Group #21425997 — subgroups

Subgroups — #21425997

Order 7552.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837838	C₂	1	1	yes	maximal, characteristic
2	#16837523	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837514	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837226	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837213	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837179	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273749	C₂²	1	1	yes	characteristic
9	#14273735	—	2	1	no	—
10	D₄×Dic₅₉	—	2	1	no	—
11	#14273671	C₄	1	2	yes	complemented
12	#14273668	C₄	1	2	yes	complemented
13	#14273667	C₂²	1	1	yes	characteristic
14	#14273659	C₂²	1	1	yes	characteristic
15	#14273653	—	2	1	no	—
16	#14273648	—	2	1	no	—
17	#14273637	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
19	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
20	#108×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₅₉	—	2	1	no	—
22	C₂²×Dic₅₉	—	4	1	no	—
23	#11745685	—	2	1	no	—
24	D₈×C₅₉	—	2	1	no	—
25	D₈×C₅₉	C₂×C₄	1	2	yes	—
26	M₁₆×C₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₅₉	—	4	1	no	—
29	C₄⋊Dic₅₉	—	2	1	no	—
30	C₄×Dic₅₉	D₄	1	1	yes	characteristic
31	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	Dic₂₃₆	C₂×C₄	1	1	yes	characteristic, complemented
33	D₄₇₂	C₂×C₄	1	1	yes	characteristic, complemented
34	#11745660	—	2	1	no	—
35	#11745659	—	4	1	no	—
36	C₈×D₅₉	—	4	1	no	—
37	C₁₆×C₅₉	—	2	1	no	—
38	C₅₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
39	C₂³×C₅₉	—	4	1	no	—
40	D₄×C₅₉	—	2	2	no	—
41	D₄×C₅₉	—	4	1	no	—
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂²⋊D₅₉	—	4	1	no	—
44	C₂×Dic₅₉	—	2	1	no	—
45	C₂×Dic₅₉	—	4	1	no	—
46	C₂×Dic₅₉	—	8	2	no	—
47	D₂₃₆	—	2	1	no	—
48	C₄×D₅₉	—	4	1	no	—
49	Dic₁₁₈	—	2	1	no	—
50	C₈×C₅₉	—	4	1	no	—
51	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₅₉	#41	1	1	yes	characteristic
53	C₅₉⋊C₈	—	2	1	no	—
54	C₂²×C₅₉	—	4	2	no	—
55	C₂²×C₅₉	—	8	1	no	—
56	C₂²×C₅₉	—	8	1	no	—
57	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₂×D₅₉	—	4	1	no	—
59	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
60	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	Dic₅₉	—	4	1	no	—
62	Dic₅₉	—	4	1	no	—
63	Dic₅₉	—	8	1	no	—
64	AGL(1,ℤ/16)	—	59	1	no	maximal, Sylow
65	C₂×C₅₉	—	2	1	no	—
66	C₂×C₅₉	—	8	2	no	—
67	C₂×C₅₉	#442	1	1	yes	characteristic
68	D₅₉	—	8	1	no	—
69	#509	—	59	1	no	—
70	#444	—	59	1	no	—
71	#442	—	59	1	no	—
72	#365	—	59	1	no	—
73	#361	D₅₉	1	1	yes	characteristic, complemented
74	#360	—	59	1	no	—
75	#347	—	59	1	no	—
76	C₅₉	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
77	#135	—	59	1	no	—
78	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
79	#131	—	118	1	no	—
80	C₄×D₄	—	118	1	no	—
81	SD₃₂	—	59	2	no	—
82	D₁₆	—	59	2	no	—
83	M₃₂	—	59	1	no	—
84	M₃₂	C₂×D₅₉	1	1	yes	characteristic
85	#108	C₂×D₅₉	1	1	yes	characteristic
86	#106	—	59	1	no	—
87	#104	—	118	1	no	—
88	#102	—	118	1	no	—
89	#97	—	59	1	no	—
90	#41	—	118	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂²×C₄	—	236	1	no	—
93	Q₁₆	—	59	1	no	—
94	SD₁₆	—	118	1	no	—
95	D₈	—	2	1	no	—
96	D₈	—	59	1	no	—
97	D₈	C₄×D₅₉	1	2	yes	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	236	1	no	—
100	C₂×C₈	—	59	1	no	—
101	C₂×C₈	—	236	1	no	—
102	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
103	C₄⋊C₄	—	118	1	no	—
104	C₂²⋊C₄	—	236	1	no	—
105	C₄²	—	59	1	no	—
106	C₁₆	—	2	1	no	—
107	C₁₆	—	59	2	no	—
108	C₂³	—	4	1	no	—
109	Q₈	—	118	1	no	—
110	D₄	—	2	2	no	—
111	D₄	—	4	1	no	—
112	D₄	—	118	1	no	—
113	D₄	—	236	1	no	—
114	C₂×C₄	—	118	1	no	—
115	C₂×C₄	—	472	2	no	—
116	C₂×C₄	—	236	1	no	—
117	C₂×C₄	—	236	1	no	—
118	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	118	1	no	—
121	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
122	C₈	#11745687	1	1	yes	characteristic
123	C₂²	—	4	2	no	—
124	C₂²	—	8	1	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	236	1	no	—
127	C₂²	#14273754	1	1	yes	characteristic
128	C₄	—	472	1	no	—
129	C₄	—	236	1	no	—
130	C₄	—	236	1	no	—
131	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
132	C₄	#14273751	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	8	2	no	—
135	C₂	—	472	1	no	—
136	C₂	#16837511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837838

C₂

yes

maximal, characteristic

#16837523

C₂

yes

maximal, characteristic, complemented

#16837514

C₂

yes

maximal, characteristic, complemented

#16837226

C₂

yes

maximal, characteristic, complemented

#16837213

C₂

yes

maximal, characteristic, complemented

#16837179

C₂

yes

maximal, characteristic, complemented

C₅₉×#361

C₂

yes

maximal, characteristic, complemented

#14273749

C₂²

yes

characteristic

#14273735

—

D₄×Dic₅₉

—

#14273671

C₄

yes

complemented

#14273668

C₄

yes

complemented

#14273667

C₂²

yes

characteristic

#14273659

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂²×Dic₅₉

—

#11745685

—

D₈×C₅₉

—

D₈×C₅₉

C₂×C₄

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

C₂×C₄

yes

characteristic, complemented

D₄₇₂

C₂×C₄

yes

characteristic, complemented

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

C₂×C₄

yes

complemented

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

AGL(1,ℤ/8)

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

AGL(1,ℤ/16)

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#442

yes

characteristic

D₅₉

—

—

—

—

—

D₅₉

yes

characteristic, complemented

#360

—

#347

—

C₅₉

AGL(1,ℤ/16)

yes

characteristic, complemented, Sylow

#135

—

C₂×D₈

C₂×D₅₉

yes

characteristic

#131

—

118

—

C₄×D₄

—

118

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

C₂×D₅₉

yes

characteristic

#108

C₂×D₅₉

yes

characteristic

#106

—

#104

—

118

—

#102

—

118

—

#97

—

#41

—

118

—

C₂×D₄

—

C₂²×C₄

—

236

—

Q₁₆

—

SD₁₆

—

118

—

D₈

—

D₈

—

D₈

C₄×D₅₉

yes

—

M₁₆

—

M₁₆

—

236

—

100

C₂×C₈

—

101

C₂×C₈

—

236

—

102

C₂×C₈

C₂²×D₅₉

yes

characteristic

103

C₄⋊C₄

—

118

—

104

C₂²⋊C₄

—

236

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₂³

—

109

Q₈

—

118

—

110

D₄

—

111

D₄

—

112

D₄

—

118

—

113

D₄

—

236

—

114

C₂×C₄

—

118

—

115

C₂×C₄

—

472

—

116

C₂×C₄

—

236

—

117

C₂×C₄

—

236

—

118

C₂×C₄

D₄×D₅₉

yes

characteristic

119

C₈

—

120

C₈

—

118

—

121

C₈

C₂×C₄×D₅₉

yes

characteristic

122

C₈

#11745687

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

236

—

127

C₂²

#14273754

yes

characteristic

128

C₄

—

472

—

129

C₄

—

236

—

130

C₄

—

236

—

131

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

132

C₄

#14273751

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

—

472

—

136

C₂

#16837511

yes

characteristic

	subgroup	G/H
center	C₂	#16837511
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#361	C₂
socle	C₂×C₅₉	#442
hypercenter	#361	D₅₉
hypocenter	C₅₉	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

subgroup

G/H

center

C₂

#16837511

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#361

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

N	H ≅ G/N	type
2. #16837523	C₂	semidirect
3. #16837514	C₂	semidirect
4. #16837226	C₂	semidirect
5. #16837213	C₂	semidirect
6. #16837179	C₂	semidirect
7. C₅₉×#361	C₂	semidirect
11. #14273671	C₄	semidirect
12. #14273668	C₄	semidirect
32. Dic₂₃₆	C₂×C₄	semidirect
33. D₄₇₂	C₂×C₄	semidirect
38. C₅₉⋊C₁₆	C₂×C₄	semidirect
73. #361	D₅₉	semidirect
76. C₅₉	AGL(1,ℤ/16)	semidirect

H ≅ G/N

type

2. #16837523

C₂

semidirect

3. #16837514

C₂

semidirect

4. #16837226

C₂

semidirect

5. #16837213

C₂

semidirect

6. #16837179

C₂

semidirect

7. C₅₉×#361

C₂

semidirect

11. #14273671

C₄

semidirect

12. #14273668

C₄

semidirect

32. Dic₂₃₆

C₂×C₄

semidirect

33. D₄₇₂

C₂×C₄

semidirect

38. C₅₉⋊C₁₆

C₂×C₄

semidirect

73. #361

D₅₉

semidirect

76. C₅₉

AGL(1,ℤ/16)

semidirect

Subgroups — #21425997

Conjugacy classes of subgroups

Complements