Group #21425006 — subgroups

Subgroups — #21425006

Order 7552.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×D₄₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273663	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837406	C₂	1	2	yes	maximal, complemented
4	C₂×#14273643	C₂	1	2	yes	maximal, complemented
5	C₅₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₃₆	C₂²	1	2	yes	—
7	C₂×D₄₇₂	—	2	2	no	—
8	C₂×D₄₇₂	—	2	1	no	—
9	C₂×D₄₇₂	—	2	1	no	—
10	C₂×D₄₇₂	C₄	1	4	yes	complemented
11	C₂×#11745664	C₂²	1	2	yes	—
12	#14273663	—	2	2	no	—
13	#14273647	—	2	4	no	—
14	#14273643	—	2	4	no	—
15	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	#117×C₅₉	C₂²	1	1	yes	characteristic
18	#97×C₅₉	—	2	1	no	—
19	C₄×C₈×C₅₉	—	2	1	no	—
20	C₂³×D₅₉	—	2	2	no	—
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂×D₂₃₆	—	2	2	no	—
23	C₂×D₂₃₆	—	2	2	no	—
24	C₂×D₂₃₆	—	2	2	no	—
25	C₂×D₂₃₆	—	2	2	no	—
26	C₂×D₂₃₆	—	2	2	no	—
27	C₂×D₂₃₆	C₂×C₄	1	4	yes	—
28	M₁₆×C₅₉	—	2	1	no	—
29	M₁₆×C₅₉	D₄	1	2	yes	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₅₉	D₄	1	2	yes	—
33	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
36	C₂²⋊C₄×C₅₉	—	2	1	no	—
37	C₄²×C₅₉	—	2	1	no	—
38	#11745664	—	2	2	no	—
39	#11745664	—	2	2	no	—
40	C₂×C₅₉⋊C₈	—	2	2	no	—
41	D₄₇₂	—	2	4	no	—
42	D₄₇₂	—	2	4	no	—
43	C₂³×C₅₉	#41	1	1	yes	characteristic
44	C₂²×D₅₉	—	2	4	no	—
45	C₂²×D₅₉	—	4	2	no	—
46	C₂²×D₅₉	—	4	2	no	—
47	C₂²×D₅₉	—	8	4	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
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×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
55	D₂₃₆	—	2	4	no	—
56	D₂₃₆	—	2	4	no	—
57	D₂₃₆	—	4	2	no	—
58	D₂₃₆	—	4	2	no	—
59	C₈×C₅₉	—	2	2	no	—
60	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
61	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
62	C₅₉⋊C₈	—	4	2	no	—
63	C₂²×C₅₉	—	2	2	no	—
64	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
65	C₂²×C₅₉	#121	1	1	yes	characteristic
66	C₂²×C₅₉	#124	1	1	yes	characteristic
67	C₂×D₅₉	—	4	4	no	—
68	C₂×D₅₉	—	8	4	no	—
69	C₂×D₅₉	—	8	4	no	—
70	C₂×D₅₉	—	8	4	no	—
71	C₄×C₅₉	—	4	2	no	—
72	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
74	C₄×C₅₉	#121	1	1	yes	characteristic
75	C₄×C₅₉	#127	1	1	yes	characteristic
76	#1978	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	—	2	1	no	—
78	C₂×C₅₉	—	2	1	no	—
79	C₂×C₅₉	#390	1	1	yes	characteristic
80	C₂×C₅₉	#472	1	2	yes	—
81	D₅₉	—	8	4	no	—
82	C₂²×D₈	—	59	1	no	—
83	C₂×#108	—	59	1	no	—
84	#418	—	59	2	no	—
85	C₂×#100	—	59	2	no	—
86	#405	D₅₉	1	1	yes	characteristic, complemented
87	C₅₉	#1978	1	1	yes	characteristic, complemented, Sylow
88	C₂²×D₄	—	59	2	no	—
89	C₂×D₈	—	59	4	no	—
90	C₂×D₈	—	118	2	no	—
91	C₂×D₈	—	118	1	no	—
92	C₂×D₈	—	118	1	no	—
93	C₂×M₁₆	—	59	2	no	—
94	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
96	#117	C₂×D₅₉	1	1	yes	characteristic
97	#108	—	118	2	no	—
98	#102	—	118	4	no	—
99	#100	—	118	4	no	—
100	#97	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₂⁴	—	118	2	no	—
103	C₂×D₄	—	59	4	no	—
104	C₂×D₄	—	118	2	no	—
105	C₂×D₄	—	118	2	no	—
106	C₂×D₄	—	118	2	no	—
107	C₂×D₄	—	118	2	no	—
108	C₂×D₄	—	118	2	no	—
109	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
110	D₈	—	118	4	no	—
111	D₈	—	118	4	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	118	2	no	—
114	M₁₆	—	118	2	no	—
115	M₁₆	D₂₃₆	1	2	yes	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	118	2	no	—
118	C₂×C₈	C₄×D₅₉	1	2	yes	—
119	C₂×C₈	D₂₃₆	1	2	yes	—
120	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
122	C₄⋊C₄	D₂₃₆	1	2	yes	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₂³	—	118	4	no	—
126	C₂³	—	472	4	no	—
127	C₂³	—	236	2	no	—
128	C₂³	—	236	2	no	—
129	C₂³	#11745685	1	1	yes	characteristic
130	D₄	—	118	4	no	—
131	D₄	—	118	4	no	—
132	D₄	—	236	2	no	—
133	D₄	—	236	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
136	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
137	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
139	C₂×C₄	#11745685	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	236	2	no	—
143	C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
144	C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	472	4	no	—
147	C₂²	—	472	4	no	—
148	C₂²	—	472	4	no	—
149	C₂²	—	236	4	no	—
150	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
151	C₂²	#14273701	1	1	yes	characteristic
152	C₂²	#14273715	1	1	yes	characteristic
153	C₄	—	4	2	no	—
154	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
155	C₄	C₄×D₂₃₆	1	1	yes	characteristic
156	C₄	#14273701	1	1	yes	characteristic
157	C₄	#14273724	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	472	4	no	—
161	C₂	#16837304	1	1	yes	characteristic
162	C₂	#16837672	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₇₂

C₂

yes

maximal, characteristic

C₂×#14273663

C₂

yes

maximal, characteristic, complemented

#16837406

C₂

yes

maximal, complemented

C₂×#14273643

C₂

yes

maximal, complemented

C₅₉×#405

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

C₄

yes

complemented

C₂×#11745664

C₂²

yes

—

#14273663

—

#14273647

—

#14273643

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

D₄₇₂

—

D₄₇₂

—

C₂³×C₅₉

#41

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#390

yes

characteristic

C₂×C₅₉

#472

yes

—

D₅₉

—

C₂²×D₈

—

—

—

—

D₅₉

yes

characteristic, complemented

C₅₉

#1978

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

118

—

C₂×D₈

—

118

—

C₂×D₈

—

118

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

C₂×D₅₉

yes

characteristic

#108

—

118

—

#102

—

118

—

#100

—

118

—

100

#97

—

101

C₄×C₈

—

102

C₂⁴

—

118

—

103

C₂×D₄

—

104

C₂×D₄

—

118

—

105

C₂×D₄

—

118

—

106

C₂×D₄

—

118

—

107

C₂×D₄

—

118

—

108

C₂×D₄

—

118

—

109

C₂²×C₄

C₂²×D₅₉

yes

characteristic

110

D₈

—

118

—

111

D₈

—

118

—

112

M₁₆

—

113

M₁₆

—

118

—

114

M₁₆

—

118

—

115

M₁₆

D₂₃₆

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

118

—

118

C₂×C₈

C₄×D₅₉

yes

—

119

C₂×C₈

D₂₃₆

yes

—

120

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

121

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

122

C₄⋊C₄

D₂₃₆

yes

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₂³

—

118

—

126

C₂³

—

472

—

127

C₂³

—

236

—

128

C₂³

—

236

—

129

C₂³

#11745685

yes

characteristic

130

D₄

—

118

—

131

D₄

—

118

—

132

D₄

—

236

—

133

D₄

—

236

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

136

C₂×C₄

C₂×D₂₃₆

yes

characteristic

137

C₂×C₄

C₂×D₂₃₆

yes

characteristic

138

C₂×C₄

C₂×D₂₃₆

yes

characteristic

139

C₂×C₄

#11745685

yes

characteristic

140

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

141

C₈

—

142

C₈

—

236

—

143

C₈

(C₂×D₅₉)⋊C₄

yes

—

144

C₈

(C₂×D₅₉)⋊C₄

yes

—

145

C₂²

—

146

C₂²

—

472

—

147

C₂²

—

472

—

148

C₂²

—

472

—

149

C₂²

—

236

—

150

C₂²

C₄×D₂₃₆

yes

characteristic

151

C₂²

#14273701

yes

characteristic

152

C₂²

#14273715

yes

characteristic

153

C₄

—

154

C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

155

C₄

C₄×D₂₃₆

yes

characteristic

156

C₄

#14273701

yes

characteristic

157

C₄

#14273724

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

472

—

161

C₂

#16837304

yes

characteristic

162

C₂

#16837672

yes

—

	subgroup	G/H
center	C₂²	#14273701
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#405	C₂
socle	C₂²×C₅₉	#121
hypercenter	#405	D₅₉
hypocenter	C₅₉	#1978
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#14273701

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#405

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
2. C₂×#14273663	C₂	semidirect
3. #16837406	C₂	semidirect
4. C₂×#14273643	C₂	semidirect
5. C₅₉×#405	C₂	semidirect
10. C₂×D₄₇₂	C₄	semidirect
86. #405	D₅₉	semidirect
87. C₅₉	#1978	semidirect

H ≅ G/N

type

2. C₂×#14273663

C₂

semidirect

3. #16837406

C₂

semidirect

4. C₂×#14273643

C₂

semidirect

5. C₅₉×#405

C₂

semidirect

10. C₂×D₄₇₂

C₄

semidirect

86. #405

D₅₉

semidirect

87. C₅₉

#1978

semidirect

Subgroups — #21425006

Conjugacy classes of subgroups

Complements