Group #43251998 — subgroups

Subgroups — #2983×C₂₂₃ #43251998

Order 28544.

356 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 268 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	#446×C₂₂₃	C₂	1	8	yes	maximal, complemented
3	#432×C₂₂₃	C₂	1	4	yes	maximal
4	#431×C₂₂₃	C₂	1	2	yes	maximal
5	C₂²×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
6	C₄×Q₈×C₂₂₃	C₄	1	4	yes	—
7	C₄×Q₈×C₂₂₃	C₂²	1	4	yes	—
8	C₂×C₄⋊C₄×C₂₂₃	C₄	1	2	yes	—
9	C₂×C₄⋊C₄×C₂₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
12	C₄⋊C₈×C₂₂₃	C₂²	1	4	yes	—
13	C₄⋊C₈×C₂₂₃	C₂²	1	8	yes	—
14	C₂²⋊C₈×C₂₂₃	C₂²	1	4	yes	—
15	#97×C₂₂₃	C₂²	1	8	yes	—
16	C₄×C₈×C₂₂₃	C₂²	1	4	yes	—
17	C₂×Q₈×C₂₂₃	—	2	4	no	—
18	C₂×Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₂₃	C₂³	1	8	yes	—
25	C₄⋊C₄×C₂₂₃	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₂₂₃	C₂×C₄	1	4	yes	—
27	C₄²×C₂₂₃	C₂×C₄	1	4	yes	—
28	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
29	C₄²×C₂₂₃	C₂³	1	2	yes	—
30	C₄²×C₂₂₃	C₂³	1	4	yes	—
31	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
32	Q₈×C₂₂₃	—	2	8	no	—
33	C₂×C₄×C₂₂₃	—	2	4	no	—
34	C₂×C₄×C₂₂₃	—	2	2	no	—
35	C₂×C₄×C₂₂₃	—	4	1	no	—
36	C₂×C₄×C₂₂₃	—	4	2	no	—
37	C₂×C₄×C₂₂₃	M₁₆	1	4	yes	—
38	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
41	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
44	C₈×C₂₂₃	—	2	8	no	—
45	C₂²×C₂₂₃	—	2	1	no	—
46	C₂²×C₂₂₃	—	2	1	no	—
47	C₂²×C₂₂₃	C₂×M₁₆	1	2	yes	—
48	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
49	C₄×C₂₂₃	—	2	1	no	—
50	C₄×C₂₂₃	—	2	4	no	—
51	C₄×C₂₂₃	—	2	2	no	—
52	C₄×C₂₂₃	—	2	4	no	—
53	C₄×C₂₂₃	C₂×M₁₆	1	4	yes	—
54	C₄×C₂₂₃	2₋¹⁺⁴	1	2	yes	—
55	C₂×C₂₂₃	—	2	2	no	—
56	C₂×C₂₂₃	#519	1	1	yes	characteristic
57	C₂×C₂₂₃	C₂²×M₁₆	1	1	yes	characteristic
58	C₂×C₂₂₃	#568	1	1	yes	characteristic
59	C₂₂₃	#2983	1	1	yes	characteristic, direct product factor, Sylow
60	#2983	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic
62	#446	C₂×C₂₂₃	1	8	yes	complemented
63	#432	C₂×C₂₂₃	1	4	yes	—
64	#431	C₂×C₂₂₃	1	2	yes	—
65	C₂²×Q₈	C₄×C₂₂₃	1	1	yes	characteristic
66	C₄×Q₈	C₄×C₂₂₃	1	4	yes	—
67	C₄×Q₈	C₂²×C₂₂₃	1	4	yes	—
68	C₂×C₄⋊C₄	C₄×C₂₂₃	1	2	yes	—
69	C₂×C₄⋊C₄	C₄×C₂₂₃	1	1	yes	characteristic
70	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
71	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
72	C₄⋊C₈	C₂²×C₂₂₃	1	4	yes	—
73	C₄⋊C₈	C₂²×C₂₂₃	1	8	yes	—
74	C₂²⋊C₈	C₂²×C₂₂₃	1	4	yes	—
75	#97	C₂²×C₂₂₃	1	8	yes	—
76	C₄×C₈	C₂²×C₂₂₃	1	4	yes	—
77	C₂×Q₈	—	2	4	no	—
78	C₂×Q₈	C₂×C₄×C₂₂₃	1	4	yes	—
79	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
80	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
82	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
83	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
84	C₂×C₈	C₂³×C₂₂₃	1	8	yes	—
85	C₄⋊C₄	C₂×C₄×C₂₂₃	1	8	yes	—
86	C₄⋊C₄	C₂×C₄×C₂₂₃	1	4	yes	—
87	C₄²	C₂×C₄×C₂₂₃	1	4	yes	—
88	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
89	C₄²	C₂³×C₂₂₃	1	2	yes	—
90	C₄²	C₂³×C₂₂₃	1	4	yes	—
91	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	M₁₆×C₂₂₃	1	4	yes	—
98	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
99	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
100	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
101	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
102	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
103	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
104	C₈	—	2	8	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×M₁₆×C₂₂₃	1	2	yes	—
108	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
109	C₄	—	2	4	no	—
110	C₄	—	2	2	no	—
111	C₄	—	2	4	no	—
112	C₄	—	2	1	no	—
113	C₄	C₂×M₁₆×C₂₂₃	1	4	yes	—
114	C₄	2₋¹⁺⁴×C₂₂₃	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	#519×C₂₂₃	1	1	yes	characteristic
117	C₂	C₂²×M₁₆×C₂₂₃	1	1	yes	characteristic
118	C₂	#568×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₂₂₃

C₂

yes

maximal, characteristic

#446×C₂₂₃

C₂

yes

maximal, complemented

#432×C₂₂₃

C₂

yes

maximal

#431×C₂₂₃

C₂

yes

maximal

C₂²×Q₈×C₂₂₃

C₄

yes

characteristic

C₄×Q₈×C₂₂₃

C₄

yes

—

C₄×Q₈×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₄

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₄

yes

characteristic

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₂₃

C₂²

yes

—

C₄⋊C₈×C₂₂₃

C₂²

yes

—

C₄⋊C₈×C₂₂₃

C₂²

yes

—

C₂²⋊C₈×C₂₂₃

C₂²

yes

—

#97×C₂₂₃

C₂²

yes

—

C₄×C₈×C₂₂₃

C₂²

yes

—

C₂×Q₈×C₂₂₃

—

C₂×Q₈×C₂₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₂₃

C₂³

yes

—

C₄⋊C₄×C₂₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

C₂³

yes

—

C₄²×C₂₂₃

C₂³

yes

—

C₂³×C₂₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

M₁₆

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₂₃

C₂⁴

yes

characteristic

C₈×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂×M₁₆

yes

—

C₂²×C₂₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×M₁₆

yes

—

C₄×C₂₂₃

2₋¹⁺⁴

yes

—

C₂×C₂₂₃

—

C₂×C₂₂₃

#519

yes

characteristic

C₂×C₂₂₃

C₂²×M₁₆

yes

C₂×C₂₂₃

yes

C₂₂₃

yes

characteristic, direct product factor, Sylow

#2983

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂×C₂₂₃

yes

characteristic

#446

C₂×C₂₂₃

yes

complemented

#432

C₂×C₂₂₃

yes

—

#431

C₂×C₂₂₃

yes

—

C₂²×Q₈

C₄×C₂₂₃

yes

characteristic

C₄×Q₈

C₄×C₂₂₃

yes

—

C₄×Q₈

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₄×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₄×C₂₂₃

yes

characteristic

C₂×C₄²

C₂²×C₂₂₃

yes

characteristic

C₂×C₄²

C₂²×C₂₂₃

yes

—

C₄⋊C₈

C₂²×C₂₂₃

yes

—

C₄⋊C₈

C₂²×C₂₂₃

yes

—

C₂²⋊C₈

C₂²×C₂₂₃

yes

—

#97

C₂²×C₂₂₃

yes

—

C₄×C₈

C₂²×C₂₂₃

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₂₃

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₂₃

yes

characteristic

C₂²×C₄

C₂³×C₂₂₃

yes

characteristic

C₂×C₈

C₂³×C₂₂₃

yes

—

C₄⋊C₄

C₂×C₄×C₂₂₃

yes

—

C₄⋊C₄

C₂×C₄×C₂₂₃

yes

—

C₄²

C₂×C₄×C₂₂₃

yes

—

C₄²

C₂×C₄×C₂₂₃

yes

—

C₄²

C₂³×C₂₂₃

yes

—

C₄²

C₂³×C₂₂₃

yes

—

C₂³

C₂²×C₄×C₂₂₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

M₁₆×C₂₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₂₃

yes

characteristic

C₂×C₄

C₂²×C₄×C₂₂₃

yes

—

100

C₂×C₄

C₂²×C₄×C₂₂₃

yes

—

101

C₂×C₄

C₂²×C₄×C₂₂₃

yes

—

102

C₂×C₄

C₂²×C₄×C₂₂₃

yes

—

103

C₂×C₄

C₂⁴×C₂₂₃

yes

characteristic

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×M₁₆×C₂₂₃

yes

—

108

C₂²

C₂³×C₄×C₂₂₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₂×M₁₆×C₂₂₃

yes

—

114

C₄

2₋¹⁺⁴×C₂₂₃

yes

—

115

C₂

—

116

C₂

#519×C₂₂₃

yes

characteristic

117

C₂

C₂²×M₁₆×C₂₂₃

yes

characteristic

118

C₂

#568×C₂₂₃

yes

characteristic

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

subgroup

G/H

center

C₂×C₄×C₂₂₃

C₂⁴

commutator

C₂²

C₂³×C₄×C₂₂₃

Frattini, cosocle

C₂×C₄

C₂⁴×C₂₂₃

socle

C₂²×C₂₂₃

C₂³×C₄

prime radical

C₂³×C₂₂₃

C₂²×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₂₃ ◁ G

N	H ≅ G/N	type
2. #446×C₂₂₃	C₂	semidirect
59. C₂₂₃	#2983	direct (N × H)
60. #2983	C₂₂₃	direct (N × H)
62. #446	C₂×C₂₂₃	semidirect

H ≅ G/N

type

2. #446×C₂₂₃

C₂

semidirect

59. C₂₂₃

#2983

direct (N × H)

60. #2983

C₂₂₃

direct (N × H)

62. #446

C₂×C₂₂₃

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₂₂₃	2
#446×C₂₂₃	2
#432×C₂₂₃	2
#431×C₂₂₃	2
#2983	223

subgroup

index

C₂×C₄×Q₈×C₂₂₃

#446×C₂₂₃

#432×C₂₂₃

#431×C₂₂₃

#2983

223

Subgroups — #2983×C₂₂₃ #43251998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2983×C223 #43251998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2983×C₂₂₃ #43251998