Group #21422796 — subgroups

Subgroups — #21422796

Order 7552.

208 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16837459	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16837148	C₂	1	2	yes	maximal, complemented
4	#16837148	C₂	1	2	yes	maximal, complemented
5	C₅₉×#533	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
11	#14273638	—	2	1	no	—
12	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	#125×C₅₉	C₄	1	2	yes	—
15	C₂²⋊Q₈×C₅₉	—	2	1	no	—
16	C₂²⋊Q₈×C₅₉	—	2	1	no	—
17	C₄×Q₈×C₅₉	—	2	1	no	—
18	C₄×Q₈×C₅₉	—	2	1	no	—
19	#117×C₅₉	C₄	1	2	yes	—
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	—	2	1	no	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
31	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₅₉	—	2	2	no	—
33	C₄²×C₅₉	—	2	2	no	—
34	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₉	D₄	1	1	yes	characteristic
36	C₄²×C₅₉	D₄	1	1	yes	characteristic
37	C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	C₂×C₅₉⋊C₈	—	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	Q₈×C₅₉	—	4	2	no	—
43	Q₈×C₅₉	—	4	1	no	—
44	Q₈×C₅₉	—	4	1	no	—
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	—	2	1	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₅₉⋊C₈	—	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	—	4	2	no	—
68	C₄×C₅₉	—	4	2	no	—
69	C₄×C₅₉	#135	1	2	yes	—
70	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₅₉	#137	1	1	yes	characteristic
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	D₅₉	1	1	yes	characteristic
78	#432	—	59	1	no	—
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	C₂×D₅₉	1	1	yes	characteristic
85	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
86	#125	Dic₅₉	1	2	yes	—
87	C₂²⋊Q₈	—	2	1	no	—
88	C₂²⋊Q₈	—	2	1	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	2	1	no	—
91	#117	Dic₅₉	1	2	yes	—
92	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
93	C₄⋊C₈	—	59	2	no	—
94	C₄⋊C₈	—	59	2	no	—
95	C₄⋊C₈	—	118	1	no	—
96	C₂²⋊C₈	—	118	1	no	—
97	#97	—	118	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
103	C₂×C₈	—	118	2	no	—
104	C₂×C₈	—	118	2	no	—
105	C₂×C₈	—	236	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
111	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
115	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
117	C₄²	C₂²×D₅₉	1	1	yes	characteristic
118	C₂³	C₂²⋊Dic₅₉	1	1	yes	characteristic
119	Q₈	—	4	2	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	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	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
134	C₈	—	236	2	no	—
135	C₈	—	236	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#14273646	1	2	yes	—
139	C₂²	C₂×C₂²⋊Dic₅₉	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	—	4	2	no	—
145	C₄	—	4	2	no	—
146	C₄	#14273747	1	2	yes	—
147	C₄	#14273752	1	1	yes	characteristic
148	C₄	#14273757	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	C₂×#14273646	1	1	yes	characteristic
151	C₂	#16837398	1	1	yes	characteristic
152	C₂	#16837403	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837459

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

#16837148

C₂

yes

maximal, complemented

#16837148

C₂

yes

maximal, complemented

C₅₉×#533

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#125×C₅₉

C₄

yes

—

C₂²⋊Q₈×C₅₉

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

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

C₅₉⋊C₈

—

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

—

C₄×C₅₉

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#432

—

C₂×C₄⋊C₈

—

#332

—

#332

—

C₅₉

#1552

yes

characteristic, complemented, Sylow

C₂²×C₈

—

118

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#125

Dic₅₉

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₂²⋊C₈

—

118

—

#97

—

118

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

101

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

102

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

103

C₂×C₈

—

118

—

104

C₂×C₈

—

118

—

105

C₂×C₈

—

236

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×Dic₅₉

yes

—

111

C₄⋊C₄

C₂×Dic₅₉

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂×Dic₅₉

yes

characteristic

115

C₄²

C₂²⋊D₅₉

yes

characteristic

116

C₄²

C₂²⋊D₅₉

yes

characteristic

117

C₄²

C₂²×D₅₉

yes

characteristic

118

C₂³

C₂²⋊Dic₅₉

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

129

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

130

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

131

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

134

C₈

—

236

—

135

C₈

—

236

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14273646

yes

—

139

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#14273747

yes

—

147

C₄

#14273752

yes

characteristic

148

C₄

#14273757

yes

characteristic

149

C₂

—

150

C₂

C₂×#14273646

yes

characteristic

151

C₂

#16837398

yes

characteristic

152

C₂

#16837403

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#533	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#533	D₅₉
hypocenter	C₅₉	#1552
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#533

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #16837148	C₂	semidirect
4. #16837148	C₂	semidirect
82. C₅₉	#1552	semidirect

H ≅ G/N

type

3. #16837148

C₂

semidirect

4. #16837148

C₂

semidirect

82. C₅₉

#1552

semidirect

Subgroups — #21422796

Conjugacy classes of subgroups

Complements