Group #21422797 — subgroups

Subgroups — #21422797

Order 7552.

208 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16837959	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16837149	C₂	1	2	yes	maximal, complemented
4	#16837147	C₂	1	2	yes	maximal, complemented
5	C₅₉×#432	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	—	2	1	no	—
7	#14273727	C₂²	1	1	yes	characteristic
8	#14273727	C₂²	1	1	yes	characteristic
9	#14273719	C₄	1	2	yes	—
10	#14273706	—	2	1	no	—
11	#14273706	—	2	1	no	—
12	C₄×Dic₁₁₈	—	2	1	no	—
13	C₄×Dic₁₁₈	—	2	1	no	—
14	#14273686	C₄	1	2	yes	—
15	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
16	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₅₉	—	2	1	no	—
20	#97×C₅₉	—	2	1	no	—
21	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₂×Dic₁₁₈	—	2	1	no	—
25	C₂×Dic₁₁₈	—	2	1	no	—
26	C₂×C₈×C₅₉	—	2	2	no	—
27	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₉	D₄	1	1	yes	characteristic
29	C₄²×C₅₉	D₄	1	1	yes	characteristic
30	C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₂²⋊Dic₅₉	—	2	2	no	—
32	C₄⋊Dic₅₉	—	2	1	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
35	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
36	Dic₅₉⋊C₄	—	2	2	no	—
37	Dic₅₉⋊C₄	—	2	2	no	—
38	C₄×Dic₅₉	—	2	2	no	—
39	C₂×C₅₉⋊C₈	—	2	2	no	—
40	C₂×C₅₉⋊C₈	—	4	2	no	—
41	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	2	1	no	—
46	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅₉	—	2	2	no	—
53	C₂×Dic₅₉	—	2	2	no	—
54	Dic₁₁₈	—	4	2	no	—
55	Dic₁₁₈	—	4	1	no	—
56	Dic₁₁₈	—	4	1	no	—
57	C₈×C₅₉	—	4	2	no	—
58	C₅₉⋊C₈	—	4	2	no	—
59	C₂²×C₅₉	—	2	1	no	—
60	C₂²×C₅₉	—	2	1	no	—
61	C₂²×C₅₉	#101	1	2	yes	—
62	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	—	2	1	no	—
65	C₄×C₅₉	—	2	1	no	—
66	C₄×C₅₉	—	2	1	no	—
67	C₄×C₅₉	#135	1	2	yes	—
68	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₅₉	#137	1	1	yes	characteristic
70	Dic₅₉	—	4	2	no	—
71	Dic₅₉	—	4	2	no	—
72	#1552	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	—	2	2	no	—
74	C₂×C₅₉	C₂×#101	1	1	yes	characteristic
75	C₂×C₅₉	#416	1	1	yes	characteristic
76	C₂×C₅₉	#417	1	1	yes	characteristic
77	#533	—	59	1	no	—
78	#432	D₅₉	1	1	yes	characteristic
79	C₂×C₄⋊C₈	—	59	1	no	—
80	#332	—	59	2	no	—
81	#332	—	59	2	no	—
82	C₅₉	#1552	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	—	118	1	no	—
84	C₄⋊Q₈	—	59	1	no	—
85	C₄⋊Q₈	—	59	1	no	—
86	#125	—	59	2	no	—
87	C₂²⋊Q₈	—	118	1	no	—
88	C₂²⋊Q₈	—	118	1	no	—
89	C₄×Q₈	—	118	1	no	—
90	C₄×Q₈	—	118	1	no	—
91	#117	—	59	2	no	—
92	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
93	C₄⋊C₈	—	59	2	no	—
94	C₄⋊C₈	—	118	1	no	—
95	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
96	C₂²⋊C₈	—	2	1	no	—
97	#97	—	2	1	no	—
98	C₂×Q₈	—	118	1	no	—
99	C₂×Q₈	—	118	1	no	—
100	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
101	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	118	2	no	—
105	C₂×C₈	—	236	2	no	—
106	C₄⋊C₄	—	59	2	no	—
107	C₄⋊C₄	—	59	2	no	—
108	C₄⋊C₄	—	118	2	no	—
109	C₄⋊C₄	—	118	2	no	—
110	C₄⋊C₄	—	118	1	no	—
111	C₄⋊C₄	—	118	1	no	—
112	C₂²⋊C₄	—	118	2	no	—
113	C₄²	—	118	2	no	—
114	C₄²	C₄×D₅₉	1	1	yes	characteristic
115	C₄²	D₂₃₆	1	1	yes	characteristic
116	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
117	C₄²	C₂²×D₅₉	1	1	yes	characteristic
118	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
119	Q₈	—	236	2	no	—
120	Q₈	—	236	1	no	—
121	Q₈	—	236	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	118	2	no	—
127	C₂×C₄	—	118	2	no	—
128	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
132	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	236	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#14273645	1	2	yes	—
139	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	236	2	no	—
145	C₄	—	236	2	no	—
146	C₄	#14273747	1	2	yes	—
147	C₄	#14273755	1	1	yes	characteristic
148	C₄	#14273760	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	C₂×#14273645	1	1	yes	characteristic
151	C₂	#16837396	1	1	yes	characteristic
152	C₂	#16837404	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal, complemented

#16837147

C₂

yes

maximal, complemented

C₅₉×#432

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

—

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273719

C₄

yes

—

#14273706

—

#14273706

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

#14273686

C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#101

yes

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1552

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

C₂×C₄⋊C₈

—

#332

—

#332

—

C₅₉

#1552

yes

characteristic, complemented, Sylow

C₂²×C₈

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

118

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

118

—

C₄×Q₈

—

118

—

#117

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

—

C₂²⋊C₈

—

#97

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

100

C₂²×C₄

C₄×D₅₉

yes

characteristic

101

C₂²×C₄

D₂₃₆

yes

characteristic

102

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

118

—

105

C₂×C₈

—

236

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

118

—

109

C₄⋊C₄

—

118

—

110

C₄⋊C₄

—

118

—

111

C₄⋊C₄

—

118

—

112

C₂²⋊C₄

—

118

—

113

C₄²

—

118

—

114

C₄²

C₄×D₅₉

yes

characteristic

115

C₄²

D₂₃₆

yes

characteristic

116

C₄²

C₂²⋊D₅₉

yes

characteristic

117

C₄²

C₂²×D₅₉

yes

characteristic

118

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

119

Q₈

—

236

—

120

Q₈

—

236

—

121

Q₈

—

236

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

118

—

127

C₂×C₄

—

118

—

128

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

129

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

130

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

131

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

132

C₂×C₄

C₂×D₂₃₆

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

134

C₈

—

135

C₈

—

236

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14273645

yes

—

139

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

236

—

145

C₄

—

236

—

146

C₄

#14273747

yes

—

147

C₄

#14273755

yes

characteristic

148

C₄

#14273760

yes

characteristic

149

C₂

—

150

C₂

C₂×#14273645

yes

characteristic

151

C₂

#16837396

yes

characteristic

152

C₂

#16837404

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₅₉)⋊C₄
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#432	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#432	D₅₉
hypocenter	C₅₉	#1552
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₅₉)⋊C₄

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#432

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #16837149	C₂	semidirect
4. #16837147	C₂	semidirect
82. C₅₉	#1552	semidirect

H ≅ G/N

type

3. #16837149

C₂

semidirect

4. #16837147

C₂

semidirect

82. C₅₉

#1552

semidirect

Subgroups — #21422797

Conjugacy classes of subgroups

Complements