Group #19156435 — subgroups

Subgroups — C₂₃×#8809 #19156435

Order 5888.

402 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 222 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×#1842	C₂	1	2	yes	maximal
2	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#334	C₂	1	4	yes	maximal
4	C₂×C₂₃×C₄⋊C₈	—	2	4	no	—
5	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
7	C₄³×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#336	C₄	1	4	yes	—
9	C₂₃×#334	—	2	8	no	—
10	C₂²×C₈×C₂₃	—	2	4	no	—
11	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₂₃	D₄	1	2	yes	—
14	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
16	C₂₃×C₄⋊C₈	—	2	8	no	—
17	C₂₃×C₄⋊C₈	—	4	8	no	—
18	C₄×C₈×C₂₃	D₄	1	4	yes	—
19	C₄×C₈×C₂₃	Q₈	1	4	yes	—
20	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
21	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₈×C₂₃	—	2	8	no	—
25	C₂×C₈×C₂₃	—	4	4	no	—
26	C₂×C₈×C₂₃	—	4	8	no	—
27	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
28	C₄²×C₂₃	—	2	4	no	—
29	C₄²×C₂₃	—	4	2	no	—
30	C₄²×C₂₃	SD₁₆	1	8	yes	—
31	C₄²×C₂₃	C₂×D₄	1	2	yes	—
32	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
33	#8809	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	#106	1	8	yes	—
39	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
42	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#131	1	4	yes	—
44	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
45	C₈×C₂₃	—	2	8	no	—
46	C₈×C₂₃	—	8	4	no	—
47	#1842	C₂×C₂₃	1	2	yes	—
48	#1744	C₂×C₂₃	1	1	yes	characteristic
49	C₂×#334	C₂×C₂₃	1	4	yes	—
50	C₂²×C₂₃	#384	1	1	yes	characteristic
51	C₂²×C₂₃	#424	1	2	yes	—
52	C₂²×C₂₃	C₂×#106	1	2	yes	—
53	C₂²×C₂₃	#430	1	2	yes	—
54	C₄×C₂₃	—	2	8	no	—
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	#424	1	4	yes	—
57	C₄×C₂₃	#492	1	2	yes	—
58	C₄×C₂₃	#498	1	2	yes	—
59	C₂×C₄⋊C₈	—	2	4	no	—
60	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
61	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
62	C₄³	C₂²×C₂₃	1	1	yes	characteristic
63	#336	C₄×C₂₃	1	4	yes	—
64	#334	—	2	8	no	—
65	C₂×C₂₃	#1570	1	4	yes	—
66	C₂×C₂₃	#1843	1	1	yes	characteristic
67	C₂×C₂₃	#1844	1	1	yes	characteristic
68	C₂×C₂₃	#1849	1	1	yes	characteristic
69	C₂²×C₈	—	2	4	no	—
70	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
71	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
72	C₂×C₄²	D₄×C₂₃	1	2	yes	—
73	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
74	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
75	C₄⋊C₈	—	2	8	no	—
76	C₄⋊C₈	—	4	8	no	—
77	C₄×C₈	D₄×C₂₃	1	4	yes	—
78	C₄×C₈	Q₈×C₂₃	1	4	yes	—
79	C₂₃	#8809	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
81	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
83	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
84	C₂×C₈	—	2	8	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	—	4	8	no	—
87	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
88	C₄²	—	2	4	no	—
89	C₄²	—	4	2	no	—
90	C₄²	SD₁₆×C₂₃	1	8	yes	—
91	C₄²	C₂×D₄×C₂₃	1	2	yes	—
92	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
93	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	C₂₃×#106	1	8	yes	—
98	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
99	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
100	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
101	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄	C₂₃×#131	1	4	yes	—
103	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
104	C₈	—	2	8	no	—
105	C₈	—	8	4	no	—
106	C₂²	C₂₃×#384	1	1	yes	characteristic
107	C₂²	C₂₃×#424	1	2	yes	—
108	C₂²	C₂×C₂₃×#106	1	2	yes	—
109	C₂²	C₂₃×#430	1	2	yes	—
110	C₄	—	2	8	no	—
111	C₄	—	2	2	no	—
112	C₄	C₂₃×#424	1	4	yes	—
113	C₄	C₂₃×#492	1	2	yes	—
114	C₄	C₂₃×#498	1	2	yes	—
115	C₂	C₂₃×#1570	1	4	yes	—
116	C₂	C₂₃×#1843	1	1	yes	characteristic
117	C₂	C₂₃×#1844	1	1	yes	characteristic
118	C₂	C₂₃×#1849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1842

C₂

yes

maximal

C₂₃×#1744

C₂

yes

maximal, characteristic

C₂×C₂₃×#334

C₂

yes

maximal

C₂×C₂₃×C₄⋊C₈

—

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₄³×C₂₃

C₂²

yes

characteristic

C₂₃×#336

C₄

yes

—

C₂₃×#334

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

Q₈

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

SD₁₆

yes

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×Q₈

yes

—

#8809

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#106

yes

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₂₃

#127

yes

characteristic

C₂×C₄×C₂₃

C₄⋊Q₈

yes

—

C₂×C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₂₃

#131

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

#1842

C₂×C₂₃

yes

—

#1744

C₂×C₂₃

yes

characteristic

C₂×#334

C₂×C₂₃

yes

—

C₂²×C₂₃

#384

yes

characteristic

C₂²×C₂₃

#424

yes

—

C₂²×C₂₃

C₂×#106

yes

—

C₂²×C₂₃

#430

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#424

yes

—

C₄×C₂₃

#492

yes

—

C₄×C₂₃

#498

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₄³

C₂²×C₂₃

yes

characteristic

#336

C₄×C₂₃

yes

—

#334

—

C₂×C₂₃

#1570

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

—

C₂×C₄²

D₄×C₂₃

yes

—

C₂×C₄²

Q₈×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

D₄×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

C₂₃

#8809

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄⋊C₄×C₂₃

yes

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

C₂²×C₄

C₂×Q₈×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₄²

—

C₄²

—

C₄²

SD₁₆×C₂₃

yes

—

C₄²

C₂×D₄×C₂₃

yes

—

C₄²

C₂×Q₈×C₂₃

yes

—

C₂³

C₂×C₄⋊C₄×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂₃×#106

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

—

C₂×C₄

C₂₃×#127

yes

characteristic

100

C₂×C₄

C₂₃×C₄⋊Q₈

yes

—

101

C₂×C₄

C₂₃×C₄⋊Q₈

yes

characteristic

102

C₂×C₄

C₂₃×#131

yes

—

103

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

104

C₈

—

105

C₈

—

106

C₂²

C₂₃×#384

yes

characteristic

107

C₂²

C₂₃×#424

yes

—

108

C₂²

C₂×C₂₃×#106

yes

—

109

C₂²

C₂₃×#430

yes

—

110

C₄

—

111

C₄

—

112

C₄

C₂₃×#424

yes

—

113

C₄

C₂₃×#492

yes

—

114

C₄

C₂₃×#498

yes

—

115

C₂

C₂₃×#1570

yes

—

116

C₂

C₂₃×#1843

yes

characteristic

117

C₂

C₂₃×#1844

yes

characteristic

118

C₂

C₂₃×#1849

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

subgroup

G/H

center

C₂²×C₄×C₂₃

C₂×D₄

commutator

C₂²×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

socle, prime radical

C₂³×C₂₃

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

N	H ≅ G/N	type
33. #8809	C₂₃	direct (N × H)
79. C₂₃	#8809	direct (N × H)

H ≅ G/N

type

33. #8809

C₂₃

direct (N × H)

79. C₂₃

#8809

direct (N × H)

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

subgroup	index
C₂₃×#1842	2
C₂₃×#1744	2
C₂×C₂₃×#334	2
#8809	23

subgroup

index

C₂₃×#1842

C₂₃×#1744

C₂×C₂₃×#334

#8809

Subgroups — C₂₃×#8809 #19156435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8809 #19156435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8809 #19156435