Group #44485296 — subgroups

Subgroups — #44485296

Order 30592.

208 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 93 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×#21439896	C₂	1	1	yes	maximal, characteristic
2	#29504810	C₂	1	1	yes	maximal, characteristic
3	#29504798	C₂	1	2	yes	maximal
4	#29504532	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838532	C₂²	1	2	yes	—
7	#21439919	C₂²	1	2	yes	—
8	#21439896	—	2	2	no	—
9	#21439894	C₄	1	2	yes	complemented
10	#21439892	C₄	1	2	yes	complemented
11	Dic₂₃₉⋊C₈	—	2	4	no	—
12	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₃₉	—	2	2	no	—
16	#16838532	—	2	2	no	—
17	#16838532	—	2	2	no	—
18	C₂×C₂₃₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₂₃₉	C₂×C₄	1	4	yes	—
20	C₂²⋊Dic₂₃₉	—	2	2	no	—
21	C₄×Dic₂₃₉	—	2	2	no	—
22	C₂²×C₄×C₂₃₉	—	2	1	no	—
23	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₃₉	—	2	1	no	—
25	C₂×C₈×C₂₃₉	—	2	1	no	—
26	C₂×C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
29	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
30	C₂×C₈×C₂₃₉	Q₈	1	2	yes	—
31	C₄²×C₂₃₉	—	2	1	no	—
32	C₄²×C₂₃₉	Q₈	1	2	yes	—
33	C₂×Dic₂₃₉	—	2	4	no	—
34	C₂₃₉⋊C₈	—	4	2	no	—
35	C₂³×C₂₃₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₂₃₉	—	2	4	no	—
37	C₂×C₄×C₂₃₉	—	2	2	no	—
38	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
44	C₈×C₂₃₉	—	2	2	no	—
45	C₈×C₂₃₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₃₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₃₉	—	2	2	no	—
48	C₂²×C₂₃₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₃₉	#125	1	1	yes	characteristic
51	C₄×C₂₃₉	—	2	4	no	—
52	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₂₃₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₃₉	#121	1	1	yes	characteristic
55	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₂₃₉	—	4	4	no	—
57	C₂×C₂₃₉	—	2	1	no	—
58	C₂×C₂₃₉	—	2	1	no	—
59	C₂×C₂₃₉	#389	1	1	yes	characteristic
60	C₂×C₂₃₉	#443	1	2	yes	—
61	C₂₃₉	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	239	1	no	maximal, Sylow
63	C₂×#108	—	239	1	no	—
64	#427	—	239	1	no	—
65	#424	—	239	2	no	—
66	C₂×C₄×C₈	D₂₃₉	1	1	yes	characteristic
67	#339	—	239	2	no	—
68	C₂×M₁₆	—	239	2	no	—
69	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
71	#117	—	239	2	no	—
72	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
73	#108	—	478	2	no	—
74	#107	—	239	2	no	—
75	#106	—	239	2	no	—
76	C₄⋊C₈	—	478	4	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
80	M₁₆	—	478	2	no	—
81	M₁₆	—	478	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	478	2	no	—
85	C₂×C₈	C₄×D₂₃₉	1	1	yes	characteristic
86	C₂×C₈	C₄×D₂₃₉	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊D₂₃₉	1	2	yes	—
88	C₂×C₈	D₉₅₆	1	2	yes	—
89	C₂×C₈	Dic₄₇₈	1	2	yes	—
90	C₄⋊C₄	—	239	4	no	—
91	C₂²⋊C₄	—	478	2	no	—
92	C₄²	—	2	1	no	—
93	C₄²	—	478	2	no	—
94	C₄²	Dic₄₇₈	1	2	yes	—
95	C₂³	#16838548	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	478	4	no	—
99	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
102	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
104	C₂×C₄	#16838548	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	956	2	no	—
107	C₈	Dic₂₃₉⋊C₄	1	2	yes	—
108	C₈	Dic₂₃₉⋊C₄	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	C₄×D₉₅₆	1	1	yes	characteristic
111	C₂²	#21439939	1	1	yes	characteristic
112	C₂²	#21439952	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	956	4	no	—
115	C₄	C₂×Dic₂₃₉⋊C₄	1	1	yes	characteristic
116	C₄	C₄×Dic₄₇₈	1	1	yes	characteristic
117	C₄	#21439934	1	1	yes	characteristic
118	C₄	#21439960	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#29504670	1	1	yes	characteristic
122	C₂	#29504890	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439896

C₂

yes

maximal, characteristic

#29504810

C₂

yes

maximal, characteristic

#29504798

C₂

yes

maximal

#29504532

C₂

yes

maximal

C₂×C₄×C₈×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838532

C₂²

yes

—

#21439919

C₂²

yes

—

#21439896

—

#21439894

C₄

yes

complemented

#21439892

C₄

yes

complemented

Dic₂₃₉⋊C₈

—

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

—

#16838532

—

#16838532

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

C₂×C₄

yes

—

C₂²⋊Dic₂₃₉

—

C₄×Dic₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

Q₈

yes

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

Q₈

yes

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

#41

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

characteristic

C₈×C₂₃₉

—

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₄×D₄

yes

characteristic

C₂²×C₂₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃₉

#125

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃₉

C₄×Q₈

yes

characteristic

C₄×C₂₃₉

#121

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#389

yes

characteristic

C₂×C₂₃₉

#443

yes

—

C₂₃₉

#1945

yes

characteristic, complemented, Sylow

#1945

—

239

maximal, Sylow

C₂×#108

—

239

—

#427

—

239

—

#424

—

239

—

C₂×C₄×C₈

D₂₃₉

yes

characteristic

#339

—

239

—

C₂×M₁₆

—

239

—

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

#117

—

239

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

#108

—

478

—

#107

—

239

—

#106

—

239

—

C₄⋊C₈

—

478

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

M₁₆

—

478

—

M₁₆

—

478

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

478

—

C₂×C₈

C₄×D₂₃₉

yes

characteristic

C₂×C₈

C₄×D₂₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₂₃₉

yes

—

C₂×C₈

D₉₅₆

yes

—

C₂×C₈

Dic₄₇₈

yes

—

C₄⋊C₄

—

239

—

C₂²⋊C₄

—

478

—

C₄²

—

C₄²

—

478

—

C₄²

Dic₄₇₈

yes

—

C₂³

#16838548

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

478

—

C₂×C₄

C₂×C₄×D₂₃₉

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

101

C₂×C₄

C₂×D₉₅₆

yes

characteristic

102

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

103

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

104

C₂×C₄

#16838548

yes

characteristic

105

C₈

—

106

C₈

—

956

—

107

C₈

Dic₂₃₉⋊C₄

yes

—

108

C₈

Dic₂₃₉⋊C₄

yes

—

109

C₂²

—

110

C₂²

C₄×D₉₅₆

yes

characteristic

111

C₂²

#21439939

yes

characteristic

112

C₂²

#21439952

yes

characteristic

113

C₄

—

114

C₄

—

956

—

115

C₄

C₂×Dic₂₃₉⋊C₄

yes

characteristic

116

C₄

C₄×Dic₄₇₈

yes

characteristic

117

C₄

#21439934

yes

characteristic

118

C₄

#21439960

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#29504670

yes

characteristic

122

C₂

#29504890

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₉₅₆
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂×C₄×C₈×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₄×D₄
prime radical	C₂³×C₂₃₉	#41
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×C₄×C₈	D₂₃₉
hypocenter	C₂₃₉	#1945
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₉₅₆

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₂×C₄×C₈×C₂₃₉

C₂

socle

C₂²×C₂₃₉

C₄×D₄

prime radical

C₂³×C₂₃₉

#41

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₂×C₄×C₈

D₂₃₉

hypocenter

C₂₃₉

#1945

LCS

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. #21439894	C₄	semidirect
10. #21439892	C₄	semidirect
61. C₂₃₉	#1945	semidirect

H ≅ G/N

type

9. #21439894

C₄

semidirect

10. #21439892

C₄

semidirect

61. C₂₃₉

#1945

semidirect

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

subgroup	index
C₂×#21439896	2
#29504810	2
#29504798	2
#29504532	2
C₂×C₄×C₈×C₂₃₉	2
#1945	239

subgroup

index

C₂×C₄×C₈×C₂₃₉

239

Subgroups — #44485296

Conjugacy classes of subgroups

Complements

Maximal subgroups