Group #43250851 — subgroups

Subgroups — #1836×C₂₂₃ #43250851

Order 28544.

224 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 120 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×#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#384×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#382×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#340×C₂₂₃	C₂	1	4	yes	maximal
5	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
6	C₂×C₄⋊C₄×C₂₂₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
10	#97×C₂₂₃	C₄	1	4	yes	complemented
11	#95×C₂₂₃	—	2	2	no	—
12	C₂²×C₄×C₂₂₃	—	2	2	no	—
13	C₂²×C₄×C₂₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₂₃	D₄	1	2	yes	—
15	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
16	C₂×C₈×C₂₂₃	—	2	2	no	—
17	C₂×C₈×C₂₂₃	—	2	2	no	—
18	C₂×C₈×C₂₂₃	C₂×C₄	1	4	yes	—
19	C₄⋊C₄×C₂₂₃	—	2	4	no	—
20	C₄⋊C₄×C₂₂₃	—	2	4	no	—
21	C₄⋊C₄×C₂₂₃	—	4	4	no	—
22	C₄²×C₂₂₃	C₂×C₄	1	2	yes	—
23	C₄²×C₂₂₃	Q₈	1	2	yes	—
24	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	—	2	2	no	—
26	C₂×C₄×C₂₂₃	—	2	2	no	—
27	C₂×C₄×C₂₂₃	—	4	2	no	—
28	C₂×C₄×C₂₂₃	—	4	2	no	—
29	C₂×C₄×C₂₂₃	—	4	4	no	—
30	C₂×C₄×C₂₂₃	—	4	2	no	—
31	C₂×C₄×C₂₂₃	—	4	2	no	—
32	C₂×C₄×C₂₂₃	C₄⋊C₄	1	4	yes	—
33	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₂₃	#41	1	2	yes	—
36	C₂×C₄×C₂₂₃	#41	1	2	yes	—
37	C₈×C₂₂₃	—	2	4	no	—
38	C₂²×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₂₃	#124	1	1	yes	characteristic
40	C₂²×C₂₂₃	#124	1	1	yes	characteristic
41	C₂²×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₂₂₃	#137	1	2	yes	—
43	C₄×C₂₂₃	—	2	4	no	—
44	C₄×C₂₂₃	—	8	2	no	—
45	C₄×C₂₂₃	—	8	2	no	—
46	C₄×C₂₂₃	#117	1	2	yes	—
47	C₄×C₂₂₃	#125	1	2	yes	—
48	C₂×C₂₂₃	#382	1	1	yes	characteristic
49	C₂×C₂₂₃	#428	1	2	yes	—
50	C₂×C₂₂₃	#489	1	2	yes	—
51	C₂×C₂₂₃	#490	1	1	yes	characteristic
52	C₂×C₂₂₃	#491	1	1	yes	characteristic
53	C₂₂₃	#1836	1	1	yes	characteristic, direct product factor, Sylow
54	#1836	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×#97	C₂×C₂₂₃	1	1	yes	characteristic
56	#384	C₂×C₂₂₃	1	1	yes	characteristic
57	#382	C₂×C₂₂₃	1	1	yes	characteristic
58	#340	C₂×C₂₂₃	1	4	yes	—
59	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
62	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
63	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
64	#97	C₄×C₂₂₃	1	4	yes	complemented
65	#95	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	D₄×C₂₂₃	1	2	yes	—
69	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×C₄×C₂₂₃	1	4	yes	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄²	C₂×C₄×C₂₂₃	1	2	yes	—
77	C₄²	Q₈×C₂₂₃	1	2	yes	—
78	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	C₄⋊C₄×C₂₂₃	1	4	yes	—
87	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
88	C₂×C₄	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
89	C₂×C₄	#41×C₂₂₃	1	2	yes	—
90	C₂×C₄	#41×C₂₂₃	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
93	C₂²	#124×C₂₂₃	1	1	yes	characteristic
94	C₂²	#124×C₂₂₃	1	1	yes	characteristic
95	C₂²	AGL(1,ℤ/8)×C₂₂₃	1	2	yes	—
96	C₂²	#137×C₂₂₃	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	8	2	no	—
100	C₄	#117×C₂₂₃	1	2	yes	—
101	C₄	#125×C₂₂₃	1	2	yes	—
102	C₂	#382×C₂₂₃	1	1	yes	characteristic
103	C₂	#428×C₂₂₃	1	2	yes	—
104	C₂	#489×C₂₂₃	1	2	yes	—
105	C₂	#490×C₂₂₃	1	1	yes	characteristic
106	C₂	#491×C₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#97×C₂₂₃

C₂

yes

maximal, characteristic

#384×C₂₂₃

C₂

yes

maximal, characteristic

#382×C₂₂₃

C₂

yes

maximal, characteristic

#340×C₂₂₃

C₂

yes

maximal

C₂²×C₈×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

#97×C₂₂₃

C₄

yes

complemented

#95×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

C₂×C₄

yes

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄²×C₂₂₃

C₂×C₄

yes

—

C₄²×C₂₂₃

Q₈

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

C₄⋊C₄

yes

—

C₂×C₄×C₂₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₂₃

#41

yes

—

C₂×C₄×C₂₂₃

#41

yes

—

C₈×C₂₂₃

—

C₂²×C₂₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

yes

C₂²×C₂₂₃

yes

—

C₂²×C₂₂₃

#137

yes

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

#117

yes

—

C₄×C₂₂₃

#125

yes

—

C₂×C₂₂₃

#382

yes

characteristic

C₂×C₂₂₃

#428

yes

—

C₂×C₂₂₃

#489

yes

—

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂₂₃

yes

characteristic, direct product factor, Sylow

#1836

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×#97

C₂×C₂₂₃

yes

characteristic

#384

C₂×C₂₂₃

yes

characteristic

#382

C₂×C₂₂₃

yes

characteristic

#340

C₂×C₂₂₃

yes

—

C₂²×C₈

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄²

C₂²×C₂₂₃

yes

characteristic

#97

C₄×C₂₂₃

yes

complemented

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₂₂₃

yes

—

C₄²

Q₈×C₂₂₃

yes

—

C₂³

C₂×D₄×C₂₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₂₂₃

yes

—

C₂×C₄

C₂²×C₄×C₂₂₃

yes

characteristic

C₂×C₄

C₂×Q₈×C₂₂₃

yes

characteristic

C₂×C₄

#41×C₂₂₃

yes

—

C₂×C₄

#41×C₂₂₃

yes

—

C₈

—

C₂²

C₂×C₄⋊C₄×C₂₂₃

yes

characteristic

C₂²

#124×C₂₂₃

yes

characteristic

C₂²

#124×C₂₂₃

yes

characteristic

C₂²

AGL(1,ℤ/8)×C₂₂₃

yes

—

C₂²

#137×C₂₂₃

yes

—

C₄

—

C₄

—

C₄

—

100

C₄

#117×C₂₂₃

yes

—

101

C₄

#125×C₂₂₃

yes

—

102

C₂

#382×C₂₂₃

yes

characteristic

103

C₂

#428×C₂₂₃

yes

—

104

C₂

#489×C₂₂₃

yes

—

105

C₂

#490×C₂₂₃

yes

characteristic

106

C₂

#491×C₂₂₃

yes

characteristic

	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
10. #97×C₂₂₃	C₄	semidirect
53. C₂₂₃	#1836	direct (N × H)
54. #1836	C₂₂₃	direct (N × H)
64. #97	C₄×C₂₂₃	semidirect

H ≅ G/N

type

10. #97×C₂₂₃

C₄

semidirect

53. C₂₂₃

#1836

direct (N × H)

54. #1836

C₂₂₃

direct (N × H)

64. #97

C₄×C₂₂₃

semidirect

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

subgroup	index
C₂×#97×C₂₂₃	2
#384×C₂₂₃	2
#382×C₂₂₃	2
#340×C₂₂₃	2
#1836	223

subgroup

index

C₂×#97×C₂₂₃

#384×C₂₂₃

#382×C₂₂₃

#340×C₂₂₃

#1836

223

Subgroups — #1836×C₂₂₃ #43250851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1836×C223 #43250851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1836×C₂₂₃ #43250851