Group #43251007 — subgroups

Subgroups — #1992×C₂₂₃ #43251007

Order 28544.

260 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 128 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×#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#384×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#340×C₂₂₃	C₂	1	4	yes	maximal
5	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
6	#125×C₂₂₃	—	2	2	no	—
7	#125×C₂₂₃	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₂₂₃	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₂₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
13	#97×C₂₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₂₃	D₄	1	2	yes	—
17	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₂₂₃	—	2	2	no	—
19	C₂×C₈×C₂₂₃	—	2	2	no	—
20	C₂×C₈×C₂₂₃	Q₈	1	4	yes	—
21	C₄⋊C₄×C₂₂₃	—	2	8	no	—
22	C₄⋊C₄×C₂₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₂₃	—	2	4	no	—
24	C₄⋊C₄×C₂₂₃	—	4	4	no	—
25	C₄⋊C₄×C₂₂₃	C₂×C₄	1	4	yes	—
26	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
27	C₄²×C₂₂₃	D₄	1	2	yes	—
28	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	—	2	2	no	—
30	C₂×C₄×C₂₂₃	—	2	2	no	—
31	C₂×C₄×C₂₂₃	—	2	4	no	—
32	C₂×C₄×C₂₂₃	—	4	2	no	—
33	C₂×C₄×C₂₂₃	—	4	2	no	—
34	C₂×C₄×C₂₂₃	—	4	2	no	—
35	C₂×C₄×C₂₂₃	—	4	4	no	—
36	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₂₂₃	#41	1	2	yes	—
41	C₈×C₂₂₃	—	2	4	no	—
42	C₂²×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₂₂₃	#124	1	1	yes	characteristic
44	C₂²×C₂₂₃	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₂₂₃	#137	1	2	yes	—
47	C₄×C₂₂₃	—	2	4	no	—
48	C₄×C₂₂₃	—	4	4	no	—
49	C₄×C₂₂₃	—	8	2	no	—
50	C₄×C₂₂₃	C₄×Q₈	1	2	yes	—
51	C₄×C₂₂₃	C₂²⋊Q₈	1	2	yes	—
52	C₂×C₂₂₃	#391	1	1	yes	characteristic
53	C₂×C₂₂₃	#417	1	2	yes	—
54	C₂×C₂₂₃	#490	1	1	yes	characteristic
55	C₂×C₂₂₃	#491	1	1	yes	characteristic
56	C₂×C₂₂₃	#501	1	2	yes	—
57	C₂₂₃	#1992	1	1	yes	characteristic, direct product factor, Sylow
58	#1992	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×#125	C₂×C₂₂₃	1	1	yes	characteristic
60	C₂×#97	C₂×C₂₂₃	1	1	yes	characteristic
61	#384	C₂×C₂₂₃	1	1	yes	characteristic
62	#340	C₂×C₂₂₃	1	4	yes	—
63	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
64	#125	—	2	2	no	—
65	#125	C₄×C₂₂₃	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
69	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
70	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	D₄×C₂₂₃	1	2	yes	—
75	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Q₈×C₂₂₃	1	4	yes	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	4	4	no	—
83	C₄⋊C₄	C₂×C₄×C₂₂₃	1	4	yes	—
84	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
85	C₄²	D₄×C₂₂₃	1	2	yes	—
86	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	4	yes	—
95	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
96	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
97	C₂×C₄	C₂×Q₈×C₂₂₃	1	2	yes	—
98	C₂×C₄	#41×C₂₂₃	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
101	C₂²	#124×C₂₂₃	1	1	yes	characteristic
102	C₂²	C₄⋊Q₈×C₂₂₃	1	1	yes	characteristic
103	C₂²	AGL(1,ℤ/8)×C₂₂₃	1	2	yes	—
104	C₂²	#137×C₂₂₃	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	8	2	no	—
108	C₄	C₄×Q₈×C₂₂₃	1	2	yes	—
109	C₄	C₂²⋊Q₈×C₂₂₃	1	2	yes	—
110	C₂	#391×C₂₂₃	1	1	yes	characteristic
111	C₂	#417×C₂₂₃	1	2	yes	—
112	C₂	#490×C₂₂₃	1	1	yes	characteristic
113	C₂	#491×C₂₂₃	1	1	yes	characteristic
114	C₂	#501×C₂₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₂₂₃

C₂

yes

maximal, characteristic

C₂×#97×C₂₂₃

C₂

yes

maximal, characteristic

#384×C₂₂₃

C₂

yes

maximal, characteristic

#340×C₂₂₃

C₂

yes

maximal

C₂²×C₈×C₂₂₃

C₂²

yes

—

#125×C₂₂₃

—

#125×C₂₂₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₂₃

—

C₂×C₄⋊C₄×C₂₂₃

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

#97×C₂₂₃

—

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

D₄

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

—

C₂×C₈×C₂₂₃

Q₈

yes

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

D₄

yes

—

C₂³×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₂₃

#41

yes

—

C₈×C₂₂₃

—

C₂²×C₂₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₂₃

#124

yes

characteristic

C₂²×C₂₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₂₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₂₃

#137

yes

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₄×Q₈

yes

—

C₄×C₂₂₃

C₂²⋊Q₈

yes

—

C₂×C₂₂₃

#391

yes

characteristic

C₂×C₂₂₃

#417

yes

—

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

—

C₂₂₃

#1992

yes

characteristic, direct product factor, Sylow

#1992

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×#125

C₂×C₂₂₃

yes

characteristic

C₂×#97

C₂×C₂₂₃

yes

characteristic

#384

C₂×C₂₂₃

yes

characteristic

#340

C₂×C₂₂₃

yes

—

C₂²×C₈

C₂²×C₂₂₃

yes

—

#125

—

#125

C₄×C₂₂₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄²

C₂²×C₂₂₃

yes

characteristic

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₈×C₂₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₂₂₃

yes

—

C₄²

C₂×C₄×C₂₂₃

yes

—

C₄²

D₄×C₂₂₃

yes

—

C₂³

C₂×D₄×C₂₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₂₃

yes

characteristic

C₂×C₄

C₂×D₄×C₂₂₃

yes

characteristic

C₂×C₄

C₂×Q₈×C₂₂₃

yes

—

C₂×C₄

#41×C₂₂₃

yes

—

C₈

—

100

C₂²

C₂×C₂²⋊C₄×C₂₂₃

yes

characteristic

101

C₂²

#124×C₂₂₃

yes

characteristic

102

C₂²

C₄⋊Q₈×C₂₂₃

yes

characteristic

103

C₂²

AGL(1,ℤ/8)×C₂₂₃

yes

—

104

C₂²

#137×C₂₂₃

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

C₄×Q₈×C₂₂₃

yes

—

109

C₄

C₂²⋊Q₈×C₂₂₃

yes

—

110

C₂

#391×C₂₂₃

yes

characteristic

111

C₂

#417×C₂₂₃

yes

—

112

C₂

#490×C₂₂₃

yes

characteristic

113

C₂

#491×C₂₂₃

yes

characteristic

114

C₂

#501×C₂₂₃

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ C₂×C₄²×C₂₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₂₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₂₃

Frattini, cosocle

C₂²×C₄

C₂³×C₂₂₃

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₂₃ ◁ C₂×C₄²×C₂₂₃ ◁ G

N	H ≅ G/N	type
7. #125×C₂₂₃	C₄	semidirect
57. C₂₂₃	#1992	direct (N × H)
58. #1992	C₂₂₃	direct (N × H)
65. #125	C₄×C₂₂₃	semidirect

H ≅ G/N

type

7. #125×C₂₂₃

C₄

semidirect

57. C₂₂₃

#1992

direct (N × H)

58. #1992

C₂₂₃

direct (N × H)

65. #125

C₄×C₂₂₃

semidirect

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

subgroup	index
C₂×#125×C₂₂₃	2
C₂×#97×C₂₂₃	2
#384×C₂₂₃	2
#340×C₂₂₃	2
#1992	223

subgroup

index

C₂×#125×C₂₂₃

C₂×#97×C₂₂₃

#384×C₂₂₃

#340×C₂₂₃

#1992

223

Subgroups — #1992×C₂₂₃ #43251007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1992×C223 #43251007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1992×C₂₂₃ #43251007