Group #21436965 — subgroups

Subgroups — #21436965

Order 7552.

352 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16838113	C₂	1	2	yes	maximal, complemented
2	#16837803	C₂	1	2	yes	maximal, complemented
3	#16837705	C₂	1	4	yes	maximal, complemented
4	#16837689	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
7	#16837484	C₂	1	2	yes	maximal
8	C₅₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
10	#14273718	C₂²	1	4	yes	—
11	#14273707	—	2	2	no	—
12	#14273703	—	2	4	no	—
13	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
14	C₄×D₂₃₆	C₂²	1	2	yes	—
15	#14273689	C₂²	1	4	yes	—
16	C₄⋊C₄×D₅₉	—	2	2	no	—
17	#14273661	C₂²	1	1	yes	characteristic
18	#14273658	C₂²	1	4	yes	—
19	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₅₉⋊C₈	C₂²	1	2	yes	—
21	#14273649	C₂²	1	2	yes	—
22	#14273647	C₂²	1	4	yes	—
23	C₈×Dic₅₉	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
25	#107×C₅₉	C₂²	1	2	yes	—
26	#107×C₅₉	C₂²	1	2	yes	—
27	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₉	—	2	2	no	—
29	C₂×D₂₃₆	C₂³	1	2	yes	—
30	C₂×C₄×D₅₉	—	2	2	no	—
31	C₂×C₄×D₅₉	—	2	4	no	—
32	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	—	2	2	no	—
36	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
37	C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	(C₂×D₅₉)⋊C₄	—	2	2	no	—
39	(C₂×D₅₉)⋊C₄	—	2	4	no	—
40	C₄⋊Dic₅₉	—	2	4	no	—
41	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₉	C₂³	1	2	yes	—
43	Dic₅₉⋊C₄	—	2	4	no	—
44	Dic₅₉⋊C₄	D₄	1	2	yes	—
45	C₄×Dic₅₉	C₂³	1	2	yes	—
46	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
47	D₄₇₂	Q₈	1	4	yes	complemented
48	C₂²×D₅₉	—	2	2	no	—
49	Q₈×C₅₉	—	8	2	no	—
50	C₂×C₄×C₅₉	—	2	4	no	—
51	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₅₉	—	2	2	no	—
55	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
56	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
57	D₂₃₆	—	2	2	no	—
58	D₂₃₆	C₂×Q₈	1	4	yes	—
59	C₄×D₅₉	—	4	4	no	—
60	C₄×D₅₉	—	8	4	no	—
61	Dic₁₁₈	—	2	2	no	—
62	C₈×C₅₉	—	2	1	no	—
63	C₈×C₅₉	C₂×Q₈	1	2	yes	—
64	C₅₉⋊C₈	—	2	2	no	—
65	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₅₉	—	2	4	no	—
67	C₂×D₅₉	—	4	2	no	—
68	C₂×D₅₉	—	4	2	no	—
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	—	4	4	no	—
71	C₄×C₅₉	#135	1	2	yes	—
72	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₅₉	—	2	2	no	—
75	Dic₅₉	—	2	2	no	—
76	Dic₅₉	—	4	2	no	—
77	#3380	—	59	1	no	maximal, Sylow
78	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
79	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
80	C₂×C₅₉	#576	1	1	yes	characteristic
81	D₅₉	—	4	4	no	—
82	#554	—	59	2	no	—
83	#500	D₅₉	1	1	yes	characteristic, complemented
84	#499	—	59	2	no	—
85	#478	—	59	4	no	—
86	#476	—	59	2	no	—
87	C₈×Q₈	—	59	1	no	—
88	C₄×D₈	—	59	2	no	—
89	C₄×D₈	—	59	1	no	—
90	C₅₉	#3380	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	59	1	no	—
92	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
93	#125	—	59	4	no	—
94	C₂²⋊Q₈	—	118	2	no	—
95	C₂²⋊Q₈	—	118	4	no	—
96	C₄×Q₈	—	59	1	no	—
97	C₄×D₄	—	59	4	no	—
98	C₄×D₄	—	59	2	no	—
99	C₂×C₄⋊C₄	—	118	2	no	—
100	#107	—	59	1	no	—
101	#107	C₂×D₅₉	1	2	yes	—
102	#107	C₂×D₅₉	1	2	yes	—
103	#106	—	59	4	no	—
104	C₄⋊C₈	—	59	2	no	—
105	C₄⋊C₈	—	59	1	no	—
106	#102	—	59	2	no	—
107	#102	—	59	4	no	—
108	C₄×C₈	—	59	2	no	—
109	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	59	1	no	—
112	C₂×D₄	—	59	2	no	—
113	C₂²×C₄	—	118	4	no	—
114	C₂²×C₄	—	118	2	no	—
115	D₈	—	59	4	no	—
116	C₂×C₈	—	59	2	no	—
117	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	59	1	no	—
121	C₄⋊C₄	—	59	2	no	—
122	C₄⋊C₄	—	59	2	no	—
123	C₄⋊C₄	—	118	4	no	—
124	C₄⋊C₄	—	118	4	no	—
125	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
126	C₂²⋊C₄	—	118	4	no	—
127	C₂²⋊C₄	—	118	2	no	—
128	C₄²	—	59	2	no	—
129	C₄²	C₂²×D₅₉	1	1	yes	characteristic
130	C₂³	—	118	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	118	2	no	—
133	D₄	—	59	4	no	—
134	D₄	—	118	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	59	2	no	—
137	C₂×C₄	—	59	2	no	—
138	C₂×C₄	—	118	2	no	—
139	C₂×C₄	—	472	4	no	—
140	C₂×C₄	—	236	4	no	—
141	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
142	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	118	2	no	—
146	C₈	Q₈×D₅₉	1	2	yes	—
147	C₂²	—	118	4	no	—
148	C₂²	—	236	2	no	—
149	C₂²	—	236	2	no	—
150	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	4	no	—
153	C₄	—	118	2	no	—
154	C₄	—	118	2	no	—
155	C₄	—	236	2	no	—
156	C₄	#14273750	1	2	yes	—
157	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
158	C₄	#14273775	1	1	yes	characteristic
159	C₂	—	236	4	no	—
160	C₂	#16838067	1	1	yes	characteristic
161	C₂	C₂×#14273750	1	1	yes	characteristic
162	C₂	#16838203	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838113

C₂

yes

maximal, complemented

#16837803

C₂

yes

maximal, complemented

#16837705

C₂

yes

maximal, complemented

#16837689

C₂

yes

maximal, complemented

C₈×Dic₁₁₈

C₂

yes

maximal, characteristic, complemented

C₄×D₄₇₂

C₂

yes

maximal, characteristic

#16837484

C₂

yes

maximal

C₅₉×#500

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₂²

yes

characteristic

#14273718

C₂²

yes

—

#14273707

—

#14273703

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

C₄×D₂₃₆

C₂²

yes

—

#14273689

C₂²

yes

—

C₄⋊C₄×D₅₉

—

#14273661

C₂²

yes

characteristic

#14273658

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

C₂²

yes

—

#14273649

C₂²

yes

—

#14273647

C₂²

yes

—

C₈×Dic₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₂₃₆

C₂³

yes

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

D₄

yes

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

D₄₇₂

Q₈

yes

complemented

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×D₄

yes

—

D₂₃₆

—

D₂₃₆

C₂×Q₈

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#135

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#3380

—

maximal, Sylow

C₂×C₅₉

D₄×Q₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#554

—

#500

D₅₉

yes

characteristic, complemented

#499

—

#478

—

#476

—

C₈×Q₈

—

C₄×D₈

—

C₄×D₈

—

C₅₉

#3380

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄⋊Q₈

C₂×D₅₉

yes

—

#125

—

C₂²⋊Q₈

—

118

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

118

—

100

#107

—

101

#107

C₂×D₅₉

yes

—

102

#107

C₂×D₅₉

yes

—

103

#106

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#102

—

107

#102

—

108

C₄×C₈

—

109

C₄×C₈

C₂×D₅₉

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂²×C₄

—

118

—

114

C₂²×C₄

—

118

—

115

D₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₅₉

yes

characteristic

118

C₂×C₈

C₂²×D₅₉

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

118

—

124

C₄⋊C₄

—

118

—

125

C₄⋊C₄

C₂²×D₅₉

yes

—

126

C₂²⋊C₄

—

118

—

127

C₂²⋊C₄

—

118

—

128

C₄²

—

129

C₄²

C₂²×D₅₉

yes

characteristic

130

C₂³

—

118

—

131

Q₈

—

132

Q₈

—

118

—

133

D₄

—

134

D₄

—

118

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

118

—

139

C₂×C₄

—

472

—

140

C₂×C₄

—

236

—

141

C₂×C₄

D₄×D₅₉

yes

characteristic

142

C₂×C₄

D₄×D₅₉

yes

characteristic

143

C₂×C₄

C₂³×D₅₉

yes

characteristic

144

C₈

—

145

C₈

—

118

—

146

C₈

Q₈×D₅₉

yes

—

147

C₂²

—

118

—

148

C₂²

—

236

—

149

C₂²

—

236

—

150

C₂²

C₂×D₄×D₅₉

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

118

—

154

C₄

—

118

—

155

C₄

—

236

—

156

C₄

#14273750

yes

—

157

C₄

C₂×Q₈×D₅₉

yes

characteristic

158

C₄

#14273775

yes

characteristic

159

C₂

—

236

—

160

C₂

#16838067

yes

characteristic

161

C₂

C₂×#14273750

yes

characteristic

162

C₂

#16838203

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#500	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#500	D₅₉
hypocenter	C₅₉	#3380
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#500

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #16838113	C₂	semidirect
2. #16837803	C₂	semidirect
3. #16837705	C₂	semidirect
4. #16837689	C₂	semidirect
5. C₈×Dic₁₁₈	C₂	semidirect
8. C₅₉×#500	C₂	semidirect
47. D₄₇₂	Q₈	semidirect
83. #500	D₅₉	semidirect
90. C₅₉	#3380	semidirect

H ≅ G/N

type

1. #16838113

C₂

semidirect

2. #16837803

C₂

semidirect

3. #16837705

C₂

semidirect

4. #16837689

C₂

semidirect

5. C₈×Dic₁₁₈

C₂

semidirect

8. C₅₉×#500

C₂

semidirect

47. D₄₇₂

Q₈

semidirect

83. #500

D₅₉

semidirect

90. C₅₉

#3380

semidirect

Subgroups — #21436965

Conjugacy classes of subgroups

Complements