Group #19154765 — subgroups

Subgroups — C₂₃×#7139 #19154765

Order 5888.

262 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₄×C₂₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×M₃₂	C₄	1	4	yes	—
4	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#346	—	2	2	no	—
7	C₄×C₁₆×C₂₃	—	2	2	no	—
8	C₂₃×#322	—	2	2	no	—
9	C₂²×C₈×C₂₃	C₈	1	4	yes	—
10	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂₃×M₃₂	—	2	4	no	—
13	C₂₃×M₃₂	C₈	1	8	yes	complemented
14	C₂×C₁₆×C₂₃	—	2	4	no	—
15	C₄×C₈×C₂₃	—	2	2	no	—
16	C₄×C₈×C₂₃	—	2	2	no	—
17	C₄×C₈×C₂₃	D₄	1	2	yes	—
18	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
19	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₂₃	—	2	4	no	—
23	C₂×C₈×C₂₃	—	4	4	no	—
24	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
27	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
28	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄²×C₂₃	M₁₆	1	2	yes	—
31	C₁₆×C₂₃	—	2	8	no	—
32	#7139	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₂₃	C₄×C₈	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₄×C₈	1	2	yes	—
37	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
38	C₂×C₄×C₂₃	#97	1	2	yes	—
39	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
40	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
41	C₈×C₂₃	—	4	4	no	—
42	C₈×C₂₃	C₂²⋊C₈	1	4	yes	—
43	C₈×C₂₃	C₄⋊C₈	1	4	yes	—
44	C₄×M₃₂	C₂×C₂₃	1	2	yes	—
45	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	C₈²	1	1	yes	characteristic
48	C₂²×C₂₃	#322	1	1	yes	characteristic
49	C₂²×C₂₃	#336	1	1	yes	characteristic
50	C₄×C₂₃	—	2	2	no	—
51	C₄×C₂₃	#322	1	2	yes	—
52	C₄×C₂₃	#336	1	2	yes	—
53	C₄×C₂₃	#349	1	4	yes	—
54	C₂×M₃₂	C₄×C₂₃	1	4	yes	—
55	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
56	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
57	#346	—	2	2	no	—
58	C₄×C₁₆	—	2	2	no	—
59	#322	—	2	2	no	—
60	C₂×C₂₃	—	2	1	no	—
61	C₂×C₂₃	—	2	1	no	—
62	C₂×C₂₃	#1273	1	1	yes	characteristic
63	C₂×C₂₃	#1377	1	2	yes	—
64	C₂²×C₈	C₈×C₂₃	1	4	yes	—
65	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
66	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
67	M₃₂	—	2	4	no	—
68	M₃₂	C₈×C₂₃	1	8	yes	complemented
69	C₂×C₁₆	—	2	4	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	D₄×C₂₃	1	2	yes	—
73	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
74	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
75	C₂₃	#7139	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
77	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
81	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
82	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
83	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
84	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
85	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
86	C₄²	M₁₆×C₂₃	1	2	yes	—
87	C₁₆	—	2	8	no	—
88	C₂³	C₄×C₈×C₂₃	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
91	C₂×C₄	C₄×C₈×C₂₃	1	2	yes	—
92	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
93	C₂×C₄	C₂₃×#97	1	2	yes	—
94	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
95	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
96	C₈	—	4	4	no	—
97	C₈	C₂₃×C₂²⋊C₈	1	4	yes	—
98	C₈	C₂₃×C₄⋊C₈	1	4	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	C₈²×C₂₃	1	1	yes	characteristic
101	C₂²	C₂₃×#322	1	1	yes	characteristic
102	C₂²	C₂₃×#336	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	C₂₃×#322	1	2	yes	—
105	C₄	C₂₃×#336	1	2	yes	—
106	C₄	C₂₃×#349	1	4	yes	—
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	C₂₃×#1273	1	1	yes	characteristic
110	C₂	C₂₃×#1377	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₂₃×M₃₂

C₂

yes

maximal

C₂×C₂₃×#322

C₂

yes

maximal, characteristic

C₂×C₂₃×M₃₂

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#346

—

C₄×C₁₆×C₂₃

—

C₂₃×#322

—

C₂²×C₈×C₂₃

C₈

yes

—

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂₃×M₃₂

—

C₂₃×M₃₂

C₈

yes

complemented

C₂×C₁₆×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

D₄

yes

characteristic

C₄×C₈×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₄²

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₂×C₈

yes

—

C₂×C₈×C₂₃

M₁₆

yes

—

C₄²×C₂₃

C₂²⋊C₄

yes

characteristic

C₄²×C₂₃

C₄⋊C₄

yes

characteristic

C₄²×C₂₃

M₁₆

yes

—

C₁₆×C₂₃

—

#7139

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₄×C₈

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#95

yes

characteristic

C₂×C₄×C₂₃

C₄×C₈

yes

—

C₂×C₄×C₂₃

#97

yes

characteristic

C₂×C₄×C₂₃

#97

yes

—

C₂×C₄×C₂₃

C₂²⋊C₈

yes

—

C₂×C₄×C₂₃

C₄⋊C₈

yes

—

C₈×C₂₃

—

C₈×C₂₃

C₂²⋊C₈

yes

—

C₈×C₂₃

C₄⋊C₈

yes

—

C₄×M₃₂

C₂×C₂₃

yes

—

C₂×#322

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

C₈²

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

#322

yes

—

C₄×C₂₃

#336

yes

—

C₄×C₂₃

#349

yes

—

C₂×M₃₂

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#346

—

C₄×C₁₆

—

#322

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1273

yes

characteristic

C₂×C₂₃

#1377

yes

—

C₂²×C₈

C₈×C₂₃

yes

—

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

M₃₂

—

M₃₂

C₈×C₂₃

yes

complemented

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

characteristic

C₄×C₈

Q₈×C₂₃

yes

characteristic

C₂₃

#7139

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄²×C₂₃

yes

characteristic

C₂²×C₄

C₂×C₈×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

C₂×C₈×C₂₃

yes

—

C₂×C₈

M₁₆×C₂₃

yes

—

C₄²

C₂²⋊C₄×C₂₃

yes

characteristic

C₄²

C₄⋊C₄×C₂₃

yes

characteristic

C₄²

M₁₆×C₂₃

yes

—

C₁₆

—

C₂³

C₄×C₈×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂₃×#95

yes

characteristic

C₂×C₄

C₄×C₈×C₂₃

yes

—

C₂×C₄

C₂₃×#97

yes

characteristic

C₂×C₄

C₂₃×#97

yes

—

C₂×C₄

C₂₃×C₂²⋊C₈

yes

—

C₂×C₄

C₂₃×C₄⋊C₈

yes

—

C₈

—

C₈

C₂₃×C₂²⋊C₈

yes

—

C₈

C₂₃×C₄⋊C₈

yes

—

C₂²

—

100

C₂²

C₈²×C₂₃

yes

characteristic

101

C₂²

C₂₃×#322

yes

characteristic

102

C₂²

C₂₃×#336

yes

characteristic

103

C₄

—

104

C₄

C₂₃×#322

yes

—

105

C₄

C₂₃×#336

yes

—

106

C₄

C₂₃×#349

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂₃×#1273

yes

characteristic

110

C₂

C₂₃×#1377

yes

—

	subgroup	G/H
center	C₄²×C₂₃	C₂²⋊C₄
commutator	C₂²	C₈²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#336
prime radical	C₂³×C₂₃	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₄²×C₂₃

C₂²⋊C₄

commutator

C₂²

C₈²×C₂₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₂₃

socle

C₂²×C₂₃

#336

prime radical

C₂³×C₂₃

C₄×C₈

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
13. C₂₃×M₃₂	C₈	semidirect
32. #7139	C₂₃	direct (N × H)
68. M₃₂	C₈×C₂₃	semidirect
75. C₂₃	#7139	direct (N × H)

H ≅ G/N

type

13. C₂₃×M₃₂

C₈

semidirect

32. #7139

C₂₃

direct (N × H)

68. M₃₂

C₈×C₂₃

semidirect

75. C₂₃

#7139

direct (N × H)

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

subgroup	index
C₄×C₂₃×M₃₂	2
C₂×C₂₃×#322	2
#7139	23

subgroup

index

C₄×C₂₃×M₃₂

C₂×C₂₃×#322

#7139

Subgroups — C₂₃×#7139 #19154765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#7139 #19154765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7139 #19154765