Group #15671233 — subgroups

Subgroups — #15671233

Order 2944.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄×#67267	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
5	C₂×#67267	C₄	1	2	yes	—
6	C₂²×C₂₃⋊C₈	—	2	1	no	—
7	C₂₃×C₄⋊C₈	C₄	1	2	yes	—
8	C₂₃×C₂²⋊C₈	—	2	1	no	—
9	C₄×C₈×C₂₃	—	2	1	no	—
10	#10624985	—	2	1	no	—
11	C₄×C₂₃⋊C₈	—	2	1	no	—
12	C₄×C₂₃⋊C₈	—	2	1	no	—
13	C₄×C₂₃⋊C₈	C₄	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₂₃	—	2	2	no	—
18	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₂₃	D₄	1	1	yes	characteristic
21	C₄²×C₂₃	D₄	1	1	yes	characteristic
22	#67267	—	2	2	no	—
23	#67267	C₈	1	4	yes	complemented
24	C₂×C₂₃⋊C₈	—	2	4	no	—
25	C₂×C₂₃⋊C₈	—	2	2	no	—
26	C₂×C₂₃⋊C₈	—	2	2	no	—
27	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	—	2	2	no	—
29	C₂×C₄×C₂₃	—	2	2	no	—
30	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
38	C₈×C₂₃	—	4	2	no	—
39	C₂₃⋊C₈	—	2	4	no	—
40	C₂₃⋊C₈	—	2	4	no	—
41	#1278	—	23	1	no	maximal, Sylow
42	C₂²×C₂₃	—	2	1	no	—
43	C₂²×C₂₃	—	2	1	no	—
44	C₂²×C₂₃	#95	1	1	yes	characteristic
45	C₂²×C₂₃	C₄⋊C₈	1	2	yes	—
46	C₄×C₂₃	—	2	1	no	—
47	C₄×C₂₃	—	2	1	no	—
48	C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
49	C₄×C₂₃	#97	1	1	yes	characteristic
50	C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
51	C₄×C₂₃	#104	1	2	yes	—
52	C₄×C₂₃	#108	1	2	yes	—
53	#431	D₂₃	1	1	yes	characteristic
54	C₄×M₁₆	—	23	1	no	—
55	C₂×C₄×C₈	—	23	1	no	—
56	C₂×C₂₃	—	2	2	no	—
57	C₂×C₂₃	#336	1	1	yes	characteristic
58	C₂×C₂₃	#339	1	1	yes	characteristic
59	C₂×C₂₃	#341	1	1	yes	characteristic
60	C₂×M₁₆	—	23	2	no	—
61	C₂²×C₈	—	46	1	no	—
62	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
63	C₄⋊C₈	Dic₂₃	1	2	yes	—
64	C₂²⋊C₈	—	2	1	no	—
65	#97	—	46	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	23	2	no	—
68	C₄×C₈	—	46	1	no	—
69	C₄×C₈	—	46	1	no	—
70	C₂₃	#1278	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
74	M₁₆	—	23	4	no	—
75	M₁₆	—	46	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	46	4	no	—
78	C₂×C₈	—	46	2	no	—
79	C₂×C₈	—	46	2	no	—
80	C₄²	C₄×D₂₃	1	1	yes	characteristic
81	C₄²	D₉₂	1	1	yes	characteristic
82	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
83	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
84	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₈×D₂₃	1	2	yes	—
88	C₂×C₄	#67262	1	2	yes	—
89	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
90	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
92	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
93	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	46	4	no	—
97	C₈	—	46	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	Dic₂₃⋊C₈	1	2	yes	—
101	C₂²	C₂₃⋊#95	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₂₃	1	1	yes	characteristic
105	C₄	#10624997	1	1	yes	characteristic
106	C₄	(C₂×D₂₃)⋊C₈	1	2	yes	—
107	C₄	#10625004	1	2	yes	—
108	C₄	#10625019	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#13092171	1	1	yes	characteristic
111	C₂	#13092176	1	1	yes	characteristic
112	C₂	#13092181	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#67267

C₂

yes

maximal, characteristic

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

C₂₃×#431

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₄

yes

—

C₂²×C₂₃⋊C₈

—

C₂₃×C₄⋊C₈

C₄

yes

—

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

—

#10624985

—

C₄×C₂₃⋊C₈

—

C₄×C₂₃⋊C₈

—

C₄×C₂₃⋊C₈

C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

Q₈

yes

characteristic

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

#67267

—

#67267

C₈

yes

complemented

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1278

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#95

yes

characteristic

C₂²×C₂₃

C₄⋊C₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×C₈

yes

characteristic

C₄×C₂₃

#97

yes

characteristic

C₄×C₂₃

C₂²⋊C₈

yes

—

C₄×C₂₃

#104

yes

—

C₄×C₂₃

#108

yes

—

#431

D₂₃

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

Dic₂₃

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#1278

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₂₃

yes

characteristic

C₄²

D₉₂

yes

characteristic

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₂³

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₂₃

yes

—

C₂×C₄

#67262

yes

—

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₂₃

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

Dic₂₃⋊C₈

yes

—

101

C₂²

C₂₃⋊#95

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₂₃

yes

characteristic

105

C₄

#10624997

yes

characteristic

106

C₄

(C₂×D₂₃)⋊C₈

yes

—

107

C₄

#10625004

yes

—

108

C₄

#10625019

yes

—

109

C₂

—

110

C₂

#13092171

yes

characteristic

111

C₂

#13092176

yes

characteristic

112

C₂

#13092181

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₂₃
commutator	C₄×C₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₃
Fitting	C₂₃×#431	C₂
socle	C₂²×C₂₃	#95
prime radical	C₂³×C₂₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₃	C₂²
hypercenter	#431	D₂₃
hypocenter	C₂₃	#1278
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₂₃

commutator

C₄×C₂₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₂₃

Fitting

C₂₃×#431

C₂

socle

C₂²×C₂₃

#95

prime radical

C₂³×C₂₃

C₄⋊C₄

cosocle

C₂×C₄²×C₂₃

C₂²

D₂₃

C₂₃

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
23. #67267	C₈	semidirect
70. C₂₃	#1278	semidirect

H ≅ G/N

type

23. #67267

C₈

semidirect

70. C₂₃

#1278

semidirect

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

subgroup	index
C₄×#67267	2
C₂×C₄×C₂₃⋊C₈	2
C₂₃×#431	2
#1278	23

subgroup

index

C₄×#67267

C₂×C₄×C₂₃⋊C₈

C₂₃×#431

#1278

Subgroups — #15671233

Conjugacy classes of subgroups

Complements

Maximal subgroups