Group #27217801 — subgroups

Subgroups — C₅₃×#10289 #27217801

Order 13568.

614 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 130 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	C₅₃×#3226	C₂	1	2	yes	maximal
2	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1546	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#532	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#500	—	2	2	no	—
10	C₅₃×#486	—	2	2	no	—
11	C₅₃×#480	—	2	4	no	—
12	C₅₃×#474	—	2	4	no	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#423	C₂²	1	1	yes	characteristic
15	C₂×#104×C₅₃	—	2	2	no	—
16	C₅₃×#384	—	2	4	no	—
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#332	—	2	4	no	—
19	C₂×#41×C₅₃	—	2	1	no	—
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	1	no	—
22	C₄⋊Q₈×C₅₃	—	2	2	no	—
23	C₄⋊Q₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	4	4	no	—
27	C₄⋊Q₈×C₅₃	—	4	4	no	—
28	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
29	#127×C₅₃	D₄	1	1	yes	characteristic
30	#124×C₅₃	—	2	1	no	—
31	#121×C₅₃	—	2	2	no	—
32	C₄×D₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
36	C₂×C₄²×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#107×C₅₃	—	2	8	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
42	#104×C₅₃	—	4	4	no	—
43	#102×C₅₃	—	2	8	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	—	2	2	no	—
46	#41×C₅₃	—	4	2	no	—
47	#41×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	2	1	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	—	4	4	no	—
52	C₂×Q₈×C₅₃	—	4	4	no	—
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	2	1	no	—
55	C₂×D₄×C₅₃	—	4	1	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	M₁₆×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	4	1	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	4	16	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₂²⋊C₄×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	—	2	8	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	D₈	1	4	yes	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	2	yes	—
76	C₄²×C₅₃	#41	1	1	yes	characteristic
77	C₂³×C₅₃	—	4	1	no	—
78	C₂³×C₅₃	#120	1	1	yes	characteristic
79	Q₈×C₅₃	—	4	2	no	—
80	Q₈×C₅₃	—	8	4	no	—
81	Q₈×C₅₃	—	8	2	no	—
82	Q₈×C₅₃	—	8	2	no	—
83	D₄×C₅₃	—	4	4	no	—
84	D₄×C₅₃	—	4	2	no	—
85	D₄×C₅₃	—	8	2	no	—
86	C₂×C₄×C₅₃	—	2	8	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	—	4	8	no	—
92	C₂×C₄×C₅₃	—	8	1	no	—
93	C₂×C₄×C₅₃	—	8	1	no	—
94	C₂×C₄×C₅₃	#120	1	2	yes	—
95	C₂×C₄×C₅₃	#121	1	2	yes	—
96	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
100	C₈×C₅₃	—	4	4	no	—
101	C₈×C₅₃	—	8	1	no	—
102	#10289	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	8	2	no	—
105	C₂²×C₅₃	—	8	1	no	—
106	C₂²×C₅₃	#392	1	1	yes	characteristic
107	C₂²×C₅₃	#447	1	2	yes	—
108	C₄×C₅₃	—	2	8	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	8	4	no	—
111	C₄×C₅₃	—	8	1	no	—
112	C₄×C₅₃	#456	1	4	yes	—
113	C₄×C₅₃	#459	1	2	yes	—
114	C₄×C₅₃	#493	1	1	yes	characteristic
115	C₄×C₅₃	#496	1	1	yes	characteristic
116	#3226	C₂×C₅₃	1	2	yes	—
117	#2868	C₂×C₅₃	1	1	yes	characteristic, complemented
118	#1962	C₂×C₅₃	1	1	yes	characteristic
119	#1546	C₂×C₅₃	1	2	yes	complemented
120	#1276	C₂×C₅₃	1	1	yes	characteristic, complemented
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	—	16	1	no	—
123	C₂×C₅₃	#2005	1	2	yes	—
124	C₂×C₅₃	#2011	1	1	yes	characteristic
125	#532	C₂²×C₅₃	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	—	2	2	no	—
127	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
128	#500	—	2	2	no	—
129	#486	—	2	2	no	—
130	#480	—	2	4	no	—
131	#474	—	2	4	no	—
132	#431	C₂²×C₅₃	1	2	yes	—
133	#423	C₂²×C₅₃	1	1	yes	characteristic
134	C₂×#104	—	2	2	no	—
135	#384	—	2	4	no	—
136	C₄³	C₂²×C₅₃	1	1	yes	characteristic
137	#332	—	2	4	no	—
138	C₅₃	#10289	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#41	—	2	1	no	—
140	C₂²×Q₈	—	2	2	no	—
141	C₂×M₁₆	—	2	1	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	4	4	no	—
147	C₄⋊Q₈	—	4	4	no	—
148	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
149	#127	D₄×C₅₃	1	1	yes	characteristic
150	#124	—	2	1	no	—
151	#121	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	8	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	D₄×C₅₃	1	2	yes	—
158	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
159	#107	—	2	8	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
162	#104	—	4	4	no	—
163	#102	—	2	8	no	—
164	C₂²⋊C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	#41	—	4	2	no	—
167	#41	—	4	2	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	1	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	16	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₄²	—	2	8	no	—
192	C₄²	—	2	2	no	—
193	C₄²	D₈×C₅₃	1	4	yes	—
194	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₅₃	1	2	yes	—
196	C₄²	#41×C₅₃	1	1	yes	characteristic
197	C₂³	—	4	1	no	—
198	C₂³	#120×C₅₃	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	Q₈	—	8	2	no	—
201	Q₈	—	8	2	no	—
202	Q₈	—	8	4	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	D₄	—	8	2	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	#120×C₅₃	1	2	yes	—
215	C₂×C₄	#121×C₅₃	1	2	yes	—
216	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
217	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
219	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
220	C₈	—	4	4	no	—
221	C₈	—	8	1	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	1	no	—
225	C₂²	C₅₃×#392	1	1	yes	characteristic
226	C₂²	C₅₃×#447	1	2	yes	—
227	C₄	—	2	8	no	—
228	C₄	—	2	2	no	—
229	C₄	—	8	4	no	—
230	C₄	—	8	1	no	—
231	C₄	C₅₃×#456	1	4	yes	—
232	C₄	C₅₃×#459	1	2	yes	—
233	C₄	C₅₃×#493	1	1	yes	characteristic
234	C₄	C₅₃×#496	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	16	1	no	—
237	C₂	C₅₃×#2005	1	2	yes	—
238	C₂	C₅₃×#2011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3226

C₂

yes

maximal

C₅₃×#2868

C₂

yes

maximal, characteristic, complemented

C₅₃×#1962

C₂

yes

maximal, characteristic

C₅₃×#1546

C₂

yes

maximal, complemented

C₅₃×#1276

C₂

yes

maximal, characteristic, complemented

C₅₃×#532

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#500

—

C₅₃×#486

—

C₅₃×#480

—

C₅₃×#474

—

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₂×#104×C₅₃

—

C₅₃×#384

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#332

—

C₂×#41×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

characteristic

#127×C₅₃

D₄

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

#104×C₅₃

—

#102×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

100

C₈×C₅₃

—

101

C₈×C₅₃

—

102

#10289

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

103

C₂²×C₅₃

—

104

C₂²×C₅₃

—

105

C₂²×C₅₃

—

106

C₂²×C₅₃

#392

yes

characteristic

107

C₂²×C₅₃

#447

yes

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

#456

yes

—

113

C₄×C₅₃

#459

yes

—

114

C₄×C₅₃

#493

yes

characteristic

115

C₄×C₅₃

#496

yes

characteristic

116

#3226

C₂×C₅₃

yes

—

117

#2868

C₂×C₅₃

yes

characteristic, complemented

118

#1962

C₂×C₅₃

yes

characteristic

119

#1546

C₂×C₅₃

yes

complemented

120

#1276

C₂×C₅₃

yes

characteristic, complemented

121

C₂×C₅₃

—

122

C₂×C₅₃

—

123

C₂×C₅₃

#2005

yes

—

124

C₂×C₅₃

#2011

yes

characteristic

125

#532

C₂²×C₅₃

yes

characteristic

126

C₂×C₄⋊Q₈

—

127

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

—

128

#500

—

129

#486

—

130

#480

—

131

#474

—

132

#431

C₂²×C₅₃

yes

—

133

#423

C₂²×C₅₃

yes

characteristic

134

C₂×#104

—

135

#384

—

136

C₄³

C₂²×C₅₃

yes

characteristic

137

#332

—

138

C₅₃

#10289

yes

characteristic, direct product factor, Sylow

139

C₂×#41

—

140

C₂²×Q₈

—

141

C₂×M₁₆

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

—

146

C₄⋊Q₈

—

147

C₄⋊Q₈

—

148

C₄⋊Q₈

D₄×C₅₃

yes

characteristic

149

#127

D₄×C₅₃

yes

characteristic

150

#124

—

151

#121

—

152

C₄×D₄

—

153

C₂×C₄⋊C₄

—

154

C₂×C₄⋊C₄

—

155

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

156

C₂×C₄²

—

157

C₂×C₄²

D₄×C₅₃

yes

—

158

C₂×C₄²

C₂³×C₅₃

yes

characteristic

159

#107

—

160

C₄⋊C₈

—

161

C₄⋊C₈

D₄×C₅₃

yes

—

162

#104

—

163

#102

—

164

C₂²⋊C₈

—

165

C₄×C₈

—

166

#41

—

167

#41

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂×D₄

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

—

179

C₂²×C₄

C₂×D₄×C₅₃

yes

—

180

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

181

M₁₆

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₂²⋊C₄

—

191

C₄²

—

192

C₄²

—

193

C₄²

D₈×C₅₃

yes

—

194

C₄²

C₂×D₄×C₅₃

yes

characteristic

195

C₄²

C₂×D₄×C₅₃

yes

—

196

C₄²

#41×C₅₃

yes

characteristic

197

C₂³

—

198

C₂³

#120×C₅₃

yes

characteristic

199

Q₈

—

200

Q₈

—

201

Q₈

—

202

Q₈

—

203

D₄

—

204

D₄

—

205

D₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

#120×C₅₃

yes

—

215

C₂×C₄

#121×C₅₃

yes

—

216

C₂×C₄

#121×C₅₃

yes

characteristic

217

C₂×C₄

#127×C₅₃

yes

characteristic

218

C₂×C₄

C₂×D₈×C₅₃

yes

—

219

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

220

C₈

—

221

C₈

—

222

C₂²

—

223

C₂²

—

224

C₂²

—

225

C₂²

C₅₃×#392

yes

characteristic

226

C₂²

C₅₃×#447

yes

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

C₅₃×#456

yes

—

232

C₄

C₅₃×#459

yes

—

233

C₄

C₅₃×#493

yes

characteristic

234

C₄

C₅₃×#496

yes

characteristic

235

C₂

—

236

C₂

—

237

C₂

C₅₃×#2005

yes

—

238

C₂

C₅₃×#2011

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₅₃×#532

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#2868	C₂	semidirect
4. C₅₃×#1546	C₂	semidirect
5. C₅₃×#1276	C₂	semidirect
102. #10289	C₅₃	direct (N × H)
117. #2868	C₂×C₅₃	semidirect
119. #1546	C₂×C₅₃	semidirect
120. #1276	C₂×C₅₃	semidirect
138. C₅₃	#10289	direct (N × H)

H ≅ G/N

type

2. C₅₃×#2868

C₂

semidirect

4. C₅₃×#1546

C₂

semidirect

5. C₅₃×#1276

C₂

semidirect

102. #10289

C₅₃

direct (N × H)

117. #2868

C₂×C₅₃

semidirect

119. #1546

C₂×C₅₃

semidirect

120. #1276

C₂×C₅₃

semidirect

138. C₅₃

#10289

direct (N × H)

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

subgroup	index
C₅₃×#3226	2
C₅₃×#2868	2
C₅₃×#1962	2
C₅₃×#1546	2
C₅₃×#1276	2
#10289	53

subgroup

index

C₅₃×#3226

C₅₃×#2868

C₅₃×#1962

C₅₃×#1546

C₅₃×#1276

#10289

Subgroups — C₅₃×#10289 #27217801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10289 #27217801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10289 #27217801