Subgroups — C₂×C₅₃×#2293 #27241532

Order 13568.

3790 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2293	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#390	C₂	1	2	yes	maximal
5	C₂×C₅₃×#389	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#388	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₅₃	C₄	1	16	yes	complemented
13	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
14	C₂×#117×C₅₃	—	2	16	no	—
15	C₂×#117×C₅₃	C₂²	1	16	yes	complemented
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
20	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#390	C₂²	1	16	yes	—
29	C₅₃×#389	C₂²	1	16	yes	complemented
30	C₅₃×#388	C₂²	1	32	yes	complemented
31	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
32	C₄×C₂²⋊C₄×C₅₃	C₂²	1	16	yes	complemented
33	C₂×#95×C₅₃	C₂²	1	4	yes	—
34	C₂×#95×C₅₃	C₂²	1	2	yes	—
35	C₄³×C₅₃	C₂²	1	8	yes	complemented
36	C₂⁵×C₅₃	C₂³	1	2	yes	—
37	C₂²×D₄×C₅₃	—	2	8	no	—
38	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	—
39	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
40	C₂³×C₄×C₅₃	—	2	8	no	—
41	C₂³×C₄×C₅₃	—	2	8	no	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×D₄×C₅₃	C₂×C₄	1	64	yes	complemented
52	#117×C₅₃	—	2	64	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
55	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
59	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
61	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
62	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
63	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
64	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
65	C₂×C₄²×C₅₃	—	2	8	no	—
66	C₂×C₄²×C₅₃	—	2	8	no	—
67	C₂×C₄²×C₅₃	—	2	8	no	—
68	C₂×C₄²×C₅₃	—	2	4	no	—
69	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
70	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
71	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
72	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
74	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
75	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
76	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
77	#95×C₅₃	C₂³	1	16	yes	—
78	#95×C₅₃	C₂³	1	8	yes	—
79	C₂⁴×C₅₃	—	2	4	no	—
80	C₂⁴×C₅₃	—	2	8	no	—
81	C₂⁴×C₅₃	—	2	4	no	—
82	C₂⁴×C₅₃	—	2	8	no	—
83	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
84	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
85	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂×D₄×C₅₃	—	2	32	no	—
87	C₂×D₄×C₅₃	—	2	16	no	—
88	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
89	C₂²×C₄×C₅₃	—	2	4	no	—
90	C₂²×C₄×C₅₃	—	2	4	no	—
91	C₂²×C₄×C₅₃	—	2	4	no	—
92	C₂²×C₄×C₅₃	—	2	4	no	—
93	C₂²×C₄×C₅₃	—	2	4	no	—
94	C₂²×C₄×C₅₃	—	2	4	no	—
95	C₂²×C₄×C₅₃	—	2	4	no	—
96	C₂²×C₄×C₅₃	—	2	64	no	—
97	C₂²×C₄×C₅₃	—	2	16	no	—
98	C₂²×C₄×C₅₃	—	2	16	no	—
99	C₂²×C₄×C₅₃	—	2	8	no	—
100	C₂²×C₄×C₅₃	—	2	8	no	—
101	C₂²×C₄×C₅₃	—	2	8	no	—
102	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
104	C₂²×C₄×C₅₃	C₂²×C₄	1	32	yes	—
105	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
106	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
109	C₂²×C₄×C₅₃	#41	1	8	yes	—
110	C₂²×C₄×C₅₃	#41	1	8	yes	—
111	C₂²×C₄×C₅₃	#41	1	4	yes	—
112	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
113	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
114	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
116	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
119	C₄⋊C₄×C₅₃	—	2	32	no	—
120	C₄⋊C₄×C₅₃	—	2	16	no	—
121	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
122	C₂²⋊C₄×C₅₃	—	2	32	no	—
123	C₂²⋊C₄×C₅₃	—	2	32	no	—
124	C₂²⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
125	C₄²×C₅₃	—	2	16	no	—
126	C₄²×C₅₃	—	2	16	no	—
127	C₄²×C₅₃	—	2	16	no	—
128	C₄²×C₅₃	—	2	8	no	—
129	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
130	C₄²×C₅₃	C₂×D₄	1	16	yes	complemented
131	C₂³×C₅₃	—	2	16	no	—
132	C₂³×C₅₃	—	2	16	no	—
133	C₂³×C₅₃	—	2	16	no	—
134	C₂³×C₅₃	—	2	4	no	—
135	C₂³×C₅₃	—	2	4	no	—
136	C₂³×C₅₃	—	2	8	no	—
137	C₂³×C₅₃	—	2	8	no	—
138	C₂³×C₅₃	—	4	16	no	—
139	C₂³×C₅₃	—	4	16	no	—
140	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
141	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
142	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
143	C₂³×C₅₃	C₂×#41	1	4	yes	—
144	C₂³×C₅₃	C₂×#41	1	2	yes	—
145	C₂³×C₅₃	C₂×#41	1	4	yes	—
146	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
147	D₄×C₅₃	—	2	32	no	—
148	C₂×C₄×C₅₃	—	2	16	no	—
149	C₂×C₄×C₅₃	—	2	8	no	—
150	C₂×C₄×C₅₃	—	2	16	no	—
151	C₂×C₄×C₅₃	—	2	16	no	—
152	C₂×C₄×C₅₃	—	2	4	no	—
153	C₂×C₄×C₅₃	—	2	4	no	—
154	C₂×C₄×C₅₃	—	2	4	no	—
155	C₂×C₄×C₅₃	—	2	4	no	—
156	C₂×C₄×C₅₃	—	2	4	no	—
157	C₂×C₄×C₅₃	—	2	4	no	—
158	C₂×C₄×C₅₃	—	2	4	no	—
159	C₂×C₄×C₅₃	—	2	64	no	—
160	C₂×C₄×C₅₃	—	2	8	no	—
161	C₂×C₄×C₅₃	—	2	8	no	—
162	C₂×C₄×C₅₃	—	2	8	no	—
163	C₂×C₄×C₅₃	—	4	16	no	—
164	C₂×C₄×C₅₃	#117	1	16	yes	complemented
165	C₂×C₄×C₅₃	C₄×D₄	1	16	yes	—
166	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
167	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
168	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
169	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
170	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
171	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
172	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
173	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
174	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
175	C₂×#2293	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂²×C₅₃	—	2	8	no	—
177	C₂²×C₅₃	—	2	8	no	—
178	C₂²×C₅₃	—	2	8	no	—
179	C₂²×C₅₃	—	4	8	no	—
180	C₂²×C₅₃	—	4	8	no	—
181	C₂²×C₅₃	—	4	32	no	—
182	C₂²×C₅₃	C₂×#117	1	4	yes	—
183	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
184	C₂²×C₅₃	C₄×#41	1	4	yes	—
185	C₂²×C₅₃	#529	1	8	yes	—
186	C₂²×C₅₃	#532	1	8	yes	—
187	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
188	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
189	C₂²×C₅₃	C₂²×#41	1	2	yes	—
190	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
191	C₂²×C₅₃	C₂²×#41	1	2	yes	—
192	C₄×C₅₃	—	2	8	no	—
193	C₄×C₅₃	—	2	4	no	—
194	C₄×C₅₃	—	2	8	no	—
195	C₄×C₅₃	—	2	8	no	—
196	C₄×C₅₃	—	4	8	no	—
197	C₄×C₅₃	C₂×#117	1	8	yes	complemented
198	C₄×C₅₃	C₂×C₄×D₄	1	8	yes	—
199	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
200	C₂²×#117	C₂×C₅₃	1	2	yes	complemented
201	#2293	C₂×C₅₃	1	16	yes	direct product factor
202	C₂×#390	C₂×C₅₃	1	2	yes	—
203	C₂×#389	C₂×C₅₃	1	2	yes	complemented
204	C₂×#388	C₂×C₅₃	1	4	yes	complemented
205	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
206	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
207	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₂×C₅₃	—	4	8	no	—
209	C₂×C₅₃	#2293	1	8	yes	direct product factor
210	C₂×C₅₃	C₂²×#117	1	1	yes	characteristic
211	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
212	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
213	C₂×C₅₃	C₂×#529	1	2	yes	—
214	C₂×C₅₃	C₂×#532	1	2	yes	—
215	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
216	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
217	C₂×C₄×D₄	C₄×C₅₃	1	16	yes	complemented
218	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
219	C₂×#117	—	2	16	no	—
220	C₂×#117	C₂²×C₅₃	1	16	yes	complemented
221	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
222	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
223	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
224	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
225	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
226	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
227	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
228	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
229	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
230	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
231	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
232	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
233	#390	C₂²×C₅₃	1	16	yes	—
234	#389	C₂²×C₅₃	1	16	yes	complemented
235	#388	C₂²×C₅₃	1	32	yes	complemented
236	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
237	C₄×C₂²⋊C₄	C₂²×C₅₃	1	16	yes	complemented
238	C₂×#95	C₂²×C₅₃	1	4	yes	—
239	C₂×#95	C₂²×C₅₃	1	2	yes	—
240	C₄³	C₂²×C₅₃	1	8	yes	complemented
241	C₅₃	C₂×#2293	1	1	yes	characteristic, direct product factor, Sylow
242	C₂⁵	C₂³×C₅₃	1	2	yes	—
243	C₂²×D₄	—	2	8	no	—
244	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	—
245	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
246	C₂³×C₄	—	2	8	no	—
247	C₂³×C₄	—	2	8	no	—
248	C₂³×C₄	C₂×C₄×C₅₃	1	16	yes	—
249	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
250	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
251	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
252	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
253	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
254	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
255	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
256	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
257	C₄×D₄	C₂×C₄×C₅₃	1	64	yes	complemented
258	#117	—	2	64	no	—
259	C₂×C₄⋊C₄	—	2	16	no	—
260	C₂×C₄⋊C₄	—	2	8	no	—
261	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
262	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
263	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
264	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
265	C₂×C₂²⋊C₄	—	2	16	no	—
266	C₂×C₂²⋊C₄	—	2	16	no	—
267	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	16	yes	—
268	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
269	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
270	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
271	C₂×C₄²	—	2	8	no	—
272	C₂×C₄²	—	2	8	no	—
273	C₂×C₄²	—	2	8	no	—
274	C₂×C₄²	—	2	4	no	—
275	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
276	C₂×C₄²	D₄×C₅₃	1	8	yes	complemented
277	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
278	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
279	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
280	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
281	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
282	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
283	#95	C₂³×C₅₃	1	16	yes	—
284	#95	C₂³×C₅₃	1	8	yes	—
285	C₂⁴	—	2	4	no	—
286	C₂⁴	—	2	8	no	—
287	C₂⁴	—	2	4	no	—
288	C₂⁴	—	2	8	no	—
289	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	—
290	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
291	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
292	C₂×D₄	—	2	32	no	—
293	C₂×D₄	—	2	16	no	—
294	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	64	no	—
303	C₂²×C₄	—	2	16	no	—
304	C₂²×C₄	—	2	16	no	—
305	C₂²×C₄	—	2	8	no	—
306	C₂²×C₄	—	2	8	no	—
307	C₂²×C₄	—	2	8	no	—
308	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
309	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
310	C₂²×C₄	C₂²×C₄×C₅₃	1	32	yes	—
311	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
312	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
313	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
314	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
315	C₂²×C₄	#41×C₅₃	1	8	yes	—
316	C₂²×C₄	#41×C₅₃	1	8	yes	—
317	C₂²×C₄	#41×C₅₃	1	4	yes	—
318	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
319	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
320	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
321	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
322	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
323	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
324	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
325	C₄⋊C₄	—	2	32	no	—
326	C₄⋊C₄	—	2	16	no	—
327	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
328	C₂²⋊C₄	—	2	32	no	—
329	C₂²⋊C₄	—	2	32	no	—
330	C₂²⋊C₄	C₂²×C₄×C₅₃	1	32	yes	—
331	C₄²	—	2	16	no	—
332	C₄²	—	2	16	no	—
333	C₄²	—	2	16	no	—
334	C₄²	—	2	8	no	—
335	C₄²	C₂²×C₄×C₅₃	1	16	yes	—
336	C₄²	C₂×D₄×C₅₃	1	16	yes	complemented
337	C₂³	—	2	16	no	—
338	C₂³	—	2	16	no	—
339	C₂³	—	2	16	no	—
340	C₂³	—	2	4	no	—
341	C₂³	—	2	8	no	—
342	C₂³	—	2	4	no	—
343	C₂³	—	2	8	no	—
344	C₂³	—	4	16	no	—
345	C₂³	—	4	16	no	—
346	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
347	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
348	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
349	C₂³	C₂×#41×C₅₃	1	4	yes	—
350	C₂³	C₂×#41×C₅₃	1	2	yes	—
351	C₂³	C₂×#41×C₅₃	1	4	yes	—
352	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
353	D₄	—	2	32	no	—
354	C₂×C₄	—	2	16	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	16	no	—
357	C₂×C₄	—	2	16	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	64	no	—
366	C₂×C₄	—	2	8	no	—
367	C₂×C₄	—	2	8	no	—
368	C₂×C₄	—	2	8	no	—
369	C₂×C₄	—	4	16	no	—
370	C₂×C₄	#117×C₅₃	1	16	yes	complemented
371	C₂×C₄	C₄×D₄×C₅₃	1	16	yes	—
372	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
373	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
374	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
375	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
376	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
377	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
378	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
379	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
380	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
381	C₂²	—	2	8	no	—
382	C₂²	—	2	8	no	—
383	C₂²	—	2	8	no	—
384	C₂²	—	4	32	no	—
385	C₂²	—	4	8	no	—
386	C₂²	—	4	8	no	—
387	C₂²	C₂×#117×C₅₃	1	4	yes	—
388	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
389	C₂²	C₄×#41×C₅₃	1	4	yes	—
390	C₂²	C₅₃×#529	1	8	yes	—
391	C₂²	C₅₃×#532	1	8	yes	—
392	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
393	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
394	C₂²	C₂²×#41×C₅₃	1	2	yes	—
395	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
396	C₂²	C₂²×#41×C₅₃	1	2	yes	—
397	C₄	—	2	8	no	—
398	C₄	—	2	4	no	—
399	C₄	—	2	8	no	—
400	C₄	—	2	8	no	—
401	C₄	—	4	8	no	—
402	C₄	C₂×#117×C₅₃	1	8	yes	complemented
403	C₄	C₂×C₄×D₄×C₅₃	1	8	yes	—
404	C₂	—	4	8	no	—
405	C₂	C₅₃×#2293	1	8	yes	direct product factor
406	C₂	C₂²×#117×C₅₃	1	1	yes	characteristic
407	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
408	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
409	C₂	C₂×C₅₃×#529	1	2	yes	—
410	C₂	C₂×C₅₃×#532	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅₃	C₂³
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×#117×C₅₃	C₂	semidirect
3. C₅₃×#2293	C₂	direct (N × H)
5. C₂×C₅₃×#389	C₂	semidirect
6. C₂×C₅₃×#388	C₂	semidirect
7. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
8. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
9. C₂×C₄³×C₅₃	C₂	semidirect
12. C₂×C₄×D₄×C₅₃	C₄	semidirect
15. C₂×#117×C₅₃	C₂²	semidirect
29. C₅₃×#389	C₂²	semidirect
30. C₅₃×#388	C₂²	semidirect
31. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
32. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
35. C₄³×C₅₃	C₂²	semidirect
51. C₄×D₄×C₅₃	C₂×C₄	semidirect
70. C₂×C₄²×C₅₃	D₄	semidirect
130. C₄²×C₅₃	C₂×D₄	semidirect
164. C₂×C₄×C₅₃	#117	semidirect
175. C₂×#2293	C₅₃	direct (N × H)
197. C₄×C₅₃	C₂×#117	semidirect
200. C₂²×#117	C₂×C₅₃	semidirect
201. #2293	C₂×C₅₃	direct (N × H)
203. C₂×#389	C₂×C₅₃	semidirect
204. C₂×#388	C₂×C₅₃	semidirect
205. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
206. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
207. C₂×C₄³	C₂×C₅₃	semidirect
209. C₂×C₅₃	#2293	direct (N × H)
217. C₂×C₄×D₄	C₄×C₅₃	semidirect
220. C₂×#117	C₂²×C₅₃	semidirect
234. #389	C₂²×C₅₃	semidirect
235. #388	C₂²×C₅₃	semidirect
236. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
237. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
240. C₄³	C₂²×C₅₃	semidirect
241. C₅₃	C₂×#2293	direct (N × H)
257. C₄×D₄	C₂×C₄×C₅₃	semidirect
276. C₂×C₄²	D₄×C₅₃	semidirect
336. C₄²	C₂×D₄×C₅₃	semidirect
370. C₂×C₄	#117×C₅₃	semidirect
402. C₄	C₂×#117×C₅₃	semidirect
405. C₂	C₅₃×#2293	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₅₃	2
C₂²×#117×C₅₃	2
C₅₃×#2293	2
C₂×C₅₃×#390	2
C₂×C₅₃×#389	2
C₂×C₅₃×#388	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×C₄³×C₅₃	2
C₂×#2293	53

Subgroups — C2×C53×#2293 #27241532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2293 #27241532