Group #21422806 — subgroups

Subgroups — #21422806

Order 7552.

198 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16837455	C₂	1	1	yes	maximal, characteristic
3	#16837151	C₂	1	2	yes	maximal, complemented
4	#16837148	C₂	1	2	yes	maximal, complemented
5	C₅₉×#533	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
10	#14273638	—	2	1	no	—
11	C₄×C₅₉⋊C₈	—	2	1	no	—
12	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	#125×C₅₉	—	2	1	no	—
15	C₂²⋊Q₈×C₅₉	—	2	1	no	—
16	C₂²⋊Q₈×C₅₉	C₄	1	2	yes	—
17	C₄×Q₈×C₅₉	—	2	1	no	—
18	C₄×Q₈×C₅₉	C₄	1	2	yes	—
19	#117×C₅₉	—	2	1	no	—
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
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	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₅₉	—	4	2	no	—
42	Q₈×C₅₉	—	4	2	no	—
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	—	2	2	no	—
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	4	1	no	—
48	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₅₉⋊C₈	—	4	2	no	—
55	C₅₉⋊C₈	—	4	2	no	—
56	C₂²×C₅₉	—	2	1	no	—
57	C₂²×C₅₉	—	2	1	no	—
58	C₂²×C₅₉	—	2	1	no	—
59	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅₉	—	2	1	no	—
61	C₄×C₅₉	—	2	1	no	—
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	—	4	2	no	—
65	C₄×C₅₉	—	4	2	no	—
66	C₄×C₅₉	#135	1	2	yes	—
67	C₄×C₅₉	#137	1	2	yes	—
68	#1554	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	—	4	1	no	—
70	C₂×C₅₉	#413	1	1	yes	characteristic
71	C₂×C₅₉	#416	1	1	yes	characteristic
72	C₂×C₅₉	#419	1	1	yes	characteristic
73	#533	D₅₉	1	1	yes	characteristic
74	#432	—	59	1	no	—
75	#431	—	59	1	no	—
76	#333	—	59	2	no	—
77	#332	—	59	2	no	—
78	C₅₉	#1554	1	1	yes	characteristic, complemented, Sylow
79	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
80	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
81	#125	—	2	1	no	—
82	C₂²⋊Q₈	—	2	1	no	—
83	C₂²⋊Q₈	Dic₅₉	1	2	yes	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	Dic₅₉	1	2	yes	—
86	#117	—	2	1	no	—
87	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
88	C₄⋊C₈	—	59	2	no	—
89	C₄⋊C₈	—	59	2	no	—
90	C₂²⋊C₈	—	118	1	no	—
91	C₂²⋊C₈	—	118	1	no	—
92	#97	—	118	1	no	—
93	C₄×C₈	—	118	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	C₂×Dic₅₉	1	2	yes	—
96	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
99	C₂×C₈	—	118	2	no	—
100	C₂×C₈	—	118	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
110	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
111	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
112	C₄²	C₂²×D₅₉	1	1	yes	characteristic
113	C₂³	C₂²⋊Dic₅₉	1	1	yes	characteristic
114	Q₈	—	4	2	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
124	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
127	C₈	—	236	2	no	—
128	C₈	—	236	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	2	no	—
138	C₄	—	4	2	no	—
139	C₄	#14273747	1	2	yes	—
140	C₄	#14273758	1	2	yes	—
141	C₂	—	4	1	no	—
142	C₂	#16837386	1	1	yes	characteristic
143	C₂	#16837398	1	1	yes	characteristic
144	C₂	#16837411	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837459

C₂

yes

maximal, characteristic

#16837455

C₂

yes

maximal, characteristic

#16837151

C₂

yes

maximal, complemented

#16837148

C₂

yes

maximal, complemented

C₅₉×#533

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#125×C₅₉

—

C₂²⋊Q₈×C₅₉

—

C₂²⋊Q₈×C₅₉

C₄

yes

—

C₄×Q₈×C₅₉

—

C₄×Q₈×C₅₉

C₄

yes

—

#117×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

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

—

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

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×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

—

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

#137

yes

—

#1554

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

—

—

—

—

C₅₉

#1554

yes

characteristic, complemented, Sylow

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₅₉

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₅₉

yes

—

#117

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

118

—

C₂²⋊C₈

—

118

—

#97

—

118

—

C₄×C₈

—

118

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

—

118

—

100

C₂×C₈

—

118

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×Dic₅₉

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

C₂×Dic₅₉

yes

characteristic

110

C₄²

C₂²⋊D₅₉

yes

characteristic

111

C₄²

C₂²⋊D₅₉

yes

characteristic

112

C₄²

C₂²×D₅₉

yes

characteristic

113

C₂³

C₂²⋊Dic₅₉

yes

characteristic

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

124

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

127

C₈

—

236

—

128

C₈

—

236

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#14273747

yes

—

140

C₄

#14273758

yes

—

141

C₂

—

142

C₂

#16837386

yes

characteristic

143

C₂

#16837398

yes

characteristic

144

C₂

#16837411

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	#1554
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. #16837151	C₂	semidirect
4. #16837148	C₂	semidirect
78. C₅₉	#1554	semidirect

H ≅ G/N

type

3. #16837151

C₂

semidirect

4. #16837148

C₂

semidirect

78. C₅₉

#1554

semidirect

Subgroups — #21422806

Conjugacy classes of subgroups

Complements