Subgroups — C₂×C₅₃×#2633 #27242102

Order 13568.

2894 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2633	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#399	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#398	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#397	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#394	C₂	1	2	yes	maximal
8	C₂×C₅₃×#385	C₂	1	4	yes	maximal, complemented
9	C₂×C₅₃×#383	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×#125×C₅₃	—	2	8	no	—
13	C₂×#125×C₅₃	C₂²	1	8	yes	complemented
14	C₂×C₄×D₄×C₅₃	—	2	8	no	—
15	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
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	1	yes	characteristic
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	4	yes	—
23	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
25	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#399	C₂²	1	16	yes	complemented
27	C₅₃×#398	C₂²	1	16	yes	complemented
28	C₅₃×#397	C₂²	1	16	yes	complemented
29	C₅₃×#394	C₂²	1	16	yes	—
30	C₅₃×#385	C₂²	1	32	yes	complemented
31	C₅₃×#383	C₂²	1	8	yes	complemented
32	C₂×#95×C₅₃	C₂²	1	2	yes	—
33	C₂×#95×C₅₃	C₂²	1	2	yes	—
34	C₂×#95×C₅₃	C₂²	1	2	yes	—
35	C₂×#95×C₅₃	C₂²	1	4	yes	—
36	C₂⁵×C₅₃	C₂³	1	2	yes	—
37	C₂²×D₄×C₅₃	—	2	8	no	—
38	C₂²×D₄×C₅₃	—	2	4	no	—
39	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
40	C₂³×C₄×C₅₃	—	2	8	no	—
41	C₂³×C₄×C₅₃	—	4	8	no	—
42	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
50	#125×C₅₃	—	2	32	no	—
51	C₄×D₄×C₅₃	—	2	32	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
57	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
64	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
65	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	4	8	no	—
67	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
68	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
69	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
70	C₂×C₄²×C₅₃	D₄	1	8	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
72	#95×C₅₃	C₂³	1	8	yes	—
73	#95×C₅₃	C₂³	1	8	yes	—
74	#95×C₅₃	C₂³	1	8	yes	—
75	#95×C₅₃	C₂³	1	16	yes	—
76	C₂⁴×C₅₃	—	2	4	no	—
77	C₂⁴×C₅₃	—	2	4	no	—
78	C₂⁴×C₅₃	—	2	8	no	—
79	C₂⁴×C₅₃	—	2	8	no	—
80	C₂⁴×C₅₃	#41	1	8	yes	—
81	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
82	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₂×D₄×C₅₃	—	2	32	no	—
84	C₂×D₄×C₅₃	—	2	16	no	—
85	C₂×D₄×C₅₃	—	2	8	no	—
86	C₂²×C₄×C₅₃	—	2	4	no	—
87	C₂²×C₄×C₅₃	—	2	4	no	—
88	C₂²×C₄×C₅₃	—	2	4	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	8	no	—
93	C₂²×C₄×C₅₃	—	2	8	no	—
94	C₂²×C₄×C₅₃	—	2	4	no	—
95	C₂²×C₄×C₅₃	—	2	8	no	—
96	C₂²×C₄×C₅₃	—	2	4	no	—
97	C₂²×C₄×C₅₃	—	2	8	no	—
98	C₂²×C₄×C₅₃	—	2	8	no	—
99	C₂²×C₄×C₅₃	—	2	16	no	—
100	C₂²×C₄×C₅₃	—	4	16	no	—
101	C₂²×C₄×C₅₃	—	4	32	no	—
102	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
106	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
107	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
108	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
112	C₄⋊C₄×C₅₃	—	2	8	no	—
113	C₄⋊C₄×C₅₃	—	2	16	no	—
114	C₄⋊C₄×C₅₃	—	2	16	no	—
115	C₄⋊C₄×C₅₃	—	2	8	no	—
116	C₄⋊C₄×C₅₃	—	2	8	no	—
117	C₄⋊C₄×C₅₃	—	4	16	no	—
118	C₂²⋊C₄×C₅₃	—	2	32	no	—
119	C₂²⋊C₄×C₅₃	—	2	16	no	—
120	C₂²⋊C₄×C₅₃	—	4	16	no	—
121	C₄²×C₅₃	C₂×D₄	1	16	yes	—
122	C₂³×C₅₃	—	2	16	no	—
123	C₂³×C₅₃	—	2	32	no	—
124	C₂³×C₅₃	—	2	4	no	—
125	C₂³×C₅₃	—	2	4	no	—
126	C₂³×C₅₃	—	2	8	no	—
127	C₂³×C₅₃	—	2	8	no	—
128	C₂³×C₅₃	—	4	8	no	—
129	C₂³×C₅₃	—	4	8	no	—
130	C₂³×C₅₃	—	4	16	no	—
131	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
132	C₂³×C₅₃	C₂×#41	1	8	yes	—
133	C₂³×C₅₃	C₂×#41	1	4	yes	—
134	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
135	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
136	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
137	D₄×C₅₃	—	2	32	no	—
138	C₂×C₄×C₅₃	—	2	8	no	—
139	C₂×C₄×C₅₃	—	2	8	no	—
140	C₂×C₄×C₅₃	—	2	8	no	—
141	C₂×C₄×C₅₃	—	2	4	no	—
142	C₂×C₄×C₅₃	—	2	4	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	4	no	—
147	C₂×C₄×C₅₃	—	2	8	no	—
148	C₂×C₄×C₅₃	—	2	8	no	—
149	C₂×C₄×C₅₃	—	2	8	no	—
150	C₂×C₄×C₅₃	—	2	4	no	—
151	C₂×C₄×C₅₃	—	2	8	no	—
152	C₂×C₄×C₅₃	—	2	4	no	—
153	C₂×C₄×C₅₃	—	2	8	no	—
154	C₂×C₄×C₅₃	—	4	32	no	—
155	C₂×C₄×C₅₃	—	4	8	no	—
156	C₂×C₄×C₅₃	—	4	8	no	—
157	C₂×C₄×C₅₃	—	4	8	no	—
158	C₂×C₄×C₅₃	—	4	16	no	—
159	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
160	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
161	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
162	C₂×#2633	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂²×C₅₃	—	2	8	no	—
164	C₂²×C₅₃	—	2	16	no	—
165	C₂²×C₅₃	—	4	32	no	—
166	C₂²×C₅₃	—	4	4	no	—
167	C₂²×C₅₃	—	4	4	no	—
168	C₂²×C₅₃	—	4	8	no	—
169	C₂²×C₅₃	#525	1	4	yes	—
170	C₂²×C₅₃	#535	1	4	yes	—
171	C₂²×C₅₃	#536	1	4	yes	—
172	C₂²×C₅₃	#539	1	16	yes	—
173	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
174	C₂²×C₅₃	C₂²×#41	1	2	yes	—
175	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
176	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
177	C₄×C₅₃	—	2	4	no	—
178	C₄×C₅₃	—	2	4	no	—
179	C₄×C₅₃	—	2	4	no	—
180	C₄×C₅₃	—	4	4	no	—
181	C₄×C₅₃	—	4	4	no	—
182	C₄×C₅₃	—	4	4	no	—
183	C₄×C₅₃	—	4	8	no	—
184	C₂²×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
185	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
186	#2633	C₂×C₅₃	1	16	yes	direct product factor
187	C₂×#399	C₂×C₅₃	1	2	yes	complemented
188	C₂×#398	C₂×C₅₃	1	2	yes	complemented
189	C₂×#397	C₂×C₅₃	1	2	yes	complemented
190	C₂×#394	C₂×C₅₃	1	2	yes	—
191	C₂×#385	C₂×C₅₃	1	4	yes	complemented
192	C₂×#383	C₂×C₅₃	1	1	yes	characteristic, complemented
193	C₂×C₅₃	—	4	8	no	—
194	C₂×C₅₃	#2633	1	8	yes	direct product factor
195	C₂×C₅₃	C₂×#525	1	1	yes	characteristic
196	C₂×C₅₃	C₂×#535	1	1	yes	characteristic
197	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
198	C₂×C₅₃	C₂×#539	1	4	yes	—
199	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
200	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
201	C₂×#125	—	2	8	no	—
202	C₂×#125	C₂²×C₅₃	1	8	yes	complemented
203	C₂×C₄×D₄	—	2	8	no	—
204	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
205	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
206	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
207	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
208	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
209	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
210	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
211	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
212	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
213	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
214	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
215	#399	C₂²×C₅₃	1	16	yes	complemented
216	#398	C₂²×C₅₃	1	16	yes	complemented
217	#397	C₂²×C₅₃	1	16	yes	complemented
218	#394	C₂²×C₅₃	1	16	yes	—
219	#385	C₂²×C₅₃	1	32	yes	complemented
220	#383	C₂²×C₅₃	1	8	yes	complemented
221	C₂×#95	C₂²×C₅₃	1	2	yes	—
222	C₂×#95	C₂²×C₅₃	1	4	yes	—
223	C₂×#95	C₂²×C₅₃	1	2	yes	—
224	C₂×#95	C₂²×C₅₃	1	2	yes	—
225	C₅₃	C₂×#2633	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁵	C₂³×C₅₃	1	2	yes	—
227	C₂²×D₄	—	2	8	no	—
228	C₂²×D₄	—	2	4	no	—
229	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
230	C₂³×C₄	—	2	8	no	—
231	C₂³×C₄	—	4	8	no	—
232	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
233	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
234	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
235	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
236	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
237	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
238	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
239	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
240	#125	—	2	32	no	—
241	C₄×D₄	—	2	32	no	—
242	C₂×C₄⋊C₄	—	2	4	no	—
243	C₂×C₄⋊C₄	—	2	4	no	—
244	C₂×C₄⋊C₄	—	2	4	no	—
245	C₂×C₄⋊C₄	—	2	8	no	—
246	C₂×C₄⋊C₄	—	2	8	no	—
247	C₂×C₄⋊C₄	—	4	8	no	—
248	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
249	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
250	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
251	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
252	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
253	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
254	C₂×C₂²⋊C₄	—	2	16	no	—
255	C₂×C₂²⋊C₄	—	2	8	no	—
256	C₂×C₂²⋊C₄	—	4	8	no	—
257	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
258	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
259	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
260	C₂×C₄²	D₄×C₅₃	1	8	yes	—
261	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
262	#95	C₂³×C₅₃	1	8	yes	—
263	#95	C₂³×C₅₃	1	8	yes	—
264	#95	C₂³×C₅₃	1	8	yes	—
265	#95	C₂³×C₅₃	1	16	yes	—
266	C₂⁴	—	2	4	no	—
267	C₂⁴	—	2	4	no	—
268	C₂⁴	—	2	8	no	—
269	C₂⁴	—	2	8	no	—
270	C₂⁴	#41×C₅₃	1	8	yes	—
271	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
272	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
273	C₂×D₄	—	2	32	no	—
274	C₂×D₄	—	2	16	no	—
275	C₂×D₄	—	2	8	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	2	8	no	—
283	C₂²×C₄	—	2	8	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	8	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	8	no	—
288	C₂²×C₄	—	2	8	no	—
289	C₂²×C₄	—	2	16	no	—
290	C₂²×C₄	—	4	16	no	—
291	C₂²×C₄	—	4	32	no	—
292	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
293	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
294	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
295	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
296	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
297	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
298	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
299	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
300	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
301	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
302	C₄⋊C₄	—	2	8	no	—
303	C₄⋊C₄	—	2	8	no	—
304	C₄⋊C₄	—	2	8	no	—
305	C₄⋊C₄	—	2	16	no	—
306	C₄⋊C₄	—	2	16	no	—
307	C₄⋊C₄	—	4	16	no	—
308	C₂²⋊C₄	—	2	32	no	—
309	C₂²⋊C₄	—	2	16	no	—
310	C₂²⋊C₄	—	4	16	no	—
311	C₄²	C₂×D₄×C₅₃	1	16	yes	—
312	C₂³	—	2	16	no	—
313	C₂³	—	2	32	no	—
314	C₂³	—	2	4	no	—
315	C₂³	—	2	4	no	—
316	C₂³	—	2	8	no	—
317	C₂³	—	2	8	no	—
318	C₂³	—	4	8	no	—
319	C₂³	—	4	8	no	—
320	C₂³	—	4	16	no	—
321	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
322	C₂³	C₂×#41×C₅₃	1	8	yes	—
323	C₂³	C₂×#41×C₅₃	1	4	yes	—
324	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
325	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
326	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
327	D₄	—	2	32	no	—
328	C₂×C₄	—	2	8	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	2	8	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	8	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	8	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	8	no	—
344	C₂×C₄	—	4	32	no	—
345	C₂×C₄	—	4	8	no	—
346	C₂×C₄	—	4	8	no	—
347	C₂×C₄	—	4	8	no	—
348	C₂×C₄	—	4	16	no	—
349	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
350	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
351	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
352	C₂²	—	2	16	no	—
353	C₂²	—	2	8	no	—
354	C₂²	—	4	4	no	—
355	C₂²	—	4	8	no	—
356	C₂²	—	4	32	no	—
357	C₂²	—	4	4	no	—
358	C₂²	C₅₃×#525	1	4	yes	—
359	C₂²	C₅₃×#535	1	4	yes	—
360	C₂²	C₅₃×#536	1	4	yes	—
361	C₂²	C₅₃×#539	1	16	yes	—
362	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
363	C₂²	C₂²×#41×C₅₃	1	2	yes	—
364	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
365	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
366	C₄	—	2	4	no	—
367	C₄	—	2	4	no	—
368	C₄	—	2	4	no	—
369	C₄	—	4	8	no	—
370	C₄	—	4	4	no	—
371	C₄	—	4	4	no	—
372	C₄	—	4	4	no	—
373	C₂	—	4	8	no	—
374	C₂	C₅₃×#2633	1	8	yes	direct product factor
375	C₂	C₂×C₅₃×#525	1	1	yes	characteristic
376	C₂	C₂×C₅₃×#535	1	1	yes	characteristic
377	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
378	C₂	C₂×C₅₃×#539	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
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×#125×C₅₃	C₂	semidirect
3. C₅₃×#2633	C₂	direct (N × H)
4. C₂×C₅₃×#399	C₂	semidirect
5. C₂×C₅₃×#398	C₂	semidirect
6. C₂×C₅₃×#397	C₂	semidirect
8. C₂×C₅₃×#385	C₂	semidirect
9. C₂×C₅₃×#383	C₂	semidirect
13. C₂×#125×C₅₃	C₂²	semidirect
26. C₅₃×#399	C₂²	semidirect
27. C₅₃×#398	C₂²	semidirect
28. C₅₃×#397	C₂²	semidirect
30. C₅₃×#385	C₂²	semidirect
31. C₅₃×#383	C₂²	semidirect
162. C₂×#2633	C₅₃	direct (N × H)
184. C₂²×#125	C₂×C₅₃	semidirect
186. #2633	C₂×C₅₃	direct (N × H)
187. C₂×#399	C₂×C₅₃	semidirect
188. C₂×#398	C₂×C₅₃	semidirect
189. C₂×#397	C₂×C₅₃	semidirect
191. C₂×#385	C₂×C₅₃	semidirect
192. C₂×#383	C₂×C₅₃	semidirect
194. C₂×C₅₃	#2633	direct (N × H)
202. C₂×#125	C₂²×C₅₃	semidirect
215. #399	C₂²×C₅₃	semidirect
216. #398	C₂²×C₅₃	semidirect
217. #397	C₂²×C₅₃	semidirect
219. #385	C₂²×C₅₃	semidirect
220. #383	C₂²×C₅₃	semidirect
225. C₅₃	C₂×#2633	direct (N × H)
374. C₂	C₅₃×#2633	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×#125×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₅₃×#2633	2
C₂×C₅₃×#399	2
C₂×C₅₃×#398	2
C₂×C₅₃×#397	2
C₂×C₅₃×#394	2
C₂×C₅₃×#385	2
C₂×C₅₃×#383	2
C₂×#2633	53

Subgroups — C2×C53×#2633 #27242102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2633 #27242102