Group #15674391 — subgroups

Subgroups — #15674391

Order 2944.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13092441	C₂	1	1	yes	maximal, characteristic
2	C₄×#67267	C₂	1	1	yes	maximal, characteristic
3	#13092186	C₂	1	4	yes	maximal
4	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#125	—	2	1	no	—
6	C₂₃×#125	—	2	1	no	—
7	C₂₃×#125	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×#67267	C₂²	1	2	yes	—
12	C₂×#67267	C₂²	1	2	yes	—
13	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
14	#10624985	—	2	1	no	—
15	C₄×C₂₃⋊C₈	—	2	1	no	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₂₃	—	2	4	no	—
20	C₄⋊C₄×C₂₃	—	2	4	no	—
21	C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄²×C₂₃	D₄	1	1	yes	characteristic
24	C₄²×C₂₃	D₄	1	1	yes	characteristic
25	C₄²×C₂₃	D₄	1	2	yes	—
26	#67267	—	2	2	no	—
27	#67267	—	2	4	no	—
28	#67267	Q₈	1	4	yes	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	—	2	4	no	—
33	C₂×C₄×C₂₃	—	4	2	no	—
34	C₂×C₄×C₂₃	—	4	4	no	—
35	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₂₃	#41	1	2	yes	—
41	C₂₃⋊C₈	—	2	4	no	—
42	C₂₃⋊C₈	—	4	2	no	—
43	#1998	—	23	1	no	maximal, Sylow
44	C₂²×C₂₃	—	2	2	no	—
45	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
47	C₄×C₂₃	—	2	2	no	—
48	C₄×C₂₃	—	2	2	no	—
49	C₄×C₂₃	—	4	4	no	—
50	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
51	C₄×C₂₃	#124	1	1	yes	characteristic
52	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×#125	D₂₃	1	1	yes	characteristic
54	#423	—	23	1	no	—
55	C₄×M₁₆	—	23	1	no	—
56	#343	—	23	4	no	—
57	C₂×C₂₃	—	2	2	no	—
58	C₂×C₂₃	#391	1	1	yes	characteristic
59	C₂×C₂₃	#413	1	2	yes	—
60	C₂×M₁₆	—	23	2	no	—
61	C₂×M₁₆	—	23	2	no	—
62	#125	—	2	1	no	—
63	#125	—	2	1	no	—
64	#125	—	2	2	no	—
65	C₂×C₄⋊C₄	Dic₂₃	1	4	yes	—
66	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
67	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
68	C₄⋊C₈	—	46	2	no	—
69	#97	—	46	1	no	—
70	C₄×C₈	—	46	1	no	—
71	C₂₃	#1998	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
73	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
75	M₁₆	—	23	4	no	—
76	M₁₆	—	46	2	no	—
77	M₁₆	—	46	4	no	—
78	C₂×C₈	—	46	2	no	—
79	C₂×C₈	—	46	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
85	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
86	C₄²	C₂²⋊D₂₃	1	2	yes	—
87	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
93	C₂×C₄	Q₈×D₂₃	1	2	yes	—
94	C₂×C₄	#67292	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
97	C₈	—	46	4	no	—
98	C₈	—	92	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	Q₈×Dic₂₃	1	2	yes	—
101	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#10625115	1	1	yes	characteristic
106	C₄	#10625117	1	2	yes	—
107	C₄	#10625118	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#13092324	1	1	yes	characteristic
110	C₂	#13092401	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092441

C₂

yes

maximal, characteristic

C₄×#67267

C₂

yes

maximal, characteristic

#13092186

C₂

yes

maximal

C₂×C₂₃×#125

C₂

yes

maximal, characteristic

C₂₃×#125

—

C₂₃×#125

—

C₂₃×#125

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

#67267

—

#67267

—

#67267

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1998

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×#125

D₂₃

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₂×C₂₃

—

C₂×C₂₃

#391

yes

characteristic

C₂×C₂₃

#413

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₂₃

yes

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂₃

#1998

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂²×Dic₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

Q₈×D₂₃

yes

—

C₂×C₄

#67292

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

C₈

—

C₂²

—

100

C₂²

Q₈×Dic₂₃

yes

—

101

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#10625115

yes

characteristic

106

C₄

#10625117

yes

—

107

C₄

#10625118

yes

characteristic

108

C₂

—

109

C₂

#13092324

yes

characteristic

110

C₂

#13092401

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator, prime radical	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#125	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	#1998
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator, prime radical

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#125

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
71. C₂₃	#1998	semidirect

H ≅ G/N

type

71. C₂₃

#1998

semidirect

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

subgroup	index
#13092441	2
C₄×#67267	2
#13092186	2
C₂×C₂₃×#125	2
#1998	23

subgroup

index

C₂×C₂₃×#125

Subgroups — #15674391

Conjugacy classes of subgroups

Complements

Maximal subgroups