Group #21425399 — subgroups

Subgroups — #21425399

Order 7552.

264 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16837430	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837398	C₂	1	2	yes	maximal, complemented
3	#16837376	C₂	1	2	yes	maximal, complemented
4	#16837159	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#575	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
7	#14273686	C₂²	1	2	yes	—
8	#14273686	C₂²	1	1	yes	characteristic
9	Dic₅₉⋊C₈	—	2	2	no	—
10	#14273651	—	2	2	no	—
11	#14273648	—	2	2	no	—
12	#14273642	—	2	4	no	—
13	C₂×#41×C₅₉	C₂²	1	2	yes	—
14	#137×C₅₉	—	2	2	no	—
15	AGL(1,ℤ/8)×C₅₉	—	2	2	no	—
16	#135×C₅₉	—	2	2	no	—
17	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	#41×C₅₉	—	2	2	no	—
19	#41×C₅₉	—	2	2	no	—
20	#41×C₅₉	—	2	2	no	—
21	#41×C₅₉	—	2	2	no	—
22	C₂×Q₈×C₅₉	D₄	1	2	yes	—
23	C₂×D₄×C₅₉	—	2	2	no	—
24	C₂×D₄×C₅₉	D₄	1	2	yes	—
25	C₂²×C₄×C₅₉	—	2	2	no	—
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	Q₁₆×C₅₉	—	2	2	no	—
28	SD₁₆×C₅₉	—	2	4	no	—
29	D₈×C₅₉	—	2	2	no	—
30	M₁₆×C₅₉	—	2	2	no	—
31	C₂×C₈×C₅₉	D₄	1	2	yes	—
32	C₂²⋊Dic₅₉	—	2	2	no	—
33	C₂²⋊Dic₅₉	D₄	1	2	yes	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	2	no	—
36	C₄×Dic₅₉	—	2	1	no	—
37	C₄×Dic₅₉	—	2	2	no	—
38	C₂×C₅₉⋊C₈	—	2	2	no	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂³×C₅₉	—	2	2	no	—
41	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₅₉	—	2	2	no	—
43	Q₈×C₅₉	—	2	2	no	—
44	D₄×C₅₉	—	2	2	no	—
45	D₄×C₅₉	—	2	2	no	—
46	D₄×C₅₉	—	4	2	no	—
47	D₄×C₅₉	—	4	2	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	4	2	no	—
50	C₂×C₄×C₅₉	—	4	2	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#41	1	2	yes	—
55	C₂×Dic₅₉	—	2	2	no	—
56	C₂×Dic₅₉	—	4	2	no	—
57	C₈×C₅₉	—	2	2	no	—
58	C₅₉⋊C₈	—	4	1	no	—
59	C₂²×C₅₉	—	4	1	no	—
60	C₂²×C₅₉	—	4	2	no	—
61	C₂²×C₅₉	—	8	2	no	—
62	C₂²×C₅₉	#121	1	1	yes	characteristic
63	C₂²×C₅₉	#121	1	2	yes	—
64	C₄×C₅₉	—	4	2	no	—
65	C₄×C₅₉	#120	1	1	yes	characteristic
66	C₄×C₅₉	#123	1	2	yes	—
67	C₄×C₅₉	#124	1	1	yes	characteristic
68	Dic₅₉	—	4	2	no	—
69	Dic₅₉	—	8	2	no	—
70	#2053	—	59	1	no	maximal, Sylow
71	C₂×C₅₉	—	2	1	no	—
72	C₂×C₅₉	—	2	2	no	—
73	C₂×C₅₉	—	8	2	no	—
74	C₂×C₅₉	#394	1	1	yes	characteristic
75	#575	D₅₉	1	1	yes	characteristic
76	#424	—	59	1	no	—
77	#416	—	59	2	no	—
78	#410	—	59	2	no	—
79	H(3,ℤ/4)	—	59	1	no	—
80	C₅₉	#2053	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	C₂×D₅₉	1	2	yes	—
82	#137	—	2	2	no	—
83	AGL(1,ℤ/8)	—	2	2	no	—
84	#135	—	2	2	no	—
85	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
86	C₂²×C₈	—	59	1	no	—
87	#117	—	59	1	no	—
88	#117	—	59	2	no	—
89	C₄⋊C₈	—	118	2	no	—
90	#103	—	118	2	no	—
91	#102	—	118	2	no	—
92	#99	—	118	4	no	—
93	#41	—	2	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂²⋊D₅₉	1	2	yes	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	C₂²⋊D₅₉	1	2	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
102	Q₁₆	—	2	2	no	—
103	SD₁₆	—	2	4	no	—
104	D₈	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	118	2	no	—
107	C₂×C₈	—	118	1	no	—
108	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
109	C₄⋊C₄	—	118	1	no	—
110	C₄⋊C₄	—	118	2	no	—
111	C₂²⋊C₄	—	59	2	no	—
112	C₂²⋊C₄	—	118	2	no	—
113	C₄²	—	118	1	no	—
114	C₄²	—	118	2	no	—
115	C₂³	—	2	2	no	—
116	C₂³	D₄×D₅₉	1	1	yes	characteristic
117	Q₈	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	D₄	—	2	2	no	—
120	D₄	—	2	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	118	2	no	—
127	C₂×C₄	—	236	2	no	—
128	C₂×C₄	#11745687	1	1	yes	characteristic
129	C₂×C₄	#11745687	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	236	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#14273698	1	1	yes	characteristic
138	C₂²	#14273699	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	472	2	no	—
141	C₄	—	236	2	no	—
142	C₄	#14273696	1	1	yes	characteristic
143	C₄	#14273708	1	2	yes	—
144	C₄	#14273714	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	8	2	no	—
148	C₂	#16837316	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837430

C₂

yes

maximal, characteristic, complemented

#16837398

C₂

yes

maximal, complemented

#16837376

C₂

yes

maximal, complemented

#16837159

C₂

yes

maximal, characteristic, complemented

C₅₉×#575

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273686

C₂²

yes

—

#14273686

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273651

—

#14273648

—

#14273642

—

C₂×#41×C₅₉

C₂²

yes

—

#137×C₅₉

—

AGL(1,ℤ/8)×C₅₉

—

#135×C₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

Q₁₆×C₅₉

—

SD₁₆×C₅₉

—

D₈×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

#121

yes

—

C₄×C₅₉

—

C₄×C₅₉

#120

yes

characteristic

C₄×C₅₉

#123

yes

—

C₄×C₅₉

#124

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2053

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#394

yes

characteristic

#575

D₅₉

yes

—

—

—

—

C₅₉

#2053

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₉

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

C₄⋊C₈

—

118

—

#103

—

118

—

#102

—

118

—

#99

—

118

—

—

—

—

—

C₂×Q₈

C₂²⋊D₅₉

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₅₉

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₅₉

yes

characteristic

102

Q₁₆

—

103

SD₁₆

—

104

D₈

—

105

M₁₆

—

106

C₂×C₈

—

118

—

107

C₂×C₈

—

118

—

108

C₂×C₈

C₂²⋊D₅₉

yes

—

109

C₄⋊C₄

—

118

—

110

C₄⋊C₄

—

118

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

118

—

113

C₄²

—

118

—

114

C₄²

—

118

—

115

C₂³

—

116

C₂³

D₄×D₅₉

yes

characteristic

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

236

—

128

C₂×C₄

#11745687

yes

characteristic

129

C₂×C₄

#11745687

yes

—

130

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

132

C₈

—

133

C₈

—

236

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#14273698

yes

characteristic

138

C₂²

#14273699

yes

—

139

C₄

—

140

C₄

—

472

—

141

C₄

—

236

—

142

C₄

#14273696

yes

characteristic

143

C₄

#14273708

yes

—

144

C₄

#14273714

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#16837316

yes

characteristic

	subgroup	G/H
center	C₄	#14273714
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#575	C₂
socle	C₂×C₅₉	#394
hypercenter	#575	D₅₉
hypocenter	C₅₉	#2053
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#14273714

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#575

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
1. #16837430	C₂	semidirect
2. #16837398	C₂	semidirect
3. #16837376	C₂	semidirect
4. #16837159	C₂	semidirect
80. C₅₉	#2053	semidirect

H ≅ G/N

type

1. #16837430

C₂

semidirect

2. #16837398

C₂

semidirect

3. #16837376

C₂

semidirect

4. #16837159

C₂

semidirect

80. C₅₉

#2053

semidirect

Subgroups — #21425399

Conjugacy classes of subgroups

Complements