Group #13093081 — subgroups

Subgroups — Q₈×C₂²⋊D₂₃ #13093081

Order 1472.

280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625117	C₂	1	3	yes	maximal, complemented
4	Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#10625115	C₂	1	3	yes	maximal, complemented
6	C₄×C₂²⋊D₂₃	C₂	1	3	yes	maximal
7	#10625095	C₂	1	3	yes	maximal, complemented
8	C₂×Q₈×C₂₃	—	2	4	no	—
9	C₂×Q₈×C₂₃	C₂²	1	3	yes	—
10	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	C₂²	1	3	yes	—
12	C₂×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
13	Q₈×D₂₃	—	2	4	no	—
14	C₂×C₄×D₂₃	C₂²	1	3	yes	—
15	C₂×Dic₄₆	C₂²	1	3	yes	—
16	C₂²⋊Dic₂₃	C₂²	1	3	yes	—
17	(C₂×D₂₃)⋊C₄	C₂²	1	3	yes	—
18	C₄⋊Dic₂₃	C₂²	1	3	yes	—
19	Dic₂₃⋊C₄	C₂²	1	6	yes	—
20	Dic₂₃⋊C₄	C₂²	1	3	yes	—
21	C₄×Dic₂₃	C₂²	1	3	yes	—
22	C₂³×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×D₂₃	C₂³	1	1	yes	characteristic
24	Q₈×C₂₃	—	2	6	no	—
25	Q₈×C₂₃	D₄	1	4	yes	complemented
26	C₂×C₄×C₂₃	—	2	6	no	—
27	C₂×C₄×C₂₃	C₂³	1	3	yes	—
28	C₂×C₄×C₂₃	C₂³	1	3	yes	—
29	C₂²⋊D₂₃	Q₈	1	4	yes	direct product factor
30	C₂×Dic₂₃	C₂³	1	3	yes	—
31	C₂×Dic₂₃	C₂³	1	3	yes	—
32	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
33	C₄×D₂₃	—	2	6	no	—
34	Dic₄₆	—	2	6	no	—
35	C₂²×C₂₃	—	2	1	no	—
36	C₂²×C₂₃	—	2	1	no	—
37	C₂²×C₂₃	C₂×Q₈	1	2	yes	complemented
38	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
39	C₂×D₂₃	—	2	1	no	—
40	C₂×D₂₃	—	2	1	no	—
41	C₂×D₂₃	C₂×Q₈	1	2	yes	complemented
42	C₄×C₂₃	—	2	3	no	—
43	C₄×C₂₃	C₂×D₄	1	6	yes	—
44	Dic₂₃	—	2	3	no	—
45	Dic₂₃	—	2	3	no	—
46	Dic₂₃	C₂×Q₈	1	2	yes	complemented
47	D₄×Q₈	—	23	1	no	maximal, Sylow
48	C₂×C₂₃	—	2	2	no	—
49	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
52	D₂₃	—	2	2	no	—
53	C₂²×Q₈	—	23	1	no	—
54	C₂²×Q₈	D₂₃	1	1	yes	characteristic, complemented
55	C₄⋊Q₈	—	23	3	no	—
56	C₂²⋊Q₈	—	23	3	no	—
57	C₂²⋊Q₈	—	23	3	no	—
58	C₄×Q₈	—	23	1	no	—
59	C₄×D₄	—	23	3	no	—
60	C₂₃	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₈	—	2	4	no	—
62	C₂×Q₈	—	23	3	no	—
63	C₂×Q₈	—	46	4	no	—
64	C₂×Q₈	C₂×D₂₃	1	3	yes	—
65	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
66	C₂×D₄	—	23	1	no	—
67	C₂²×C₄	—	23	3	no	—
68	C₂²×C₄	C₂×D₂₃	1	3	yes	—
69	C₄⋊C₄	—	23	3	no	—
70	C₄⋊C₄	—	23	3	no	—
71	C₄⋊C₄	—	23	6	no	—
72	C₂²⋊C₄	—	23	3	no	—
73	C₂²⋊C₄	—	23	3	no	—
74	C₄²	—	23	3	no	—
75	C₂³	—	23	1	no	—
76	C₂³	C₂²×D₂₃	1	1	yes	characteristic
77	Q₈	—	2	6	no	—
78	Q₈	—	46	6	no	—
79	Q₈	C₂²⋊D₂₃	1	4	yes	direct product factor
80	D₄	—	23	4	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	23	1	no	—
83	C₂×C₄	—	23	3	no	—
84	C₂×C₄	—	23	3	no	—
85	C₂×C₄	—	46	6	no	—
86	C₂×C₄	C₂²×D₂₃	1	3	yes	—
87	C₂×C₄	C₂²×D₂₃	1	3	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	23	2	no	—
91	C₂²	—	46	1	no	—
92	C₂²	—	46	1	no	—
93	C₂²	Q₈×D₂₃	1	2	yes	complemented
94	C₂²	C₂³×D₂₃	1	1	yes	characteristic
95	C₄	—	2	3	no	—
96	C₄	—	23	2	no	—
97	C₄	—	46	3	no	—
98	C₄	—	46	3	no	—
99	C₄	C₂×C₂²⋊D₂₃	1	6	yes	—
100	C₂	—	2	2	no	—
101	C₂	—	46	2	no	—
102	C₂	C₂×Q₈×D₂₃	1	1	yes	characteristic
103	C₂	#10625158	1	1	yes	characteristic
104	C₂	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₈×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₂₃

C₂

yes

maximal, characteristic, complemented

#10625117

C₂

yes

maximal, complemented

Q₈×Dic₂₃

C₂

yes

maximal, characteristic, complemented

#10625115

C₂

yes

maximal, complemented

C₄×C₂²⋊D₂₃

C₂

yes

maximal

#10625095

C₂

yes

maximal, complemented

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂²

yes

—

C₂×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

characteristic

Q₈×D₂₃

—

C₂×C₄×D₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

C₂³

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

D₄

yes

complemented

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂²⋊D₂₃

Q₈

yes

direct product factor

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₄×D₂₃

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×Q₈

yes

complemented

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

C₂×Q₈

yes

complemented

C₄×C₂₃

—

C₄×C₂₃

C₂×D₄

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₂×Q₈

yes

complemented

D₄×Q₈

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₃

C₂²×Q₈

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

D₂₃

—

C₂²×Q₈

—

C₂²×Q₈

D₂₃

yes

characteristic, complemented

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₂₃

D₄×Q₈

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₂₃

yes

—

C₂×Q₈

C₂×D₂₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×D₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

C₂²⋊D₂₃

yes

direct product factor

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₂₃

yes

—

C₂×C₄

C₂²×D₂₃

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

Q₈×D₂₃

yes

complemented

C₂²

C₂³×D₂₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×C₂²⋊D₂₃

yes

—

100

C₂

—

101

C₂

—

102

C₂

C₂×Q₈×D₂₃

yes

characteristic

103

C₂

#10625158

yes

characteristic

104

C₂

C₂²×C₂²⋊D₂₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂²×Q₈×C₂₃	C₂
prime radical	C₂×C₂²⋊D₂₃	C₂²
hypercenter	C₂²×Q₈	D₂₃
hypocenter	C₂₃	D₄×Q₈
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂²×Q₈×C₂₃

C₂

prime radical

C₂×C₂²⋊D₂₃

C₂²

hypercenter

C₂²×Q₈

D₂₃

hypocenter

C₂₃

D₄×Q₈

LCS

G ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. C₂²×Q₈×C₂₃	C₂	semidirect
2. C₂×Q₈×D₂₃	C₂	semidirect
3. #10625117	C₂	semidirect
4. Q₈×Dic₂₃	C₂	semidirect
5. #10625115	C₂	semidirect
7. #10625095	C₂	semidirect
25. Q₈×C₂₃	D₄	semidirect
29. C₂²⋊D₂₃	Q₈	direct (N × H)
37. C₂²×C₂₃	C₂×Q₈	semidirect
41. C₂×D₂₃	C₂×Q₈	semidirect
46. Dic₂₃	C₂×Q₈	semidirect
54. C₂²×Q₈	D₂₃	semidirect
60. C₂₃	D₄×Q₈	semidirect
79. Q₈	C₂²⋊D₂₃	direct (N × H)
93. C₂²	Q₈×D₂₃	semidirect

H ≅ G/N

type

1. C₂²×Q₈×C₂₃

C₂

semidirect

2. C₂×Q₈×D₂₃

C₂

semidirect

3. #10625117

C₂

semidirect

4. Q₈×Dic₂₃

C₂

semidirect

5. #10625115

C₂

semidirect

7. #10625095

C₂

semidirect

25. Q₈×C₂₃

D₄

semidirect

29. C₂²⋊D₂₃

Q₈

direct (N × H)

37. C₂²×C₂₃

C₂×Q₈

semidirect

41. C₂×D₂₃

C₂×Q₈

semidirect

46. Dic₂₃

C₂×Q₈

semidirect

54. C₂²×Q₈

D₂₃

semidirect

60. C₂₃

D₄×Q₈

semidirect

79. Q₈

C₂²⋊D₂₃

direct (N × H)

93. C₂²

Q₈×D₂₃

semidirect

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

subgroup	index
C₂²×Q₈×C₂₃	2
C₂×Q₈×D₂₃	2
#10625117	2
Q₈×Dic₂₃	2
#10625115	2
C₄×C₂²⋊D₂₃	2
#10625095	2
D₄×Q₈	23

subgroup

index

C₂²×Q₈×C₂₃

C₂×Q₈×D₂₃

#10625117

Q₈×Dic₂₃

#10625115

C₄×C₂²⋊D₂₃

#10625095

D₄×Q₈

Subgroups — Q₈×C₂²⋊D₂₃ #13093081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C22⋊D23 #13093081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂²⋊D₂₃ #13093081