Group #15673151 — subgroups

Subgroups — #15673151

Order 2944.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13092375	C₂	1	1	yes	maximal, characteristic
2	#13092373	C₂	1	2	yes	maximal
3	C₂₃⋊#338	C₂	1	1	yes	maximal, characteristic
4	#13092173	C₂	1	2	yes	maximal
5	C₂₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#117	C₂²	1	2	yes	—
8	#10625097	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	#10624997	C₄	1	2	yes	complemented
12	C₈×Dic₂₃	C₄	1	2	yes	complemented
13	#10624985	C₄	1	4	yes	complemented
14	C₄×C₂₃⋊C₈	C₄	1	4	yes	complemented
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	M₁₆×C₂₃	—	2	2	no	—
17	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₄⋊C₄×C₂₃	—	2	2	no	—
19	C₂²⋊C₄×C₂₃	—	2	2	no	—
20	C₄²×C₂₃	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₂₃	—	2	1	no	—
22	C₄⋊Dic₂₃	—	2	1	no	—
23	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
24	#67267	—	2	2	no	—
25	#67267	—	2	2	no	—
26	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₂₃	#41	1	1	yes	characteristic
32	C₂×C₄×C₂₃	—	2	4	no	—
33	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	#41	1	2	yes	—
36	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
37	C₂×Dic₂₃	—	2	2	no	—
38	C₈×C₂₃	—	2	2	no	—
39	C₂₃⋊C₈	—	2	4	no	—
40	C₂₃⋊C₈	C₄²	1	1	yes	characteristic
41	C₂₃⋊C₈	C₄²	1	2	yes	—
42	C₂₃⋊C₈	C₄²	1	1	yes	characteristic
43	#1747	—	23	1	no	maximal, Sylow
44	C₂²×C₂₃	—	4	1	no	—
45	C₂²×C₂₃	#117	1	2	yes	—
46	C₂²×C₂₃	#117	1	1	yes	characteristic
47	C₄×C₂₃	—	4	4	no	—
48	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
49	C₄×C₂₃	#117	1	1	yes	characteristic
50	C₄×C₂₃	#117	1	2	yes	—
51	Dic₂₃	—	4	2	no	—
52	#405	—	23	1	no	—
53	#405	—	23	2	no	—
54	#338	—	23	1	no	—
55	H(3,ℤ/4)	—	23	2	no	—
56	H(3,ℤ/4)	D₂₃	1	1	yes	characteristic
57	C₂×C₂₃	—	2	1	no	—
58	C₂×C₂₃	—	2	2	no	—
59	C₂×C₂₃	#376	1	1	yes	characteristic
60	C₂×M₁₆	—	23	2	no	—
61	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
62	C₂²×C₈	—	23	1	no	—
63	#117	—	23	1	no	—
64	#117	C₂×D₂₃	1	2	yes	—
65	#97	—	23	2	no	—
66	#97	—	23	4	no	—
67	C₄×C₈	—	23	2	no	—
68	C₄×C₈	—	23	4	no	—
69	C₂₃	#1747	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	46	2	no	—
73	M₁₆	—	46	2	no	—
74	C₂×C₈	—	23	1	no	—
75	C₂×C₈	—	23	2	no	—
76	C₂×C₈	—	23	1	no	—
77	C₂×C₈	—	23	2	no	—
78	C₂×C₈	—	23	4	no	—
79	C₂×C₈	C₄×D₂₃	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	46	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	46	1	no	—
84	C₄²	—	23	2	no	—
85	C₄²	C₄×D₂₃	1	4	yes	—
86	C₂³	#67290	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	46	2	no	—
89	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
91	C₂×C₄	#67290	1	2	yes	—
92	C₂×C₄	#67292	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	23	1	no	—
95	C₈	—	23	1	no	—
96	C₈	—	23	2	no	—
97	C₈	—	46	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#10625048	1	2	yes	—
100	C₂²	#10625063	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	92	2	no	—
103	C₄	C₄²×D₂₃	1	1	yes	characteristic
104	C₄	#10625048	1	1	yes	characteristic
105	C₄	#10625063	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#13092262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092375

C₂

yes

maximal, characteristic

#13092373

C₂

yes

maximal

C₂₃⋊#338

C₂

yes

maximal, characteristic

#13092173

C₂

yes

maximal

C₂₃×H(3,ℤ/4)

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

#10625097

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10624997

C₄

yes

complemented

C₈×Dic₂₃

C₄

yes

complemented

#10624985

C₄

yes

complemented

C₄×C₂₃⋊C₈

C₄

yes

complemented

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₄²

yes

characteristic

C₂₃⋊C₈

C₄²

yes

—

C₂₃⋊C₈

C₄²

yes

characteristic

#1747

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

#117

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄²

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#376

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×D₂₃

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#1747

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₂₃

yes

—

C₂³

#67290

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃

yes

—

C₂×C₄

#67290

yes

—

C₂×C₄

#67292

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#10625048

yes

—

100

C₂²

#10625063

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

C₄²×D₂₃

yes

characteristic

104

C₄

#10625048

yes

characteristic

105

C₄

#10625063

yes

—

106

C₂

—

107

C₂

—

108

C₂

#13092262

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₂₃
commutator	C₄×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×H(3,ℤ/4)	C₂
socle	C₂×C₂₃	#376
prime radical	C₂³×C₂₃	#41
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	H(3,ℤ/4)	D₂₃
hypocenter	C₂₃	#1747
LCS	G ▷ C₄×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₂₃

commutator

C₄×C₂₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×H(3,ℤ/4)

C₂

C₂×C₂₃

C₂³×C₂₃

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₄×C₂₃ ▷ C₂₃

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
11. #10624997	C₄	semidirect
12. C₈×Dic₂₃	C₄	semidirect
13. #10624985	C₄	semidirect
14. C₄×C₂₃⋊C₈	C₄	semidirect
69. C₂₃	#1747	semidirect

H ≅ G/N

type

11. #10624997

C₄

semidirect

12. C₈×Dic₂₃

C₄

semidirect

13. #10624985

C₄

semidirect

14. C₄×C₂₃⋊C₈

C₄

semidirect

69. C₂₃

#1747

semidirect

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

subgroup	index
#13092375	2
#13092373	2
C₂₃⋊#338	2
#13092173	2
C₂₃×H(3,ℤ/4)	2
#1747	23

subgroup

index

C₂₃⋊#338

C₂₃×H(3,ℤ/4)

Subgroups — #15673151

Conjugacy classes of subgroups

Complements

Maximal subgroups