Group #19212064 — subgroups

Subgroups — #19212064

Order 5888.

186 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 27 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	#15671560	C₂	1	2	yes	maximal, complemented
2	C₂₃×#1712	C₂	1	1	yes	maximal, characteristic
3	#13092210	—	2	2	no	—
4	C₂₃×#493	—	2	2	no	—
5	C₂₃×#493	C₄	1	1	yes	characteristic, complemented
6	C₂₃×#493	C₂²	1	1	yes	characteristic
7	C₈²×C₂₃	C₄	1	1	yes	characteristic, complemented
8	C₂×D₈×C₂₃	—	4	1	no	—
9	C₂×D₈×C₂₃	—	4	1	no	—
10	C₂×D₈×C₂₃	—	4	4	no	—
11	C₂₃×#127	—	2	1	no	—
12	C₂₃×#127	D₄	1	2	yes	—
13	C₄×C₈×C₂₃	—	2	1	no	—
14	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	#10625016	—	4	2	no	—
16	C₂×D₄×C₂₃	—	2	2	no	—
17	C₂×D₄×C₂₃	—	4	2	no	—
18	C₂×D₄×C₂₃	—	4	2	no	—
19	C₂×D₄×C₂₃	—	4	1	no	—
20	D₈×C₂₃	—	8	2	no	—
21	D₈×C₂₃	—	8	2	no	—
22	D₈×C₂₃	—	8	8	no	—
23	C₂×C₈×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	2	1	no	—
25	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²⋊Dic₂₃	—	8	2	no	—
27	#7529	—	23	1	no	maximal, Sylow
28	C₂³×C₂₃	—	4	2	no	—
29	C₂³×C₂₃	—	8	1	no	—
30	D₄×C₂₃	—	8	2	no	—
31	D₄×C₂₃	—	8	4	no	—
32	D₄×C₂₃	—	8	4	no	—
33	D₄×C₂₃	—	8	2	no	—
34	C₂×C₄×C₂₃	—	2	1	no	—
35	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
36	C₂×Dic₂₃	—	16	2	no	—
37	C₈×C₂₃	—	2	4	no	—
38	C₈×C₂₃	—	2	2	no	—
39	#1712	D₂₃	1	1	yes	characteristic
40	#1408	—	23	2	no	—
41	C₂²×C₂₃	—	8	2	no	—
42	C₂²×C₂₃	—	16	2	no	—
43	C₂²×C₂₃	—	16	2	no	—
44	C₂²×C₂₃	—	16	1	no	—
45	C₂²×C₂₃	#353	1	1	yes	characteristic
46	C₄×C₂₃	—	2	2	no	—
47	C₄×C₂₃	—	2	1	no	—
48	Dic₂₃	—	32	2	no	—
49	#493	—	2	2	no	—
50	#493	Dic₂₃	1	1	yes	characteristic, complemented
51	#493	C₂×D₂₃	1	1	yes	characteristic
52	#353	—	46	2	no	—
53	C₈²	Dic₂₃	1	1	yes	characteristic, complemented
54	C₂×C₂₃	—	2	1	no	—
55	C₂×C₂₃	—	16	2	no	—
56	C₂×C₂₃	—	32	1	no	—
57	C₂×C₂₃	#1408	1	1	yes	characteristic
58	C₂×D₈	—	4	1	no	—
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	—	4	4	no	—
61	#127	—	2	1	no	—
62	#127	C₂²⋊D₂₃	1	2	yes	—
63	#99	—	92	2	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	C₂×Dic₂₃	1	1	yes	characteristic
66	C₂₃	#7529	1	1	yes	characteristic, complemented, Sylow
67	C₂×D₄	—	2	2	no	—
68	C₂×D₄	—	4	2	no	—
69	C₂×D₄	—	4	2	no	—
70	C₂×D₄	—	4	1	no	—
71	D₈	—	8	2	no	—
72	D₈	—	8	2	no	—
73	D₈	—	8	8	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂²⋊C₄	—	184	2	no	—
77	C₄²	C₂²⋊Dic₂₃	1	1	yes	characteristic
78	C₂³	—	4	2	no	—
79	C₂³	—	8	1	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	8	4	no	—
83	D₄	—	8	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	368	2	no	—
86	C₂×C₄	#10625016	1	1	yes	characteristic
87	C₈	—	2	4	no	—
88	C₈	—	2	2	no	—
89	C₂²	—	8	2	no	—
90	C₂²	—	16	2	no	—
91	C₂²	—	16	2	no	—
92	C₂²	—	16	1	no	—
93	C₂²	#13092210	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	736	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	16	2	no	—
99	C₂	—	32	1	no	—
100	C₂	#15671560	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15671560

C₂

yes

maximal, complemented

C₂₃×#1712

C₂

yes

maximal, characteristic

#13092210

—

C₂₃×#493

—

C₂₃×#493

C₄

yes

characteristic, complemented

C₂₃×#493

C₂²

yes

characteristic

C₈²×C₂₃

C₄

yes

characteristic, complemented

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

—

C₂₃×#127

—

C₂₃×#127

D₄

yes

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

#10625016

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

D₈×C₂₃

—

D₈×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₂₃

—

#7529

—

maximal, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#99

yes

characteristic

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1712

D₂₃

yes

characteristic

#1408

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#353

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

#493

—

#493

Dic₂₃

yes

characteristic, complemented

#493

C₂×D₂₃

yes

characteristic

#353

—

C₈²

Dic₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1408

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

C₂²⋊D₂₃

yes

—

#99

—

C₄×C₈

—

C₄×C₈

C₂×Dic₂₃

yes

characteristic

C₂₃

#7529

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

184

—

C₄²

C₂²⋊Dic₂₃

yes

characteristic

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

368

—

C₂×C₄

#10625016

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#13092210

yes

characteristic

C₄

—

C₄

—

C₄

—

736

—

C₂

—

C₂

—

C₂

—

100

C₂

#15671560

yes

characteristic

	subgroup	G/H
center	C₂	#15671560
commutator	C₄×C₈×C₂₃	C₂×C₄
Frattini	#493	C₂×D₂₃
Fitting	C₂₃×#1712	C₂
socle	C₂×C₂₃	#1408
hypercenter	#1712	D₂₃
hypocenter	C₂₃	#7529
LCS	G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1712

subgroup

G/H

center

C₂

#15671560

commutator

C₄×C₈×C₂₃

C₂×C₄

Frattini

#493

C₂×D₂₃

Fitting

C₂₃×#1712

C₂

C₂×C₂₃

D₂₃

C₂₃

G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1712

N	H ≅ G/N	type
1. #15671560	C₂	semidirect
5. C₂₃×#493	C₄	semidirect
7. C₈²×C₂₃	C₄	semidirect
50. #493	Dic₂₃	semidirect
53. C₈²	Dic₂₃	semidirect
66. C₂₃	#7529	semidirect

H ≅ G/N

type

1. #15671560

C₂

semidirect

5. C₂₃×#493

C₄

semidirect

7. C₈²×C₂₃

C₄

semidirect

50. #493

Dic₂₃

semidirect

53. C₈²

Dic₂₃

semidirect

66. C₂₃

#7529

semidirect

Subgroups — #19212064

Conjugacy classes of subgroups

Complements