Group #21426177 — subgroups

Subgroups — #21426177

Order 7552.

150 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16837842	C₂	1	2	yes	maximal
2	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837223	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837217	C₂	1	2	yes	maximal, complemented
5	C₅₉×#443	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	—	2	1	no	—
7	C₂×Dic₂₃₆	C₂²	1	2	yes	—
8	#14273727	—	2	1	no	—
9	#14273673	—	2	2	no	—
10	#14273671	—	2	2	no	—
11	#14273667	C₂²	1	2	yes	—
12	#135×C₅₉	C₂²	1	1	yes	characteristic
13	#131×C₅₉	—	2	1	no	—
14	#108×C₅₉	C₂²	1	1	yes	characteristic
15	#104×C₅₉	—	2	1	no	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	#41×C₅₉	—	2	1	no	—
18	C₂×Dic₁₁₈	—	2	2	no	—
19	Q₁₆×C₅₉	D₄	1	1	yes	characteristic
20	SD₁₆×C₅₉	—	2	1	no	—
21	D₈×C₅₉	D₄	1	1	yes	characteristic
22	M₁₆×C₅₉	—	2	1	no	—
23	M₁₆×C₅₉	—	4	1	no	—
24	C₂×C₈×C₅₉	—	4	1	no	—
25	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₄²×C₅₉	D₄	1	1	yes	characteristic
28	C₄⋊Dic₅₉	—	4	2	no	—
29	Dic₂₃₆	—	2	2	no	—
30	Dic₂₃₆	—	2	2	no	—
31	Dic₂₃₆	—	4	1	no	—
32	Dic₂₃₆	—	4	1	no	—
33	C₅₉⋊C₁₆	—	2	2	no	—
34	Q₈×C₅₉	—	2	1	no	—
35	D₄×C₅₉	—	2	1	no	—
36	D₄×C₅₉	—	4	1	no	—
37	C₂×C₄×C₅₉	—	2	1	no	—
38	C₂×C₄×C₅₉	—	4	1	no	—
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₉	—	4	2	no	—
41	Dic₁₁₈	—	4	2	no	—
42	Dic₁₁₈	—	4	2	no	—
43	C₈×C₅₉	—	4	1	no	—
44	C₈×C₅₉	D₈	1	1	yes	characteristic
45	C₈×C₅₉	D₈	1	1	yes	characteristic
46	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₈×C₅₉	#41	1	1	yes	characteristic
48	C₂²×C₅₉	—	4	1	no	—
49	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₄×C₅₉	—	2	2	no	—
51	C₄×C₅₉	—	4	1	no	—
52	C₄×C₅₉	#121	1	1	yes	characteristic
53	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
54	Dic₅₉	—	8	2	no	—
55	#2214	—	59	1	no	maximal, Sylow
56	C₂×C₅₉	—	2	1	no	—
57	C₂×C₅₉	—	8	1	no	—
58	C₂×C₅₉	#459	1	1	yes	characteristic
59	#510	—	59	2	no	—
60	#494	—	59	1	no	—
61	#443	D₅₉	1	1	yes	characteristic
62	#364	—	59	1	no	—
63	#362	—	59	2	no	—
64	C₅₉	#2214	1	1	yes	characteristic, complemented, Sylow
65	#135	C₂×D₅₉	1	1	yes	characteristic
66	C₂×Q₁₆	—	59	2	no	—
67	C₂×Q₁₆	—	118	1	no	—
68	#131	—	2	1	no	—
69	C₄⋊Q₈	—	118	1	no	—
70	Q₃₂	—	118	2	no	—
71	SD₃₂	—	118	2	no	—
72	M₃₂	—	59	2	no	—
73	#108	C₂×D₅₉	1	1	yes	characteristic
74	#104	—	2	1	no	—
75	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
76	#41	—	2	1	no	—
77	C₂×Q₈	—	118	2	no	—
78	Q₁₆	—	118	2	no	—
79	Q₁₆	—	118	2	no	—
80	Q₁₆	—	236	1	no	—
81	Q₁₆	—	236	1	no	—
82	Q₁₆	D₂₃₆	1	1	yes	characteristic
83	SD₁₆	—	2	1	no	—
84	D₈	D₂₃₆	1	1	yes	characteristic
85	M₁₆	—	2	1	no	—
86	M₁₆	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
90	C₄⋊C₄	—	236	2	no	—
91	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
92	C₁₆	—	118	2	no	—
93	Q₈	—	2	1	no	—
94	Q₈	—	236	2	no	—
95	Q₈	—	236	2	no	—
96	D₄	—	2	1	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	236	2	no	—
101	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
102	C₈	—	4	1	no	—
103	C₈	#11745669	1	1	yes	characteristic
104	C₈	#11745669	1	1	yes	characteristic
105	C₈	C₂×D₂₃₆	1	1	yes	characteristic
106	C₈	#11745685	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	#14273752	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	4	1	no	—
111	C₄	—	472	2	no	—
112	C₄	#14273701	1	1	yes	characteristic
113	C₄	C₂×#11745669	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	8	1	no	—
116	C₂	#16837597	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837842

C₂

yes

maximal

#16837782

C₂

yes

maximal, characteristic, complemented

#16837223

C₂

yes

maximal, characteristic, complemented

#16837217

C₂

yes

maximal, complemented

C₅₉×#443

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₂²

yes

—

—

—

—

C₂²

yes

—

#135×C₅₉

C₂²

yes

characteristic

#131×C₅₉

—

#108×C₅₉

C₂²

yes

characteristic

#104×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

D₄

yes

characteristic

SD₁₆×C₅₉

—

D₈×C₅₉

D₄

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

D₈

yes

characteristic

C₈×C₅₉

D₈

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

#2214

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

—

D₅₉

yes

characteristic

#364

—

#362

—

C₅₉

#2214

yes

characteristic, complemented, Sylow

#135

C₂×D₅₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

#131

—

C₄⋊Q₈

—

118

—

Q₃₂

—

118

—

SD₃₂

—

118

—

M₃₂

—

#108

C₂×D₅₉

yes

characteristic

#104

—

C₄×C₈

C₂×D₅₉

yes

characteristic

#41

—

C₂×Q₈

—

118

—

Q₁₆

—

118

—

Q₁₆

—

118

—

Q₁₆

—

236

—

Q₁₆

—

236

—

Q₁₆

D₂₃₆

yes

characteristic

SD₁₆

—

D₈

D₂₃₆

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

236

—

C₄²

C₂²⋊D₅₉

yes

characteristic

C₁₆

—

118

—

Q₈

—

Q₈

—

236

—

Q₈

—

236

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

236

—

101

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

102

C₈

—

103

C₈

#11745669

yes

characteristic

104

C₈

#11745669

yes

characteristic

105

C₈

C₂×D₂₃₆

yes

characteristic

106

C₈

#11745685

yes

characteristic

107

C₂²

—

108

C₂²

#14273752

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

472

—

112

C₄

#14273701

yes

characteristic

113

C₄

C₂×#11745669

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#16837597

yes

characteristic

	subgroup	G/H
center	C₂	#16837597
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#443	C₂
socle	C₂×C₅₉	#459
hypercenter	#443	D₅₉
hypocenter	C₅₉	#2214
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#16837597

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#443

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
2. #16837782	C₂	semidirect
3. #16837223	C₂	semidirect
4. #16837217	C₂	semidirect
64. C₅₉	#2214	semidirect

H ≅ G/N

type

2. #16837782

C₂

semidirect

3. #16837223

C₂

semidirect

4. #16837217

C₂

semidirect

64. C₅₉

#2214

semidirect

Subgroups — #21426177

Conjugacy classes of subgroups

Complements