Group #15681947 — subgroups

Subgroups — C₂²×Dic₂₃⋊C₈ #15681947

Order 2944.

640 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 389 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×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₂₃⋊C₈	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₂₃	C₄	1	1	yes	characteristic
7	C₂³×Dic₂₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	—	2	4	no	—
9	C₂²×C₈×C₂₃	C₂²	1	6	yes	—
10	C₂×C₄×Dic₂₃	C₄	1	6	yes	—
11	C₂×C₄×Dic₂₃	C₂²	1	6	yes	—
12	C₂²×C₂₃⋊C₈	—	2	4	no	—
13	C₂²×C₂₃⋊C₈	C₂²	1	6	yes	—
14	Dic₂₃⋊C₈	C₂²	1	16	yes	direct product factor
15	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₂₃	D₄	1	4	yes	—
18	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
19	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
20	C₂²×Dic₂₃	—	2	4	no	—
21	C₂²×Dic₂₃	C₈	1	8	yes	complemented
22	C₂²×Dic₂₃	C₂×C₄	1	6	yes	—
23	C₂²×Dic₂₃	C₂×C₄	1	6	yes	—
24	C₂×C₈×C₂₃	—	2	12	no	—
25	C₂×C₈×C₂₃	C₂³	1	4	yes	—
26	C₄×Dic₂₃	C₂×C₄	1	12	yes	—
27	C₄×Dic₂₃	C₂³	1	4	yes	—
28	C₂×C₂₃⋊C₈	—	2	12	no	—
29	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
30	C₂³×C₂₃	C₄⋊C₄	1	4	yes	—
31	C₂³×C₂₃	C₂×C₈	1	4	yes	—
32	C₂³×C₂₃	M₁₆	1	4	yes	—
33	C₂³×C₂₃	C₂²×C₄	1	3	yes	—
34	C₂×C₄×C₂₃	C₄⋊C₄	1	12	yes	—
35	C₂×C₄×C₂₃	C₂²×C₄	1	3	yes	—
36	C₂×C₄×C₂₃	C₂×D₄	1	6	yes	—
37	C₂×C₄×C₂₃	C₂×Q₈	1	6	yes	—
38	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
39	C₂×Dic₂₃	—	2	12	no	—
40	C₂×Dic₂₃	C₂×C₈	1	24	yes	complemented
41	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
42	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
43	C₈×C₂₃	—	2	4	no	—
44	C₂₃⋊C₈	—	2	4	no	—
45	C₂²×C₄⋊C₈	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	C₄⋊C₈	1	16	yes	complemented
47	C₂²×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
48	C₂²×C₂₃	C₂²×C₈	1	6	yes	—
49	C₂²×C₂₃	C₂×M₁₆	1	6	yes	—
50	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
51	C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
52	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
53	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
54	Dic₂₃	—	2	4	no	—
55	Dic₂₃	C₂²×C₈	1	8	yes	complemented
56	C₂³×C₈	—	23	1	no	—
57	C₂³×C₈	D₂₃	1	1	yes	characteristic
58	C₂²×C₄²	—	23	1	no	—
59	C₂×C₄⋊C₈	—	23	12	no	—
60	C₂×C₂₃	C₂×C₄⋊C₈	1	12	yes	complemented
61	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₂₃	C₂³×C₈	1	1	yes	characteristic
63	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
64	C₂³×C₄	—	23	1	no	—
65	C₂³×C₄	—	23	1	no	—
66	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	—	23	6	no	—
69	C₂²×C₈	—	46	4	no	—
70	C₂²×C₈	C₂×D₂₃	1	6	yes	—
71	C₂×C₄²	—	23	6	no	—
72	C₂×C₄²	—	23	6	no	—
73	C₄⋊C₈	—	23	16	no	—
74	C₂₃	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
75	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
76	C₂²×C₄	—	23	8	no	—
77	C₂²×C₄	—	23	6	no	—
78	C₂²×C₄	—	23	6	no	—
79	C₂²×C₄	—	46	4	no	—
80	C₂²×C₄	Dic₄₆	1	4	yes	—
81	C₂²×C₄	C₄×D₂₃	1	3	yes	—
82	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
83	C₂²×C₄	C₂²×D₂₃	1	3	yes	—
84	C₂×C₈	—	2	12	no	—
85	C₂×C₈	—	23	4	no	—
86	C₂×C₈	—	46	12	no	—
87	C₂×C₈	C₂²×D₂₃	1	4	yes	—
88	C₄²	—	23	12	no	—
89	C₄²	—	23	4	no	—
90	C₂³	C₈×D₂₃	1	4	yes	—
91	C₂³	#67262	1	4	yes	—
92	C₂³	Dic₂₃⋊C₄	1	4	yes	—
93	C₂³	C₂×C₄×D₂₃	1	3	yes	—
94	C₂×C₄	—	23	24	no	—
95	C₂×C₄	—	23	4	no	—
96	C₂×C₄	—	23	4	no	—
97	C₂×C₄	—	46	12	no	—
98	C₂×C₄	Dic₂₃⋊C₄	1	12	yes	—
99	C₂×C₄	C₂×Dic₄₆	1	6	yes	—
100	C₂×C₄	C₂×C₄×D₂₃	1	3	yes	—
101	C₂×C₄	C₂×C₂²⋊D₂₃	1	6	yes	—
102	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	46	4	no	—
105	C₂²	Dic₂₃⋊C₈	1	16	yes	direct product factor
106	C₂²	C₂×C₈×D₂₃	1	6	yes	—
107	C₂²	C₂×#67262	1	6	yes	—
108	C₂²	C₂×Dic₂₃⋊C₄	1	6	yes	—
109	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
110	C₄	—	23	8	no	—
111	C₄	—	46	4	no	—
112	C₄	C₂×Dic₂₃⋊C₄	1	6	yes	—
113	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
114	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
115	C₂	C₂×Dic₂₃⋊C₈	1	12	yes	direct product factor
116	C₂	C₂²×Dic₂₃⋊C₄	1	1	yes	characteristic
117	C₂	C₂²×C₈×D₂₃	1	1	yes	characteristic
118	C₂	C₂²×#67262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₂₃⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₂₃

C₂

yes

maximal, characteristic

C₂×Dic₂₃⋊C₈

C₂

yes

maximal, direct product factor

C₂³×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂³×Dic₂₃

C₄

yes

characteristic

C₂³×Dic₂₃

C₄

yes

characteristic

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

Dic₂₃⋊C₈

C₂²

yes

direct product factor

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

C₈

yes

complemented

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂×C₈

yes

—

C₂³×C₂₃

M₁₆

yes

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×C₈

yes

complemented

C₂×Dic₂₃

C₂²×C₄

yes

—

C₂×Dic₂₃

C₂²×C₄

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂²×C₄⋊C₈

—

maximal, Sylow

C₂²×C₂₃

C₄⋊C₈

yes

complemented

C₂²×C₂₃

C₂×C₄⋊C₄

yes

—

C₂²×C₂₃

C₂²×C₈

yes

—

C₂²×C₂₃

C₂×M₁₆

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₄×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

C₂²×Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

C₂²×C₈

yes

complemented

C₂³×C₈

—

C₂³×C₈

D₂₃

yes

characteristic

C₂²×C₄²

—

C₂×C₄⋊C₈

—

C₂×C₂₃

C₂×C₄⋊C₈

yes

complemented

C₂×C₂₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₂₃

C₂³×C₈

yes

characteristic

C₂×C₂₃

C₂²×M₁₆

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂₃

C₂²×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂⁴

C₄×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₄₆

yes

—

C₂²×C₄

C₄×D₂₃

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₂₃

yes

—

C₄²

—

C₄²

—

C₂³

C₈×D₂₃

yes

—

C₂³

#67262

yes

—

C₂³

Dic₂₃⋊C₄

yes

—

C₂³

C₂×C₄×D₂₃

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₂₃⋊C₄

yes

—

C₂×C₄

C₂×Dic₄₆

yes

—

100

C₂×C₄

C₂×C₄×D₂₃

yes

—

101

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

102

C₂×C₄

C₂³×D₂₃

yes

characteristic

103

C₈

—

104

C₈

—

105

C₂²

Dic₂₃⋊C₈

yes

direct product factor

106

C₂²

C₂×C₈×D₂₃

yes

—

107

C₂²

C₂×#67262

yes

—

108

C₂²

C₂×Dic₂₃⋊C₄

yes

—

109

C₂²

C₂²×C₄×D₂₃

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₂×Dic₂₃⋊C₄

yes

—

113

C₄

C₂²×Dic₄₆

yes

characteristic

114

C₄

C₂²×C₂²⋊D₂₃

yes

characteristic

115

C₂

C₂×Dic₂₃⋊C₈

yes

direct product factor

116

C₂

C₂²×Dic₂₃⋊C₄

yes

characteristic

117

C₂

C₂²×C₈×D₂₃

yes

characteristic

118

C₂

C₂²×#67262

yes

characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₂₃
commutator	C₂×C₂₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂³×C₈×C₂₃	C₂
socle, prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂³×C₈	D₂₃
hypocenter	C₂₃	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂³×C₄

C₂×D₂₃

commutator

C₂×C₂₃

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂³×C₈×C₂₃

C₂

socle, prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂×C₄×C₂₃

C₂⁴

hypercenter

C₂³×C₈

D₂₃

hypocenter

C₂₃

C₂²×C₄⋊C₈

LCS

G ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
3. C₂×Dic₂₃⋊C₈	C₂	direct (N × H)
14. Dic₂₃⋊C₈	C₂²	direct (N × H)
21. C₂²×Dic₂₃	C₈	semidirect
40. C₂×Dic₂₃	C₂×C₈	semidirect
46. C₂²×C₂₃	C₄⋊C₈	semidirect
55. Dic₂₃	C₂²×C₈	semidirect
60. C₂×C₂₃	C₂×C₄⋊C₈	semidirect
74. C₂₃	C₂²×C₄⋊C₈	semidirect
105. C₂²	Dic₂₃⋊C₈	direct (N × H)
115. C₂	C₂×Dic₂₃⋊C₈	direct (N × H)

H ≅ G/N

type

3. C₂×Dic₂₃⋊C₈

C₂

direct (N × H)

14. Dic₂₃⋊C₈

C₂²

direct (N × H)

21. C₂²×Dic₂₃

C₈

semidirect

40. C₂×Dic₂₃

C₂×C₈

semidirect

46. C₂²×C₂₃

C₄⋊C₈

semidirect

55. Dic₂₃

C₂²×C₈

semidirect

60. C₂×C₂₃

C₂×C₄⋊C₈

semidirect

74. C₂₃

C₂²×C₄⋊C₈

semidirect

105. C₂²

Dic₂₃⋊C₈

direct (N × H)

115. C₂

C₂×Dic₂₃⋊C₈

direct (N × H)

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

subgroup	index
C₂³×C₂₃⋊C₈	2
C₂²×C₄×Dic₂₃	2
C₂×Dic₂₃⋊C₈	2
C₂³×C₈×C₂₃	2
C₂²×C₄⋊C₈	23

subgroup

index

C₂³×C₂₃⋊C₈

C₂²×C₄×Dic₂₃

C₂×Dic₂₃⋊C₈

C₂³×C₈×C₂₃

C₂²×C₄⋊C₈

Subgroups — C₂²×Dic₂₃⋊C₈ #15681947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×Dic23⋊C8 #15681947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₂₃⋊C₈ #15681947