Group #44483624 — subgroups

Subgroups — #44483624

Order 30592.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#29505502	C₂	1	1	yes	maximal, characteristic, complemented
2	#29504969	C₂	1	2	yes	maximal
3	#29504521	C₂	1	1	yes	maximal, characteristic, complemented
4	#29504504	C₂	1	2	yes	maximal, complemented
5	#492×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838536	—	2	2	no	—
7	#21439959	—	2	2	no	—
8	#21439958	—	2	2	no	—
9	C₄×Dic₄₇₈	C₂²	1	2	yes	—
10	#21439925	—	2	2	no	—
11	C₄⋊(C₂₃₉⋊C₈)	C₂²	1	2	yes	—
12	#21439881	—	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	—	2	2	no	—
18	C₂×Dic₄₇₈	D₄	1	2	yes	—
19	#16838536	—	4	4	no	—
20	C₂×C₂₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₃₉	D₄	1	2	yes	—
22	Dic₂₃₉⋊C₄	—	2	4	no	—
23	Dic₂₃₉⋊C₄	—	2	2	no	—
24	C₄×Dic₂₃₉	—	2	2	no	—
25	C₄×Dic₂₃₉	—	2	2	no	—
26	C₂×Q₈×C₂₃₉	D₄	1	2	yes	—
27	C₂×D₄×C₂₃₉	—	2	1	no	—
28	C₂×D₄×C₂₃₉	—	2	2	no	—
29	SD₁₆×C₂₃₉	—	4	4	no	—
30	C₂×C₈×C₂₃₉	—	2	1	no	—
31	C₄⋊C₄×C₂₃₉	—	2	2	no	—
32	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
33	Dic₄₇₈	—	2	2	no	—
34	Dic₄₇₈	—	8	2	no	—
35	Dic₄₇₈	SD₁₆	1	4	yes	complemented
36	C₂×Dic₂₃₉	—	2	2	no	—
37	C₂×Dic₂₃₉	—	2	2	no	—
38	C₂₃₉⋊C₈	—	4	2	no	—
39	C₂³×C₂₃₉	—	4	1	no	—
40	Q₈×C₂₃₉	—	4	2	no	—
41	D₄×C₂₃₉	—	4	2	no	—
42	D₄×C₂₃₉	—	4	2	no	—
43	D₄×C₂₃₉	—	4	2	no	—
44	C₂×C₄×C₂₃₉	—	2	2	no	—
45	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
47	C₈×C₂₃₉	—	2	2	no	—
48	C₂²×C₂₃₉	—	8	2	no	—
49	C₂²×C₂₃₉	—	8	1	no	—
50	C₂²×C₂₃₉	#120	1	1	yes	characteristic
51	C₄×C₂₃₉	—	2	1	no	—
52	C₄×C₂₃₉	—	4	2	no	—
53	C₄×C₂₃₉	C₂×SD₁₆	1	2	yes	—
54	C₄×C₂₃₉	#137	1	2	yes	—
55	Dic₂₃₉	—	2	4	no	—
56	Dic₂₃₉	—	4	2	no	—
57	C₂×C₂₃₉	—	16	1	no	—
58	C₂×C₂₃₉	#448	1	2	yes	—
59	C₂×C₂₃₉	#453	1	1	yes	characteristic
60	C₂₃₉	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	239	1	no	maximal, Sylow
62	Q₈²	—	239	1	no	—
63	#492	D₂₃₉	1	1	yes	characteristic
64	#460	—	239	2	no	—
65	#333	—	239	1	no	—
66	Q₈⋊C₈	—	239	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	478	2	no	—
69	C₄⋊Q₈	—	478	2	no	—
70	C₄⋊Q₈	—	478	2	no	—
71	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
72	#127	C₂×D₂₃₉	1	1	yes	characteristic
73	C₄×Q₈	—	478	2	no	—
74	C₄×Q₈	—	239	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	—	478	2	no	—
79	C₂×Q₈	—	239	2	no	—
80	C₂×Q₈	C₂²⋊D₂₃₉	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	956	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	478	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	478	4	no	—
89	C₄⋊C₄	—	478	2	no	—
90	C₄⋊C₄	—	239	2	no	—
91	C₄²	—	478	2	no	—
92	C₄²	—	478	2	no	—
93	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	1912	2	no	—
97	Q₈	—	478	2	no	—
98	Q₈	—	239	4	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	478	2	no	—
104	C₂×C₄	—	478	2	no	—
105	C₂×C₄	D₄×D₂₃₉	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	956	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#21439928	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	956	2	no	—
115	C₄	—	478	4	no	—
116	C₄	SD₁₆×D₂₃₉	1	2	yes	—
117	C₄	#21439989	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#29504909	1	2	yes	—
120	C₂	#29504939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29505502

C₂

yes

maximal, characteristic, complemented

#29504969

C₂

yes

maximal

#29504521

C₂

yes

maximal, characteristic, complemented

#29504504

C₂

yes

maximal, complemented

#492×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838536

—

#21439959

—

#21439958

—

C₄×Dic₄₇₈

C₂²

yes

—

#21439925

—

C₄⋊(C₂₃₉⋊C₈)

C₂²

yes

—

#21439881

—

C₂×SD₁₆×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

#127×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

D₄

yes

—

#16838536

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

D₄

yes

—

Dic₂₃₉⋊C₄

—

Dic₂₃₉⋊C₄

—

C₄×Dic₂₃₉

—

C₄×Dic₂₃₉

—

C₂×Q₈×C₂₃₉

D₄

yes

—

C₂×D₄×C₂₃₉

—

C₂×D₄×C₂₃₉

—

SD₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

C₂³

yes

characteristic

Dic₄₇₈

—

Dic₄₇₈

—

Dic₄₇₈

SD₁₆

yes

complemented

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

C₂×SD₁₆

yes

—

C₄×C₂₃₉

#137

yes

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#448

yes

—

C₂×C₂₃₉

#453

yes

characteristic

C₂₃₉

#1623

yes

characteristic, complemented, Sylow

#1623

—

239

maximal, Sylow

Q₈²

—

239

—

#492

D₂₃₉

yes

characteristic

#460

—

239

—

#333

—

239

—

Q₈⋊C₈

—

239

—

C₂×SD₁₆

—

C₂×SD₁₆

—

478

—

C₄⋊Q₈

—

478

—

C₄⋊Q₈

—

478

—

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

#127

C₂×D₂₃₉

yes

characteristic

C₄×Q₈

—

478

—

C₄×Q₈

—

239

—

C₄⋊C₈

—

239

—

#102

—

478

—

C₄×C₈

C₂×D₂₃₉

yes

characteristic

C₂×Q₈

—

478

—

C₂×Q₈

—

239

—

C₂×Q₈

C₂²⋊D₂₃₉

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

956

—

C₂×C₈

—

C₂×C₈

—

478

—

C₄⋊C₄

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

239

—

C₄²

—

478

—

C₄²

—

478

—

C₄²

C₂²×D₂₃₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

1912

—

Q₈

—

478

—

Q₈

—

239

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

478

—

104

C₂×C₄

—

478

—

105

C₂×C₄

D₄×D₂₃₉

yes

—

106

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

107

C₈

—

108

C₈

—

956

—

109

C₂²

—

110

C₂²

—

111

C₂²

#21439928

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

956

—

115

C₄

—

478

—

116

C₄

SD₁₆×D₂₃₉

yes

—

117

C₄

#21439989

yes

—

118

C₂

—

119

C₂

#29504909

yes

—

120

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	#1623
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
1. #29505502	C₂	semidirect
3. #29504521	C₂	semidirect
4. #29504504	C₂	semidirect
35. Dic₄₇₈	SD₁₆	semidirect
60. C₂₃₉	#1623	semidirect

H ≅ G/N

type

1. #29505502

C₂

semidirect

3. #29504521

C₂

semidirect

4. #29504504

C₂

semidirect

35. Dic₄₇₈

SD₁₆

semidirect

60. C₂₃₉

#1623

semidirect

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

subgroup	index
#29505502	2
#29504969	2
#29504521	2
#29504504	2
#492×C₂₃₉	2
#1623	239

subgroup

index

#492×C₂₃₉

239

Subgroups — #44483624

Conjugacy classes of subgroups

Complements

Maximal subgroups