Group #15673298 — subgroups

Subgroups — #15673298

Order 2944.

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

normal subgroups only

This group has 228 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×#10625004	C₂	1	2	yes	maximal
2	M₁₆×Dic₂₃	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092182	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	#10625004	C₄	1	8	yes	complemented
12	#10624997	—	2	2	no	—
13	C₈×Dic₂₃	—	2	2	no	—
14	C₈×Dic₂₃	C₄	1	4	yes	—
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₂₃	Q₈	1	2	yes	—
17	M₁₆×C₂₃	—	2	2	no	—
18	M₁₆×C₂₃	C₂×C₄	1	4	yes	—
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	—	2	2	no	—
21	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
23	C₄×Dic₂₃	D₄	1	1	yes	characteristic
24	C₄×Dic₂₃	D₄	1	1	yes	characteristic
25	#67267	—	2	2	no	—
26	#67267	C₂×C₄	1	4	yes	—
27	C₂×C₂₃⋊C₈	—	2	2	no	—
28	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	#41	1	2	yes	—
35	C₂×Dic₂₃	—	2	4	no	—
36	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
38	C₈×C₂₃	—	2	2	no	—
39	C₈×C₂₃	—	2	4	no	—
40	C₂₃⋊C₈	—	2	4	no	—
41	C₂₃⋊C₈	C₄²	1	4	yes	—
42	C₄×#108	—	23	1	no	maximal, Sylow
43	C₂²×C₂₃	—	2	2	no	—
44	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
46	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₂₃	#117	1	1	yes	characteristic
48	C₄×C₂₃	C₄×D₄	1	2	yes	—
49	Dic₂₃	—	2	2	no	—
50	Dic₂₃	#108	1	4	yes	complemented
51	C₂×#108	—	23	2	no	—
52	C₄×M₁₆	—	23	2	no	—
53	C₂×C₄×C₈	—	23	1	no	—
54	#341	—	23	1	no	—
55	#341	D₂₃	1	1	yes	characteristic
56	C₂×C₂₃	—	2	2	no	—
57	C₂×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₂₃	C₂×#108	1	2	yes	—
59	C₂×M₁₆	—	23	2	no	—
60	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
61	C₂²×C₈	—	23	1	no	—
62	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
63	C₂×C₄²	—	23	1	no	—
64	#108	—	23	8	no	—
65	#97	—	46	2	no	—
66	C₄×C₈	—	23	4	no	—
67	C₄×C₈	—	46	2	no	—
68	C₂₃	C₄×#108	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₄	—	23	2	no	—
70	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	23	4	no	—
73	M₁₆	—	46	2	no	—
74	M₁₆	C₄×D₂₃	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	23	2	no	—
78	C₂×C₈	—	23	4	no	—
79	C₂×C₈	—	46	2	no	—
80	C₂×C₈	C₄×D₂₃	1	2	yes	—
81	C₄²	—	23	2	no	—
82	C₄²	—	23	1	no	—
83	C₄²	—	23	1	no	—
84	C₂³	Q₈×D₂₃	1	1	yes	characteristic
85	C₂×C₄	—	23	2	no	—
86	C₂×C₄	—	23	2	no	—
87	C₂×C₄	—	46	4	no	—
88	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
90	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
91	C₂×C₄	#67290	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	23	4	no	—
95	C₈	—	46	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	#10625058	1	2	yes	—
98	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
99	C₄	—	23	4	no	—
100	C₄	—	46	2	no	—
101	C₄	C₄²×D₂₃	1	1	yes	characteristic
102	C₄	#10625048	1	1	yes	characteristic
103	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#13092268	1	1	yes	characteristic
106	C₂	#108×D₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625004

C₂

yes

maximal

M₁₆×Dic₂₃

C₂

yes

maximal

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

#13092182

C₂

yes

maximal, characteristic

C₂₃×#341

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10625004

C₄

yes

complemented

#10624997

—

C₈×Dic₂₃

—

C₈×Dic₂₃

C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

Q₈

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

D₄

yes

characteristic

C₄×Dic₂₃

D₄

yes

characteristic

#67267

—

#67267

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₄²

yes

—

C₄×#108

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

C₂×C₄²

yes

characteristic

C₄×C₂₃

#117

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

—

Dic₂₃

—

Dic₂₃

#108

yes

complemented

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

#341

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₃

C₂×#108

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂₃

C₄×#108

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₄×D₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃

yes

—

C₂×C₄

D₄×D₂₃

yes

characteristic

C₂×C₄

#67290

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#10625058

yes

—

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

C₄

—

100

C₄

—

101

C₄

C₄²×D₂₃

yes

characteristic

102

C₄

#10625048

yes

characteristic

103

C₄

(C₂²⋊D₂₃)⋊C₄

yes

—

104

C₂

—

105

C₂

#13092268

yes

characteristic

106

C₂

#108×D₂₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#341	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃	C₂×Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#341	D₂₃
hypocenter	C₂₃	C₄×#108
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator

C₄×C₂₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#341

C₂

socle

C₂²×C₂₃

C₂×C₄⋊C₄

prime radical

C₂³×C₂₃

C₂×Q₈

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
11. #10625004	C₄	semidirect
50. Dic₂₃	#108	semidirect
68. C₂₃	C₄×#108	semidirect

H ≅ G/N

type

11. #10625004

C₄

semidirect

50. Dic₂₃

#108

semidirect

68. C₂₃

C₄×#108

semidirect

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

subgroup	index
C₂×#10625004	2
M₁₆×Dic₂₃	2
C₂×C₈×Dic₂₃	2
#13092182	2
C₂₃×#341	2
C₄×#108	23

subgroup

index

C₂×#10625004

M₁₆×Dic₂₃

C₂×C₈×Dic₂₃

#13092182

C₂₃×#341

C₄×#108

Subgroups — #15673298

Conjugacy classes of subgroups

Complements

Maximal subgroups