Group #43251973 — subgroups

Subgroups — C₂×C₈×Q₈×C₂₂₃ #43251973

Order 28544.

440 conjugacy classes of subgroups; only the 392 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 392 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×C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₂₂₃	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₈×C₂₂₃	C₂	1	3	yes	maximal
4	C₂×C₄×C₈×C₂₂₃	C₂	1	3	yes	maximal
5	C₂²×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₂₂₃	C₂²	1	3	yes	—
7	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₄×Q₈×C₂₂₃	C₄	1	4	yes	—
9	C₄×Q₈×C₂₂₃	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₂₂₃	C₄	1	3	yes	—
11	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	—
12	C₄⋊C₈×C₂₂₃	C₂²	1	12	yes	—
13	C₄×C₈×C₂₂₃	C₂²	1	12	yes	—
14	C₂×Q₈×C₂₂₃	C₈	1	8	yes	direct product factor
15	C₂×Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₂₂₃	—	2	6	no	—
20	C₂×C₈×C₂₂₃	Q₈	1	4	yes	direct product factor
21	C₂×C₈×C₂₂₃	C₂³	1	6	yes	—
22	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
23	C₄⋊C₄×C₂₂₃	C₂×C₄	1	12	yes	—
24	C₄²×C₂₂₃	C₂×C₄	1	6	yes	—
25	C₄²×C₂₂₃	C₂³	1	6	yes	—
26	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
27	Q₈×C₂₂₃	C₂×C₈	1	16	yes	direct product factor
28	C₂×C₄×C₂₂₃	—	2	6	no	—
29	C₂×C₄×C₂₂₃	C₂×C₈	1	12	yes	—
30	C₂×C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
31	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
33	C₂×C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₂₂₃	#41	1	2	yes	—
35	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
36	C₈×C₂₂₃	—	2	6	no	—
37	C₈×C₂₂₃	C₂×Q₈	1	4	yes	direct product factor
38	C₂²×C₂₂₃	C₄×Q₈	1	2	yes	—
39	C₂²×C₂₂₃	C₂²×C₈	1	2	yes	—
40	C₂²×C₂₂₃	#131	1	2	yes	—
41	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
42	C₄×C₂₂₃	—	2	6	no	—
43	C₄×C₂₂₃	C₄×Q₈	1	2	yes	—
44	C₄×C₂₂₃	C₂²×C₈	1	12	yes	—
45	C₄×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₂₂₃	C₂×#41	1	1	yes	characteristic
47	C₂×C₂₂₃	C₈×Q₈	1	4	yes	direct product factor
48	C₂×C₂₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
49	C₂×C₂₂₃	C₂³×C₈	1	1	yes	characteristic
50	C₂×C₂₂₃	C₂×#131	1	1	yes	characteristic
51	C₂₂₃	C₂×C₈×Q₈	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₈×Q₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic
54	C₈×Q₈	C₂×C₂₂₃	1	8	yes	direct product factor
55	C₂×C₄⋊C₈	C₂×C₂₂₃	1	3	yes	—
56	C₂×C₄×C₈	C₂×C₂₂₃	1	3	yes	—
57	C₂²×Q₈	C₄×C₂₂₃	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₂₂₃	1	3	yes	—
59	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
60	C₄×Q₈	C₄×C₂₂₃	1	4	yes	—
61	C₄×Q₈	C₂²×C₂₂₃	1	4	yes	—
62	C₂×C₄⋊C₄	C₄×C₂₂₃	1	3	yes	—
63	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	—
64	C₄⋊C₈	C₂²×C₂₂₃	1	12	yes	—
65	C₄×C₈	C₂²×C₂₂₃	1	12	yes	—
66	C₂×Q₈	C₈×C₂₂₃	1	8	yes	direct product factor
67	C₂×Q₈	C₂×C₄×C₂₂₃	1	4	yes	—
68	C₂²×C₄	C₂×C₄×C₂₂₃	1	3	yes	—
69	C₂²×C₄	C₂×C₄×C₂₂₃	1	3	yes	—
70	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	Q₈×C₂₂₃	1	4	yes	direct product factor
73	C₂×C₈	C₂³×C₂₂₃	1	6	yes	—
74	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
75	C₄⋊C₄	C₂×C₄×C₂₂₃	1	12	yes	—
76	C₄²	C₂×C₄×C₂₂₃	1	6	yes	—
77	C₄²	C₂³×C₂₂₃	1	6	yes	—
78	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
79	Q₈	C₂×C₈×C₂₂₃	1	16	yes	direct product factor
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	C₂×C₈×C₂₂₃	1	12	yes	—
82	C₂×C₄	C₂²×C₄×C₂₂₃	1	6	yes	—
83	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
84	C₂×C₄	C₂²×C₄×C₂₂₃	1	6	yes	—
85	C₂×C₄	C₂×Q₈×C₂₂₃	1	2	yes	—
86	C₂×C₄	#41×C₂₂₃	1	2	yes	—
87	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
88	C₈	—	2	6	no	—
89	C₈	C₂×Q₈×C₂₂₃	1	4	yes	direct product factor
90	C₂²	C₄×Q₈×C₂₂₃	1	2	yes	—
91	C₂²	C₂²×C₈×C₂₂₃	1	2	yes	—
92	C₂²	#131×C₂₂₃	1	2	yes	—
93	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
94	C₄	—	2	6	no	—
95	C₄	C₄×Q₈×C₂₂₃	1	2	yes	—
96	C₄	C₂²×C₈×C₂₂₃	1	12	yes	—
97	C₄	C₂²×Q₈×C₂₂₃	1	1	yes	characteristic
98	C₄	C₂×#41×C₂₂₃	1	1	yes	characteristic
99	C₂	C₈×Q₈×C₂₂₃	1	4	yes	direct product factor
100	C₂	C₂×C₄×Q₈×C₂₂₃	1	1	yes	characteristic
101	C₂	C₂³×C₈×C₂₂₃	1	1	yes	characteristic
102	C₂	C₂×#131×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₂₂₃

C₂

yes

maximal, characteristic

C₈×Q₈×C₂₂₃

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₈×C₂₂₃

C₂

yes

maximal

C₂×C₄×C₈×C₂₂₃

C₂

yes

maximal

C₂²×Q₈×C₂₂₃

C₄

yes

characteristic

C₂²×C₈×C₂₂₃

C₂²

yes

—

C₂²×C₈×C₂₂₃

C₂²

yes

characteristic

C₄×Q₈×C₂₂₃

C₄

yes

—

C₄×Q₈×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₄

yes

—

C₂×C₄²×C₂₂₃

C₂²

yes

—

C₄⋊C₈×C₂₂₃

C₂²

yes

—

C₄×C₈×C₂₂₃

C₂²

yes

—

C₂×Q₈×C₂₂₃

C₈

yes

direct product factor

C₂×Q₈×C₂₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

Q₈

yes

direct product factor

C₂×C₈×C₂₂₃

C₂³

yes

—

C₂×C₈×C₂₂₃

C₂³

yes

—

C₄⋊C₄×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

C₂³

yes

—

C₂³×C₂₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₂₃

C₂×C₈

yes

direct product factor

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₂₃

#41

yes

—

C₂×C₄×C₂₂₃

C₂⁴

yes

characteristic

C₈×C₂₂₃

—

C₈×C₂₂₃

C₂×Q₈

yes

direct product factor

C₂²×C₂₂₃

C₄×Q₈

yes

—

C₂²×C₂₂₃

C₂²×C₈

yes

—

C₂²×C₂₂₃

#131

yes

—

C₂²×C₂₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

C₄×Q₈

yes

—

C₄×C₂₂₃

C₂²×C₈

yes

—

C₄×C₂₂₃

C₂²×Q₈

yes

characteristic

C₄×C₂₂₃

C₂×#41

yes

characteristic

C₂×C₂₂₃

C₈×Q₈

yes

direct product factor

C₂×C₂₂₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₂₂₃

C₂³×C₈

yes

characteristic

C₂×C₂₂₃

C₂×#131

yes

characteristic

C₂₂₃

C₂×C₈×Q₈

yes

characteristic, direct product factor, Sylow

C₂×C₈×Q₈

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂×C₂₂₃

yes

characteristic

C₈×Q₈

C₂×C₂₂₃

yes

direct product factor

C₂×C₄⋊C₈

C₂×C₂₂₃

yes

—

C₂×C₄×C₈

C₂×C₂₂₃

yes

—

C₂²×Q₈

C₄×C₂₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₂₃

yes

—

C₂²×C₈

C₂²×C₂₂₃

yes

characteristic

C₄×Q₈

C₄×C₂₂₃

yes

—

C₄×Q₈

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₄×C₂₂₃

yes

—

C₂×C₄²

C₂²×C₂₂₃

yes

—

C₄⋊C₈

C₂²×C₂₂₃

yes

—

C₄×C₈

C₂²×C₂₂₃

yes

—

C₂×Q₈

C₈×C₂₂₃

yes

direct product factor

C₂×Q₈

C₂×C₄×C₂₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

Q₈×C₂₂₃

yes

direct product factor

C₂×C₈

C₂³×C₂₂₃

yes

—

C₂×C₈

C₂³×C₂₂₃

yes

—

C₄⋊C₄

C₂×C₄×C₂₂₃

yes

—

C₄²

C₂×C₄×C₂₂₃

yes

—

C₄²

C₂³×C₂₂₃

yes

—

C₂³

C₂²×C₄×C₂₂₃

yes

characteristic

Q₈

C₂×C₈×C₂₂₃

yes

direct product factor

C₂×C₄

—

C₂×C₄

C₂×C₈×C₂₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₂₃

yes

characteristic

C₂×C₄

C₂²×C₄×C₂₂₃

yes

—

C₂×C₄

C₂×Q₈×C₂₂₃

yes

—

C₂×C₄

#41×C₂₂₃

yes

—

C₂×C₄

C₂⁴×C₂₂₃

yes

characteristic

C₈

—

C₈

C₂×Q₈×C₂₂₃

yes

direct product factor

C₂²

C₄×Q₈×C₂₂₃

yes

—

C₂²

C₂²×C₈×C₂₂₃

yes

—

C₂²

#131×C₂₂₃

yes

—

C₂²

C₂³×C₄×C₂₂₃

yes

characteristic

C₄

—

C₄

C₄×Q₈×C₂₂₃

yes

—

C₄

C₂²×C₈×C₂₂₃

yes

—

C₄

C₂²×Q₈×C₂₂₃

yes

characteristic

C₄

C₂×#41×C₂₂₃

yes

characteristic

C₂

C₈×Q₈×C₂₂₃

yes

direct product factor

100

C₂

C₂×C₄×Q₈×C₂₂₃

yes

characteristic

101

C₂

C₂³×C₈×C₂₂₃

yes

characteristic

102

C₂

C₂×#131×C₂₂₃

yes

characteristic

	subgroup	G/H
center	C₂²×C₈×C₂₂₃	C₂²
commutator	C₂	C₂³×C₈×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle, prime radical	C₂³×C₂₂₃	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₂₃ ◁ G

subgroup

G/H

center

C₂²×C₈×C₂₂₃

C₂²

commutator

C₂

C₂³×C₈×C₂₂₃

Frattini, cosocle

C₂×C₄

C₂⁴×C₂₂₃

socle, prime radical

C₂³×C₂₂₃

C₂²×C₄

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₂₂₃ ◁ G

N	H ≅ G/N	type
2. C₈×Q₈×C₂₂₃	C₂	direct (N × H)
14. C₂×Q₈×C₂₂₃	C₈	direct (N × H)
20. C₂×C₈×C₂₂₃	Q₈	direct (N × H)
27. Q₈×C₂₂₃	C₂×C₈	direct (N × H)
37. C₈×C₂₂₃	C₂×Q₈	direct (N × H)
47. C₂×C₂₂₃	C₈×Q₈	direct (N × H)
51. C₂₂₃	C₂×C₈×Q₈	direct (N × H)
52. C₂×C₈×Q₈	C₂₂₃	direct (N × H)
54. C₈×Q₈	C₂×C₂₂₃	direct (N × H)
66. C₂×Q₈	C₈×C₂₂₃	direct (N × H)
72. C₂×C₈	Q₈×C₂₂₃	direct (N × H)
79. Q₈	C₂×C₈×C₂₂₃	direct (N × H)
89. C₈	C₂×Q₈×C₂₂₃	direct (N × H)
99. C₂	C₈×Q₈×C₂₂₃	direct (N × H)

H ≅ G/N

type

2. C₈×Q₈×C₂₂₃

C₂

direct (N × H)

14. C₂×Q₈×C₂₂₃

C₈

direct (N × H)

20. C₂×C₈×C₂₂₃

Q₈

direct (N × H)

27. Q₈×C₂₂₃

C₂×C₈

direct (N × H)

37. C₈×C₂₂₃

C₂×Q₈

direct (N × H)

47. C₂×C₂₂₃

C₈×Q₈

direct (N × H)

51. C₂₂₃

C₂×C₈×Q₈

direct (N × H)

52. C₂×C₈×Q₈

C₂₂₃

direct (N × H)

54. C₈×Q₈

C₂×C₂₂₃

direct (N × H)

66. C₂×Q₈

C₈×C₂₂₃

direct (N × H)

72. C₂×C₈

Q₈×C₂₂₃

direct (N × H)

79. Q₈

C₂×C₈×C₂₂₃

direct (N × H)

89. C₈

C₂×Q₈×C₂₂₃

direct (N × H)

99. C₂

C₈×Q₈×C₂₂₃

direct (N × H)

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

subgroup	index
C₂×C₄×Q₈×C₂₂₃	2
C₈×Q₈×C₂₂₃	2
C₂×C₄⋊C₈×C₂₂₃	2
C₂×C₄×C₈×C₂₂₃	2
C₂×C₈×Q₈	223

subgroup

index

C₂×C₄×Q₈×C₂₂₃

C₈×Q₈×C₂₂₃

C₂×C₄⋊C₈×C₂₂₃

C₂×C₄×C₈×C₂₂₃

C₂×C₈×Q₈

223

Subgroups — C₂×C₈×Q₈×C₂₂₃ #43251973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C8×Q8×C223 #43251973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Q₈×C₂₂₃ #43251973