Group #19158238 — subgroups

Subgroups — C₂₃×#10612 #19158238

Order 5888.

328 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 98 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×#1574	C₂	1	3	yes	maximal, complemented
2	C₂₃×#1545	C₂	1	3	yes	maximal
3	C₂₃×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
5	C₂₃×#431	C₂²	1	3	yes	—
6	C₂₃×#423	C₂²	1	3	yes	—
7	C₂₃×#335	—	2	3	no	—
8	C₂₃×#334	—	2	3	no	—
9	C₂₃×#331	—	2	3	no	—
10	C₂₃×#331	—	2	3	no	—
11	C₂²×D₄×C₂₃	—	2	3	no	—
12	C₂×M₁₆×C₂₃	—	2	3	no	—
13	C₂₃×#127	—	2	3	no	—
14	C₂₃×#127	Q₈	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
16	C₂₃×C₄⋊C₈	—	2	3	no	—
17	C₂₃×C₄⋊C₈	—	2	3	no	—
18	C₂₃×C₄⋊C₈	D₄	1	3	yes	—
19	C₂₃×C₄⋊C₈	D₄	1	3	yes	—
20	C₂₃×C₂²⋊C₈	—	2	3	no	—
21	C₄×C₈×C₂₃	—	2	3	no	—
22	C₂⁴×C₂₃	—	4	1	no	—
23	C₂×D₄×C₂₃	—	2	6	no	—
24	C₂×D₄×C₂₃	—	4	3	no	—
25	C₂×D₄×C₂₃	—	8	3	no	—
26	C₂×D₄×C₂₃	—	8	3	no	—
27	C₂²×C₄×C₂₃	#41	1	3	yes	—
28	M₁₆×C₂₃	—	8	3	no	—
29	C₂×C₈×C₂₃	—	2	3	no	—
30	C₂×C₈×C₂₃	—	2	3	no	—
31	C₂×C₈×C₂₃	—	4	3	no	—
32	C₄²×C₂₃	C₂×D₄	1	3	yes	—
33	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	#10612	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₂₃	—	4	2	no	—
36	C₂³×C₂₃	—	8	6	no	—
37	C₂³×C₂₃	#126	1	1	yes	characteristic
38	D₄×C₂₃	—	8	6	no	—
39	D₄×C₂₃	—	8	3	no	—
40	D₄×C₂₃	—	8	3	no	—
41	C₂×C₄×C₂₃	—	2	3	no	—
42	C₂×C₄×C₂₃	—	2	3	no	—
43	C₂×C₄×C₂₃	#121	1	3	yes	—
44	C₂×C₄×C₂₃	C₂²⋊Q₈	1	3	yes	—
45	C₈×C₂₃	—	4	3	no	—
46	C₈×C₂₃	—	4	3	no	—
47	C₈×C₂₃	—	8	3	no	—
48	#1574	C₂×C₂₃	1	3	yes	complemented
49	#1545	C₂×C₂₃	1	3	yes	—
50	#1309	C₂×C₂₃	1	1	yes	characteristic, complemented
51	C₂²×C₂₃	—	2	3	no	—
52	C₂²×C₂₃	—	8	6	no	—
53	C₂²×C₂₃	—	16	4	no	—
54	C₂²×C₂₃	#396	1	1	yes	characteristic
55	C₄×C₂₃	—	2	3	no	—
56	C₄×C₂₃	#471	1	3	yes	—
57	C₄×C₂₃	#472	1	3	yes	—
58	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
59	#431	C₂²×C₂₃	1	3	yes	—
60	#423	C₂²×C₂₃	1	3	yes	—
61	#335	—	2	3	no	—
62	#334	—	2	3	no	—
63	#331	—	2	3	no	—
64	#331	—	2	3	no	—
65	C₂×C₂₃	—	4	1	no	—
66	C₂×C₂₃	—	16	2	no	—
67	C₂×C₂₃	#2063	1	3	yes	—
68	C₂²×D₄	—	2	3	no	—
69	C₂×M₁₆	—	2	3	no	—
70	#127	—	2	3	no	—
71	#127	Q₈×C₂₃	1	2	yes	—
72	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
73	C₄⋊C₈	—	2	3	no	—
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	D₄×C₂₃	1	3	yes	—
76	C₄⋊C₈	D₄×C₂₃	1	3	yes	—
77	C₂²⋊C₈	—	2	3	no	—
78	C₄×C₈	—	2	3	no	—
79	C₂₃	#10612	1	1	yes	characteristic, direct product factor, Sylow
80	C₂⁴	—	4	1	no	—
81	C₂×D₄	—	2	6	no	—
82	C₂×D₄	—	4	3	no	—
83	C₂×D₄	—	8	3	no	—
84	C₂×D₄	—	8	3	no	—
85	C₂²×C₄	#41×C₂₃	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	2	3	no	—
88	C₂×C₈	—	2	3	no	—
89	C₂×C₈	—	4	3	no	—
90	C₄²	C₂×D₄×C₂₃	1	3	yes	—
91	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
92	C₂³	—	4	2	no	—
93	C₂³	—	8	6	no	—
94	C₂³	C₂₃×#126	1	1	yes	characteristic
95	D₄	—	8	6	no	—
96	D₄	—	8	3	no	—
97	D₄	—	8	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	C₂₃×#121	1	3	yes	—
101	C₂×C₄	C₂₃×C₂²⋊Q₈	1	3	yes	—
102	C₈	—	4	3	no	—
103	C₈	—	4	3	no	—
104	C₈	—	8	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	8	6	no	—
107	C₂²	—	16	4	no	—
108	C₂²	C₂₃×#396	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	C₂₃×#471	1	3	yes	—
111	C₄	C₂₃×#472	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	16	2	no	—
114	C₂	C₂₃×#2063	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1574

C₂

yes

maximal, complemented

C₂₃×#1545

C₂

yes

maximal

C₂₃×#1309

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#127

C₂²

yes

characteristic

C₂₃×#431

C₂²

yes

—

C₂₃×#423

C₂²

yes

—

C₂₃×#335

—

C₂₃×#334

—

C₂₃×#331

—

C₂₃×#331

—

C₂²×D₄×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂₃×#127

—

C₂₃×#127

Q₈

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

D₄

yes

—

C₂₃×C₄⋊C₈

D₄

yes

—

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

—

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

#41

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×Q₈

yes

characteristic

#10612

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

#126

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1574

C₂×C₂₃

yes

complemented

#1545

C₂×C₂₃

yes

—

#1309

C₂×C₂₃

yes

characteristic, complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#396

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#471

yes

—

C₄×C₂₃

#472

yes

—

C₂×#127

C₂²×C₂₃

yes

characteristic

#431

C₂²×C₂₃

yes

—

#423

C₂²×C₂₃

yes

—

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#2063

yes

—

C₂²×D₄

—

C₂×M₁₆

—

#127

—

#127

Q₈×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄×C₂₃

yes

—

C₄⋊C₈

D₄×C₂₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂₃

#10612

yes

characteristic, direct product factor, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

#41×C₂₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂×D₄×C₂₃

yes

—

C₄²

C₂×Q₈×C₂₃

yes

characteristic

C₂³

—

C₂³

—

C₂³

C₂₃×#126

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂₃×#121

yes

—

101

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂₃×#396

yes

characteristic

109

C₄

—

110

C₄

C₂₃×#471

yes

—

111

C₄

C₂₃×#472

yes

—

112

C₂

—

113

C₂

—

114

C₂

C₂₃×#2063

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#396

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂×C₂₃×#127

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₂₃×#1574	C₂	semidirect
3. C₂₃×#1309	C₂	semidirect
34. #10612	C₂₃	direct (N × H)
48. #1574	C₂×C₂₃	semidirect
50. #1309	C₂×C₂₃	semidirect
79. C₂₃	#10612	direct (N × H)

H ≅ G/N

type

1. C₂₃×#1574

C₂

semidirect

3. C₂₃×#1309

C₂

semidirect

34. #10612

C₂₃

direct (N × H)

48. #1574

C₂×C₂₃

semidirect

50. #1309

C₂×C₂₃

semidirect

79. C₂₃

#10612

direct (N × H)

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

subgroup	index
C₂₃×#1574	2
C₂₃×#1545	2
C₂₃×#1309	2
#10612	23

subgroup

index

C₂₃×#1574

C₂₃×#1545

C₂₃×#1309

#10612

Subgroups — C₂₃×#10612 #19158238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10612 #19158238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10612 #19158238