Group #15670521 — subgroups

Subgroups — C₂²×C₂₃×C₄⋊C₈ #15670521

Order 2944.

640 conjugacy classes of subgroups; only the 520 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 520 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×C₈×C₂₃	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×C₄⋊C₈	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	—	2	8	no	—
8	C₂²×C₈×C₂₃	C₂²	1	12	yes	—
9	C₂×C₄²×C₂₃	C₄	1	6	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	6	yes	—
11	C₂₃×C₄⋊C₈	C₂²	1	16	yes	direct product factor
12	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	—	2	4	no	—
14	C₂²×C₄×C₂₃	C₈	1	8	yes	complemented
15	C₂²×C₄×C₂₃	C₂×C₄	1	6	yes	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	6	yes	—
18	C₂²×C₄×C₂₃	D₄	1	4	yes	—
19	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
20	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
21	C₂×C₈×C₂₃	—	2	24	no	—
22	C₂×C₈×C₂₃	C₂³	1	8	yes	—
23	C₄²×C₂₃	C₂×C₄	1	12	yes	—
24	C₄²×C₂₃	C₂³	1	4	yes	—
25	C₂³×C₂₃	C₄⋊C₄	1	4	yes	—
26	C₂³×C₂₃	C₂×C₈	1	4	yes	—
27	C₂³×C₂₃	M₁₆	1	4	yes	—
28	C₂³×C₂₃	C₂²×C₄	1	3	yes	—
29	C₂×C₄×C₂₃	—	2	12	no	—
30	C₂×C₄×C₂₃	C₄⋊C₄	1	12	yes	—
31	C₂×C₄×C₂₃	C₂×C₈	1	24	yes	complemented
32	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
33	C₂×C₄×C₂₃	C₂²×C₄	1	3	yes	—
34	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
35	C₂×C₄×C₂₃	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₂₃	C₂×Q₈	1	6	yes	—
37	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
38	C₈×C₂₃	—	2	8	no	—
39	C₂²×C₄⋊C₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₂₃	C₄⋊C₈	1	16	yes	direct product factor
41	C₂²×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
42	C₂²×C₂₃	C₂²×C₈	1	6	yes	—
43	C₂²×C₂₃	C₂×M₁₆	1	6	yes	—
44	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
45	C₄×C₂₃	—	2	4	no	—
46	C₄×C₂₃	C₂×C₄⋊C₄	1	6	yes	—
47	C₄×C₂₃	C₂²×C₈	1	8	yes	complemented
48	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
50	C₂³×C₈	C₂×C₂₃	1	2	yes	—
51	C₂²×C₄²	C₂×C₂₃	1	1	yes	characteristic
52	C₂×C₄⋊C₈	C₂×C₂₃	1	12	yes	direct product factor
53	C₂×C₂₃	C₂×C₄⋊C₈	1	12	yes	direct product factor
54	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₂₃	C₂³×C₈	1	1	yes	characteristic
56	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
57	C₂³×C₄	C₄×C₂₃	1	1	yes	characteristic
58	C₂³×C₄	C₄×C₂₃	1	1	yes	characteristic
59	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
60	C₂²×C₈	—	2	8	no	—
61	C₂²×C₈	C₂²×C₂₃	1	12	yes	—
62	C₂×C₄²	C₄×C₂₃	1	6	yes	—
63	C₂×C₄²	C₂²×C₂₃	1	6	yes	—
64	C₄⋊C₈	C₂²×C₂₃	1	16	yes	direct product factor
65	C₂₃	C₂²×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
66	C₂⁴	C₂×C₄×C₂₃	1	1	yes	characteristic
67	C₂²×C₄	—	2	4	no	—
68	C₂²×C₄	C₈×C₂₃	1	8	yes	complemented
69	C₂²×C₄	C₂×C₄×C₂₃	1	6	yes	—
70	C₂²×C₄	C₂×C₄×C₂₃	1	3	yes	—
71	C₂²×C₄	C₂×C₄×C₂₃	1	6	yes	—
72	C₂²×C₄	D₄×C₂₃	1	4	yes	—
73	C₂²×C₄	Q₈×C₂₃	1	4	yes	—
74	C₂²×C₄	C₂³×C₂₃	1	3	yes	—
75	C₂×C₈	—	2	24	no	—
76	C₂×C₈	C₂³×C₂₃	1	8	yes	—
77	C₄²	C₂×C₄×C₂₃	1	12	yes	—
78	C₄²	C₂³×C₂₃	1	4	yes	—
79	C₂³	C₄⋊C₄×C₂₃	1	4	yes	—
80	C₂³	C₂×C₈×C₂₃	1	4	yes	—
81	C₂³	M₁₆×C₂₃	1	4	yes	—
82	C₂³	C₂²×C₄×C₂₃	1	3	yes	—
83	C₂×C₄	—	2	12	no	—
84	C₂×C₄	C₄⋊C₄×C₂₃	1	12	yes	—
85	C₂×C₄	C₂×C₈×C₂₃	1	24	yes	complemented
86	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
87	C₂×C₄	C₂²×C₄×C₂₃	1	3	yes	—
88	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
89	C₂×C₄	C₂×D₄×C₂₃	1	6	yes	—
90	C₂×C₄	C₂×Q₈×C₂₃	1	6	yes	—
91	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
92	C₈	—	2	8	no	—
93	C₂²	C₂₃×C₄⋊C₈	1	16	yes	direct product factor
94	C₂²	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
95	C₂²	C₂²×C₈×C₂₃	1	6	yes	—
96	C₂²	C₂×M₁₆×C₂₃	1	6	yes	—
97	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	C₂×C₄⋊C₄×C₂₃	1	6	yes	—
100	C₄	C₂²×C₈×C₂₃	1	8	yes	complemented
101	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
102	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
103	C₂	C₂×C₂₃×C₄⋊C₈	1	12	yes	direct product factor
104	C₂	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
105	C₂	C₂³×C₈×C₂₃	1	1	yes	characteristic
106	C₂	C₂²×M₁₆×C₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₈×C₂₃

C₂

yes

maximal

C₂²×C₄²×C₂₃

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, direct product factor

C₂³×C₄×C₂₃

C₄

yes

characteristic

C₂³×C₄×C₂₃

C₄

yes

characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂₃×C₄⋊C₈

C₂²

yes

direct product factor

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₈

yes

complemented

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂³

yes

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂×C₈

yes

—

C₂³×C₂₃

M₁₆

yes

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

complemented

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×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

C₈×C₂₃

—

C₂²×C₄⋊C₈

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

C₄⋊C₈

yes

direct product factor

C₂²×C₂₃

C₂×C₄⋊C₄

yes

—

C₂²×C₂₃

C₂²×C₈

yes

—

C₂²×C₂₃

C₂×M₁₆

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₄×C₂₃

C₂²×C₈

yes

complemented

C₄×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

C₂²×Q₈

yes

characteristic

C₂³×C₈

C₂×C₂₃

yes

—

C₂²×C₄²

C₂×C₂₃

yes

characteristic

C₂×C₄⋊C₈

C₂×C₂₃

yes

direct product factor

C₂×C₂₃

C₂×C₄⋊C₈

yes

direct product factor

C₂×C₂₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₂₃

C₂³×C₈

yes

characteristic

C₂×C₂₃

C₂²×M₁₆

yes

characteristic

C₂³×C₄

C₄×C₂₃

yes

characteristic

C₂³×C₄

C₄×C₂₃

yes

characteristic

C₂³×C₄

C₂²×C₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²×C₂₃

yes

—

C₂×C₄²

C₄×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

C₄⋊C₈

C₂²×C₂₃

yes

direct product factor

C₂₃

C₂²×C₄⋊C₈

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₈×C₂₃

yes

complemented

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

Q₈×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

C₂³×C₂₃

yes

—

C₄²

C₂×C₄×C₂₃

yes

—

C₄²

C₂³×C₂₃

yes

—

C₂³

C₄⋊C₄×C₂₃

yes

—

C₂³

C₂×C₈×C₂₃

yes

—

C₂³

M₁₆×C₂₃

yes

—

C₂³

C₂²×C₄×C₂₃

yes

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₂₃

yes

—

C₂×C₄

C₂×C₈×C₂₃

yes

complemented

C₂×C₄

C₂²×C₄×C₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₃

yes

—

C₂×C₄

C₂×D₄×C₂₃

yes

—

C₂×C₄

C₂×Q₈×C₂₃

yes

—

C₂×C₄

C₂⁴×C₂₃

yes

characteristic

C₈

—

C₂²

C₂₃×C₄⋊C₈

yes

direct product factor

C₂²

C₂×C₄⋊C₄×C₂₃

yes

—

C₂²

C₂²×C₈×C₂₃

yes

—

C₂²

C₂×M₁₆×C₂₃

yes

—

C₂²

C₂³×C₄×C₂₃

yes

characteristic

C₄

—

C₄

C₂×C₄⋊C₄×C₂₃

yes

—

100

C₄

C₂²×C₈×C₂₃

yes

complemented

101

C₄

C₂²×D₄×C₂₃

yes

characteristic

102

C₄

C₂²×Q₈×C₂₃

yes

characteristic

103

C₂

C₂×C₂₃×C₄⋊C₈

yes

direct product factor

104

C₂

C₂²×C₄⋊C₄×C₂₃

yes

characteristic

105

C₂

C₂³×C₈×C₂₃

yes

characteristic

106

C₂

C₂²×M₁₆×C₂₃

yes

characteristic

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

subgroup

G/H

center

C₂³×C₄×C₂₃

C₂²

commutator

C₂

C₂³×C₈×C₂₃

Frattini, cosocle

C₂×C₄

C₂⁴×C₂₃

socle, prime radical

C₂⁴×C₂₃

C₂×C₄

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
3. C₂×C₂₃×C₄⋊C₈	C₂	direct (N × H)
11. C₂₃×C₄⋊C₈	C₂²	direct (N × H)
14. C₂²×C₄×C₂₃	C₈	semidirect
31. C₂×C₄×C₂₃	C₂×C₈	semidirect
39. C₂²×C₄⋊C₈	C₂₃	direct (N × H)
40. C₂²×C₂₃	C₄⋊C₈	direct (N × H)
47. C₄×C₂₃	C₂²×C₈	semidirect
52. C₂×C₄⋊C₈	C₂×C₂₃	direct (N × H)
53. C₂×C₂₃	C₂×C₄⋊C₈	direct (N × H)
64. C₄⋊C₈	C₂²×C₂₃	direct (N × H)
65. C₂₃	C₂²×C₄⋊C₈	direct (N × H)
68. C₂²×C₄	C₈×C₂₃	semidirect
85. C₂×C₄	C₂×C₈×C₂₃	semidirect
93. C₂²	C₂₃×C₄⋊C₈	direct (N × H)
100. C₄	C₂²×C₈×C₂₃	semidirect
103. C₂	C₂×C₂₃×C₄⋊C₈	direct (N × H)

H ≅ G/N

type

3. C₂×C₂₃×C₄⋊C₈

C₂

direct (N × H)

11. C₂₃×C₄⋊C₈

C₂²

direct (N × H)

14. C₂²×C₄×C₂₃

C₈

semidirect

31. C₂×C₄×C₂₃

C₂×C₈

semidirect

39. C₂²×C₄⋊C₈

C₂₃

direct (N × H)

40. C₂²×C₂₃

C₄⋊C₈

direct (N × H)

47. C₄×C₂₃

C₂²×C₈

semidirect

52. C₂×C₄⋊C₈

C₂×C₂₃

direct (N × H)

53. C₂×C₂₃

C₂×C₄⋊C₈

direct (N × H)

64. C₄⋊C₈

C₂²×C₂₃

direct (N × H)

65. C₂₃

C₂²×C₄⋊C₈

direct (N × H)

68. C₂²×C₄

C₈×C₂₃

semidirect

85. C₂×C₄

C₂×C₈×C₂₃

semidirect

93. C₂²

C₂₃×C₄⋊C₈

direct (N × H)

100. C₄

C₂²×C₈×C₂₃

semidirect

103. C₂

C₂×C₂₃×C₄⋊C₈

direct (N × H)

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

subgroup	index
C₂³×C₈×C₂₃	2
C₂²×C₄²×C₂₃	2
C₂×C₂₃×C₄⋊C₈	2
C₂²×C₄⋊C₈	23

subgroup

index

C₂³×C₈×C₂₃

C₂²×C₄²×C₂₃

C₂×C₂₃×C₄⋊C₈

C₂²×C₄⋊C₈

Subgroups — C₂²×C₂₃×C₄⋊C₈ #15670521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C23×C4⋊C8 #15670521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×C₄⋊C₈ #15670521