Group #19156451 — subgroups

Subgroups — C₂₃×#8825 #19156451

Order 5888.

338 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 198 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	C₂₃×#1848	C₂	1	2	yes	maximal
2	C₂₃×#1566	C₂	1	4	yes	maximal
3	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#108	—	2	4	no	—
5	C₂₃×#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂₃×#334	—	2	8	no	—
9	C₂₃×#322	C₄	1	4	yes	—
10	C₂×M₁₆×C₂₃	—	2	4	no	—
11	C₂²×C₈×C₂₃	D₄	1	2	yes	—
12	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
13	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
14	C₂₃×#108	—	4	8	no	—
15	C₂₃×C₄⋊C₈	—	2	8	no	—
16	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₂₃	D₄	1	4	yes	—
19	C₄×C₈×C₂₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
22	M₁₆×C₂₃	—	4	8	no	—
23	C₂×C₈×C₂₃	—	4	4	no	—
24	C₂×C₈×C₂₃	—	4	4	no	—
25	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₂₃	SD₁₆	1	8	yes	—
28	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₂₃	C₂×D₄	1	2	yes	—
30	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
31	#8825	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	4	no	—
34	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
38	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
39	C₈×C₂₃	—	2	8	no	—
40	C₈×C₂₃	—	8	4	no	—
41	C₈×C₂₃	#106	1	8	yes	—
42	#1848	C₂×C₂₃	1	2	yes	—
43	#1566	C₂×C₂₃	1	4	yes	—
44	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
45	C₂²×C₂₃	—	2	2	no	—
46	C₂²×C₂₃	#384	1	1	yes	characteristic
47	C₂²×C₂₃	#498	1	2	yes	—
48	C₄×C₂₃	—	2	2	no	—
49	C₄×C₂₃	C₂×#106	1	2	yes	—
50	C₄×C₂₃	#430	1	4	yes	—
51	C₄×C₂₃	#492	1	2	yes	—
52	C₂×#108	—	2	4	no	—
53	#423	C₂²×C₂₃	1	4	yes	—
54	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
55	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
56	#334	—	2	8	no	—
57	#322	C₄×C₂₃	1	4	yes	—
58	C₂×C₂₃	—	2	2	no	—
59	C₂×C₂₃	#1844	1	1	yes	characteristic
60	C₂×C₂₃	#1849	1	2	yes	—
61	C₂×M₁₆	—	2	4	no	—
62	C₂²×C₈	D₄×C₂₃	1	2	yes	—
63	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
64	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
65	#108	—	4	8	no	—
66	C₄⋊C₈	—	2	8	no	—
67	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
68	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
69	C₄×C₈	D₄×C₂₃	1	4	yes	—
70	C₄×C₈	Q₈×C₂₃	1	2	yes	—
71	C₂₃	#8825	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
73	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
74	M₁₆	—	4	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
78	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
79	C₂×C₈	SD₁₆×C₂₃	1	8	yes	—
80	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
81	C₄²	C₂×D₄×C₂₃	1	2	yes	—
82	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
83	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
87	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
88	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
89	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	8	4	no	—
92	C₈	C₂₃×#106	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	C₂₃×#384	1	1	yes	characteristic
95	C₂²	C₂₃×#498	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	C₂×C₂₃×#106	1	2	yes	—
98	C₄	C₂₃×#430	1	4	yes	—
99	C₄	C₂₃×#492	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	C₂₃×#1844	1	1	yes	characteristic
102	C₂	C₂₃×#1849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1848

C₂

yes

maximal

C₂₃×#1566

C₂

yes

maximal

C₂×C₂₃×#322

C₂

yes

maximal, characteristic

C₂×C₂₃×#108

—

C₂₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂₃×#334

—

C₂₃×#322

C₄

yes

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

Q₈

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#108

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×Q₈

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₄²×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×Q₈

yes

characteristic

#8825

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₂₃

#127

yes

characteristic

C₂×C₄×C₂₃

C₄⋊Q₈

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#106

yes

—

#1848

C₂×C₂₃

yes

—

#1566

C₂×C₂₃

yes

—

C₂×#322

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

#384

yes

characteristic

C₂²×C₂₃

#498

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×#106

yes

—

C₄×C₂₃

#430

yes

—

C₄×C₂₃

#492

yes

—

C₂×#108

—

#423

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#334

—

#322

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

#1844

yes

characteristic

C₂×C₂₃

#1849

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₂₃

yes

—

C₂²×C₈

Q₈×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

C₂₃

#8825

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

C₂²×C₄

C₂×Q₈×C₂₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

SD₁₆×C₂₃

yes

—

C₄²

C₂²×C₄×C₂₃

yes

characteristic

C₄²

C₂×D₄×C₂₃

yes

—

C₄²

C₂×Q₈×C₂₃

yes

characteristic

C₂³

C₂₃×C₄⋊Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

—

C₂×C₄

C₂₃×#127

yes

characteristic

C₂×C₄

C₂₃×C₄⋊Q₈

yes

—

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

C₈

—

C₈

—

C₈

C₂₃×#106

yes

—

C₂²

—

C₂²

C₂₃×#384

yes

characteristic

C₂²

C₂₃×#498

yes

—

C₄

—

C₄

C₂×C₂₃×#106

yes

—

C₄

C₂₃×#430

yes

—

C₄

C₂₃×#492

yes

—

100

C₂

—

101

C₂

C₂₃×#1844

yes

characteristic

102

C₂

C₂₃×#1849

yes

—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#384
prime radical	C₂³×C₂₃	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₃

#127

commutator

C₄²

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

socle

C₂²×C₂₃

#384

prime radical

C₂³×C₂₃

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
31. #8825	C₂₃	direct (N × H)
71. C₂₃	#8825	direct (N × H)

H ≅ G/N

type

31. #8825

C₂₃

direct (N × H)

71. C₂₃

#8825

direct (N × H)

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

subgroup	index
C₂₃×#1848	2
C₂₃×#1566	2
C₂×C₂₃×#322	2
#8825	23

subgroup

index

C₂₃×#1848

C₂₃×#1566

C₂×C₂₃×#322

#8825

Subgroups — C₂₃×#8825 #19156451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8825 #19156451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8825 #19156451