Group #27216579 — subgroups

Subgroups — C₅₃×#9067 #27216579

Order 13568.

444 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 126 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×#3131	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2979	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1525	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1485	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1278	C₂	1	2	yes	maximal, complemented
6	C₅₃×#533	C₂²	1	1	yes	characteristic
7	C₅₃×#532	C₂²	1	1	yes	characteristic
8	C₅₃×#488	—	2	2	no	—
9	C₅₃×#487	C₄	1	2	yes	—
10	C₅₃×#486	C₄	1	2	yes	—
11	C₅₃×#436	—	2	2	no	—
12	C₈×D₄×C₅₃	—	2	2	no	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#416	—	2	2	no	—
15	C₅₃×#408	—	2	2	no	—
16	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#329	—	2	2	no	—
19	C₅₃×Q₈⋊C₈	—	2	2	no	—
20	C₅₃×Q₈⋊C₈	—	2	2	no	—
21	C₂×#41×C₅₃	D₄	1	2	yes	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#125×C₅₃	—	2	1	no	—
30	#124×C₅₃	D₄	1	2	yes	—
31	C₂²⋊Q₈×C₅₃	—	2	2	no	—
32	#121×C₅₃	—	2	2	no	—
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	C₄×D₄×C₅₃	—	2	2	no	—
35	#117×C₅₃	—	2	1	no	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
38	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
39	#103×C₅₃	—	2	4	no	—
40	#102×C₅₃	—	2	4	no	—
41	C₂²⋊C₈×C₅₃	—	2	4	no	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	#97×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	—	2	1	no	—
45	C₄×C₈×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	#41×C₅₃	—	4	2	no	—
49	#41×C₅₃	—	4	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	2	yes	—
58	M₁₆×C₅₃	—	2	4	no	—
59	M₁₆×C₅₃	—	4	2	no	—
60	M₁₆×C₅₃	—	8	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	4	1	no	—
65	C₂×C₈×C₅₃	—	4	1	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	8	2	no	—
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	4	2	no	—
72	C₂²⋊C₄×C₅₃	—	2	4	no	—
73	C₂²⋊C₄×C₅₃	—	4	1	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	2	yes	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	#123	1	1	yes	characteristic
80	Q₈×C₅₃	—	4	2	no	—
81	Q₈×C₅₃	—	4	2	no	—
82	Q₈×C₅₃	—	4	2	no	—
83	D₄×C₅₃	—	4	2	no	—
84	D₄×C₅₃	—	4	2	no	—
85	D₄×C₅₃	—	4	2	no	—
86	D₄×C₅₃	—	8	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	2	yes	—
99	C₈×C₅₃	—	4	2	no	—
100	C₈×C₅₃	—	4	4	no	—
101	C₈×C₅₃	—	4	2	no	—
102	C₈×C₅₃	—	4	2	no	—
103	#9067	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	4	4	no	—
107	C₂²×C₅₃	—	4	2	no	—
108	C₂²×C₅₃	#386	1	1	yes	characteristic
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	4	2	no	—
112	C₄×C₅₃	—	8	2	no	—
113	C₄×C₅₃	#443	1	2	yes	—
114	C₄×C₅₃	#444	1	2	yes	—
115	C₄×C₅₃	#455	1	1	yes	characteristic
116	C₄×C₅₃	#455	1	1	yes	characteristic
117	#3131	C₂×C₅₃	1	1	yes	characteristic
118	#2979	C₂×C₅₃	1	1	yes	characteristic
119	#1525	C₂×C₅₃	1	1	yes	characteristic
120	#1485	C₂×C₅₃	1	2	yes	complemented
121	#1278	C₂×C₅₃	1	2	yes	complemented
122	C₂×C₅₃	—	4	1	no	—
123	C₂×C₅₃	—	8	2	no	—
124	C₂×C₅₃	#1876	1	2	yes	—
125	C₂×C₅₃	#1885	1	1	yes	characteristic
126	#533	C₂²×C₅₃	1	1	yes	characteristic
127	#532	C₂²×C₅₃	1	1	yes	characteristic
128	#488	—	2	2	no	—
129	#487	C₄×C₅₃	1	2	yes	—
130	#486	C₄×C₅₃	1	2	yes	—
131	#436	—	2	2	no	—
132	C₈×D₄	—	2	2	no	—
133	#431	C₂²×C₅₃	1	2	yes	—
134	#416	—	2	2	no	—
135	#408	—	2	2	no	—
136	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
138	#329	—	2	2	no	—
139	Q₈⋊C₈	—	2	2	no	—
140	Q₈⋊C₈	—	2	2	no	—
141	C₅₃	#9067	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#41	D₄×C₅₃	1	2	yes	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
148	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
149	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
150	#125	—	2	1	no	—
151	#124	D₄×C₅₃	1	2	yes	—
152	C₂²⋊Q₈	—	2	2	no	—
153	#121	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×D₄	—	2	2	no	—
156	#117	—	2	1	no	—
157	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
158	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
159	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
160	#103	—	2	4	no	—
161	#102	—	2	4	no	—
162	C₂²⋊C₈	—	2	4	no	—
163	C₂²⋊C₈	—	2	2	no	—
164	#97	—	2	2	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
169	#41	—	4	2	no	—
170	#41	—	4	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	C₂²⋊C₄×C₅₃	1	2	yes	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	#41×C₅₃	1	2	yes	—
179	M₁₆	—	2	4	no	—
180	M₁₆	—	4	2	no	—
181	M₁₆	—	8	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	8	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	4	1	no	—
195	C₄²	—	4	1	no	—
196	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
197	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
198	C₄²	C₂×D₄×C₅₃	1	2	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	#123×C₅₃	1	1	yes	characteristic
201	Q₈	—	4	2	no	—
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	D₄	—	4	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	4	2	no	—
207	D₄	—	8	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
218	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
219	C₂×C₄	#121×C₅₃	1	2	yes	—
220	C₈	—	4	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	4	2	no	—
223	C₈	—	4	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₅₃×#386	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	8	2	no	—
233	C₄	C₅₃×#443	1	2	yes	—
234	C₄	C₅₃×#444	1	2	yes	—
235	C₄	C₅₃×#455	1	1	yes	characteristic
236	C₄	C₅₃×#455	1	1	yes	characteristic
237	C₂	—	4	1	no	—
238	C₂	—	8	2	no	—
239	C₂	C₅₃×#1876	1	2	yes	—
240	C₂	C₅₃×#1885	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3131

C₂

yes

maximal, characteristic

C₅₃×#2979

C₂

yes

maximal, characteristic

C₅₃×#1525

C₂

yes

maximal, characteristic

C₅₃×#1485

C₂

yes

maximal, complemented

C₅₃×#1278

C₂

yes

maximal, complemented

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#488

—

C₅₃×#487

C₄

yes

—

C₅₃×#486

C₄

yes

—

C₅₃×#436

—

C₈×D₄×C₅₃

—

C₅₃×#431

C₂²

yes

—

C₅₃×#416

—

C₅₃×#408

—

C₄×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#329

—

C₅₃×Q₈⋊C₈

—

C₅₃×Q₈⋊C₈

—

C₂×#41×C₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

characteristic

#127×C₅₃

C₂×C₄

yes

characteristic

#125×C₅₃

—

#124×C₅₃

D₄

yes

—

C₂²⋊Q₈×C₅₃

—

#121×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×D₄×C₅₃

—

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

D₄

yes

—

C₄⋊C₈×C₅₃

D₄

yes

—

#103×C₅₃

—

#102×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂²⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×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

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₈×C₅₃

—

100

C₈×C₅₃

—

101

C₈×C₅₃

—

102

C₈×C₅₃

—

103

#9067

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

104

C₂²×C₅₃

—

105

C₂²×C₅₃

—

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

#386

yes

characteristic

109

C₄×C₅₃

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

#443

yes

—

114

C₄×C₅₃

#444

yes

—

115

C₄×C₅₃

#455

yes

characteristic

116

C₄×C₅₃

#455

yes

characteristic

117

#3131

C₂×C₅₃

yes

characteristic

118

#2979

C₂×C₅₃

yes

characteristic

119

#1525

C₂×C₅₃

yes

characteristic

120

#1485

C₂×C₅₃

yes

complemented

121

#1278

C₂×C₅₃

yes

complemented

122

C₂×C₅₃

—

123

C₂×C₅₃

—

124

C₂×C₅₃

#1876

yes

—

125

C₂×C₅₃

#1885

yes

characteristic

126

#533

C₂²×C₅₃

yes

characteristic

127

#532

C₂²×C₅₃

yes

characteristic

128

#488

—

129

#487

C₄×C₅₃

yes

—

130

#486

C₄×C₅₃

yes

—

131

#436

—

132

C₈×D₄

—

133

#431

C₂²×C₅₃

yes

—

134

#416

—

135

#408

—

136

C₄×M₁₆

C₂²×C₅₃

yes

—

137

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

138

#329

—

139

Q₈⋊C₈

—

140

Q₈⋊C₈

—

141

C₅₃

#9067

yes

characteristic, direct product factor, Sylow

142

C₂×#41

D₄×C₅₃

yes

—

143

C₂×M₁₆

—

144

C₂×M₁₆

—

145

C₂²×C₈

—

146

C₂²×C₈

—

147

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

148

C₄⋊Q₈

C₂×C₄×C₅₃

yes

characteristic

149

#127

C₂×C₄×C₅₃

yes

characteristic

150

#125

—

151

#124

D₄×C₅₃

yes

—

152

C₂²⋊Q₈

—

153

#121

—

154

C₄×Q₈

—

155

C₄×D₄

—

156

#117

—

157

C₂×C₄²

C₂³×C₅₃

yes

characteristic

158

C₄⋊C₈

D₄×C₅₃

yes

—

159

C₄⋊C₈

D₄×C₅₃

yes

—

160

#103

—

161

#102

—

162

C₂²⋊C₈

—

163

C₂²⋊C₈

—

164

#97

—

165

C₄×C₈

—

166

C₄×C₈

—

167

C₄×C₈

—

168

C₄×C₈

C₂×C₄×C₅₃

yes

—

169

#41

—

170

#41

—

171

C₂×Q₈

—

172

C₂×Q₈

C₂²⋊C₄×C₅₃

yes

—

173

C₂×D₄

—

174

C₂×D₄

—

175

C₂×D₄

C₂²⋊C₄×C₅₃

yes

—

176

C₂²×C₄

—

177

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

178

C₂²×C₄

#41×C₅₃

yes

—

179

M₁₆

—

180

M₁₆

—

181

M₁₆

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₂×C₈

—

188

C₂×C₈

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₂²⋊C₄

—

194

C₂²⋊C₄

—

195

C₄²

—

196

C₄²

C₂²×C₄×C₅₃

yes

characteristic

197

C₄²

C₂×D₄×C₅₃

yes

characteristic

198

C₄²

C₂×D₄×C₅₃

yes

—

199

C₂³

—

200

C₂³

#123×C₅₃

yes

characteristic

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

D₄

—

205

D₄

—

206

D₄

—

207

D₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

217

C₂×C₄

C₄×D₄×C₅₃

yes

—

218

C₂×C₄

#120×C₅₃

yes

characteristic

219

C₂×C₄

#121×C₅₃

yes

—

220

C₈

—

221

C₈

—

222

C₈

—

223

C₈

—

224

C₂²

—

225

C₂²

—

226

C₂²

—

227

C₂²

—

228

C₂²

C₅₃×#386

yes

characteristic

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

C₅₃×#443

yes

—

234

C₄

C₅₃×#444

yes

—

235

C₄

C₅₃×#455

yes

characteristic

236

C₄

C₅₃×#455

yes

characteristic

237

C₂

—

238

C₂

—

239

C₂

C₅₃×#1876

yes

—

240

C₂

C₅₃×#1885

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#386
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#120

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle

C₂²×C₅₃

#386

prime radical

C₅₃×#532

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
4. C₅₃×#1485	C₂	semidirect
5. C₅₃×#1278	C₂	semidirect
103. #9067	C₅₃	direct (N × H)
120. #1485	C₂×C₅₃	semidirect
121. #1278	C₂×C₅₃	semidirect
141. C₅₃	#9067	direct (N × H)

H ≅ G/N

type

4. C₅₃×#1485

C₂

semidirect

5. C₅₃×#1278

C₂

semidirect

103. #9067

C₅₃

direct (N × H)

120. #1485

C₂×C₅₃

semidirect

121. #1278

C₂×C₅₃

semidirect

141. C₅₃

#9067

direct (N × H)

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

subgroup	index
C₅₃×#3131	2
C₅₃×#2979	2
C₅₃×#1525	2
C₅₃×#1485	2
C₅₃×#1278	2
#9067	53

subgroup

index

C₅₃×#3131

C₅₃×#2979

C₅₃×#1525

C₅₃×#1485

C₅₃×#1278

#9067

Subgroups — C₅₃×#9067 #27216579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9067 #27216579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9067 #27216579