Group #44486625 — subgroups

Subgroups — #44486625

Order 30592.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#29505210	C₂	1	2	yes	maximal, complemented
2	#29505149	C₂	1	1	yes	maximal, characteristic, complemented
3	#29504890	C₂	1	1	yes	maximal, characteristic, complemented
4	#29504586	C₂	1	2	yes	maximal, complemented
5	#364×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#21439982	C₂²	1	1	yes	characteristic
7	C₂×D₁₉₁₂	—	2	1	no	—
8	C₂×D₁₉₁₂	C₂²	1	2	yes	—
9	#21439966	—	2	1	no	—
10	#21439957	—	2	1	no	—
11	#21439905	—	2	2	no	—
12	D₃₈₂₄	—	2	2	no	—
13	#21439896	C₂²	1	1	yes	characteristic
14	#21439888	—	2	1	no	—
15	M₃₂×C₂₃₉	C₂²	1	2	yes	—
16	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	#16838548	—	2	1	no	—
18	C₂×D₉₅₆	—	2	2	no	—
19	C₂×D₉₅₆	—	4	2	no	—
20	Dic₉₅₆	D₄	1	1	yes	characteristic, complemented
21	#16838537	—	2	1	no	—
22	D₁₉₁₂	—	2	2	no	—
23	D₁₉₁₂	—	2	2	no	—
24	D₁₉₁₂	—	4	1	no	—
25	D₁₉₁₂	—	4	1	no	—
26	D₁₉₁₂	D₄	1	1	yes	characteristic, complemented
27	#16838532	—	2	1	no	—
28	#16838531	—	4	1	no	—
29	C₈×D₂₃₉	—	4	1	no	—
30	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
32	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
33	C₁₆×C₂₃₉	—	2	2	no	—
34	C₂²×D₂₃₉	—	4	2	no	—
35	Dic₄₇₈	—	2	1	no	—
36	D₉₅₆	—	2	1	no	—
37	D₉₅₆	—	4	2	no	—
38	D₉₅₆	—	4	2	no	—
39	D₉₅₆	—	8	4	no	—
40	C₂²⋊D₂₃₉	—	4	1	no	—
41	C₄×D₂₃₉	—	4	1	no	—
42	C₂₃₉⋊C₈	—	4	1	no	—
43	C₂×C₄×C₂₃₉	—	2	1	no	—
44	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
45	C₈×C₂₃₉	D₈	1	1	yes	characteristic
46	C₈×C₂₃₉	D₈	1	1	yes	characteristic
47	C₈×C₂₃₉	C₂×D₄	1	1	yes	characteristic
48	C₈×C₂₃₉	#41	1	1	yes	characteristic
49	C₂²×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₂₃₉	—	4	1	no	—
51	C₂×D₂₃₉	—	8	4	no	—
52	C₂×D₂₃₉	—	8	2	no	—
53	C₄×C₂₃₉	—	2	2	no	—
54	C₄×C₂₃₉	#121	1	1	yes	characteristic
55	C₄×C₂₃₉	C₂×D₈	1	1	yes	characteristic
56	Dic₂₃₉	—	4	1	no	—
57	C₂×C₂₃₉	—	2	1	no	—
58	C₂×C₂₃₉	#459	1	1	yes	characteristic
59	D₂₃₉	—	8	1	no	—
60	D₂₃₉	—	16	2	no	—
61	C₂₃₉	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	239	1	no	maximal, Sylow
63	#509	—	239	2	no	—
64	#493	—	239	1	no	—
65	#443	—	239	1	no	—
66	#364	D₂₃₉	1	1	yes	characteristic, complemented
67	#361	—	239	2	no	—
68	#135	—	239	1	no	—
69	C₂×D₈	—	478	1	no	—
70	C₂×D₈	—	239	2	no	—
71	#131	—	478	1	no	—
72	#127	—	478	1	no	—
73	SD₃₂	—	478	2	no	—
74	D₁₆	—	478	2	no	—
75	M₃₂	C₂×D₂₃₉	1	2	yes	—
76	#108	—	239	1	no	—
77	#104	—	478	1	no	—
78	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
79	#41	—	478	1	no	—
80	C₂×D₄	—	956	2	no	—
81	C₂×D₄	—	478	2	no	—
82	Q₁₆	—	239	1	no	—
83	SD₁₆	—	478	1	no	—
84	D₈	—	956	1	no	—
85	D₈	—	956	1	no	—
86	D₈	—	478	2	no	—
87	D₈	—	478	2	no	—
88	D₈	—	239	1	no	—
89	M₁₆	—	956	1	no	—
90	M₁₆	—	478	1	no	—
91	C₂×C₈	—	956	1	no	—
92	C₂×C₈	D₉₅₆	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
94	C₄²	D₉₅₆	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	956	2	no	—
97	Q₈	—	478	1	no	—
98	D₄	—	1912	4	no	—
99	D₄	—	956	1	no	—
100	D₄	—	956	2	no	—
101	D₄	—	956	2	no	—
102	D₄	—	478	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	956	1	no	—
105	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
106	C₈	—	956	1	no	—
107	C₈	D₁₉₁₂	1	1	yes	characteristic
108	C₈	D₁₉₁₂	1	1	yes	characteristic
109	C₈	D₄×D₂₃₉	1	1	yes	characteristic
110	C₈	#16838549	1	1	yes	characteristic
111	C₂²	—	1912	4	no	—
112	C₂²	—	1912	2	no	—
113	C₂²	—	956	1	no	—
114	C₂²	#21439988	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	956	1	no	—
117	C₄	#21439935	1	1	yes	characteristic
118	C₄	C₂×D₁₉₁₂	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	1912	1	no	—
121	C₂	—	3824	2	no	—
122	C₂	#29504968	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29505210

C₂

yes

maximal, complemented

#29505149

C₂

yes

maximal, characteristic, complemented

#29504890

C₂

yes

maximal, characteristic, complemented

#29504586

C₂

yes

maximal, complemented

#364×C₂₃₉

C₂

yes

maximal, characteristic, complemented

#21439982

C₂²

yes

characteristic

C₂×D₁₉₁₂

—

C₂×D₁₉₁₂

C₂²

yes

—

#21439966

—

#21439957

—

#21439905

—

D₃₈₂₄

—

#21439896

C₂²

yes

characteristic

#21439888

—

M₃₂×C₂₃₉

C₂²

yes

—

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

#16838548

—

C₂×D₉₅₆

—

C₂×D₉₅₆

—

Dic₉₅₆

D₄

yes

characteristic, complemented

#16838537

—

D₁₉₁₂

—

D₁₉₁₂

—

D₁₉₁₂

—

D₁₉₁₂

—

D₁₉₁₂

D₄

yes

characteristic, complemented

#16838532

—

#16838531

—

C₈×D₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

characteristic

C₂×C₈×C₂₃₉

C₂³

yes

characteristic

C₄²×C₂₃₉

D₄

yes

characteristic

C₁₆×C₂₃₉

—

C₂²×D₂₃₉

—

Dic₄₇₈

—

D₉₅₆

—

D₉₅₆

—

D₉₅₆

—

D₉₅₆

—

C₂²⋊D₂₃₉

—

C₄×D₂₃₉

—

C₂₃₉⋊C₈

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₈×C₂₃₉

D₈

yes

characteristic

C₈×C₂₃₉

D₈

yes

characteristic

C₈×C₂₃₉

C₂×D₄

yes

C₈×C₂₃₉

yes

C₂²×C₂₃₉

yes

C₂×D₂₃₉

—

C₂×D₂₃₉

—

C₂×D₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#121

yes

characteristic

C₄×C₂₃₉

C₂×D₈

yes

characteristic

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#459

yes

characteristic

D₂₃₉

—

D₂₃₉

—

C₂₃₉

#2213

yes

characteristic, complemented, Sylow

#2213

—

239

maximal, Sylow

#509

—

239

—

#493

—

239

—

#443

—

239

—

#364

D₂₃₉

yes

characteristic, complemented

#361

—

239

—

#135

—

239

—

C₂×D₈

—

478

—

C₂×D₈

—

239

—

#131

—

478

—

#127

—

478

—

SD₃₂

—

478

—

D₁₆

—

478

—

M₃₂

C₂×D₂₃₉

yes

—

#108

—

239

—

#104

—

478

—

C₄×C₈

C₂×D₂₃₉

yes

characteristic

#41

—

478

—

C₂×D₄

—

956

—

C₂×D₄

—

478

—

Q₁₆

—

239

—

SD₁₆

—

478

—

D₈

—

956

—

D₈

—

956

—

D₈

—

478

—

D₈

—

478

—

D₈

—

239

—

M₁₆

—

956

—

M₁₆

—

478

—

C₂×C₈

—

956

—

C₂×C₈

D₉₅₆

yes

characteristic

C₂×C₈

C₂²×D₂₃₉

yes

characteristic

C₄²

D₉₅₆

yes

characteristic

C₁₆

—

C₂³

—

956

—

Q₈

—

478

—

D₄

—

1912

—

D₄

—

956

—

100

D₄

—

956

—

101

D₄

—

956

—

102

D₄

—

478

—

103

C₂×C₄

—

104

C₂×C₄

—

956

—

105

C₂×C₄

C₂×D₉₅₆

yes

characteristic

106

C₈

—

956

—

107

C₈

D₁₉₁₂

yes

characteristic

108

C₈

D₁₉₁₂

yes

characteristic

109

C₈

D₄×D₂₃₉

yes

characteristic

110

C₈

#16838549

yes

characteristic

111

C₂²

—

1912

—

112

C₂²

—

1912

—

113

C₂²

—

956

—

114

C₂²

#21439988

yes

characteristic

115

C₄

—

116

C₄

—

956

—

117

C₄

#21439935

yes

characteristic

118

C₄

C₂×D₁₉₁₂

yes

characteristic

119

C₂

—

120

C₂

—

1912

—

121

C₂

—

3824

—

122

C₂

#29504968

yes

characteristic

	subgroup	G/H
center	C₂	#29504968
commutator, cosocle	C₂×C₈×C₂₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	#364×C₂₃₉	C₂
socle	C₂×C₂₃₉	#459
hypercenter	#364	D₂₃₉
hypocenter	C₂₃₉	#2213
LCS	G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#29504968

commutator, cosocle

C₂×C₈×C₂₃₉

C₂³

Frattini

C₂×C₈

C₂²×D₂₃₉

Fitting

#364×C₂₃₉

C₂

C₂×C₂₃₉

D₂₃₉

C₂₃₉

G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #29505210	C₂	semidirect
2. #29505149	C₂	semidirect
3. #29504890	C₂	semidirect
4. #29504586	C₂	semidirect
5. #364×C₂₃₉	C₂	semidirect
20. Dic₉₅₆	D₄	semidirect
26. D₁₉₁₂	D₄	semidirect
61. C₂₃₉	#2213	semidirect
66. #364	D₂₃₉	semidirect

H ≅ G/N

type

1. #29505210

C₂

semidirect

2. #29505149

C₂

semidirect

3. #29504890

C₂

semidirect

4. #29504586

C₂

semidirect

5. #364×C₂₃₉

C₂

semidirect

20. Dic₉₅₆

D₄

semidirect

26. D₁₉₁₂

D₄

semidirect

61. C₂₃₉

#2213

semidirect

66. #364

D₂₃₉

semidirect

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

subgroup	index
#29505210	2
#29505149	2
#29504890	2
#29504586	2
#364×C₂₃₉	2
#2213	239

subgroup

index

#364×C₂₃₉

239

Subgroups — #44486625

Conjugacy classes of subgroups

Complements

Maximal subgroups