Group #21426309 — subgroups

Subgroups — #21426309

Order 7552.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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	#16837652	C₂	1	1	yes	maximal, characteristic
2	#16837449	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837226	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837225	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837215	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837183	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745659	C₂²	1	1	yes	characteristic
9	M₁₆×D₅₉	—	2	1	no	—
10	#14273698	—	2	1	no	—
11	#14273667	C₂²	1	1	yes	characteristic
12	#14273663	C₂²	1	1	yes	characteristic
13	#14273662	—	2	1	no	—
14	#14273659	C₂²	1	1	yes	characteristic
15	#14273648	—	2	1	no	—
16	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
17	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
18	#108×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₅₉	—	2	1	no	—
20	C₂×C₂²⋊D₅₉	—	4	1	no	—
21	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
22	D₈×C₅₉	—	2	1	no	—
23	D₈×C₅₉	—	2	1	no	—
24	M₁₆×C₅₉	—	2	1	no	—
25	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²⋊Dic₅₉	—	4	1	no	—
27	C₄⋊Dic₅₉	—	2	1	no	—
28	#11745664	—	2	1	no	—
29	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
30	#11745659	—	2	1	no	—
31	#11745659	—	2	1	no	—
32	#11745659	—	4	1	no	—
33	C₈×D₅₉	—	4	1	no	—
34	C₁₆×C₅₉	—	2	1	no	—
35	C₅₉⋊C₁₆	—	2	1	no	—
36	C₂³×C₅₉	—	4	1	no	—
37	C₂²×D₅₉	SD₁₆	1	1	yes	characteristic
38	D₄×C₅₉	—	4	1	no	—
39	D₄×C₅₉	—	4	1	no	—
40	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²⋊D₅₉	—	8	2	no	—
42	C₂×Dic₅₉	—	4	1	no	—
43	C₂×Dic₅₉	SD₁₆	1	1	yes	characteristic
44	C₄×D₅₉	—	2	1	no	—
45	C₄×D₅₉	—	2	1	no	—
46	C₈×C₅₉	—	4	1	no	—
47	C₈×C₅₉	#41	1	1	yes	characteristic
48	C₈×C₅₉	#41	1	1	yes	characteristic
49	C₅₉⋊C₈	—	2	1	no	—
50	C₅₉⋊C₈	—	4	1	no	—
51	C₂²×C₅₉	—	8	1	no	—
52	C₂²×C₅₉	—	8	1	no	—
53	C₂²×C₅₉	—	8	1	no	—
54	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×D₅₉	—	2	1	no	—
56	C₂×D₅₉	—	4	1	no	—
57	C₄×C₅₉	#123	1	1	yes	characteristic
58	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₅₉	—	2	1	no	—
60	Dic₅₉	—	8	1	no	—
61	#2238	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	—	16	1	no	—
64	C₂×C₅₉	#481	1	1	yes	characteristic
65	D₅₉	—	4	1	no	—
66	#469	—	59	1	no	—
67	#430	—	59	1	no	—
68	#365	—	59	1	no	—
69	#365	—	59	1	no	—
70	#361	—	59	1	no	—
71	#361	D₅₉	1	1	yes	characteristic, complemented
72	#349	—	59	1	no	—
73	C₅₉	#2238	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
75	C₂×M₁₆	—	59	1	no	—
76	C₂×M₁₆	—	118	1	no	—
77	#121	—	118	1	no	—
78	M₃₂	—	59	1	no	—
79	M₃₂	C₂×D₅₉	1	1	yes	characteristic
80	#108	—	59	1	no	—
81	#108	—	118	1	no	—
82	#108	C₂×D₅₉	1	1	yes	characteristic
83	#106	—	59	1	no	—
84	#102	—	118	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	236	1	no	—
87	C₂²×C₄	—	59	1	no	—
88	D₈	—	2	1	no	—
89	D₈	—	2	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	118	1	no	—
92	M₁₆	—	118	1	no	—
93	M₁₆	—	118	1	no	—
94	M₁₆	—	236	1	no	—
95	C₂×C₈	—	59	1	no	—
96	C₂×C₈	—	236	1	no	—
97	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
98	C₄⋊C₄	—	118	1	no	—
99	C₂²⋊C₄	—	236	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	118	1	no	—
102	C₂³	—	4	1	no	—
103	C₂³	—	59	1	no	—
104	D₄	—	4	1	no	—
105	D₄	—	4	1	no	—
106	D₄	—	472	2	no	—
107	C₂×C₄	—	59	1	no	—
108	C₂×C₄	—	118	1	no	—
109	C₂×C₄	—	118	1	no	—
110	C₂×C₄	—	236	1	no	—
111	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	118	1	no	—
114	C₈	—	236	1	no	—
115	C₈	#11745685	1	1	yes	characteristic
116	C₈	#11745687	1	1	yes	characteristic
117	C₂²	—	8	1	no	—
118	C₂²	—	8	1	no	—
119	C₂²	—	8	1	no	—
120	C₂²	—	118	1	no	—
121	C₂²	—	236	1	no	—
122	C₂²	SD₁₆×D₅₉	1	1	yes	characteristic
123	C₄	—	118	1	no	—
124	C₄	—	472	1	no	—
125	C₄	#14273709	1	1	yes	characteristic
126	C₄	#14273751	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	16	1	no	—
129	C₂	—	236	1	no	—
130	C₂	#16837719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837652

C₂

yes

maximal, characteristic

#16837449

C₂

yes

maximal, characteristic, complemented

#16837226

C₂

yes

maximal, characteristic, complemented

#16837225

C₂

yes

maximal, characteristic, complemented

#16837215

C₂

yes

maximal, characteristic, complemented

#16837183

C₂

yes

maximal, characteristic, complemented

C₅₉×#361

C₂

yes

maximal, characteristic, complemented

C₂×#11745659

C₂²

yes

characteristic

M₁₆×D₅₉

—

#14273698

—

#14273667

C₂²

yes

characteristic

#14273663

C₂²

yes

characteristic

#14273662

—

#14273659

C₂²

yes

characteristic

#14273648

—

C₂×D₈×C₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×C₂²⋊D₅₉

—

C₂×C₄×D₅₉

D₄

yes

characteristic

D₈×C₅₉

—

D₈×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

D₄

yes

—

—

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂²×D₅₉

SD₁₆

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

SD₁₆

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#2238

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#481

yes

characteristic

D₅₉

—

—

—

—

—

—

D₅₉

yes

characteristic, complemented

#349

—

C₅₉

#2238

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

118

—

#121

—

118

—

M₃₂

—

M₃₂

C₂×D₅₉

yes

characteristic

#108

—

#108

—

118

—

#108

C₂×D₅₉

yes

characteristic

#106

—

#102

—

118

—

C₂×D₄

—

C₂×D₄

—

236

—

C₂²×C₄

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

236

—

100

C₁₆

—

101

C₁₆

—

118

—

102

C₂³

—

103

C₂³

—

104

D₄

—

105

D₄

—

106

D₄

—

472

—

107

C₂×C₄

—

108

C₂×C₄

—

118

—

109

C₂×C₄

—

118

—

110

C₂×C₄

—

236

—

111

C₂×C₄

D₄×D₅₉

yes

characteristic

112

C₈

—

113

C₈

—

118

—

114

C₈

—

236

—

115

C₈

#11745685

yes

characteristic

116

C₈

#11745687

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

118

—

121

C₂²

—

236

—

122

C₂²

SD₁₆×D₅₉

yes

characteristic

123

C₄

—

118

—

124

C₄

—

472

—

125

C₄

#14273709

yes

characteristic

126

C₄

#14273751

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

236

—

130

C₂

#16837719

yes

characteristic

	subgroup	G/H
center	C₂	#16837719
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#361	C₂
socle	C₂×C₅₉	#481
hypercenter	#361	D₅₉
hypocenter	C₅₉	#2238
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

subgroup

G/H

center

C₂

#16837719

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#361

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

N	H ≅ G/N	type
2. #16837449	C₂	semidirect
3. #16837226	C₂	semidirect
4. #16837225	C₂	semidirect
5. #16837215	C₂	semidirect
6. #16837183	C₂	semidirect
7. C₅₉×#361	C₂	semidirect
71. #361	D₅₉	semidirect
73. C₅₉	#2238	semidirect

H ≅ G/N

type

2. #16837449

C₂

semidirect

3. #16837226

C₂

semidirect

4. #16837225

C₂

semidirect

5. #16837215

C₂

semidirect

6. #16837183

C₂

semidirect

7. C₅₉×#361

C₂

semidirect

71. #361

D₅₉

semidirect

73. C₅₉

#2238

semidirect

Subgroups — #21426309

Conjugacy classes of subgroups

Complements