Group #19156441 — subgroups

Subgroups — C₂₃×#8815 #19156441

Order 5888.

422 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 238 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×#1844	C₂	1	2	yes	maximal
2	C₂₃×#1834	C₂	1	4	yes	maximal
3	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#106	—	2	4	no	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#384	C₂²	1	4	yes	—
8	C₂₃×#340	—	2	8	no	—
9	C₈²×C₂₃	C₄	1	4	yes	complemented
10	C₂²×C₈×C₂₃	D₄	1	4	yes	—
11	C₂²×C₈×C₂₃	D₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
13	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
14	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
15	C₂₃×#106	—	4	8	no	—
16	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₂₃	Q₈	1	4	yes	—
19	C₄×C₈×C₂₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₂₃	—	4	4	no	—
21	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
23	C₂×C₈×C₂₃	—	2	8	no	—
24	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₂₃	SD₁₆	1	8	yes	—
27	C₄⋊C₄×C₂₃	—	4	8	no	—
28	C₄⋊C₄×C₂₃	—	4	16	no	—
29	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
32	#8815	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₂₃	#127	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	8	4	no	—
35	C₂×C₄×C₂₃	—	8	8	no	—
36	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
40	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
41	C₂×C₄×C₂₃	#135	1	8	yes	—
42	C₈×C₂₃	—	2	8	no	—
43	C₈×C₂₃	#106	1	8	yes	—
44	#1844	C₂×C₂₃	1	2	yes	—
45	#1834	C₂×C₂₃	1	4	yes	—
46	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
47	C₂²×C₂₃	#384	1	1	yes	characteristic
48	C₂²×C₂₃	#492	1	2	yes	—
49	C₂²×C₂₃	#495	1	4	yes	—
50	C₄×C₂₃	—	16	4	no	—
51	C₄×C₂₃	C₂×#106	1	2	yes	—
52	C₄×C₂₃	#427	1	4	yes	—
53	C₄×C₂₃	#498	1	2	yes	—
54	C₄×C₂₃	#499	1	4	yes	—
55	C₂×#106	—	2	4	no	—
56	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
57	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
58	#384	C₂²×C₂₃	1	4	yes	—
59	#340	—	2	8	no	—
60	C₈²	C₄×C₂₃	1	4	yes	complemented
61	C₂×C₂₃	#1708	1	4	yes	—
62	C₂×C₂₃	#1844	1	1	yes	characteristic
63	C₂×C₂₃	#1846	1	2	yes	—
64	C₂²×C₈	D₄×C₂₃	1	4	yes	—
65	C₂²×C₈	D₄×C₂₃	1	2	yes	—
66	C₂×C₄⋊C₄	—	2	8	no	—
67	C₂×C₄⋊C₄	—	2	4	no	—
68	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
69	#106	—	4	8	no	—
70	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
71	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
72	C₄×C₈	Q₈×C₂₃	1	4	yes	—
73	C₄×C₈	Q₈×C₂₃	1	2	yes	—
74	C₂₃	#8815	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₄	—	4	4	no	—
76	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
80	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
81	C₂×C₈	SD₁₆×C₂₃	1	8	yes	—
82	C₄⋊C₄	—	4	8	no	—
83	C₄⋊C₄	—	4	16	no	—
84	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
85	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
86	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
87	C₂³	C₂₃×#127	1	1	yes	characteristic
88	C₂×C₄	—	8	4	no	—
89	C₂×C₄	—	8	8	no	—
90	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
92	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
94	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
95	C₂×C₄	C₂₃×#135	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	C₂₃×#106	1	8	yes	—
98	C₂²	C₂₃×#384	1	1	yes	characteristic
99	C₂²	C₂₃×#492	1	2	yes	—
100	C₂²	C₂₃×#495	1	4	yes	—
101	C₄	—	16	4	no	—
102	C₄	C₂×C₂₃×#106	1	2	yes	—
103	C₄	C₂₃×#427	1	4	yes	—
104	C₄	C₂₃×#498	1	2	yes	—
105	C₄	C₂₃×#499	1	4	yes	—
106	C₂	C₂₃×#1708	1	4	yes	—
107	C₂	C₂₃×#1844	1	1	yes	characteristic
108	C₂	C₂₃×#1846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1844

C₂

yes

maximal

C₂₃×#1834

C₂

yes

maximal

C₂×C₈²×C₂₃

C₂

yes

maximal, characteristic

C₂×C₂₃×#106

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#384

C₂²

yes

—

C₂₃×#340

—

C₈²×C₂₃

C₄

yes

complemented

C₂²×C₈×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#106

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

Q₈

yes

—

C₄×C₈×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×Q₈

yes

characteristic

C₄²×C₂₃

C₂×Q₈

yes

—

#8815

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#127

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₂₃

C₄⋊Q₈

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₂×C₄×C₂₃

#135

yes

—

C₈×C₂₃

—

C₈×C₂₃

#106

yes

—

#1844

C₂×C₂₃

yes

—

#1834

C₂×C₂₃

yes

—

C₂×C₈²

C₂×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#495

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×#106

yes

—

C₄×C₂₃

#427

yes

—

C₄×C₂₃

#498

yes

—

C₄×C₂₃

#499

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#384

C₂²×C₂₃

yes

—

#340

—

C₈²

C₄×C₂₃

yes

complemented

C₂×C₂₃

#1708

yes

—

C₂×C₂₃

#1844

yes

characteristic

C₂×C₂₃

#1846

yes

—

C₂²×C₈

D₄×C₂₃

yes

—

C₂²×C₈

D₄×C₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

#106

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

C₂₃

#8815

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

C₂²×C₄

C₂×D₄×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

SD₁₆×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₂₃

yes

characteristic

C₄²

C₂×Q₈×C₂₃

yes

characteristic

C₄²

C₂×Q₈×C₂₃

yes

—

C₂³

C₂₃×#127

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

—

C₂×C₄

C₂₃×C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂₃×C₄⋊Q₈

yes

—

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

C₂×C₄

C₂₃×#135

yes

—

C₈

—

C₈

C₂₃×#106

yes

—

C₂²

C₂₃×#384

yes

characteristic

C₂²

C₂₃×#492

yes

—

100

C₂²

C₂₃×#495

yes

—

101

C₄

—

102

C₄

C₂×C₂₃×#106

yes

—

103

C₄

C₂₃×#427

yes

—

104

C₄

C₂₃×#498

yes

—

105

C₄

C₂₃×#499

yes

—

106

C₂

C₂₃×#1708

yes

—

107

C₂

C₂₃×#1844

yes

characteristic

108

C₂

C₂₃×#1846

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

#127

commutator

C₄²

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
9. C₈²×C₂₃	C₄	semidirect
32. #8815	C₂₃	direct (N × H)
60. C₈²	C₄×C₂₃	semidirect
74. C₂₃	#8815	direct (N × H)

H ≅ G/N

type

9. C₈²×C₂₃

C₄

semidirect

32. #8815

C₂₃

direct (N × H)

60. C₈²

C₄×C₂₃

semidirect

74. C₂₃

#8815

direct (N × H)

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

subgroup	index
C₂₃×#1844	2
C₂₃×#1834	2
C₂×C₈²×C₂₃	2
#8815	23

subgroup

index

C₂₃×#1844

C₂₃×#1834

C₂×C₈²×C₂₃

#8815

Subgroups — C₂₃×#8815 #19156441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8815 #19156441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8815 #19156441