Group #19168418 — subgroups

Subgroups — C₂₃×#20792 #19168418

Order 5888.

1118 conjugacy classes of subgroups; only the 622 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 622 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	C₂×C₂₃×#423	C₂	1	6	yes	maximal
2	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1843	C₂	1	8	yes	maximal, complemented
4	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
5	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
6	C₂₃×#423	—	2	24	no	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	24	yes	—
8	C₂₃×#376	C₄	1	4	yes	—
9	C₂₃×#376	C₂²	1	4	yes	—
10	C₂×C₂₃×#95	C₄	1	4	yes	—
11	C₂³×C₄×C₂₃	C₂×C₄	1	6	yes	—
12	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	—	2	24	no	—
14	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
15	C₂×C₄²×C₂₃	C₂×C₄	1	6	yes	—
16	C₂×C₄²×C₂₃	D₄	1	12	yes	—
17	C₂×C₄²×C₂₃	Q₈	1	12	yes	—
18	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
19	C₂₃×C₄⋊C₈	—	2	48	no	—
20	C₂₃×#95	C₂×C₄	1	16	yes	—
21	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	—	4	12	no	—
23	C₂²×C₄×C₂₃	C₄⋊C₄	1	24	yes	—
24	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
25	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
27	C₂²×C₄×C₂₃	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
29	M₁₆×C₂₃	—	4	16	no	—
30	C₂×C₈×C₂₃	—	2	24	no	—
31	C₄²×C₂₃	C₄⋊C₄	1	24	yes	—
32	C₄²×C₂₃	C₂×D₄	1	12	yes	—
33	C₄²×C₂₃	C₂×Q₈	1	12	yes	—
34	#20792	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₂₃	—	2	4	no	—
36	C₂³×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
37	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	—	2	24	no	—
39	C₂×C₄×C₂₃	—	4	24	no	—
40	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	24	yes	—
41	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
42	C₂×C₄×C₂₃	#127	1	4	yes	—
43	C₂×C₄×C₂₃	C₄⋊Q₈	1	12	yes	—
44	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
45	C₂×C₄×C₂₃	C₂²×Q₈	1	3	yes	—
46	C₈×C₂₃	—	4	8	no	—
47	C₂×#423	C₂×C₂₃	1	6	yes	—
48	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
49	#1843	C₂×C₂₃	1	8	yes	complemented
50	C₂²×C₂₃	—	2	12	no	—
51	C₂²×C₂₃	#384	1	4	yes	—
52	C₂²×C₂₃	C₂²×C₄⋊C₄	1	3	yes	—
53	C₂²×C₂₃	#568	1	4	yes	—
54	C₄×C₂₃	—	2	24	no	—
55	C₄×C₂₃	#384	1	4	yes	—
56	C₄×C₂₃	C₂×#127	1	1	yes	characteristic
57	C₄×C₂₃	C₂×C₄⋊Q₈	1	3	yes	—
58	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
59	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
60	#423	—	2	24	no	—
61	C₂×C₄⋊C₈	C₂²×C₂₃	1	24	yes	—
62	#376	C₄×C₂₃	1	4	yes	—
63	#376	C₂²×C₂₃	1	4	yes	—
64	C₂×#95	C₄×C₂₃	1	4	yes	—
65	C₂×C₂₃	—	2	4	no	—
66	C₂×C₂₃	C₂×#384	1	1	yes	characteristic
67	C₂×C₂₃	#2905	1	6	yes	—
68	C₂³×C₄	C₂×C₄×C₂₃	1	6	yes	—
69	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
70	C₂×M₁₆	—	2	24	no	—
71	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
72	C₂×C₄²	C₂×C₄×C₂₃	1	6	yes	—
73	C₂×C₄²	D₄×C₂₃	1	12	yes	—
74	C₂×C₄²	Q₈×C₂₃	1	12	yes	—
75	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
76	C₄⋊C₈	—	2	48	no	—
77	#95	C₂×C₄×C₂₃	1	16	yes	—
78	C₂₃	#20792	1	1	yes	characteristic, direct product factor, Sylow
79	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
80	C₂²×C₄	—	4	12	no	—
81	C₂²×C₄	C₄⋊C₄×C₂₃	1	24	yes	—
82	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
83	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
85	C₂²×C₄	C₂×Q₈×C₂₃	1	6	yes	—
86	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
87	M₁₆	—	4	16	no	—
88	C₂×C₈	—	2	24	no	—
89	C₄²	C₄⋊C₄×C₂₃	1	24	yes	—
90	C₄²	C₂×D₄×C₂₃	1	12	yes	—
91	C₄²	C₂×Q₈×C₂₃	1	12	yes	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
94	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	4	24	no	—
97	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	24	yes	—
98	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
99	C₂×C₄	C₂₃×#127	1	4	yes	—
100	C₂×C₄	C₂₃×C₄⋊Q₈	1	12	yes	—
101	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
102	C₂×C₄	C₂²×Q₈×C₂₃	1	3	yes	—
103	C₈	—	4	8	no	—
104	C₂²	—	2	12	no	—
105	C₂²	C₂₃×#384	1	4	yes	—
106	C₂²	C₂²×C₄⋊C₄×C₂₃	1	3	yes	—
107	C₂²	C₂₃×#568	1	4	yes	—
108	C₄	—	2	24	no	—
109	C₄	C₂₃×#384	1	4	yes	—
110	C₄	C₂×C₂₃×#127	1	1	yes	characteristic
111	C₄	C₂×C₂₃×C₄⋊Q₈	1	3	yes	—
112	C₂	—	2	4	no	—
113	C₂	C₂×C₂₃×#384	1	1	yes	characteristic
114	C₂	C₂₃×#2905	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#423

C₂

yes

maximal

C₂×C₂₃×#376

C₂

yes

maximal, characteristic

C₂₃×#1843

C₂

yes

maximal, complemented

C₂²×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₄²×C₂₃

C₂²

yes

—

C₂₃×#423

—

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂₃×#376

C₄

yes

—

C₂₃×#376

C₂²

yes

—

C₂×C₂₃×#95

C₄

yes

—

C₂³×C₄×C₂₃

C₂×C₄

yes

—

C₂³×C₄×C₂₃

C₂³

yes

characteristic

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

C₂³

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

Q₈

yes

—

C₂×C₄²×C₂₃

C₂³

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×#95

C₂×C₄

yes

—

C₂⁴×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂²×C₄×C₂₃

C₂×Q₈

yes

—

C₂²×C₄×C₂₃

C₂⁴

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×Q₈

yes

—

#20792

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

C₂×C₄⋊C₄

yes

—

C₂³×C₂₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₂₃

#127

yes

—

C₂×C₄×C₂₃

C₄⋊Q₈

yes

—

C₂×C₄×C₂₃

C₂²×D₄

yes

—

C₂×C₄×C₂₃

C₂²×Q₈

yes

—

C₈×C₂₃

—

C₂×#423

C₂×C₂₃

yes

—

C₂×#376

C₂×C₂₃

yes

characteristic

#1843

C₂×C₂₃

yes

complemented

C₂²×C₂₃

—

C₂²×C₂₃

#384

yes

—

C₂²×C₂₃

C₂²×C₄⋊C₄

yes

—

C₂²×C₂₃

#568

yes

—

C₄×C₂₃

—

C₄×C₂₃

#384

yes

—

C₄×C₂₃

C₂×#127

yes

characteristic

C₄×C₂₃

C₂×C₄⋊Q₈

yes

—

C₂²×M₁₆

C₂²×C₂₃

yes

—

C₂²×C₄²

C₂²×C₂₃

yes

—

#423

—

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

#376

C₄×C₂₃

yes

—

#376

C₂²×C₂₃

yes

—

C₂×#95

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#384

yes

characteristic

C₂×C₂₃

#2905

yes

—

C₂³×C₄

C₂×C₄×C₂₃

yes

—

C₂³×C₄

C₂³×C₂₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂³×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

—

C₂×C₄²

D₄×C₂₃

yes

—

C₂×C₄²

Q₈×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

—

C₄⋊C₈

—

#95

C₂×C₄×C₂₃

yes

—

C₂₃

#20792

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂²×C₄×C₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄×C₂₃

yes

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂²×C₄

C₂×D₄×C₂₃

yes

—

C₂²×C₄

C₂×Q₈×C₂₃

yes

—

C₂²×C₄

C₂⁴×C₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄²

C₄⋊C₄×C₂₃

yes

—

C₄²

C₂×D₄×C₂₃

yes

—

C₄²

C₂×Q₈×C₂₃

yes

—

C₂³

—

C₂³

C₂×C₄⋊C₄×C₂₃

yes

—

C₂³

C₂³×C₄×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

—

C₂×C₄

C₂₃×#127

yes

—

100

C₂×C₄

C₂₃×C₄⋊Q₈

yes

—

101

C₂×C₄

C₂²×D₄×C₂₃

yes

—

102

C₂×C₄

C₂²×Q₈×C₂₃

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₂₃×#384

yes

—

106

C₂²

C₂²×C₄⋊C₄×C₂₃

yes

—

107

C₂²

C₂₃×#568

yes

—

108

C₄

—

109

C₄

C₂₃×#384

yes

—

110

C₄

C₂×C₂₃×#127

yes

characteristic

111

C₄

C₂×C₂₃×C₄⋊Q₈

yes

—

112

C₂

—

113

C₂

C₂×C₂₃×#384

yes

characteristic

114

C₂

C₂₃×#2905

yes

—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

subgroup

G/H

center

C₂²×C₄×C₂₃

C₂⁴

commutator

C₂³

C₂³×C₄×C₂₃

Frattini, cosocle

C₂²×C₄

C₂⁴×C₂₃

socle

C₂³×C₂₃

C₂³×C₄

prime radical

C₂⁴×C₂₃

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
3. C₂₃×#1843	C₂	semidirect
34. #20792	C₂₃	direct (N × H)
49. #1843	C₂×C₂₃	semidirect
78. C₂₃	#20792	direct (N × H)

H ≅ G/N

type

3. C₂₃×#1843

C₂

semidirect

34. #20792

C₂₃

direct (N × H)

49. #1843

C₂×C₂₃

semidirect

78. C₂₃

#20792

direct (N × H)

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

subgroup	index
C₂×C₂₃×#423	2
C₂×C₂₃×#376	2
C₂₃×#1843	2
#20792	23

subgroup

index

C₂×C₂₃×#423

C₂×C₂₃×#376

C₂₃×#1843

#20792

Subgroups — C₂₃×#20792 #19168418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#20792 #19168418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20792 #19168418