Group #43251164 — subgroups

Subgroups — C₂×C₄⋊C₁₆×C₂₂₃ #43251164

Order 28544.

184 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 152 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	C₂²×C₁₆×C₂₂₃	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₂₃	—	2	4	no	—
8	C₂×C₁₆×C₂₂₃	C₂²	1	4	yes	—
9	C₄×C₈×C₂₂₃	C₄	1	2	yes	—
10	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₂₃	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₂₂₃	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	—	2	2	no	—
15	C₂×C₈×C₂₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
18	C₂×C₈×C₂₂₃	Q₈	1	2	yes	—
19	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₄²×C₂₂₃	C₈	1	2	yes	—
21	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₁₆×C₂₂₃	—	2	4	no	—
23	C₂³×C₂₂₃	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	—	2	2	no	—
25	C₂×C₄×C₂₂₃	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₂₂₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₂₂₃	—	2	2	no	—
33	C₈×C₂₂₃	C₄⋊C₄	1	2	yes	—
34	C₈×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₂₂₃	C₄⋊C₈	1	2	yes	—
37	C₂²×C₂₂₃	C₂×C₁₆	1	2	yes	—
38	C₂²×C₂₂₃	M₃₂	1	2	yes	—
39	C₂²×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₂₂₃	—	2	2	no	—
41	C₄×C₂₂₃	C₄⋊C₈	1	2	yes	—
42	C₄×C₂₂₃	C₂×C₁₆	1	4	yes	complemented
43	C₄×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₂₂₃	C₄⋊C₁₆	1	4	yes	direct product factor
46	C₂×C₂₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₂₂₃	C₂²×C₁₆	1	1	yes	characteristic
48	C₂×C₂₂₃	C₂×M₃₂	1	1	yes	characteristic
49	C₂₂₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₄⋊C₁₆	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₁₆	C₂×C₂₂₃	1	2	yes	—
52	C₂×C₄×C₈	C₂×C₂₂₃	1	1	yes	characteristic
53	C₄⋊C₁₆	C₂×C₂₂₃	1	4	yes	direct product factor
54	C₂²×C₈	C₄×C₂₂₃	1	1	yes	characteristic
55	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
56	C₂×C₄²	C₄×C₂₂₃	1	1	yes	characteristic
57	C₂×C₁₆	—	2	4	no	—
58	C₂×C₁₆	C₂²×C₂₂₃	1	4	yes	—
59	C₄×C₈	C₄×C₂₂₃	1	2	yes	—
60	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
61	C₂²×C₄	C₈×C₂₂₃	1	1	yes	characteristic
62	C₂²×C₄	C₈×C₂₂₃	1	1	yes	characteristic
63	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	C₂×C₄×C₂₂₃	1	2	yes	—
66	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
67	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
68	C₂×C₈	Q₈×C₂₂₃	1	2	yes	—
69	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
70	C₄²	C₈×C₂₂₃	1	2	yes	—
71	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
72	C₁₆	—	2	4	no	—
73	C₂³	C₂×C₈×C₂₂₃	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	C₁₆×C₂₂₃	1	4	yes	complemented
76	C₂×C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
77	C₂×C₄	C₂×C₈×C₂₂₃	1	2	yes	—
78	C₂×C₄	C₂×C₈×C₂₂₃	1	1	yes	characteristic
79	C₂×C₄	C₂×C₈×C₂₂₃	1	2	yes	—
80	C₂×C₄	M₁₆×C₂₂₃	1	2	yes	—
81	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	C₄⋊C₄×C₂₂₃	1	2	yes	—
84	C₈	C₂×D₄×C₂₂₃	1	1	yes	characteristic
85	C₈	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
86	C₂²	C₄⋊C₈×C₂₂₃	1	2	yes	—
87	C₂²	C₂×C₁₆×C₂₂₃	1	2	yes	—
88	C₂²	M₃₂×C₂₂₃	1	2	yes	—
89	C₂²	C₂²×C₈×C₂₂₃	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	C₄⋊C₈×C₂₂₃	1	2	yes	—
92	C₄	C₂×C₁₆×C₂₂₃	1	4	yes	complemented
93	C₄	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
94	C₄	C₂×M₁₆×C₂₂₃	1	1	yes	characteristic
95	C₂	C₄⋊C₁₆×C₂₂₃	1	4	yes	direct product factor
96	C₂	C₂×C₄⋊C₈×C₂₂₃	1	1	yes	characteristic
97	C₂	C₂²×C₁₆×C₂₂₃	1	1	yes	characteristic
98	C₂	C₂×M₃₂×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×C₂₂₃

C₂

yes

maximal

C₂×C₄×C₈×C₂₂₃

C₂

yes

maximal, characteristic

C₄⋊C₁₆×C₂₂₃

C₂

yes

maximal, direct product factor

C₂²×C₈×C₂₂₃

C₄

yes

characteristic

C₂²×C₈×C₂₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₂₃

C₄

yes

characteristic

C₂×C₁₆×C₂₂₃

—

C₂×C₁₆×C₂₂₃

C₂²

yes

—

C₄×C₈×C₂₂₃

C₄

yes

—

C₄×C₈×C₂₂₃

C₂²

yes

—

C₂²×C₄×C₂₂₃

C₈

yes

characteristic

C₂²×C₄×C₂₂₃

C₈

yes

characteristic

C₂²×C₄×C₂₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₂₃

D₄

yes

—

C₂×C₈×C₂₂₃

Q₈

yes

—

C₂×C₈×C₂₂₃

C₂³

yes

characteristic

C₄²×C₂₂₃

C₈

yes

—

C₄²×C₂₂₃

C₂×C₄

yes

—

C₁₆×C₂₂₃

—

C₂³×C₂₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₁₆

yes

complemented

C₂×C₄×C₂₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₂₃

M₁₆

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₂₃

—

C₈×C₂₂₃

C₄⋊C₄

yes

—

C₈×C₂₂₃

C₂×D₄

yes

characteristic

C₈×C₂₂₃

C₂×Q₈

yes

characteristic

C₂²×C₂₂₃

C₄⋊C₈

yes

—

C₂²×C₂₂₃

C₂×C₁₆

yes

—

C₂²×C₂₂₃

M₃₂

yes

—

C₂²×C₂₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

C₄⋊C₈

yes

—

C₄×C₂₂₃

C₂×C₁₆

yes

complemented

C₄×C₂₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₂₃

C₂×M₁₆

yes

characteristic

C₂×C₂₂₃

C₄⋊C₁₆

yes

direct product factor

C₂×C₂₂₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₂₂₃

C₂²×C₁₆

yes

characteristic

C₂×C₂₂₃

C₂×M₃₂

yes

characteristic

C₂₂₃

C₂×C₄⋊C₁₆

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊C₁₆

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂×C₂₂₃

yes

—

C₂×C₄×C₈

C₂×C₂₂₃

yes

characteristic

C₄⋊C₁₆

C₂×C₂₂₃

yes

direct product factor

C₂²×C₈

C₄×C₂₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₂₃

yes

characteristic

C₂×C₄²

C₄×C₂₂₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₂₂₃

yes

—

C₄×C₈

C₄×C₂₂₃

yes

—

C₄×C₈

C₂²×C₂₂₃

yes

—

C₂²×C₄

C₈×C₂₂₃

yes

characteristic

C₂²×C₄

C₈×C₂₂₃

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₂₃

yes

—

C₂×C₈

C₂×C₄×C₂₂₃

yes

characteristic

C₂×C₈

D₄×C₂₂₃

yes

—

C₂×C₈

Q₈×C₂₂₃

yes

—

C₂×C₈

C₂³×C₂₂₃

yes

characteristic

C₄²

C₈×C₂₂₃

yes

—

C₄²

C₂×C₄×C₂₂₃

yes

—

C₁₆

—

C₂³

C₂×C₈×C₂₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₁₆×C₂₂₃

yes

complemented

C₂×C₄

C₄⋊C₄×C₂₂₃

yes

—

C₂×C₄

C₂×C₈×C₂₂₃

yes

—

C₂×C₄

C₂×C₈×C₂₂₃

yes

characteristic

C₂×C₄

C₂×C₈×C₂₂₃

yes

—

C₂×C₄

M₁₆×C₂₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₂₃

yes

characteristic

C₈

—

C₈

C₄⋊C₄×C₂₂₃

yes

—

C₈

C₂×D₄×C₂₂₃

yes

characteristic

C₈

C₂×Q₈×C₂₂₃

yes

characteristic

C₂²

C₄⋊C₈×C₂₂₃

yes

—

C₂²

C₂×C₁₆×C₂₂₃

yes

—

C₂²

M₃₂×C₂₂₃

yes

—

C₂²

C₂²×C₈×C₂₂₃

yes

characteristic

C₄

—

C₄

C₄⋊C₈×C₂₂₃

yes

—

C₄

C₂×C₁₆×C₂₂₃

yes

complemented

C₄

C₂×C₄⋊C₄×C₂₂₃

yes

characteristic

C₄

C₂×M₁₆×C₂₂₃

yes

characteristic

C₂

C₄⋊C₁₆×C₂₂₃

yes

direct product factor

C₂

C₂×C₄⋊C₈×C₂₂₃

yes

characteristic

C₂

C₂²×C₁₆×C₂₂₃

yes

characteristic

C₂

C₂×M₃₂×C₂₂₃

yes

characteristic

	subgroup	G/H
center	C₂²×C₈×C₂₂₃	C₂²
commutator	C₂	C₂²×C₁₆×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
socle, prime radical	C₂³×C₂₂₃	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₂₃ ◁ G

subgroup

G/H

center

C₂²×C₈×C₂₂₃

C₂²

commutator

C₂

C₂²×C₁₆×C₂₂₃

Frattini, cosocle

C₂×C₈

C₂³×C₂₂₃

socle, prime radical

C₂³×C₂₂₃

C₂×C₈

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₂₂₃ ◁ G

N	H ≅ G/N	type
3. C₄⋊C₁₆×C₂₂₃	C₂	direct (N × H)
25. C₂×C₄×C₂₂₃	C₁₆	semidirect
42. C₄×C₂₂₃	C₂×C₁₆	semidirect
45. C₂×C₂₂₃	C₄⋊C₁₆	direct (N × H)
49. C₂₂₃	C₂×C₄⋊C₁₆	direct (N × H)
50. C₂×C₄⋊C₁₆	C₂₂₃	direct (N × H)
53. C₄⋊C₁₆	C₂×C₂₂₃	direct (N × H)
75. C₂×C₄	C₁₆×C₂₂₃	semidirect
92. C₄	C₂×C₁₆×C₂₂₃	semidirect
95. C₂	C₄⋊C₁₆×C₂₂₃	direct (N × H)

H ≅ G/N

type

3. C₄⋊C₁₆×C₂₂₃

C₂

direct (N × H)

25. C₂×C₄×C₂₂₃

C₁₆

semidirect

42. C₄×C₂₂₃

C₂×C₁₆

semidirect

45. C₂×C₂₂₃

C₄⋊C₁₆

direct (N × H)

49. C₂₂₃

C₂×C₄⋊C₁₆

direct (N × H)

50. C₂×C₄⋊C₁₆

C₂₂₃

direct (N × H)

53. C₄⋊C₁₆

C₂×C₂₂₃

direct (N × H)

75. C₂×C₄

C₁₆×C₂₂₃

semidirect

92. C₄

C₂×C₁₆×C₂₂₃

semidirect

95. C₂

C₄⋊C₁₆×C₂₂₃

direct (N × H)

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

subgroup	index
C₂²×C₁₆×C₂₂₃	2
C₂×C₄×C₈×C₂₂₃	2
C₄⋊C₁₆×C₂₂₃	2
C₂×C₄⋊C₁₆	223

subgroup

index

C₂²×C₁₆×C₂₂₃

C₂×C₄×C₈×C₂₂₃

C₄⋊C₁₆×C₂₂₃

C₂×C₄⋊C₁₆

223

Subgroups — C₂×C₄⋊C₁₆×C₂₂₃ #43251164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C4⋊C16×C223 #43251164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₁₆×C₂₂₃ #43251164