2
#10625086	2
#10625083	2
C₂×Dic₉₂	2
#10625072	2
#578	23

subgroup

index

C₂₃×#135

Q₁₆×D₂₃

C₂×Dic₉₂

Subgroups — #13093231

Conjugacy classes of subgroups

Complements

Maximal subgroups