Group #21425402 — subgroups

Subgroups — #21425402

Order 7552.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16838202	C₂	1	1	yes	maximal, characteristic
2	#16837399	C₂	1	2	yes	maximal, complemented
3	#16837375	C₂	1	2	yes	maximal, complemented
4	#16837159	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745685	C₂²	1	2	yes	—
7	#14273760	—	2	2	no	—
8	#14273755	—	2	2	no	—
9	#14273749	—	2	2	no	—
10	#14273686	C₂²	1	2	yes	—
11	#14273652	—	2	2	no	—
12	#14273649	—	2	2	no	—
13	#14273641	—	2	4	no	—
14	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	#117×C₅₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₉	—	2	2	no	—
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₅₉	—	2	2	no	—
20	#11745685	—	2	2	no	—
21	#11745685	—	2	2	no	—
22	#11745685	—	2	2	no	—
23	#11745685	—	2	2	no	—
24	C₂×D₂₃₆	D₄	1	2	yes	—
25	C₂×C₄×D₅₉	—	2	2	no	—
26	C₂×Dic₁₁₈	D₄	1	2	yes	—
27	M₁₆×C₅₉	—	2	2	no	—
28	C₂×C₈×C₅₉	—	2	1	no	—
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	D₄	1	2	yes	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
33	C₄²×C₅₉	—	2	1	no	—
34	C₂²⋊Dic₅₉	—	2	2	no	—
35	C₄⋊Dic₅₉	—	2	2	no	—
36	C₄×Dic₅₉	—	2	2	no	—
37	Dic₂₃₆	—	2	2	no	—
38	D₄₇₂	—	2	2	no	—
39	#11745660	—	2	4	no	—
40	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	2	yes	—
47	C₂²⋊D₅₉	—	4	2	no	—
48	C₂²⋊D₅₉	—	4	2	no	—
49	C₂×Dic₅₉	—	2	2	no	—
50	C₂×Dic₅₉	—	4	2	no	—
51	D₂₃₆	—	2	2	no	—
52	D₂₃₆	—	2	2	no	—
53	C₄×D₅₉	—	4	2	no	—
54	C₄×D₅₉	—	4	2	no	—
55	Dic₁₁₈	—	2	2	no	—
56	Dic₁₁₈	—	2	2	no	—
57	C₈×C₅₉	—	2	2	no	—
58	C₈×C₅₉	—	4	1	no	—
59	C₂²×C₅₉	—	4	1	no	—
60	C₂²×C₅₉	#121	1	1	yes	characteristic
61	C₂²×C₅₉	#121	1	2	yes	—
62	C₂×D₅₉	—	4	2	no	—
63	C₂×D₅₉	—	8	2	no	—
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	2	no	—
72	C₂×C₅₉	—	2	1	no	—
73	C₂×C₅₉	#394	1	1	yes	characteristic
74	D₅₉	—	8	2	no	—
75	#575	—	59	1	no	—
76	#424	D₅₉	1	1	yes	characteristic, complemented
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	—	59	2	no	—
82	#137	—	118	2	no	—
83	AGL(1,ℤ/8)	—	118	2	no	—
84	#135	—	118	2	no	—
85	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
87	#117	—	59	2	no	—
88	#117	C₂×D₅₉	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	#103	—	118	2	no	—
91	#102	—	118	2	no	—
92	#99	—	118	4	no	—
93	#41	—	118	2	no	—
94	#41	—	118	2	no	—
95	#41	—	118	2	no	—
96	#41	—	118	2	no	—
97	C₂×Q₈	—	59	2	no	—
98	C₂×D₄	—	59	2	no	—
99	C₂×D₄	—	118	2	no	—
100	C₂²×C₄	—	118	2	no	—
101	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
102	Q₁₆	—	118	2	no	—
103	SD₁₆	—	118	4	no	—
104	D₈	—	118	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	118	2	no	—
111	C₂²⋊C₄	—	118	2	no	—
112	C₂²⋊C₄	D₂₃₆	1	2	yes	—
113	C₄²	—	2	1	no	—
114	C₄²	—	118	2	no	—
115	C₂³	—	118	2	no	—
116	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
117	Q₈	—	118	2	no	—
118	Q₈	—	118	2	no	—
119	D₄	—	118	2	no	—
120	D₄	—	118	2	no	—
121	D₄	—	236	2	no	—
122	D₄	—	236	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	118	2	no	—
125	C₂×C₄	—	236	2	no	—
126	C₂×C₄	—	236	2	no	—
127	C₂×C₄	—	236	2	no	—
128	C₂×C₄	#11745685	1	1	yes	characteristic
129	C₂×C₄	D₄×D₅₉	1	2	yes	—
130	C₂×C₄	#11745689	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	472	2	no	—
136	C₂²	—	236	2	no	—
137	C₂²	#14273701	1	1	yes	characteristic
138	C₂²	#14273702	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	472	2	no	—
141	C₄	—	236	2	no	—
142	C₄	#14273695	1	1	yes	characteristic
143	C₄	#14273712	1	2	yes	—
144	C₄	#14273716	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	472	2	no	—
148	C₂	#16837319	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838202

C₂

yes

maximal, characteristic

#16837399

C₂

yes

maximal, complemented

#16837375

C₂

yes

maximal, complemented

#16837159

C₂

yes

maximal, characteristic, complemented

C₅₉×#424

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

—

—

—

C₂²

yes

—

#14273652

—

#14273649

—

#14273641

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

—

—

—

—

C₂×D₂₃₆

D₄

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

D₄

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

Dic₂₃₆

—

D₄₇₂

—

#11745660

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂²⋊D₅₉

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

#121

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

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

#394

yes

characteristic

D₅₉

—

#575

—

#424

D₅₉

yes

characteristic, complemented

#416

—

#410

—

H(3,ℤ/4)

—

C₅₉

#2053

yes

characteristic, complemented, Sylow

C₂×#41

—

#137

—

118

—

AGL(1,ℤ/8)

—

118

—

#135

—

118

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

#117

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

#103

—

118

—

#102

—

118

—

#99

—

118

—

#41

—

118

—

#41

—

118

—

#41

—

118

—

#41

—

118

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

118

—

100

C₂²×C₄

—

118

—

101

C₂²×C₄

C₂²×D₅₉

yes

characteristic

102

Q₁₆

—

118

—

103

SD₁₆

—

118

—

104

D₈

—

118

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₅₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

118

—

111

C₂²⋊C₄

—

118

—

112

C₂²⋊C₄

D₂₃₆

yes

—

113

C₄²

—

114

C₄²

—

118

—

115

C₂³

—

118

—

116

C₂³

C₂×D₂₃₆

yes

characteristic

117

Q₈

—

118

—

118

Q₈

—

118

—

119

D₄

—

118

—

120

D₄

—

118

—

121

D₄

—

236

—

122

D₄

—

236

—

123

C₂×C₄

—

124

C₂×C₄

—

118

—

125

C₂×C₄

—

236

—

126

C₂×C₄

—

236

—

127

C₂×C₄

—

236

—

128

C₂×C₄

#11745685

yes

characteristic

129

C₂×C₄

D₄×D₅₉

yes

—

130

C₂×C₄

#11745689

yes

—

131

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

472

—

136

C₂²

—

236

—

137

C₂²

#14273701

yes

characteristic

138

C₂²

#14273702

yes

—

139

C₄

—

140

C₄

—

472

—

141

C₄

—

236

—

142

C₄

#14273695

yes

characteristic

143

C₄

#14273712

yes

—

144

C₄

#14273716

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

472

—

148

C₂

#16837319

yes

characteristic

	subgroup	G/H
center	C₄	#14273716
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#424	C₂
socle	C₂×C₅₉	#394
hypercenter	#424	D₅₉
hypocenter	C₅₉	#2053
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₄

#14273716

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#424

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #16837399	C₂	semidirect
3. #16837375	C₂	semidirect
4. #16837159	C₂	semidirect
5. C₅₉×#424	C₂	semidirect
76. #424	D₅₉	semidirect
80. C₅₉	#2053	semidirect

H ≅ G/N

type

2. #16837399

C₂

semidirect

3. #16837375

C₂

semidirect

4. #16837159

C₂

semidirect

5. C₅₉×#424

C₂

semidirect

76. #424

D₅₉

semidirect

80. C₅₉

#2053

semidirect

Subgroups — #21425402

Conjugacy classes of subgroups

Complements