Subgroups — C₅₃×#2438 #20271375

Order 6784.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#387	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#387	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#385	C₂	1	1	yes	maximal, characteristic
12	C₅₃×#385	C₂	1	1	yes	maximal, characteristic
13	C₅₃×#381	C₂	1	1	yes	maximal, characteristic
14	C₅₃×#380	C₂	1	1	yes	maximal, characteristic
15	C₂×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²⋊Q₈×C₅₃	—	2	4	no	—
21	#117×C₅₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
39	#95×C₅₃	C₂²	1	1	yes	characteristic
40	#95×C₅₃	C₂²	1	1	yes	characteristic
41	#95×C₅₃	C₂²	1	1	yes	characteristic
42	#95×C₅₃	C₂²	1	1	yes	characteristic
43	#95×C₅₃	C₂²	1	1	yes	characteristic
44	#95×C₅₃	C₂²	1	1	yes	characteristic
45	#95×C₅₃	C₂²	1	1	yes	characteristic
46	#95×C₅₃	C₂²	1	1	yes	characteristic
47	#95×C₅₃	C₂²	1	1	yes	characteristic
48	#95×C₅₃	C₂²	1	1	yes	characteristic
49	#95×C₅₃	C₂²	1	1	yes	characteristic
50	#95×C₅₃	C₂²	1	1	yes	characteristic
51	#95×C₅₃	C₂²	1	1	yes	characteristic
52	#95×C₅₃	C₂²	1	1	yes	characteristic
53	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₅₃	—	2	2	no	—
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂×Q₈×C₅₃	—	4	1	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	4	1	no	—
64	C₂²×C₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	Q₈	1	4	yes	complemented
66	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	2	2	no	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	4	1	no	—
89	C₄⋊C₄×C₅₃	—	4	1	no	—
90	C₄⋊C₄×C₅₃	—	4	1	no	—
91	C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₂²⋊C₄×C₅₃	—	2	2	no	—
93	C₂²⋊C₄×C₅₃	—	2	2	no	—
94	C₂²⋊C₄×C₅₃	D₄	1	4	yes	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₂³×C₅₃	—	2	1	no	—
98	C₂³×C₅₃	—	2	1	no	—
99	C₂³×C₅₃	—	2	1	no	—
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	C₂×D₄	1	2	yes	—
102	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
103	C₂³×C₅₃	#41	1	2	yes	—
104	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
105	Q₈×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	2	8	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	4	4	no	—
144	C₂×C₄×C₅₃	—	4	4	no	—
145	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
146	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
147	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
148	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
149	C₂×C₄×C₅₃	#41	1	2	yes	—
150	C₂×C₄×C₅₃	#41	1	2	yes	—
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	—	2	2	no	—
154	C₂²×C₅₃	—	2	2	no	—
155	C₂²×C₅₃	—	2	2	no	—
156	C₂²×C₅₃	—	2	2	no	—
157	C₂²×C₅₃	C₂²⋊Q₈	1	4	yes	—
158	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
161	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
162	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
163	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
165	C₄×C₅₃	—	2	2	no	—
166	C₄×C₅₃	—	2	2	no	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	2	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	4	1	no	—
172	C₄×C₅₃	—	4	1	no	—
173	C₄×C₅₃	—	4	1	no	—
174	C₄×C₅₃	—	4	1	no	—
175	C₄×C₅₃	—	4	1	no	—
176	C₄×C₅₃	—	4	1	no	—
177	C₄×C₅₃	—	4	1	no	—
178	C₄×C₅₃	—	4	1	no	—
179	#2438	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂×C₅₃	—	2	4	no	—
181	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
182	C₂×C₅₃	#533	1	1	yes	characteristic
183	C₂×C₅₃	#544	1	1	yes	characteristic
184	C₂×C₅₃	#546	1	1	yes	characteristic
185	C₂×C₅₃	#547	1	1	yes	characteristic
186	C₂×C₅₃	#551	1	1	yes	characteristic
187	C₂×C₅₃	#552	1	1	yes	characteristic
188	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
189	C₂×#117	C₂×C₅₃	1	1	yes	characteristic
190	#400	C₂×C₅₃	1	1	yes	characteristic, complemented
191	#398	C₂×C₅₃	1	1	yes	characteristic, complemented
192	#398	C₂×C₅₃	1	1	yes	characteristic, complemented
193	#395	C₂×C₅₃	1	1	yes	characteristic, complemented
194	#391	C₂×C₅₃	1	1	yes	characteristic, complemented
195	#389	C₂×C₅₃	1	1	yes	characteristic, complemented
196	#387	C₂×C₅₃	1	1	yes	characteristic, complemented
197	#387	C₂×C₅₃	1	1	yes	characteristic, complemented
198	#385	C₂×C₅₃	1	1	yes	characteristic
199	#385	C₂×C₅₃	1	1	yes	characteristic
200	#381	C₂×C₅₃	1	1	yes	characteristic
201	#380	C₂×C₅₃	1	1	yes	characteristic
202	C₂×#95	C₂×C₅₃	1	1	yes	characteristic
203	C₅₃	#2438	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
205	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
206	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
207	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
208	C₂²⋊Q₈	—	2	4	no	—
209	#117	—	2	4	no	—
210	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
220	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
227	#95	C₂²×C₅₃	1	1	yes	characteristic
228	#95	C₂²×C₅₃	1	1	yes	characteristic
229	#95	C₂²×C₅₃	1	1	yes	characteristic
230	#95	C₂²×C₅₃	1	1	yes	characteristic
231	#95	C₂²×C₅₃	1	1	yes	characteristic
232	#95	C₂²×C₅₃	1	1	yes	characteristic
233	#95	C₂²×C₅₃	1	1	yes	characteristic
234	#95	C₂²×C₅₃	1	1	yes	characteristic
235	#95	C₂²×C₅₃	1	1	yes	characteristic
236	#95	C₂²×C₅₃	1	1	yes	characteristic
237	#95	C₂²×C₅₃	1	1	yes	characteristic
238	#95	C₂²×C₅₃	1	1	yes	characteristic
239	#95	C₂²×C₅₃	1	1	yes	characteristic
240	#95	C₂²×C₅₃	1	1	yes	characteristic
241	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	Q₈×C₅₃	1	4	yes	complemented
254	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	D₄×C₅₃	1	4	yes	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₂×D₄×C₅₃	1	2	yes	—
290	C₂³	C₂×Q₈×C₅₃	1	2	yes	—
291	C₂³	#41×C₅₃	1	2	yes	—
292	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
293	Q₈	—	4	4	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
334	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
335	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
336	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
337	C₂×C₄	#41×C₅₃	1	2	yes	—
338	C₂×C₄	#41×C₅₃	1	2	yes	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	C₂²⋊Q₈×C₅₃	1	4	yes	—
346	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
347	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
348	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
349	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
350	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
351	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
352	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	4	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	4	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₂	—	2	4	no	—
368	C₂	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
369	C₂	C₅₃×#533	1	1	yes	characteristic
370	C₂	C₅₃×#544	1	1	yes	characteristic
371	C₂	C₅₃×#546	1	1	yes	characteristic
372	C₂	C₅₃×#547	1	1	yes	characteristic
373	C₂	C₅₃×#551	1	1	yes	characteristic
374	C₂	C₅₃×#552	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#400	C₂	semidirect
4. C₅₃×#398	C₂	semidirect
5. C₅₃×#398	C₂	semidirect
6. C₅₃×#395	C₂	semidirect
7. C₅₃×#391	C₂	semidirect
8. C₅₃×#389	C₂	semidirect
9. C₅₃×#387	C₂	semidirect
10. C₅₃×#387	C₂	semidirect
65. C₂²×C₄×C₅₃	Q₈	semidirect
179. #2438	C₅₃	direct (N × H)
190. #400	C₂×C₅₃	semidirect
191. #398	C₂×C₅₃	semidirect
192. #398	C₂×C₅₃	semidirect
193. #395	C₂×C₅₃	semidirect
194. #391	C₂×C₅₃	semidirect
195. #389	C₂×C₅₃	semidirect
196. #387	C₂×C₅₃	semidirect
197. #387	C₂×C₅₃	semidirect
203. C₅₃	#2438	direct (N × H)
253. C₂²×C₄	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₅₃	2
C₂×#117×C₅₃	2
C₅₃×#400	2
C₅₃×#398	2
C₅₃×#395	2
C₅₃×#391	2
C₅₃×#389	2
C₅₃×#387	2
C₅₃×#385	2
C₅₃×#381	2
C₅₃×#380	2
C₂×#95×C₅₃	2
#2438	53

Subgroups — C53×#2438 #20271375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2438 #20271375