Group #15669174 — subgroups

Subgroups — C₂₃×#1555 #15669174

Order 2944.

332 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 96 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×#127	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#432	C₂	1	2	yes	maximal, complemented
3	C₂₃×#331	C₂	1	4	yes	maximal, complemented
4	C₂²×D₄×C₂₃	—	2	2	no	—
5	C₂²×D₄×C₂₃	C₄	1	2	yes	—
6	C₂₃×#127	—	2	2	no	—
7	C₂₃×#127	C₄	1	2	yes	—
8	C₂₃×#127	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
11	C₂₃×C₂²⋊C₈	—	2	2	no	—
12	C₂₃×#97	—	2	2	no	—
13	C₂⁴×C₂₃	—	2	2	no	—
14	C₂×D₄×C₂₃	—	2	2	no	—
15	C₂×D₄×C₂₃	—	2	4	no	—
16	C₂×D₄×C₂₃	—	2	4	no	—
17	C₂×D₄×C₂₃	—	4	4	no	—
18	C₂×D₄×C₂₃	—	4	8	no	—
19	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂×C₈×C₂₃	—	2	4	no	—
23	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₃	D₄	1	2	yes	—
25	C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂³×C₂₃	—	2	4	no	—
27	C₂³×C₂₃	—	4	8	no	—
28	C₂³×C₂₃	—	4	4	no	—
29	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₂₃	—	4	4	no	—
31	D₄×C₂₃	—	4	4	no	—
32	D₄×C₂₃	—	4	8	no	—
33	D₄×C₂₃	—	4	8	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
40	C₈×C₂₃	—	4	4	no	—
41	#1555	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₂₃	—	2	2	no	—
43	C₂²×C₂₃	—	4	4	no	—
44	C₂²×C₂₃	—	4	8	no	—
45	C₂²×C₂₃	—	8	8	no	—
46	C₂²×C₂₃	#100	1	2	yes	—
47	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₂₃	—	2	1	no	—
49	C₄×C₂₃	—	2	1	no	—
50	C₄×C₂₃	—	2	2	no	—
51	C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×#127	C₂×C₂₃	1	1	yes	characteristic
53	#432	C₂×C₂₃	1	2	yes	complemented
54	#331	C₂×C₂₃	1	4	yes	complemented
55	C₂×C₂₃	—	2	2	no	—
56	C₂×C₂₃	—	8	4	no	—
57	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
58	C₂×C₂₃	#418	1	2	yes	—
59	C₂²×D₄	—	2	2	no	—
60	C₂²×D₄	C₄×C₂₃	1	2	yes	—
61	#127	—	2	2	no	—
62	#127	C₄×C₂₃	1	2	yes	—
63	#127	C₂²×C₂₃	1	2	yes	—
64	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
65	C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
66	C₂²⋊C₈	—	2	2	no	—
67	#97	—	2	2	no	—
68	C₂₃	#1555	1	1	yes	characteristic, direct product factor, Sylow
69	C₂⁴	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
76	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
77	C₂²×C₄	D₄×C₂₃	1	2	yes	—
78	C₂×C₈	—	2	4	no	—
79	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
80	C₄²	D₄×C₂₃	1	2	yes	—
81	C₄²	C₂³×C₂₃	1	1	yes	characteristic
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	8	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
93	C₂×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
94	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
95	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
96	C₈	—	4	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	4	no	—
99	C₂²	—	4	8	no	—
100	C₂²	—	8	8	no	—
101	C₂²	C₂₃×#100	1	2	yes	—
102	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×C₂₃×#100	1	1	yes	characteristic
110	C₂	C₂₃×#418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#127

C₂

yes

maximal, characteristic

C₂₃×#432

C₂

yes

maximal, complemented

C₂₃×#331

C₂

yes

maximal, complemented

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

C₄

yes

—

C₂₃×#127

—

C₂₃×#127

C₄

yes

—

C₂₃×#127

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

complemented

C₂₃×C₂²⋊C₈

—

C₂₃×#97

—

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

—

#1555

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#100

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₂×#127

C₂×C₂₃

yes

characteristic

#432

C₂×C₂₃

yes

complemented

#331

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#100

yes

characteristic

C₂×C₂₃

#418

yes

—

C₂²×D₄

—

C₂²×D₄

C₄×C₂₃

yes

—

#127

—

#127

C₄×C₂₃

yes

—

#127

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

C₄⋊C₈

C₂²×C₂₃

yes

complemented

C₂²⋊C₈

—

#97

—

C₂₃

#1555

yes

characteristic, direct product factor, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₄

D₄×C₂₃

yes

—

C₂×C₈

—

C₄²

C₂×C₄×C₂₃

yes

characteristic

C₄²

D₄×C₂₃

yes

—

C₄²

C₂³×C₂₃

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₂₃

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₃

yes

—

C₂×C₄

C₂²⋊C₄×C₂₃

yes

characteristic

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂×C₄

C₂×D₄×C₂₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

C₂₃×#100

yes

—

102

C₂²

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₂₃×AGL(1,ℤ/8)

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×C₂₃×#100

yes

characteristic

110

C₂

C₂₃×#418

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₄²

C₂³×C₂₃

prime radical

C₂×C₂₃×#127

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#432	C₂	semidirect
3. C₂₃×#331	C₂	semidirect
10. C₂₃×C₄⋊C₈	C₂²	semidirect
41. #1555	C₂₃	direct (N × H)
53. #432	C₂×C₂₃	semidirect
54. #331	C₂×C₂₃	semidirect
65. C₄⋊C₈	C₂²×C₂₃	semidirect
68. C₂₃	#1555	direct (N × H)

H ≅ G/N

type

2. C₂₃×#432

C₂

semidirect

3. C₂₃×#331

C₂

semidirect

10. C₂₃×C₄⋊C₈

C₂²

semidirect

41. #1555

C₂₃

direct (N × H)

53. #432

C₂×C₂₃

semidirect

54. #331

C₂×C₂₃

semidirect

65. C₄⋊C₈

C₂²×C₂₃

semidirect

68. C₂₃

#1555

direct (N × H)

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

subgroup	index
C₂×C₂₃×#127	2
C₂₃×#432	2
C₂₃×#331	2
#1555	23

subgroup

index

C₂×C₂₃×#127

C₂₃×#432

C₂₃×#331

#1555

Subgroups — C₂₃×#1555 #15669174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1555 #15669174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1555 #15669174