Group #44398875 — subgroups

Subgroups — #44398875

Order 29056.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#28409762	C₂	1	1	yes	maximal, characteristic
2	#28409411	C₂	1	1	yes	maximal, characteristic
3	#28409328	C₂	1	2	yes	maximal
4	#28408762	C₂	1	2	yes	maximal
5	#333×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#16813874	—	2	2	no	—
7	#20300288	C₂²	1	1	yes	characteristic
8	#20300287	—	2	2	no	—
9	#20300286	—	2	2	no	—
10	C₄×Dic₄₅₄	C₂²	1	2	yes	—
11	#20300253	—	2	2	no	—
12	#20300222	—	2	2	no	—
13	#20300213	—	2	2	no	—
14	C₄×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₂₇	C₂²	1	2	yes	—
17	C₂×Dic₄₅₄	—	2	2	no	—
18	C₂×Dic₄₅₄	—	2	1	no	—
19	C₂×Dic₄₅₄	D₄	1	2	yes	—
20	#16813874	—	4	4	no	—
21	C₂×C₂₂₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₂₂₇	—	2	2	no	—
23	C₄⋊Dic₂₂₇	D₄	1	2	yes	—
24	Dic₂₂₇⋊C₄	—	2	4	no	—
25	Dic₂₂₇⋊C₄	—	2	2	no	—
26	C₄×Dic₂₂₇	—	2	2	no	—
27	C₄×Dic₂₂₇	—	2	2	no	—
28	C₂×Q₈×C₂₂₇	D₄	1	2	yes	—
29	C₂×C₈×C₂₂₇	—	2	2	no	—
30	C₄⋊C₄×C₂₂₇	—	2	2	no	—
31	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
32	Dic₄₅₄	—	2	2	no	—
33	Dic₄₅₄	—	4	2	no	—
34	Dic₄₅₄	—	8	2	no	—
35	Dic₄₅₄	Q₁₆	1	4	yes	complemented
36	C₂×Dic₂₂₇	—	2	2	no	—
37	C₂×Dic₂₂₇	—	2	2	no	—
38	C₂×Dic₂₂₇	—	4	1	no	—
39	C₂₂₇⋊C₈	—	2	2	no	—
40	Q₈×C₂₂₇	—	4	2	no	—
41	C₂×C₄×C₂₂₇	—	2	2	no	—
42	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
44	C₈×C₂₂₇	—	4	2	no	—
45	C₂²×C₂₂₇	#120	1	1	yes	characteristic
46	C₄×C₂₂₇	—	2	1	no	—
47	C₄×C₂₂₇	—	4	2	no	—
48	C₄×C₂₂₇	C₂×Q₁₆	1	2	yes	—
49	C₄×C₂₂₇	#137	1	2	yes	—
50	Dic₂₂₇	—	2	4	no	—
51	Dic₂₂₇	—	4	2	no	—
52	Dic₂₂₇	—	8	1	no	—
53	C₂×C₂₂₇	#451	1	2	yes	—
54	C₂×C₂₂₇	#453	1	1	yes	characteristic
55	C₂₂₇	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	227	1	no	maximal, Sylow
57	Q₈²	—	227	1	no	—
58	#494	—	227	1	no	—
59	#477	—	227	2	no	—
60	#333	D₂₂₇	1	1	yes	characteristic
61	Q₈⋊C₈	—	227	2	no	—
62	C₂×Q₁₆	—	454	2	no	—
63	C₄⋊Q₈	—	454	2	no	—
64	C₄⋊Q₈	—	454	2	no	—
65	C₄⋊Q₈	—	227	1	no	—
66	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
67	C₄×Q₈	—	454	2	no	—
68	C₄×Q₈	—	227	2	no	—
69	#107	—	454	2	no	—
70	C₄⋊C₈	C₂×D₂₂₇	1	2	yes	—
71	#103	—	454	2	no	—
72	C₄×C₈	—	227	1	no	—
73	C₂×Q₈	—	454	2	no	—
74	C₂×Q₈	—	454	1	no	—
75	C₂×Q₈	—	227	2	no	—
76	C₂×Q₈	D₉₀₈	1	2	yes	—
77	Q₁₆	—	908	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	454	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	454	4	no	—
82	C₄⋊C₄	—	454	2	no	—
83	C₄⋊C₄	—	454	2	no	—
84	C₄⋊C₄	—	227	2	no	—
85	C₄²	—	454	2	no	—
86	C₄²	—	454	2	no	—
87	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	1816	2	no	—
90	Q₈	—	908	2	no	—
91	Q₈	—	454	2	no	—
92	Q₈	—	227	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	908	1	no	—
95	C₂×C₄	—	454	2	no	—
96	C₂×C₄	—	454	2	no	—
97	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
98	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
99	C₈	—	4	2	no	—
100	C₈	—	454	2	no	—
101	C₂²	#20300258	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	1816	1	no	—
105	C₄	—	908	2	no	—
106	C₄	—	454	4	no	—
107	C₄	Q₁₆×D₂₂₇	1	2	yes	—
108	C₄	#20300317	1	2	yes	—
109	C₂	#28409189	1	2	yes	—
110	C₂	#28409202	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409762

C₂

yes

maximal, characteristic

#28409411

C₂

yes

maximal, characteristic

#28409328

C₂

yes

maximal

#28408762

C₂

yes

maximal

#333×C₂₂₇

C₂

yes

maximal, characteristic

C₂×#16813874

—

#20300288

C₂²

yes

characteristic

#20300287

—

#20300286

—

C₄×Dic₄₅₄

C₂²

yes

—

#20300253

—

#20300222

—

#20300213

—

C₄×C₂₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₇

C₂²

yes

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

—

C₂×Dic₄₅₄

D₄

yes

—

#16813874

—

C₂×C₂₂₇⋊C₈

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

D₄

yes

—

Dic₂₂₇⋊C₄

—

Dic₂₂₇⋊C₄

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₂×Q₈×C₂₂₇

D₄

yes

—

C₂×C₈×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

C₂³

yes

characteristic

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

—

Dic₄₅₄

Q₁₆

yes

complemented

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂₂₇⋊C₈

—

Q₈×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

#120

yes

characteristic

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

C₂×Q₁₆

yes

—

C₄×C₂₂₇

#137

yes

—

Dic₂₂₇

—

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

#451

yes

—

C₂×C₂₂₇

#453

yes

characteristic

C₂₂₇

#1633

yes

characteristic, complemented, Sylow

#1633

—

227

maximal, Sylow

Q₈²

—

227

—

#494

—

227

—

#477

—

227

—

#333

D₂₂₇

yes

characteristic

Q₈⋊C₈

—

227

—

C₂×Q₁₆

—

454

—

C₄⋊Q₈

—

454

—

C₄⋊Q₈

—

454

—

C₄⋊Q₈

—

227

—

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

C₄×Q₈

—

454

—

C₄×Q₈

—

227

—

#107

—

454

—

C₄⋊C₈

C₂×D₂₂₇

yes

—

#103

—

454

—

C₄×C₈

—

227

—

C₂×Q₈

—

454

—

C₂×Q₈

—

454

—

C₂×Q₈

—

227

—

C₂×Q₈

D₉₀₈

yes

—

Q₁₆

—

908

—

C₂×C₈

—

C₂×C₈

—

454

—

C₄⋊C₄

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

227

—

C₄²

—

454

—

C₄²

—

454

—

C₄²

C₂²×D₂₂₇

yes

characteristic

Q₈

—

Q₈

—

1816

—

Q₈

—

908

—

Q₈

—

454

—

Q₈

—

227

—

C₂×C₄

—

C₂×C₄

—

908

—

C₂×C₄

—

454

—

C₂×C₄

—

454

—

C₂×C₄

D₄×D₂₂₇

yes

—

C₂×C₄

C₂×D₉₀₈

yes

characteristic

C₈

—

100

C₈

—

454

—

101

C₂²

#20300258

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

1816

—

105

C₄

—

908

—

106

C₄

—

454

—

107

C₄

Q₁₆×D₂₂₇

yes

—

108

C₄

#20300317

yes

—

109

C₂

#28409189

yes

—

110

C₂

#28409202

yes

characteristic

	subgroup	G/H
center	C₂²	#20300258
commutator, cosocle	C₄²×C₂₂₇	C₂³
Frattini	C₄²	C₂²×D₂₂₇
Fitting	#333×C₂₂₇	C₂
socle, prime radical	C₂²×C₂₂₇	#120
hypercenter	#333	D₂₂₇
hypocenter	C₂₂₇	#1633
LCS	G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#20300258

commutator, cosocle

C₄²×C₂₂₇

C₂³

Frattini

C₄²

C₂²×D₂₂₇

Fitting

#333×C₂₂₇

C₂

C₂²×C₂₂₇

D₂₂₇

C₂₂₇

G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
35. Dic₄₅₄	Q₁₆	semidirect
55. C₂₂₇	#1633	semidirect

H ≅ G/N

type

35. Dic₄₅₄

Q₁₆

semidirect

55. C₂₂₇

#1633

semidirect

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

subgroup	index
#28409762	2
#28409411	2
#28409328	2
#28408762	2
#333×C₂₂₇	2
#1633	227

subgroup

index

#333×C₂₂₇

227

Subgroups — #44398875

Conjugacy classes of subgroups

Complements

Maximal subgroups