Group #19156442 — subgroups

Subgroups — C₂₃×#8816 #19156442

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×#1845	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×#107	—	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×#107	—	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	D₈	1	4	yes	—
27	C₂×C₈×C₂₃	Q₁₆	1	4	yes	—
28	C₄⋊C₄×C₂₃	—	4	8	no	—
29	C₄⋊C₄×C₂₃	—	4	16	no	—
30	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
33	#8816	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂³×C₂₃	#127	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	8	4	no	—
36	C₂×C₄×C₂₃	—	8	8	no	—
37	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
40	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
43	C₂×C₄×C₂₃	#135	1	8	yes	—
44	C₈×C₂₃	—	2	8	no	—
45	C₈×C₂₃	#107	1	8	yes	—
46	#1845	C₂×C₂₃	1	2	yes	—
47	#1834	C₂×C₂₃	1	4	yes	—
48	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
49	C₂²×C₂₃	#384	1	1	yes	characteristic
50	C₂²×C₂₃	#493	1	1	yes	characteristic
51	C₂²×C₂₃	#494	1	1	yes	characteristic
52	C₂²×C₂₃	#495	1	4	yes	—
53	C₄×C₂₃	—	16	4	no	—
54	C₄×C₂₃	C₂×#107	1	2	yes	—
55	C₄×C₂₃	#427	1	4	yes	—
56	C₄×C₂₃	#499	1	4	yes	—
57	C₄×C₂₃	#500	1	2	yes	—
58	C₂×#107	—	2	4	no	—
59	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
60	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
61	#384	C₂²×C₂₃	1	4	yes	—
62	#340	—	2	8	no	—
63	C₈²	C₄×C₂₃	1	4	yes	complemented
64	C₂×C₂₃	#1709	1	2	yes	—
65	C₂×C₂₃	#1710	1	2	yes	—
66	C₂×C₂₃	#1845	1	1	yes	characteristic
67	C₂×C₂₃	#1846	1	2	yes	—
68	C₂²×C₈	D₄×C₂₃	1	4	yes	—
69	C₂²×C₈	D₄×C₂₃	1	2	yes	—
70	C₂×C₄⋊C₄	—	2	8	no	—
71	C₂×C₄⋊C₄	—	2	4	no	—
72	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
73	#107	—	4	8	no	—
74	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
75	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
76	C₄×C₈	Q₈×C₂₃	1	4	yes	—
77	C₄×C₈	Q₈×C₂₃	1	2	yes	—
78	C₂₃	#8816	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²×C₄	—	4	4	no	—
80	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
84	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
85	C₂×C₈	D₈×C₂₃	1	4	yes	—
86	C₂×C₈	Q₁₆×C₂₃	1	4	yes	—
87	C₄⋊C₄	—	4	8	no	—
88	C₄⋊C₄	—	4	16	no	—
89	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
90	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
91	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
92	C₂³	C₂₃×#127	1	1	yes	characteristic
93	C₂×C₄	—	8	4	no	—
94	C₂×C₄	—	8	8	no	—
95	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
97	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
98	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
100	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
101	C₂×C₄	C₂₃×#135	1	8	yes	—
102	C₈	—	2	8	no	—
103	C₈	C₂₃×#107	1	8	yes	—
104	C₂²	C₂₃×#384	1	1	yes	characteristic
105	C₂²	C₂₃×#493	1	1	yes	characteristic
106	C₂²	C₂₃×#494	1	1	yes	characteristic
107	C₂²	C₂₃×#495	1	4	yes	—
108	C₄	—	16	4	no	—
109	C₄	C₂×C₂₃×#107	1	2	yes	—
110	C₄	C₂₃×#427	1	4	yes	—
111	C₄	C₂₃×#499	1	4	yes	—
112	C₄	C₂₃×#500	1	2	yes	—
113	C₂	C₂₃×#1709	1	2	yes	—
114	C₂	C₂₃×#1710	1	2	yes	—
115	C₂	C₂₃×#1845	1	1	yes	characteristic
116	C₂	C₂₃×#1846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1845

C₂

yes

maximal

C₂₃×#1834

C₂

yes

maximal

C₂×C₈²×C₂₃

C₂

yes

maximal, characteristic

C₂×C₂₃×#107

—

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×#107

—

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

D₈

yes

—

C₂×C₈×C₂₃

Q₁₆

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×Q₈

yes

characteristic

C₄²×C₂₃

C₂×Q₈

yes

—

#8816

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

—

C₂×C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×D₈

yes

—

C₂×C₄×C₂₃

C₂×Q₁₆

yes

—

C₂×C₄×C₂₃

#135

yes

—

C₈×C₂₃

—

C₈×C₂₃

#107

yes

—

#1845

C₂×C₂₃

yes

—

#1834

C₂×C₂₃

yes

—

C₂×C₈²

C₂×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×#107

yes

—

C₄×C₂₃

#427

yes

—

C₄×C₂₃

#499

yes

—

C₄×C₂₃

#500

yes

—

C₂×#107

—

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

#1709

yes

—

C₂×C₂₃

#1710

yes

—

C₂×C₂₃

#1845

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

#107

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

C₂₃

#8816

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

D₈×C₂₃

yes

—

C₂×C₈

Q₁₆×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

—

C₂×C₄

C₂₃×C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂×D₈×C₂₃

yes

—

100

C₂×C₄

C₂×Q₁₆×C₂₃

yes

—

101

C₂×C₄

C₂₃×#135

yes

—

102

C₈

—

103

C₈

C₂₃×#107

yes

—

104

C₂²

C₂₃×#384

yes

characteristic

105

C₂²

C₂₃×#493

yes

characteristic

106

C₂²

C₂₃×#494

yes

characteristic

107

C₂²

C₂₃×#495

yes

—

108

C₄

—

109

C₄

C₂×C₂₃×#107

yes

—

110

C₄

C₂₃×#427

yes

—

111

C₄

C₂₃×#499

yes

—

112

C₄

C₂₃×#500

yes

—

113

C₂

C₂₃×#1709

yes

—

114

C₂

C₂₃×#1710

yes

—

115

C₂

C₂₃×#1845

yes

characteristic

116

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
33. #8816	C₂₃	direct (N × H)
63. C₈²	C₄×C₂₃	semidirect
78. C₂₃	#8816	direct (N × H)

H ≅ G/N

type

9. C₈²×C₂₃

C₄

semidirect

33. #8816

C₂₃

direct (N × H)

63. C₈²

C₄×C₂₃

semidirect

78. C₂₃

#8816

direct (N × H)

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

subgroup	index
C₂₃×#1845	2
C₂₃×#1834	2
C₂×C₈²×C₂₃	2
#8816	23

subgroup

index

C₂₃×#1845

C₂₃×#1834

C₂×C₈²×C₂₃

#8816

Subgroups — C₂₃×#8816 #19156442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8816 #19156442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8816 #19156442