Group #44400710 — subgroups

Subgroups — #44400710

Order 29056.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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	#28409589	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
3	#28408793	C₂	1	4	yes	maximal
4	#423×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	#20300288	C₄	1	2	yes	complemented
6	#20300280	C₄	1	2	yes	complemented
7	#20300267	—	2	2	no	—
8	C₄×Dic₄₅₄	—	2	2	no	—
9	#20300247	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₂₇	—	2	2	no	—
11	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
12	C₂×M₁₆×C₂₂₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₂₇	—	2	2	no	—
15	C₂×Dic₄₅₄	—	2	2	no	—
16	C₂²×Dic₂₂₇	—	2	2	no	—
17	C₄⋊Dic₂₂₇	—	2	2	no	—
18	C₄⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
19	Dic₂₂₇⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₂₂₇	—	2	2	no	—
21	C₄×Dic₂₂₇	—	2	8	no	—
22	C₄×Dic₂₂₇	—	2	2	no	—
23	C₄×Dic₂₂₇	—	2	2	no	—
24	C₄×Dic₂₂₇	Q₈	1	4	yes	complemented
25	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
26	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₂₇	—	2	4	no	—
28	C₂×C₈×C₂₂₇	—	2	2	no	—
29	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₂₂₇	D₄	1	2	yes	—
32	Dic₄₅₄	—	4	4	no	—
33	C₂×Dic₂₂₇	—	2	8	no	—
34	C₂×Dic₂₂₇	—	2	4	no	—
35	C₂×Dic₂₂₇	—	2	4	no	—
36	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₂₇	—	2	4	no	—
38	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₂₂₇	#41	1	2	yes	—
44	C₈×C₂₂₇	—	4	2	no	—
45	C₂²×C₂₂₇	—	2	2	no	—
46	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₂₇	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₂₂₇	—	2	2	no	—
49	C₄×C₂₂₇	#104	1	4	yes	—
50	C₄×C₂₂₇	C₄×Q₈	1	2	yes	—
51	C₄×C₂₂₇	#124	1	1	yes	characteristic
52	C₄×C₂₂₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₂₂₇	—	2	8	no	—
54	Dic₂₂₇	—	4	4	no	—
55	C₂×C₂₂₇	—	2	2	no	—
56	C₂×C₂₂₇	#391	1	1	yes	characteristic
57	C₂×C₂₂₇	C₂×#104	1	2	yes	—
58	C₂₂₇	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	227	1	no	maximal, Sylow
60	#533	—	227	1	no	—
61	#423	D₂₂₇	1	1	yes	characteristic
62	C₄³	—	227	1	no	—
63	#339	—	227	4	no	—
64	C₂×M₁₆	C₂×D₂₂₇	1	2	yes	—
65	C₄⋊Q₈	—	227	2	no	—
66	#125	—	227	2	no	—
67	C₂²⋊Q₈	—	454	2	no	—
68	C₄×Q₈	—	454	2	no	—
69	#117	—	227	2	no	—
70	C₂×C₄²	—	454	2	no	—
71	C₂×C₄²	—	227	2	no	—
72	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	454	2	no	—
75	C₂²×C₄	—	454	2	no	—
76	C₂²×C₄	D₉₀₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	454	4	no	—
81	C₄⋊C₄	—	454	2	no	—
82	C₄⋊C₄	—	227	4	no	—
83	C₂²⋊C₄	—	454	2	no	—
84	C₄²	—	454	8	no	—
85	C₄²	—	454	2	no	—
86	C₄²	—	454	2	no	—
87	C₄²	—	227	4	no	—
88	C₄²	C₄×D₂₂₇	1	1	yes	characteristic
89	C₄²	C₄×D₂₂₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₂₂₇	1	2	yes	—
91	C₂³	C₂×D₉₀₈	1	1	yes	characteristic
92	Q₈	—	908	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	454	4	no	—
95	C₂×C₄	—	454	8	no	—
96	C₂×C₄	—	454	4	no	—
97	C₂×C₄	(C₂×D₂₂₇)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₂₂₇)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₂₂₇	1	2	yes	—
102	C₂×C₄	#16813883	1	2	yes	—
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×(C₂×D₂₂₇)⋊C₄	1	1	yes	characteristic
106	C₂²	#20300268	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	908	4	no	—
109	C₄	—	454	8	no	—
110	C₄	#20300214	1	4	yes	—
111	C₄	#20300253	1	2	yes	—
112	C₄	#20300275	1	1	yes	characteristic
113	C₄	#20300286	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#28408937	1	1	yes	characteristic
116	C₂	C₂×#20300214	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#28409589

C₂

yes

maximal, characteristic

C₄²×Dic₂₂₇

C₂

yes

maximal, characteristic

#28408793

C₂

yes

maximal

#423×C₂₂₇

C₂

yes

maximal, characteristic

#20300288

C₄

yes

complemented

#20300280

C₄

yes

complemented

#20300267

—

C₄×Dic₄₅₄

—

#20300247

C₂²

yes

—

C₂×C₄×Dic₂₂₇

—

C₂×C₄×Dic₂₂₇

C₂²

yes

—

C₂×M₁₆×C₂₂₇

C₂²

yes

—

C₂×C₄²×C₂₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₂₇

—

C₂×Dic₄₅₄

—

C₂²×Dic₂₂₇

—

C₄⋊Dic₂₂₇

—

C₄⋊Dic₂₂₇

C₂×C₄

yes

—

Dic₂₂₇⋊C₄

—

C₂²⋊Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

—

C₄×Dic₂₂₇

Q₈

yes

complemented

C₂²×C₄×C₂₂₇

D₄

yes

—

C₂²×C₄×C₂₂₇

C₂³

yes

characteristic

M₁₆×C₂₂₇

—

C₂×C₈×C₂₂₇

—

C₄²×C₂₂₇

C₂×C₄

yes

characteristic

C₄²×C₂₂₇

C₂×C₄

yes

characteristic

C₄²×C₂₂₇

D₄

yes

—

Dic₄₅₄

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂×Dic₂₂₇

—

C₂³×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₂₇

C₂×Q₈

yes

—

C₂×C₄×C₂₂₇

#41

yes

—

C₈×C₂₂₇

—

C₂²×C₂₂₇

—

C₂²×C₂₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₂₇

C₂²⋊Q₈

yes

—

C₄×C₂₂₇

—

C₄×C₂₂₇

#104

yes

—

C₄×C₂₂₇

C₄×Q₈

yes

—

C₄×C₂₂₇

#124

yes

characteristic

C₄×C₂₂₇

C₄⋊Q₈

yes

characteristic

Dic₂₂₇

—

Dic₂₂₇

—

C₂×C₂₂₇

—

C₂×C₂₂₇

#391

yes

characteristic

C₂×C₂₂₇

C₂×#104

yes

—

C₂₂₇

#1994

yes

characteristic, complemented, Sylow

#1994

—

227

maximal, Sylow

#533

—

227

—

#423

D₂₂₇

yes

characteristic

C₄³

—

227

—

#339

—

227

—

C₂×M₁₆

C₂×D₂₂₇

yes

—

C₄⋊Q₈

—

227

—

#125

—

227

—

C₂²⋊Q₈

—

454

—

C₄×Q₈

—

454

—

#117

—

227

—

C₂×C₄²

—

454

—

C₂×C₄²

—

227

—

C₂×C₄²

C₂×D₂₂₇

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

454

—

C₂²×C₄

—

454

—

C₂²×C₄

D₉₀₈

yes

—

C₂²×C₄

C₂²×D₂₂₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

454

—

C₄⋊C₄

—

227

—

C₂²⋊C₄

—

454

—

C₄²

—

454

—

C₄²

—

454

—

C₄²

—

454

—

C₄²

—

227

—

C₄²

C₄×D₂₂₇

yes

characteristic

C₄²

C₄×D₂₂₇

yes

characteristic

C₄²

C₂²⋊D₂₂₇

yes

—

C₂³

C₂×D₉₀₈

yes

characteristic

Q₈

—

908

—

C₂×C₄

—

C₂×C₄

—

454

—

C₂×C₄

—

454

—

C₂×C₄

—

454

—

C₂×C₄

(C₂×D₂₂₇)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₂₇)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₂₇

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₂₂₇

yes

characteristic

101

C₂×C₄

Q₈×D₂₂₇

yes

—

102

C₂×C₄

#16813883

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₂×(C₂×D₂₂₇)⋊C₄

yes

characteristic

106

C₂²

#20300268

yes

—

107

C₄

—

108

C₄

—

908

—

109

C₄

—

454

—

110

C₄

#20300214

yes

—

111

C₄

#20300253

yes

—

112

C₄

#20300275

yes

characteristic

113

C₄

#20300286

yes

characteristic

114

C₂

—

115

C₂

#28408937

yes

characteristic

116

C₂

C₂×#20300214

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₂₇
commutator	C₂×C₄×C₂₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₂₇
Fitting	#423×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₂₇	C₂×D₄
cosocle	C₂²×C₄×C₂₂₇	C₂³
hypercenter	#423	D₂₂₇
hypocenter	C₂₂₇	#1994
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₂₇

commutator

C₂×C₄×C₂₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₂₇

Fitting

#423×C₂₂₇

C₂

socle

C₂²×C₂₂₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₂₇

C₂×D₄

cosocle

C₂²×C₄×C₂₂₇

C₂³

D₂₂₇

C₂₂₇

G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #20300288	C₄	semidirect
6. #20300280	C₄	semidirect
24. C₄×Dic₂₂₇	Q₈	semidirect
58. C₂₂₇	#1994	semidirect

H ≅ G/N

type

5. #20300288

C₄

semidirect

6. #20300280

C₄

semidirect

24. C₄×Dic₂₂₇

Q₈

semidirect

58. C₂₂₇

#1994

semidirect

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

subgroup	index
#28409589	2
C₄²×Dic₂₂₇	2
#28408793	2
#423×C₂₂₇	2
#1994	227

subgroup

index

#28409589

C₄²×Dic₂₂₇

#28408793

#423×C₂₂₇

#1994

227

Subgroups — #44400710

Conjugacy classes of subgroups

Complements

Maximal subgroups