Group #44398877 — subgroups

Subgroups — #44398877

Order 29056.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#28409748	C₂	1	1	yes	maximal, characteristic
2	#28409319	C₂	1	2	yes	maximal, complemented
3	#28408781	C₂	1	1	yes	maximal, characteristic, complemented
4	#28408761	C₂	1	2	yes	maximal, complemented
5	#494×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20300277	—	2	2	no	—
7	#20300263	—	2	2	no	—
8	C₄×D₉₀₈	C₂²	1	2	yes	—
9	#20300246	—	2	2	no	—
10	#20300219	—	2	2	no	—
11	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
12	#20300208	—	2	2	no	—
13	C₂×Q₁₆×C₂₂₇	—	2	2	no	—
14	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×D₉₀₈	—	2	2	no	—
18	C₂×D₉₀₈	D₄	1	2	yes	—
19	C₂×C₄×D₂₂₇	—	2	2	no	—
20	C₂×C₂₂₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₂₇	D₄	1	2	yes	—
22	(C₂×D₂₂₇)⋊C₄	—	2	4	no	—
23	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
24	C₄×Dic₂₂₇	—	2	2	no	—
25	C₂×Q₈×C₂₂₇	—	2	1	no	—
26	C₂×Q₈×C₂₂₇	D₄	1	2	yes	—
27	Q₁₆×C₂₂₇	—	4	4	no	—
28	C₂×C₈×C₂₂₇	—	2	1	no	—
29	C₄⋊C₄×C₂₂₇	—	2	2	no	—
30	C₄⋊C₄×C₂₂₇	—	2	2	no	—
31	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
32	C₂²×D₂₂₇	—	2	2	no	—
33	D₉₀₈	—	2	2	no	—
34	D₉₀₈	—	2	2	no	—
35	D₉₀₈	—	8	2	no	—
36	C₂×Dic₂₂₇	—	2	2	no	—
37	C₄×D₂₂₇	—	4	2	no	—
38	C₄×D₂₂₇	—	4	2	no	—
39	C₂₂₇⋊C₈	—	4	2	no	—
40	Q₈×C₂₂₇	—	4	2	no	—
41	Q₈×C₂₂₇	—	4	2	no	—
42	C₂×C₄×C₂₂₇	—	2	2	no	—
43	C₂×C₄×C₂₂₇	—	4	1	no	—
44	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
46	C₈×C₂₂₇	—	2	2	no	—
47	C₂²×C₂₂₇	#120	1	1	yes	characteristic
48	C₂×D₂₂₇	—	4	2	no	—
49	C₂×D₂₂₇	—	4	2	no	—
50	C₂×D₂₂₇	—	4	2	no	—
51	C₄×C₂₂₇	—	2	1	no	—
52	C₄×C₂₂₇	—	4	2	no	—
53	C₄×C₂₂₇	—	8	1	no	—
54	C₄×C₂₂₇	#135	1	2	yes	—
55	C₄×C₂₂₇	#137	1	2	yes	—
56	Dic₂₂₇	—	4	2	no	—
57	C₂×C₂₂₇	#452	1	2	yes	—
58	C₂×C₂₂₇	#453	1	1	yes	characteristic
59	D₂₂₇	—	8	2	no	—
60	C₂₂₇	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	227	1	no	maximal, Sylow
62	#555	—	227	1	no	—
63	#494	D₂₂₇	1	1	yes	characteristic, complemented
64	#476	—	227	2	no	—
65	#333	—	227	1	no	—
66	D₄⋊C₈	—	227	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₂₂₇	1	1	yes	characteristic
70	#124	—	454	2	no	—
71	#121	—	454	2	no	—
72	C₄×D₄	—	227	2	no	—
73	#117	—	454	2	no	—
74	#106	—	454	2	no	—
75	C₄⋊C₈	—	227	2	no	—
76	#102	—	454	2	no	—
77	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₂₂₇	1	2	yes	—
80	C₂×D₄	—	454	2	no	—
81	C₂×D₄	—	227	2	no	—
82	C₂²×C₄	—	454	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	454	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	227	2	no	—
89	C₂²⋊C₄	—	454	4	no	—
90	C₂²⋊C₄	—	454	2	no	—
91	C₄²	—	454	2	no	—
92	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
93	C₂³	—	454	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	1816	2	no	—
97	D₄	—	454	2	no	—
98	D₄	—	454	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	908	2	no	—
102	C₂×C₄	—	908	2	no	—
103	C₂×C₄	—	454	2	no	—
104	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	908	2	no	—
108	C₂²	—	908	2	no	—
109	C₂²	—	908	2	no	—
110	C₂²	—	908	2	no	—
111	C₂²	#20300256	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	908	2	no	—
116	C₄	#20300311	1	2	yes	—
117	C₄	#20300320	1	2	yes	—
118	C₂	—	1816	2	no	—
119	C₂	#28409193	1	2	yes	—
120	C₂	#28409199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28409748

C₂

yes

maximal, characteristic

#28409319

C₂

yes

maximal, complemented

#28408781

C₂

yes

maximal, characteristic, complemented

#28408761

C₂

yes

maximal, complemented

#494×C₂₂₇

C₂

yes

maximal, characteristic, complemented

#20300277

—

#20300263

—

C₄×D₉₀₈

C₂²

yes

—

#20300246

—

#20300219

—

C₄⋊(C₂₂₇⋊C₈)

C₂²

yes

—

#20300208

—

C₂×Q₁₆×C₂₂₇

—

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

C₄⋊Q₈×C₂₂₇

C₂²

yes

characteristic

C₄×C₈×C₂₂₇

C₂²

yes

characteristic

C₂×D₉₀₈

—

C₂×D₉₀₈

D₄

yes

—

C₂×C₄×D₂₂₇

—

C₂×C₂₂₇⋊C₈

—

C₄⋊Dic₂₂₇

D₄

yes

—

(C₂×D₂₂₇)⋊C₄

—

(C₂×D₂₂₇)⋊C₄

—

C₄×Dic₂₂₇

—

C₂×Q₈×C₂₂₇

—

C₂×Q₈×C₂₂₇

D₄

yes

—

Q₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄⋊C₄×C₂₂₇

—

C₄²×C₂₂₇

C₂³

yes

characteristic

C₂²×D₂₂₇

—

D₉₀₈

—

D₉₀₈

—

D₉₀₈

—

C₂×Dic₂₂₇

—

C₄×D₂₂₇

—

C₄×D₂₂₇

—

C₂₂₇⋊C₈

—

Q₈×C₂₂₇

—

Q₈×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

#120

yes

characteristic

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₂×D₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#135

yes

—

C₄×C₂₂₇

#137

yes

—

Dic₂₂₇

—

C₂×C₂₂₇

#452

yes

—

C₂×C₂₂₇

#453

yes

characteristic

D₂₂₇

—

C₂₂₇

#1634

yes

characteristic, complemented, Sylow

#1634

—

227

maximal, Sylow

#555

—

227

—

#494

D₂₂₇

yes

characteristic, complemented

#476

—

227

—

#333

—

227

—

D₄⋊C₈

—

227

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

C₄⋊Q₈

C₂×D₂₂₇

yes

characteristic

#124

—

454

—

#121

—

454

—

C₄×D₄

—

227

—

#117

—

454

—

#106

—

454

—

C₄⋊C₈

—

227

—

#102

—

454

—

C₄×C₈

C₂×D₂₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₂₇

yes

—

C₂×D₄

—

454

—

C₂×D₄

—

227

—

C₂²×C₄

—

454

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

454

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

227

—

C₂²⋊C₄

—

454

—

C₂²⋊C₄

—

454

—

C₄²

—

454

—

C₄²

C₂²×D₂₂₇

yes

characteristic

C₂³

—

454

—

Q₈

—

Q₈

—

D₄

—

1816

—

D₄

—

454

—

D₄

—

454

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

908

—

102

C₂×C₄

—

908

—

103

C₂×C₄

—

454

—

104

C₂×C₄

D₄×D₂₂₇

yes

—

105

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

106

C₈

—

107

C₈

—

908

—

108

C₂²

—

908

—

109

C₂²

—

908

—

110

C₂²

—

908

—

111

C₂²

#20300256

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

908

—

116

C₄

#20300311

yes

—

117

C₄

#20300320

yes

—

118

C₂

—

1816

—

119

C₂

#28409193

yes

—

120

C₂

#28409199

yes

characteristic

	subgroup	G/H
center	C₂²	#20300256
commutator, cosocle	C₄²×C₂₂₇	C₂³
Frattini	C₄²	C₂²×D₂₂₇
Fitting	#494×C₂₂₇	C₂
socle	C₂²×C₂₂₇	#120
prime radical	#28409748	C₂
hypercenter	#494	D₂₂₇
hypocenter	C₂₂₇	#1634
LCS	G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#20300256

commutator, cosocle

C₄²×C₂₂₇

C₂³

Frattini

C₄²

C₂²×D₂₂₇

Fitting

#494×C₂₂₇

C₂

C₂²×C₂₂₇

C₂

D₂₂₇

C₂₂₇

G ▷ C₄²×C₂₂₇ ▷ C₂²×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #28409319	C₂	semidirect
3. #28408781	C₂	semidirect
4. #28408761	C₂	semidirect
5. #494×C₂₂₇	C₂	semidirect
60. C₂₂₇	#1634	semidirect
63. #494	D₂₂₇	semidirect

H ≅ G/N

type

2. #28409319

C₂

semidirect

3. #28408781

C₂

semidirect

4. #28408761

C₂

semidirect

5. #494×C₂₂₇

C₂

semidirect

60. C₂₂₇

#1634

semidirect

63. #494

D₂₂₇

semidirect

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

subgroup	index
#28409748	2
#28409319	2
#28408781	2
#28408761	2
#494×C₂₂₇	2
#1634	227

subgroup

index

#494×C₂₂₇

227

Subgroups — #44398877

Conjugacy classes of subgroups

Complements

Maximal subgroups