Group #44483667 — subgroups

Subgroups — #44483667

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	#29505508	C₂	1	1	yes	maximal, characteristic
2	#29505064	C₂	1	2	yes	maximal, complemented
3	#29504516	C₂	1	1	yes	maximal, characteristic
4	#29504504	C₂	1	2	yes	maximal, complemented
5	#492×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	#21439947	—	2	2	no	—
7	#21439941	—	2	2	no	—
8	C₄×Dic₄₇₈	C₂²	1	2	yes	—
9	D₄×Dic₂₃₉	—	2	2	no	—
10	#21439893	—	2	2	no	—
11	C₄⋊(C₂₃₉⋊C₈)	C₂²	1	2	yes	—
12	#21439883	—	2	2	no	—
13	C₂×SD₁₆×C₂₃₉	—	2	2	no	—
14	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	#127×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇₈	D₄	1	2	yes	—
18	C₂²×Dic₂₃₉	—	2	2	no	—
19	C₂×C₂₃₉⋊C₈	—	2	2	no	—
20	C₄⋊Dic₂₃₉	D₄	1	2	yes	—
21	Dic₂₃₉⋊C₄	—	2	2	no	—
22	C₂²⋊Dic₂₃₉	—	2	4	no	—
23	C₂²⋊Dic₂₃₉	—	2	2	no	—
24	C₄×Dic₂₃₉	—	2	2	no	—
25	C₂×Q₈×C₂₃₉	—	2	1	no	—
26	C₂×D₄×C₂₃₉	—	2	2	no	—
27	C₂×D₄×C₂₃₉	D₄	1	2	yes	—
28	SD₁₆×C₂₃₉	—	4	4	no	—
29	C₂×C₈×C₂₃₉	—	2	1	no	—
30	C₄⋊C₄×C₂₃₉	—	2	2	no	—
31	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
32	Dic₄₇₈	—	2	2	no	—
33	Dic₄₇₈	—	2	2	no	—
34	C₂×Dic₂₃₉	—	2	2	no	—
35	C₂×Dic₂₃₉	—	2	2	no	—
36	C₂×Dic₂₃₉	—	8	2	no	—
37	C₂₃₉⋊C₈	—	4	2	no	—
38	C₂³×C₂₃₉	—	2	2	no	—
39	Q₈×C₂₃₉	—	4	2	no	—
40	D₄×C₂₃₉	—	4	2	no	—
41	D₄×C₂₃₉	—	4	2	no	—
42	D₄×C₂₃₉	—	4	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	—	4	4	no	—
48	C₂²×C₂₃₉	—	4	2	no	—
49	C₂²×C₂₃₉	#120	1	1	yes	characteristic
50	C₄×C₂₃₉	—	2	1	no	—
51	C₄×C₂₃₉	—	8	1	no	—
52	C₄×C₂₃₉	#135	1	2	yes	—
53	C₄×C₂₃₉	#137	1	2	yes	—
54	Dic₂₃₉	—	4	2	no	—
55	Dic₂₃₉	—	4	2	no	—
56	C₂×C₂₃₉	—	8	2	no	—
57	C₂×C₂₃₉	#452	1	2	yes	—
58	C₂×C₂₃₉	#453	1	1	yes	characteristic
59	C₂₃₉	#1630	1	1	yes	characteristic, complemented, Sylow
60	#1630	—	239	1	no	maximal, Sylow
61	#559	—	239	1	no	—
62	#492	D₂₃₉	1	1	yes	characteristic
63	#477	—	239	2	no	—
64	#331	—	239	1	no	—
65	Q₈⋊C₈	—	239	2	no	—
66	C₂×SD₁₆	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
68	#127	C₂×D₂₃₉	1	1	yes	characteristic
69	#124	—	478	2	no	—
70	#123	—	478	2	no	—
71	C₄×Q₈	—	239	2	no	—
72	C₄×D₄	—	478	2	no	—
73	#107	—	478	2	no	—
74	C₄⋊C₈	—	239	2	no	—
75	#103	—	478	2	no	—
76	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	239	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₂₃₉	1	2	yes	—
81	C₂²×C₄	—	478	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	478	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	478	2	no	—
87	C₄⋊C₄	—	239	2	no	—
88	C₂²⋊C₄	—	478	4	no	—
89	C₂²⋊C₄	—	478	2	no	—
90	C₄²	—	478	2	no	—
91	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	478	2	no	—
95	Q₈	—	478	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	1912	2	no	—
101	C₂×C₄	—	478	2	no	—
102	C₂×C₄	—	478	2	no	—
103	C₂×C₄	D₄×D₂₃₉	1	2	yes	—
104	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	956	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#21439928	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	8	1	no	—
112	C₄	—	956	2	no	—
113	C₄	—	956	2	no	—
114	C₄	#21439981	1	2	yes	—
115	C₄	#21439989	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#29504931	1	2	yes	—
118	C₂	#29504939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29505508

C₂

yes

maximal, characteristic

#29505064

C₂

yes

maximal, complemented

#29504516

C₂

yes

maximal, characteristic

#29504504

C₂

yes

maximal, complemented

#492×C₂₃₉

C₂

yes

maximal, characteristic

#21439947

—

#21439941

—

C₄×Dic₄₇₈

C₂²

yes

—

D₄×Dic₂₃₉

—

#21439893

—

C₄⋊(C₂₃₉⋊C₈)

C₂²

yes

—

#21439883

—

C₂×SD₁₆×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

#127×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×Dic₄₇₈

D₄

yes

—

C₂²×Dic₂₃₉

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

D₄

yes

—

Dic₂₃₉⋊C₄

—

C₂²⋊Dic₂₃₉

—

C₂²⋊Dic₂₃₉

—

C₄×Dic₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×D₄×C₂₃₉

—

C₂×D₄×C₂₃₉

D₄

yes

—

SD₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

C₂³

yes

characteristic

Dic₄₇₈

—

Dic₄₇₈

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

—

Q₈×C₂₃₉

—

D₄×C₂₃₉

—

D₄×C₂₃₉

—

D₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

#120

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#135

yes

—

C₄×C₂₃₉

#137

yes

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#452

yes

—

C₂×C₂₃₉

#453

yes

characteristic

C₂₃₉

#1630

yes

characteristic, complemented, Sylow

#1630

—

239

maximal, Sylow

#559

—

239

—

#492

D₂₃₉

yes

characteristic

#477

—

239

—

#331

—

239

—

Q₈⋊C₈

—

239

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

#127

C₂×D₂₃₉

yes

characteristic

#124

—

478

—

#123

—

478

—

C₄×Q₈

—

239

—

C₄×D₄

—

478

—

#107

—

478

—

C₄⋊C₈

—

239

—

#103

—

478

—

C₄×C₈

C₂×D₂₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

239

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₃₉

yes

—

C₂²×C₄

—

478

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

478

—

C₄⋊C₄

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

239

—

C₂²⋊C₄

—

478

—

C₂²⋊C₄

—

478

—

C₄²

—

478

—

C₄²

C₂²×D₂₃₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

478

—

Q₈

—

478

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

1912

—

101

C₂×C₄

—

478

—

102

C₂×C₄

—

478

—

103

C₂×C₄

D₄×D₂₃₉

yes

—

104

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

105

C₈

—

106

C₈

—

956

—

107

C₂²

—

108

C₂²

—

109

C₂²

#21439928

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

956

—

113

C₄

—

956

—

114

C₄

#21439981

yes

—

115

C₄

#21439989

yes

—

116

C₂

—

117

C₂

#29504931

yes

—

118

C₂

#29504939

yes

characteristic

	subgroup	G/H
center	C₂²	#21439928
commutator, cosocle	C₄²×C₂₃₉	C₂³
Frattini	C₄²	C₂²×D₂₃₉
Fitting	#492×C₂₃₉	C₂
socle	C₂²×C₂₃₉	#120
prime radical	#127×C₂₃₉	C₂²
hypercenter	#492	D₂₃₉
hypocenter	C₂₃₉	#1630
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#21439928

commutator, cosocle

C₄²×C₂₃₉

C₂³

Frattini

C₄²

C₂²×D₂₃₉

Fitting

#492×C₂₃₉

C₂

socle

C₂²×C₂₃₉

#120

prime radical

#127×C₂₃₉

C₂²

D₂₃₉

C₂₃₉

G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #29505064	C₂	semidirect
4. #29504504	C₂	semidirect
59. C₂₃₉	#1630	semidirect

H ≅ G/N

type

2. #29505064

C₂

semidirect

4. #29504504

C₂

semidirect

59. C₂₃₉

#1630

semidirect

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

subgroup	index
#29505508	2
#29505064	2
#29504516	2
#29504504	2
#492×C₂₃₉	2
#1630	239

subgroup

index

#492×C₂₃₉

239

Subgroups — #44483667

Conjugacy classes of subgroups

Complements

Maximal subgroups