Group #44483691 — subgroups

Subgroups — #44483691

Order 30592.

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	#29505488	C₂	1	1	yes	maximal, characteristic
2	#29505059	C₂	1	2	yes	maximal, complemented
3	#29504521	C₂	1	1	yes	maximal, characteristic, complemented
4	#29504501	C₂	1	2	yes	maximal, complemented
5	#494×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#21439949	—	2	2	no	—
7	#21439935	—	2	2	no	—
8	C₄×D₉₅₆	C₂²	1	2	yes	—
9	#21439918	—	2	2	no	—
10	#21439891	—	2	2	no	—
11	C₄⋊(C₂₃₉⋊C₈)	C₂²	1	2	yes	—
12	#21439880	—	2	2	no	—
13	C₂×Q₁₆×C₂₃₉	—	2	2	no	—
14	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×D₉₅₆	—	2	2	no	—
18	C₂×D₉₅₆	D₄	1	2	yes	—
19	C₂×C₄×D₂₃₉	—	2	2	no	—
20	C₂×C₂₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₃₉	D₄	1	2	yes	—
22	(C₂×D₂₃₉)⋊C₄	—	2	4	no	—
23	(C₂×D₂₃₉)⋊C₄	—	2	2	no	—
24	C₄×Dic₂₃₉	—	2	2	no	—
25	C₂×Q₈×C₂₃₉	—	2	1	no	—
26	C₂×Q₈×C₂₃₉	D₄	1	2	yes	—
27	Q₁₆×C₂₃₉	—	4	4	no	—
28	C₂×C₈×C₂₃₉	—	2	1	no	—
29	C₄⋊C₄×C₂₃₉	—	2	2	no	—
30	C₄⋊C₄×C₂₃₉	—	2	2	no	—
31	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
32	C₂²×D₂₃₉	—	2	2	no	—
33	D₉₅₆	—	2	2	no	—
34	D₉₅₆	—	2	2	no	—
35	D₉₅₆	—	8	2	no	—
36	C₂×Dic₂₃₉	—	2	2	no	—
37	C₄×D₂₃₉	—	4	2	no	—
38	C₄×D₂₃₉	—	4	2	no	—
39	C₂₃₉⋊C₈	—	4	2	no	—
40	Q₈×C₂₃₉	—	4	2	no	—
41	Q₈×C₂₃₉	—	4	2	no	—
42	C₂×C₄×C₂₃₉	—	2	2	no	—
43	C₂×C₄×C₂₃₉	—	4	1	no	—
44	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
46	C₈×C₂₃₉	—	2	2	no	—
47	C₂²×C₂₃₉	#120	1	1	yes	characteristic
48	C₂×D₂₃₉	—	4	2	no	—
49	C₂×D₂₃₉	—	4	2	no	—
50	C₂×D₂₃₉	—	4	2	no	—
51	C₄×C₂₃₉	—	2	1	no	—
52	C₄×C₂₃₉	—	4	2	no	—
53	C₄×C₂₃₉	—	8	1	no	—
54	C₄×C₂₃₉	#135	1	2	yes	—
55	C₄×C₂₃₉	#137	1	2	yes	—
56	Dic₂₃₉	—	4	2	no	—
57	C₂×C₂₃₉	#452	1	2	yes	—
58	C₂×C₂₃₉	#453	1	1	yes	characteristic
59	D₂₃₉	—	8	2	no	—
60	C₂₃₉	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	239	1	no	maximal, Sylow
62	#555	—	239	1	no	—
63	#494	D₂₃₉	1	1	yes	characteristic, complemented
64	#476	—	239	2	no	—
65	#333	—	239	1	no	—
66	D₄⋊C₈	—	239	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
70	#124	—	478	2	no	—
71	#121	—	478	2	no	—
72	C₄×D₄	—	239	2	no	—
73	#117	—	478	2	no	—
74	#106	—	478	2	no	—
75	C₄⋊C₈	—	239	2	no	—
76	#102	—	478	2	no	—
77	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₂₃₉	1	2	yes	—
80	C₂×D₄	—	478	2	no	—
81	C₂×D₄	—	239	2	no	—
82	C₂²×C₄	—	478	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	478	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	239	2	no	—
89	C₂²⋊C₄	—	478	4	no	—
90	C₂²⋊C₄	—	478	2	no	—
91	C₄²	—	478	2	no	—
92	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
93	C₂³	—	478	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	1912	2	no	—
97	D₄	—	478	2	no	—
98	D₄	—	478	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	956	2	no	—
102	C₂×C₄	—	956	2	no	—
103	C₂×C₄	—	478	2	no	—
104	C₂×C₄	D₄×D₂₃₉	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	956	2	no	—
108	C₂²	—	956	2	no	—
109	C₂²	—	956	2	no	—
110	C₂²	—	956	2	no	—
111	C₂²	#21439928	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	956	2	no	—
116	C₄	#21439983	1	2	yes	—
117	C₄	#21439992	1	2	yes	—
118	C₂	—	1912	2	no	—
119	C₂	#29504933	1	2	yes	—
120	C₂	#29504939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29505488

C₂

yes

maximal, characteristic

#29505059

C₂

yes

maximal, complemented

#29504521

C₂

yes

maximal, characteristic, complemented

#29504501

C₂

yes

maximal, complemented

#494×C₂₃₉

C₂

yes

maximal, characteristic, complemented

#21439949

—

#21439935

—

C₄×D₉₅₆

C₂²

yes

—

#21439918

—

#21439891

—

C₄⋊(C₂₃₉⋊C₈)

C₂²

yes

—

#21439880

—

C₂×Q₁₆×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×D₉₅₆

—

C₂×D₉₅₆

D₄

yes

—

C₂×C₄×D₂₃₉

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

D₄

yes

—

(C₂×D₂₃₉)⋊C₄

—

(C₂×D₂₃₉)⋊C₄

—

C₄×Dic₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

D₄

yes

—

Q₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

C₂³

yes

characteristic

C₂²×D₂₃₉

—

D₉₅₆

—

D₉₅₆

—

D₉₅₆

—

C₂×Dic₂₃₉

—

C₄×D₂₃₉

—

C₄×D₂₃₉

—

C₂₃₉⋊C₈

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

#120

yes

characteristic

C₂×D₂₃₉

—

C₂×D₂₃₉

—

C₂×D₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#135

yes

—

C₄×C₂₃₉

#137

yes

—

Dic₂₃₉

—

C₂×C₂₃₉

#452

yes

—

C₂×C₂₃₉

#453

yes

characteristic

D₂₃₉

—

C₂₃₉

#1634

yes

characteristic, complemented, Sylow

#1634

—

239

maximal, Sylow

#555

—

239

—

#494

D₂₃₉

yes

characteristic, complemented

#476

—

239

—

#333

—

239

—

D₄⋊C₈

—

239

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

#124

—

478

—

#121

—

478

—

C₄×D₄

—

239

—

#117

—

478

—

#106

—

478

—

C₄⋊C₈

—

239

—

#102

—

478

—

C₄×C₈

C₂×D₂₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃₉

yes

—

C₂×D₄

—

478

—

C₂×D₄

—

239

—

C₂²×C₄

—

478

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

478

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

239

—

C₂²⋊C₄

—

478

—

C₂²⋊C₄

—

478

—

C₄²

—

478

—

C₄²

C₂²×D₂₃₉

yes

characteristic

C₂³

—

478

—

Q₈

—

Q₈

—

D₄

—

1912

—

D₄

—

478

—

D₄

—

478

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

956

—

102

C₂×C₄

—

956

—

103

C₂×C₄

—

478

—

104

C₂×C₄

D₄×D₂₃₉

yes

—

105

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

106

C₈

—

107

C₈

—

956

—

108

C₂²

—

956

—

109

C₂²

—

956

—

110

C₂²

—

956

—

111

C₂²

#21439928

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

956

—

116

C₄

#21439983

yes

—

117

C₄

#21439992

yes

—

118

C₂

—

1912

—

119

C₂

#29504933

yes

—

120

C₂

#29504939

yes

characteristic

	subgroup	G/H
center	C₂²	#21439928
commutator, cosocle	C₄²×C₂₃₉	C₂³
Frattini	C₄²	C₂²×D₂₃₉
Fitting	#494×C₂₃₉	C₂
socle	C₂²×C₂₃₉	#120
prime radical	#29505488	C₂
hypercenter	#494	D₂₃₉
hypocenter	C₂₃₉	#1634
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#21439928

commutator, cosocle

C₄²×C₂₃₉

C₂³

Frattini

C₄²

C₂²×D₂₃₉

Fitting

#494×C₂₃₉

C₂

C₂²×C₂₃₉

C₂

D₂₃₉

C₂₃₉

G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #29505059	C₂	semidirect
3. #29504521	C₂	semidirect
4. #29504501	C₂	semidirect
5. #494×C₂₃₉	C₂	semidirect
60. C₂₃₉	#1634	semidirect
63. #494	D₂₃₉	semidirect

H ≅ G/N

type

2. #29505059

C₂

semidirect

3. #29504521

C₂

semidirect

4. #29504501

C₂

semidirect

5. #494×C₂₃₉

C₂

semidirect

60. C₂₃₉

#1634

semidirect

63. #494

D₂₃₉

semidirect

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

subgroup	index
#29505488	2
#29505059	2
#29504521	2
#29504501	2
#494×C₂₃₉	2
#1634	239

subgroup

index

#494×C₂₃₉

239

Subgroups — #44483691

Conjugacy classes of subgroups

Complements

Maximal subgroups