Group #15674366 — subgroups

Subgroups — #15674366

Order 2944.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#10625041	C₂	1	1	yes	maximal, characteristic
2	#13092287	C₂	1	1	yes	maximal, characteristic
3	#13092180	C₂	1	4	yes	maximal
4	C₂×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₃	—	2	2	no	—
8	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
10	C₂×Dic₂₃⋊C₄	—	2	2	no	—
11	#10625041	—	2	2	no	—
12	#10625041	C₄	1	4	yes	complemented
13	C₂₃×#97	—	2	2	no	—
14	C₂²×C₄×C₂₃	D₄	1	2	yes	—
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₂₃	—	2	2	no	—
17	C₂²×Dic₂₃	—	2	2	no	—
18	C₂×C₈×C₂₃	—	2	2	no	—
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	Q₈	1	4	yes	—
21	C₄²×C₂₃	C₂×C₄	1	2	yes	—
22	C₄²×C₂₃	D₄	1	2	yes	—
23	C₄⋊Dic₂₃	—	2	2	no	—
24	C₄⋊Dic₂₃	—	2	4	no	—
25	C₄⋊Dic₂₃	—	4	4	no	—
26	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
27	Dic₂₃⋊C₄	—	2	8	no	—
28	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃	—	2	2	no	—
30	C₂×C₄×C₂₃	—	2	2	no	—
31	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₂₃	#41	1	2	yes	—
36	C₂×Dic₂₃	—	2	4	no	—
37	C₂×Dic₂₃	—	4	2	no	—
38	C₂×Dic₂₃	—	4	2	no	—
39	C₂×Dic₂₃	—	4	2	no	—
40	C₂×Dic₂₃	—	4	4	no	—
41	C₈×C₂₃	—	2	4	no	—
42	#1992	—	23	1	no	maximal, Sylow
43	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₂₃	#124	1	1	yes	characteristic
45	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
46	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₂²×C₂₃	#137	1	2	yes	—
48	C₄×C₂₃	—	2	4	no	—
49	C₄×C₂₃	C₄×Q₈	1	2	yes	—
50	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
51	Dic₂₃	—	4	4	no	—
52	Dic₂₃	—	8	2	no	—
53	C₂×#125	—	23	1	no	—
54	C₂×#97	D₂₃	1	1	yes	characteristic
55	#384	—	23	1	no	—
56	#340	—	23	4	no	—
57	C₂×C₂₃	#391	1	1	yes	characteristic
58	C₂×C₂₃	#417	1	2	yes	—
59	C₂×C₂₃	#490	1	1	yes	characteristic
60	C₂×C₂₃	#491	1	1	yes	characteristic
61	C₂×C₂₃	#501	1	2	yes	—
62	C₂²×C₈	C₂×D₂₃	1	2	yes	—
63	#125	—	23	4	no	—
64	#125	—	46	2	no	—
65	C₂×C₄⋊C₄	—	23	2	no	—
66	C₂×C₄⋊C₄	—	23	2	no	—
67	C₂×C₄⋊C₄	—	46	2	no	—
68	C₂×C₄⋊C₄	—	46	2	no	—
69	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
70	#97	—	2	2	no	—
71	C₂₃	#1992	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₄	—	46	2	no	—
73	C₂²×C₄	—	46	2	no	—
74	C₂²×C₄	D₉₂	1	2	yes	—
75	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₄₆	1	4	yes	—
79	C₄⋊C₄	—	23	4	no	—
80	C₄⋊C₄	—	46	8	no	—
81	C₄⋊C₄	—	46	2	no	—
82	C₄⋊C₄	—	46	4	no	—
83	C₄⋊C₄	—	92	4	no	—
84	C₄²	C₄×D₂₃	1	2	yes	—
85	C₄²	C₂²⋊D₂₃	1	2	yes	—
86	C₂³	C₂×D₉₂	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	46	4	no	—
90	C₂×C₄	—	92	2	no	—
91	C₂×C₄	—	92	2	no	—
92	C₂×C₄	—	92	2	no	—
93	C₂×C₄	—	92	4	no	—
94	C₂×C₄	(C₂×D₂₃)⋊C₄	1	4	yes	—
95	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
96	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
97	C₂×C₄	#67288	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₂²	#10625040	1	1	yes	characteristic
101	C₂²	#10625046	1	1	yes	characteristic
102	C₂²	#10625079	1	2	yes	—
103	C₂²	#10625080	1	2	yes	—
104	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
105	C₄	—	2	4	no	—
106	C₄	—	92	4	no	—
107	C₄	—	184	2	no	—
108	C₄	C₄×Dic₄₆	1	2	yes	—
109	C₄	#10625095	1	2	yes	—
110	C₂	#13092323	1	1	yes	characteristic
111	C₂	#13092419	1	2	yes	—
112	C₂	#13092782	1	1	yes	characteristic
113	C₂	#13092786	1	1	yes	characteristic
114	C₂	#13092826	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625041

C₂

yes

maximal, characteristic

#13092287

C₂

yes

maximal, characteristic

#13092180

C₂

yes

maximal

C₂×C₂₃×#97

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

#10625041

—

#10625041

C₄

yes

complemented

C₂₃×#97

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

Q₈

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

#1992

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#124

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×#125

—

C₂×#97

D₂₃

yes

characteristic

#384

—

#340

—

C₂×C₂₃

#391

yes

characteristic

C₂×C₂₃

#417

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂²×C₈

C₂×D₂₃

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#97

—

C₂₃

#1992

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×D₉₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

C₂×C₄

C₂×Dic₄₆

yes

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

#67288

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

100

C₂²

#10625040

yes

characteristic

101

C₂²

#10625046

yes

characteristic

102

C₂²

#10625079

yes

—

103

C₂²

#10625080

yes

—

104

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

184

—

108

C₄

C₄×Dic₄₆

yes

—

109

C₄

#10625095

yes

—

110

C₂

#13092323

yes

characteristic

111

C₂

#13092419

yes

—

112

C₂

#13092782

yes

characteristic

113

C₂

#13092786

yes

characteristic

114

C₂

#13092826

yes

—

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#97	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#97	D₂₃
hypocenter	C₂₃	#1992
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#97

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×#97

D₂₃

hypocenter

C₂₃

#1992

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. #10625041	C₄	semidirect
71. C₂₃	#1992	semidirect

H ≅ G/N

type

12. #10625041

C₄

semidirect

71. C₂₃

#1992

semidirect

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

subgroup	index
C₂×#10625041	2
#13092287	2
#13092180	2
C₂×C₂₃×#97	2
#1992	23

subgroup

index

C₂×C₂₃×#97

Subgroups — #15674366

Conjugacy classes of subgroups

Complements

Maximal subgroups