Group #19156424 — subgroups

Subgroups — C₂₃×#8798 #19156424

Order 5888.

220 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 110 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×#1568	C₂	1	2	yes	maximal
2	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1287	C₂	1	4	yes	maximal
4	C₂₃×#432	C₂²	1	4	yes	—
5	C₂₃×#431	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#335	—	2	2	no	—
8	C₂₃×#334	—	2	2	no	—
9	C₂₃×#322	C₄	1	4	yes	—
10	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
12	C₂₃×C₄⋊C₈	—	2	4	no	—
13	C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂₃×C₂²⋊C₈	—	2	2	no	—
15	C₂₃×C₂²⋊C₈	—	2	4	no	—
16	C₂₃×#97	—	2	4	no	—
17	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	#41	1	2	yes	—
23	C₂×C₈×C₂₃	—	2	4	no	—
24	C₂×C₈×C₂₃	—	4	2	no	—
25	C₂×C₈×C₂₃	—	4	4	no	—
26	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
28	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₄²×C₂₃	#41	1	2	yes	—
31	#8798	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₂₃	#125	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	4	2	no	—
35	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	#117	1	2	yes	—
37	C₂×C₄×C₂₃	#124	1	2	yes	—
38	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
39	C₈×C₂₃	—	2	2	no	—
40	C₈×C₂₃	—	2	2	no	—
41	C₈×C₂₃	—	4	4	no	—
42	C₈×C₂₃	—	8	4	no	—
43	#1568	C₂×C₂₃	1	2	yes	—
44	#1452	C₂×C₂₃	1	1	yes	characteristic
45	#1287	C₂×C₂₃	1	4	yes	—
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	—	2	1	no	—
48	C₂²×C₂₃	#382	1	1	yes	characteristic
49	C₄×C₂₃	—	2	2	no	—
50	C₄×C₂₃	—	2	2	no	—
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	#430	1	2	yes	—
53	#432	C₂²×C₂₃	1	4	yes	—
54	#431	C₂²×C₂₃	1	2	yes	—
55	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
56	#335	—	2	2	no	—
57	#334	—	2	2	no	—
58	#322	C₄×C₂₃	1	4	yes	—
59	C₂×C₂₃	—	4	1	no	—
60	C₂×C₂₃	#1833	1	2	yes	—
61	C₂×C₂₃	#1838	1	1	yes	characteristic
62	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
63	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
64	C₄⋊C₈	—	2	4	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	—	2	4	no	—
68	#97	—	2	4	no	—
69	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
70	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
71	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
72	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
73	C₂₃	#8798	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
75	C₂²×C₄	#41×C₂₃	1	2	yes	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
80	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
81	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
82	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
83	C₄²	#41×C₂₃	1	2	yes	—
84	C₂³	C₂₃×#125	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
88	C₂×C₄	C₂₃×#117	1	2	yes	—
89	C₂×C₄	C₂₃×#124	1	2	yes	—
90	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	4	4	no	—
94	C₈	—	8	4	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	C₂₃×#382	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	C₂₃×#430	1	2	yes	—
102	C₂	—	4	1	no	—
103	C₂	C₂₃×#1833	1	2	yes	—
104	C₂	C₂₃×#1838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1568

C₂

yes

maximal

C₂₃×#1452

C₂

yes

maximal, characteristic

C₂₃×#1287

C₂

yes

maximal

C₂₃×#432

C₂²

yes

—

C₂₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#335

—

C₂₃×#334

—

C₂₃×#322

C₄

yes

—

C₂²×C₈×C₂₃

Q₈

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₂₃×#97

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

D₄

yes

characteristic

C₄×C₈×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₂₃

#41

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₄²×C₂₃

#41

yes

—

#8798

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#125

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

#124

yes

—

C₂×C₄×C₂₃

#125

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1568

C₂×C₂₃

yes

—

#1452

C₂×C₂₃

yes

characteristic

#1287

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#382

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#430

yes

—

#432

C₂²×C₂₃

yes

—

#431

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

—

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

#1833

yes

—

C₂×C₂₃

#1838

yes

characteristic

C₂²×C₈

Q₈×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

characteristic

C₄×C₈

D₄×C₂₃

yes

characteristic

C₂₃

#8798

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₂×Q₈×C₂₃

yes

characteristic

C₂²×C₄

#41×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₄²

C₂²×C₄×C₂₃

yes

characteristic

C₄²

C₂×D₄×C₂₃

yes

characteristic

C₄²

#41×C₂₃

yes

—

C₂³

C₂₃×#125

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

C₂×C₄

C₂₃×#117

yes

—

C₂×C₄

C₂₃×#124

yes

—

C₂×C₄

C₂₃×#125

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

C₂₃×#382

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

C₂₃×#430

yes

—

102

C₂

—

103

C₂

C₂₃×#1833

yes

—

104

C₂

C₂₃×#1838

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#382
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#382

commutator

C₄²

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

C₂³×C₂₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
31. #8798	C₂₃	direct (N × H)
73. C₂₃	#8798	direct (N × H)

H ≅ G/N

type

31. #8798

C₂₃

direct (N × H)

73. C₂₃

#8798

direct (N × H)

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

subgroup	index
C₂₃×#1568	2
C₂₃×#1452	2
C₂₃×#1287	2
#8798	23

subgroup

index

C₂₃×#1568

C₂₃×#1452

C₂₃×#1287

#8798

Subgroups — C₂₃×#8798 #19156424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8798 #19156424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8798 #19156424