Subgroups — C₂×C₅₃×#2821 #27244198

Order 13568.

2370 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#126×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2821	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#400	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#399	C₂	1	2	yes	maximal
6	C₂×C₅₃×#398	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#397	C₂	1	2	yes	maximal
8	C₂×C₅₃×#395	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#393	C₂	1	2	yes	maximal
10	C₂×C₅₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₅₃	—	2	8	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
14	C₂×#126×C₅₃	—	2	8	no	—
15	C₂×#126×C₅₃	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#400	C₂²	1	16	yes	complemented
28	C₅₃×#399	C₂²	1	16	yes	—
29	C₅₃×#398	C₂²	1	16	yes	complemented
30	C₅₃×#397	C₂²	1	16	yes	—
31	C₅₃×#395	C₂²	1	16	yes	complemented
32	C₅₃×#393	C₂²	1	16	yes	—
33	C₅₃×#383	C₂²	1	8	yes	complemented
34	C₂×#95×C₅₃	C₂²	1	2	yes	—
35	C₂×#95×C₅₃	C₂²	1	2	yes	—
36	C₂×#95×C₅₃	C₂²	1	2	yes	—
37	C₂×#95×C₅₃	C₂²	1	4	yes	—
38	C₂×#95×C₅₃	C₂²	1	4	yes	—
39	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×Q₈×C₅₃	—	2	8	no	—
41	C₂²×Q₈×C₅₃	—	4	8	no	—
42	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	—	2	32	no	—
52	#126×C₅₃	—	2	32	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
58	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
68	C₂×C₂²⋊C₄×C₅₃	—	4	8	no	—
69	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
70	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
71	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
72	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
73	C₂×C₄²×C₅₃	D₄	1	8	yes	—
74	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
75	#95×C₅₃	C₂³	1	8	yes	—
76	#95×C₅₃	C₂³	1	8	yes	—
77	#95×C₅₃	C₂³	1	8	yes	—
78	#95×C₅₃	C₂³	1	16	yes	—
79	#95×C₅₃	C₂³	1	16	yes	—
80	C₂⁴×C₅₃	—	2	2	no	—
81	C₂⁴×C₅₃	—	2	4	no	—
82	C₂⁴×C₅₃	—	2	4	no	—
83	C₂⁴×C₅₃	—	2	4	no	—
84	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
85	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂×Q₈×C₅₃	—	2	16	no	—
87	C₂×Q₈×C₅₃	—	4	16	no	—
88	C₂×Q₈×C₅₃	—	4	32	no	—
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	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	8	no	—
96	C₂²×C₄×C₅₃	—	2	8	no	—
97	C₂²×C₄×C₅₃	—	2	8	no	—
98	C₂²×C₄×C₅₃	—	2	4	no	—
99	C₂²×C₄×C₅₃	—	2	8	no	—
100	C₂²×C₄×C₅₃	—	2	4	no	—
101	C₂²×C₄×C₅₃	—	2	4	no	—
102	C₂²×C₄×C₅₃	—	2	8	no	—
103	C₂²×C₄×C₅₃	—	2	8	no	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
107	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
108	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
114	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
115	C₄⋊C₄×C₅₃	—	2	8	no	—
116	C₄⋊C₄×C₅₃	—	2	8	no	—
117	C₄⋊C₄×C₅₃	—	2	8	no	—
118	C₄⋊C₄×C₅₃	—	2	16	no	—
119	C₄⋊C₄×C₅₃	—	2	16	no	—
120	C₄⋊C₄×C₅₃	—	4	16	no	—
121	C₂²⋊C₄×C₅₃	—	2	8	no	—
122	C₂²⋊C₄×C₅₃	—	2	8	no	—
123	C₂²⋊C₄×C₅₃	—	2	8	no	—
124	C₂²⋊C₄×C₅₃	—	4	16	no	—
125	C₄²×C₅₃	C₂×D₄	1	16	yes	—
126	C₂³×C₅₃	—	2	2	no	—
127	C₂³×C₅₃	—	2	4	no	—
128	C₂³×C₅₃	—	2	4	no	—
129	C₂³×C₅₃	—	2	4	no	—
130	C₂³×C₅₃	—	4	4	no	—
131	C₂³×C₅₃	—	4	4	no	—
132	C₂³×C₅₃	—	4	4	no	—
133	C₂³×C₅₃	—	4	16	no	—
134	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
135	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
136	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
137	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
138	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
139	Q₈×C₅₃	—	4	32	no	—
140	C₂×C₄×C₅₃	—	2	8	no	—
141	C₂×C₄×C₅₃	—	2	8	no	—
142	C₂×C₄×C₅₃	—	2	8	no	—
143	C₂×C₄×C₅₃	—	2	4	no	—
144	C₂×C₄×C₅₃	—	2	4	no	—
145	C₂×C₄×C₅₃	—	2	4	no	—
146	C₂×C₄×C₅₃	—	2	8	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	8	no	—
151	C₂×C₄×C₅₃	—	2	8	no	—
152	C₂×C₄×C₅₃	—	2	8	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	8	no	—
157	C₂×C₄×C₅₃	—	2	8	no	—
158	C₂×C₄×C₅₃	—	4	4	no	—
159	C₂×C₄×C₅₃	—	4	16	no	—
160	C₂×C₄×C₅₃	—	4	4	no	—
161	C₂×C₄×C₅₃	—	4	4	no	—
162	C₂×C₄×C₅₃	—	4	16	no	—
163	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
164	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
165	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
166	C₂×#2821	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	—	4	2	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	8	no	—
171	C₂²×C₅₃	—	8	8	no	—
172	C₂²×C₅₃	#535	1	4	yes	—
173	C₂²×C₅₃	#536	1	4	yes	—
174	C₂²×C₅₃	#536	1	4	yes	—
175	C₂²×C₅₃	#563	1	16	yes	—
176	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
177	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
178	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
179	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
180	C₄×C₅₃	—	2	4	no	—
181	C₄×C₅₃	—	2	4	no	—
182	C₄×C₅₃	—	2	4	no	—
183	C₄×C₅₃	—	4	2	no	—
184	C₄×C₅₃	—	4	8	no	—
185	C₄×C₅₃	—	4	2	no	—
186	C₄×C₅₃	—	4	2	no	—
187	C₄×C₅₃	—	4	8	no	—
188	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
189	C₂²×#126	C₂×C₅₃	1	1	yes	characteristic
190	#2821	C₂×C₅₃	1	16	yes	direct product factor
191	C₂×#400	C₂×C₅₃	1	2	yes	complemented
192	C₂×#399	C₂×C₅₃	1	2	yes	—
193	C₂×#398	C₂×C₅₃	1	2	yes	complemented
194	C₂×#397	C₂×C₅₃	1	2	yes	—
195	C₂×#395	C₂×C₅₃	1	2	yes	complemented
196	C₂×#393	C₂×C₅₃	1	2	yes	—
197	C₂×#383	C₂×C₅₃	1	1	yes	characteristic, complemented
198	C₂×C₅₃	—	8	2	no	—
199	C₂×C₅₃	#2821	1	8	yes	direct product factor
200	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
201	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
203	C₂×C₅₃	C₂×#563	1	4	yes	—
204	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
205	C₂×C₄⋊Q₈	—	2	8	no	—
206	C₂×C₄⋊Q₈	C₂²×C₅₃	1	8	yes	complemented
207	C₂×#126	—	2	8	no	—
208	C₂×#126	C₂²×C₅₃	1	8	yes	—
209	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
210	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
211	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
212	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
213	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
214	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
215	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
216	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
217	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
218	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
219	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
220	#400	C₂²×C₅₃	1	16	yes	complemented
221	#399	C₂²×C₅₃	1	16	yes	—
222	#398	C₂²×C₅₃	1	16	yes	complemented
223	#397	C₂²×C₅₃	1	16	yes	—
224	#395	C₂²×C₅₃	1	16	yes	complemented
225	#393	C₂²×C₅₃	1	16	yes	—
226	#383	C₂²×C₅₃	1	8	yes	complemented
227	C₂×#95	C₂²×C₅₃	1	2	yes	—
228	C₂×#95	C₂²×C₅₃	1	4	yes	—
229	C₂×#95	C₂²×C₅₃	1	4	yes	—
230	C₂×#95	C₂²×C₅₃	1	2	yes	—
231	C₂×#95	C₂²×C₅₃	1	2	yes	—
232	C₅₃	C₂×#2821	1	1	yes	characteristic, direct product factor, Sylow
233	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
234	C₂²×Q₈	—	2	8	no	—
235	C₂²×Q₈	—	4	8	no	—
236	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
237	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
238	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
239	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
240	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
241	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
242	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
243	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
244	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
245	C₄⋊Q₈	—	2	32	no	—
246	#126	—	2	32	no	—
247	C₂×C₄⋊C₄	—	2	4	no	—
248	C₂×C₄⋊C₄	—	2	4	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	—	2	8	no	—
251	C₂×C₄⋊C₄	—	2	8	no	—
252	C₂×C₄⋊C₄	—	4	8	no	—
253	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
254	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
255	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
256	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
257	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
258	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
259	C₂×C₂²⋊C₄	—	2	4	no	—
260	C₂×C₂²⋊C₄	—	2	4	no	—
261	C₂×C₂²⋊C₄	—	2	4	no	—
262	C₂×C₂²⋊C₄	—	4	8	no	—
263	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
264	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
265	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
266	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
267	C₂×C₄²	D₄×C₅₃	1	8	yes	—
268	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
269	#95	C₂³×C₅₃	1	8	yes	—
270	#95	C₂³×C₅₃	1	8	yes	—
271	#95	C₂³×C₅₃	1	8	yes	—
272	#95	C₂³×C₅₃	1	16	yes	—
273	#95	C₂³×C₅₃	1	16	yes	—
274	C₂⁴	—	2	2	no	—
275	C₂⁴	—	2	4	no	—
276	C₂⁴	—	2	4	no	—
277	C₂⁴	—	2	4	no	—
278	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
279	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
280	C₂×Q₈	—	2	16	no	—
281	C₂×Q₈	—	4	16	no	—
282	C₂×Q₈	—	4	32	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	8	no	—
290	C₂²×C₄	—	2	8	no	—
291	C₂²×C₄	—	2	8	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	8	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	8	no	—
297	C₂²×C₄	—	2	8	no	—
298	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
299	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
300	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
301	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
302	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
303	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
304	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
305	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
306	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
307	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
308	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
309	C₄⋊C₄	—	2	8	no	—
310	C₄⋊C₄	—	2	8	no	—
311	C₄⋊C₄	—	2	8	no	—
312	C₄⋊C₄	—	2	16	no	—
313	C₄⋊C₄	—	2	16	no	—
314	C₄⋊C₄	—	4	16	no	—
315	C₂²⋊C₄	—	2	8	no	—
316	C₂²⋊C₄	—	2	8	no	—
317	C₂²⋊C₄	—	2	8	no	—
318	C₂²⋊C₄	—	4	16	no	—
319	C₄²	C₂×D₄×C₅₃	1	16	yes	—
320	C₂³	—	2	2	no	—
321	C₂³	—	2	4	no	—
322	C₂³	—	2	4	no	—
323	C₂³	—	2	4	no	—
324	C₂³	—	4	4	no	—
325	C₂³	—	4	4	no	—
326	C₂³	—	4	4	no	—
327	C₂³	—	4	16	no	—
328	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
329	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
330	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
331	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
332	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
333	Q₈	—	4	32	no	—
334	C₂×C₄	—	2	8	no	—
335	C₂×C₄	—	2	8	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	8	no	—
344	C₂×C₄	—	2	8	no	—
345	C₂×C₄	—	2	8	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	8	no	—
351	C₂×C₄	—	2	8	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	4	16	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	16	no	—
357	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
358	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
359	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	8	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	8	no	—
365	C₂²	C₅₃×#535	1	4	yes	—
366	C₂²	C₅₃×#536	1	4	yes	—
367	C₂²	C₅₃×#536	1	4	yes	—
368	C₂²	C₅₃×#563	1	16	yes	—
369	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
370	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
371	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
372	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
373	C₄	—	2	4	no	—
374	C₄	—	2	4	no	—
375	C₄	—	2	4	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	8	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	8	no	—
381	C₂	—	8	2	no	—
382	C₂	C₅₃×#2821	1	8	yes	direct product factor
383	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
384	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
385	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
386	C₂	C₂×C₅₃×#563	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
3. C₅₃×#2821	C₂	direct (N × H)
4. C₂×C₅₃×#400	C₂	semidirect
6. C₂×C₅₃×#398	C₂	semidirect
8. C₂×C₅₃×#395	C₂	semidirect
10. C₂×C₅₃×#383	C₂	semidirect
13. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
27. C₅₃×#400	C₂²	semidirect
29. C₅₃×#398	C₂²	semidirect
31. C₅₃×#395	C₂²	semidirect
33. C₅₃×#383	C₂²	semidirect
166. C₂×#2821	C₅₃	direct (N × H)
188. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
190. #2821	C₂×C₅₃	direct (N × H)
191. C₂×#400	C₂×C₅₃	semidirect
193. C₂×#398	C₂×C₅₃	semidirect
195. C₂×#395	C₂×C₅₃	semidirect
197. C₂×#383	C₂×C₅₃	semidirect
199. C₂×C₅₃	#2821	direct (N × H)
206. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
220. #400	C₂²×C₅₃	semidirect
222. #398	C₂²×C₅₃	semidirect
224. #395	C₂²×C₅₃	semidirect
226. #383	C₂²×C₅₃	semidirect
232. C₅₃	C₂×#2821	direct (N × H)
382. C₂	C₅₃×#2821	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⋊Q₈×C₅₃	2
C₂²×#126×C₅₃	2
C₅₃×#2821	2
C₂×C₅₃×#400	2
C₂×C₅₃×#399	2
C₂×C₅₃×#398	2
C₂×C₅₃×#397	2
C₂×C₅₃×#395	2
C₂×C₅₃×#393	2
C₂×C₅₃×#383	2
C₂×#2821	53

Subgroups — C2×C53×#2821 #27244198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2821 #27244198