Group #27288831 — subgroups

Subgroups — #27288831

Order 13568.

402 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20274843	C₂	1	1	yes	maximal, characteristic, complemented
2	#20274715	C₂	1	2	yes	maximal
3	#20272599	C₂	1	2	yes	maximal, complemented
4	#20272550	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1954	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133138	—	2	2	no	—
8	#16810493	C₂²	1	2	yes	—
9	#16810386	—	2	2	no	—
10	#16810385	—	2	2	no	—
11	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	—	2	2	no	—
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	—	2	2	no	—
21	#13133138	—	4	4	no	—
22	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	C₄×D₄×C₅₃	—	2	2	no	—
27	C₄×D₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₂²⋊C₈×C₅₃	—	4	4	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₂⁴×C₅₃	—	2	1	no	—
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	2	1	no	—
37	C₂×D₄×C₅₃	—	2	2	no	—
38	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
39	C₂²×C₄×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂²⋊C₄×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	#41	1	2	yes	—
56	C₄²×C₅₃	#41	1	2	yes	—
57	C₄⋊Dic₅₃	—	4	2	no	—
58	C₄⋊Dic₅₃	—	4	2	no	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	C₄⋊Dic₅₃	—	4	2	no	—
61	C₄⋊Dic₅₃	—	8	2	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂³×C₅₃	—	2	2	no	—
66	C₂³×C₅₃	—	4	1	no	—
67	C₂³×C₅₃	—	4	2	no	—
68	C₂³×C₅₃	—	4	1	no	—
69	C₂³×C₅₃	—	4	2	no	—
70	C₂³×C₅₃	#121	1	1	yes	characteristic
71	D₄×C₅₃	—	2	4	no	—
72	D₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	4	1	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	—	8	1	no	—
78	C₂×C₄×C₅₃	—	8	1	no	—
79	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#123	1	2	yes	—
81	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#126	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#135	1	2	yes	—
85	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
86	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
87	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×Dic₅₃	—	8	2	no	—
89	C₂×Dic₅₃	—	8	2	no	—
90	C₂×Dic₅₃	—	8	2	no	—
91	C₈×C₅₃	—	4	2	no	—
92	C₈×C₅₃	—	8	1	no	—
93	#88322	—	8	2	no	—
94	#10622	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	4	2	no	—
96	C₂²×C₅₃	—	4	4	no	—
97	C₂²×C₅₃	—	8	1	no	—
98	C₂²×C₅₃	—	8	1	no	—
99	C₂²×C₅₃	—	8	2	no	—
100	C₂²×C₅₃	#397	1	1	yes	characteristic
101	C₂²×C₅₃	#463	1	2	yes	—
102	C₂²×C₅₃	#467	1	1	yes	characteristic
103	C₂²×C₅₃	#469	1	1	yes	characteristic
104	C₂²×C₅₃	#471	1	2	yes	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	8	1	no	—
107	C₄×C₅₃	#482	1	2	yes	—
108	C₄×C₅₃	#482	1	2	yes	—
109	C₄×C₅₃	#484	1	2	yes	—
110	C₄×C₅₃	#488	1	1	yes	characteristic
111	C₄×C₅₃	#490	1	1	yes	characteristic
112	Dic₅₃	—	16	2	no	—
113	#1954	D₅₃	1	1	yes	characteristic
114	#1834	—	53	1	no	—
115	#1802	—	53	2	no	—
116	#1296	—	53	2	no	—
117	#1277	—	53	1	no	—
118	C₂×C₅₃	—	8	2	no	—
119	C₂×C₅₃	#1683	1	2	yes	—
120	C₂×C₅₃	#1701	1	2	yes	—
121	C₂×C₅₃	#2073	1	1	yes	characteristic
122	C₂×C₅₃	#2076	1	1	yes	characteristic
123	C₂×C₅₃	#2079	1	1	yes	characteristic
124	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	53	2	no	—
126	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₂×#102	—	106	2	no	—
128	C₂×C₂²⋊C₈	—	2	2	no	—
129	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
130	#384	—	53	2	no	—
131	#340	—	106	2	no	—
132	#340	—	106	2	no	—
133	C₅₃	#10622	1	1	yes	characteristic, complemented, Sylow
134	C₂²×D₄	D₂₁₂	1	1	yes	characteristic
135	C₂³×C₄	—	2	1	no	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	106	2	no	—
138	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
139	C₄×D₄	—	2	2	no	—
140	C₄×D₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	106	2	no	—
142	C₂×C₄⋊C₄	—	106	2	no	—
143	C₂×C₄⋊C₄	—	106	2	no	—
144	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	106	2	no	—
149	C₄⋊C₈	—	106	2	no	—
150	#102	—	212	4	no	—
151	C₂²⋊C₈	—	4	4	no	—
152	C₄×C₈	—	2	2	no	—
153	C₂⁴	—	2	1	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	Dic₂₁₂	1	2	yes	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	4	1	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	212	2	no	—
164	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
165	C₂²×C₄	#11719945	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	—	212	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	424	2	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₄⋊C₄	—	212	2	no	—
179	C₄⋊C₄	—	212	2	no	—
180	C₄⋊C₄	—	212	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	#11719947	1	2	yes	—
183	C₄²	#11719947	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	4	1	no	—
186	C₂³	—	4	2	no	—
187	C₂³	—	4	1	no	—
188	C₂³	—	4	2	no	—
189	C₂³	#13133191	1	1	yes	characteristic
190	D₄	—	2	4	no	—
191	D₄	—	4	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	—	8	1	no	—
198	C₂×C₄	—	424	2	no	—
199	C₂×C₄	—	424	2	no	—
200	C₂×C₄	—	424	2	no	—
201	C₂×C₄	#13133198	1	1	yes	characteristic
202	C₂×C₄	#13133201	1	2	yes	—
203	C₂×C₄	#13133204	1	1	yes	characteristic
204	C₂×C₄	#13133210	1	2	yes	—
205	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
206	C₂×C₄	#13133237	1	2	yes	—
207	C₂×C₄	#13133239	1	1	yes	characteristic
208	C₂×C₄	#13133243	1	2	yes	—
209	C₂×C₄	#13133245	1	2	yes	—
210	C₈	—	4	2	no	—
211	C₈	—	8	1	no	—
212	C₈	—	424	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	4	no	—
215	C₂²	—	8	2	no	—
216	C₂²	—	8	1	no	—
217	C₂²	—	8	1	no	—
218	C₂²	#16810548	1	1	yes	characteristic
219	C₂²	#16810838	1	2	yes	—
220	C₂²	#16810864	1	1	yes	characteristic
221	C₂²	#16810876	1	1	yes	characteristic
222	C₂²	#16810886	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	8	1	no	—
225	C₄	—	848	2	no	—
226	C₄	#16810944	1	2	yes	—
227	C₄	#16810948	1	2	yes	—
228	C₄	#16810959	1	2	yes	—
229	C₄	#16810977	1	1	yes	characteristic
230	C₄	#16810987	1	1	yes	characteristic
231	C₂	—	8	2	no	—
232	C₂	#20274166	1	2	yes	—
233	C₂	#20274276	1	2	yes	—
234	C₂	#20276128	1	1	yes	characteristic
235	C₂	#20276143	1	1	yes	characteristic
236	C₂	#20276160	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274843

C₂

yes

maximal, characteristic, complemented

#20274715

C₂

yes

maximal

#20272599

C₂

yes

maximal, complemented

#20272550

C₂

yes

maximal, characteristic, complemented

C₅₃×#1954

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

C₂×#13133138

—

#16810493

C₂²

yes

—

#16810386

—

#16810385

—

C₂×C₄×D₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₄⋊#88322

—

C₄⋊#88322

—

#13133138

—

C₂²×D₄×C₅₃

D₄

yes

characteristic

C₂³×C₄×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#121

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#126

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#10622

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

100

C₂²×C₅₃

#397

yes

characteristic

101

C₂²×C₅₃

#463

yes

—

102

C₂²×C₅₃

#467

yes

characteristic

103

C₂²×C₅₃

#469

yes

characteristic

104

C₂²×C₅₃

#471

yes

—

105

C₄×C₅₃

—

106

C₄×C₅₃

—

107

C₄×C₅₃

#482

yes

—

108

C₄×C₅₃

#482

yes

—

109

C₄×C₅₃

#484

yes

—

110

C₄×C₅₃

#488

yes

characteristic

111

C₄×C₅₃

#490

yes

characteristic

112

Dic₅₃

—

113

#1954

D₅₃

yes

characteristic

114

#1834

—

115

#1802

—

116

#1296

—

117

#1277

—

118

C₂×C₅₃

—

119

C₂×C₅₃

#1683

yes

—

120

C₂×C₅₃

#1701

yes

—

121

C₂×C₅₃

#2073

yes

characteristic

122

C₂×C₅₃

#2076

yes

characteristic

123

C₂×C₅₃

#2079

yes

characteristic

124

C₂×C₄×D₄

C₂×D₅₃

yes

characteristic

125

C₂×C₄⋊C₈

—

126

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

127

C₂×#102

—

106

—

128

C₂×C₂²⋊C₈

—

129

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

130

#384

—

131

#340

—

106

—

132

#340

—

106

—

133

C₅₃

#10622

yes

characteristic, complemented, Sylow

134

C₂²×D₄

D₂₁₂

yes

characteristic

135

C₂³×C₄

—

136

C₂²×C₈

—

137

C₂²×C₈

—

106

—

138

C₂²×C₈

C₂²⋊D₅₃

yes

—

139

C₄×D₄

—

140

C₄×D₄

—

141

C₂×C₄⋊C₄

—

106

—

142

C₂×C₄⋊C₄

—

106

—

143

C₂×C₄⋊C₄

—

106

—

144

C₂×C₄⋊C₄

D₂₁₂

yes

characteristic

145

C₂×C₂²⋊C₄

—

146

C₂×C₄²

C₂²×D₅₃

yes

characteristic

147

C₄⋊C₈

—

148

C₄⋊C₈

—

106

—

149

C₄⋊C₈

—

106

—

150

#102

—

212

—

151

C₂²⋊C₈

—

152

C₄×C₈

—

153

C₂⁴

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

Dic₂₁₂

yes

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

212

—

164

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

165

C₂²×C₄

#11719945

yes

characteristic

166

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

212

—

173

C₂×C₈

—

212

—

174

C₂×C₈

—

212

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

424

—

177

C₄⋊C₄

—

212

—

178

C₄⋊C₄

—

212

—

179

C₄⋊C₄

—

212

—

180

C₄⋊C₄

—

212

—

181

C₂²⋊C₄

—

182

C₄²

#11719947

yes

—

183

C₄²

#11719947

yes

—

184

C₂³

—

185

C₂³

—

186

C₂³

—

187

C₂³

—

188

C₂³

—

189

C₂³

#13133191

yes

characteristic

190

D₄

—

191

D₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

424

—

199

C₂×C₄

—

424

—

200

C₂×C₄

—

424

—

201

C₂×C₄

#13133198

yes

characteristic

202

C₂×C₄

#13133201

yes

—

203

C₂×C₄

#13133204

yes

characteristic

204

C₂×C₄

#13133210

yes

—

205

C₂×C₄

C₂×Dic₂₁₂

yes

characteristic

206

C₂×C₄

#13133237

yes

—

207

C₂×C₄

#13133239

yes

characteristic

208

C₂×C₄

#13133243

yes

—

209

C₂×C₄

#13133245

yes

—

210

C₈

—

211

C₈

—

212

C₈

—

424

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

—

218

C₂²

#16810548

yes

characteristic

219

C₂²

#16810838

yes

—

220

C₂²

#16810864

yes

characteristic

221

C₂²

#16810876

yes

characteristic

222

C₂²

#16810886

yes

—

223

C₄

—

224

C₄

—

225

C₄

—

848

—

226

C₄

#16810944

yes

—

227

C₄

#16810948

yes

—

228

C₄

#16810959

yes

—

229

C₄

#16810977

yes

characteristic

230

C₄

#16810987

yes

characteristic

231

C₂

—

232

C₂

#20274166

yes

—

233

C₂

#20274276

yes

—

234

C₂

#20276128

yes

characteristic

235

C₂

#20276143

yes

characteristic

236

C₂

#20276160

yes

characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1954	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₄×C₅₃	D₄
hypercenter	#1954	D₅₃
hypocenter	C₅₃	#10622
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1954

subgroup

G/H

center

C₂³

#13133191

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1954

C₂

socle

C₂³×C₅₃

#121

prime radical

C₂²×D₄×C₅₃

D₄

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1954

N	H ≅ G/N	type
1. #20274843	C₂	semidirect
3. #20272599	C₂	semidirect
4. #20272550	C₂	semidirect
133. C₅₃	#10622	semidirect

H ≅ G/N

type

1. #20274843

C₂

semidirect

3. #20272599

C₂

semidirect

4. #20272550

C₂

semidirect

133. C₅₃

#10622

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274843	2
#20274715	2
#20272599	2
#20272550	2
C₅₃×#1954	2
#10622	53

subgroup

index

C₅₃×#1954

Subgroups — #27288831

Conjugacy classes of subgroups

Complements

Maximal subgroups