Subgroups — C₅₃×#3488 #20272425

Order 6784.

1658 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#550	C₂	1	2	yes	maximal, complemented
4	C₅₃×#548	C₂	1	2	yes	maximal, complemented
5	C₅₃×#546	C₂	1	6	yes	maximal, complemented
6	D₄²×C₅₃	C₂	1	6	yes	maximal, complemented
7	C₅₃×#537	C₂	1	3	yes	maximal, complemented
8	C₅₃×#535	C₂	1	3	yes	maximal, complemented
9	C₅₃×#525	C₂	1	3	yes	maximal, complemented
10	C₂×#121×C₅₃	C₂	1	3	yes	maximal, complemented
11	C₅₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	H(5,𝔽₂)×C₅₃	—	2	8	no	—
13	C₂×#41×C₅₃	—	2	8	no	—
14	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#41×C₅₃	C₂²	1	3	yes	complemented
16	C₂×#41×C₅₃	C₂²	1	3	yes	complemented
17	C₂×#41×C₅₃	C₂²	1	3	yes	complemented
18	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
19	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
20	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₄×C₅₃	C₂²	1	3	yes	complemented
22	C₂²×D₄×C₅₃	C₂²	1	3	yes	complemented
23	C₂²×D₄×C₅₃	C₂²	1	6	yes	complemented
24	C₂²×D₄×C₅₃	C₂²	1	6	yes	complemented
25	C₂³×C₄×C₅₃	C₂²	1	3	yes	complemented
26	#127×C₅₃	C₂²	1	6	yes	complemented
27	#124×C₅₃	C₂²	1	6	yes	complemented
28	#123×C₅₃	C₂²	1	6	yes	complemented
29	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
31	C₂²⋊Q₈×C₅₃	C₂²	1	6	yes	complemented
32	#121×C₅₃	C₂²	1	3	yes	complemented
33	#121×C₅₃	C₂²	1	6	yes	complemented
34	#121×C₅₃	C₂²	1	6	yes	complemented
35	#121×C₅₃	C₂²	1	6	yes	complemented
36	#121×C₅₃	C₂²	1	3	yes	complemented
37	#121×C₅₃	C₂²	1	12	yes	complemented
38	#121×C₅₃	C₂²	1	6	yes	complemented
39	#120×C₅₃	C₂²	1	6	yes	complemented
40	#120×C₅₃	C₂²	1	12	yes	complemented
41	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
42	C₄×D₄×C₅₃	C₂²	1	6	yes	complemented
43	C₄×D₄×C₅₃	C₂²	1	6	yes	complemented
44	C₄×D₄×C₅₃	C₂²	1	6	yes	complemented
45	#117×C₅₃	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	complemented
47	C₂×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	complemented
48	C₂⁴×C₅₃	C₂³	1	3	yes	—
49	C₂⁴×C₅₃	C₂³	1	6	yes	—
50	#41×C₅₃	—	2	4	no	—
51	#41×C₅₃	—	2	8	no	—
52	#41×C₅₃	—	2	12	no	—
53	#41×C₅₃	—	2	12	no	—
54	#41×C₅₃	—	2	12	no	—
55	#41×C₅₃	D₄	1	8	yes	complemented
56	C₂×Q₈×C₅₃	—	2	4	no	—
57	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₅₃	C₂³	1	3	yes	—
59	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₅₃	—	2	12	no	—
61	C₂×D₄×C₅₃	—	2	24	no	—
62	C₂×D₄×C₅₃	—	2	12	no	—
63	C₂×D₄×C₅₃	—	4	12	no	—
64	C₂×D₄×C₅₃	C₂³	1	6	yes	—
65	C₂×D₄×C₅₃	C₂³	1	6	yes	—
66	C₂×D₄×C₅₃	C₂³	1	6	yes	—
67	C₂×D₄×C₅₃	C₂³	1	3	yes	—
68	C₂×D₄×C₅₃	C₂³	1	6	yes	—
69	C₂×D₄×C₅₃	C₂³	1	3	yes	—
70	C₂×D₄×C₅₃	C₂³	1	6	yes	—
71	C₂×D₄×C₅₃	C₂³	1	12	yes	—
72	C₂×D₄×C₅₃	C₂³	1	3	yes	—
73	C₂²×C₄×C₅₃	—	2	12	no	—
74	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
75	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
76	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
77	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
78	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
80	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
81	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
82	C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
83	C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
84	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
86	C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
87	C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
88	C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
89	C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
90	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
91	C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
92	C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
93	C₄²×C₅₃	C₂³	1	6	yes	—
94	C₂³×C₅₃	—	2	6	no	—
95	C₂³×C₅₃	—	2	6	no	—
96	C₂³×C₅₃	—	2	12	no	—
97	C₂³×C₅₃	—	4	3	no	—
98	C₂³×C₅₃	—	4	6	no	—
99	C₂³×C₅₃	—	4	6	no	—
100	C₂³×C₅₃	C₂⁴	1	3	yes	—
101	C₂³×C₅₃	C₂⁴	1	6	yes	—
102	C₂³×C₅₃	C₂⁴	1	3	yes	—
103	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
104	Q₈×C₅₃	—	2	6	no	—
105	Q₈×C₅₃	—	2	2	no	—
106	Q₈×C₅₃	C₂×D₄	1	4	yes	complemented
107	D₄×C₅₃	—	2	12	no	—
108	D₄×C₅₃	—	2	12	no	—
109	D₄×C₅₃	—	2	12	no	—
110	D₄×C₅₃	—	2	6	no	—
111	D₄×C₅₃	—	2	12	no	—
112	D₄×C₅₃	—	2	6	no	—
113	D₄×C₅₃	—	4	12	no	—
114	D₄×C₅₃	—	4	6	no	—
115	D₄×C₅₃	C₂×D₄	1	12	yes	complemented
116	C₂×C₄×C₅₃	—	2	6	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	6	no	—
119	C₂×C₄×C₅₃	—	2	6	no	—
120	C₂×C₄×C₅₃	—	2	6	no	—
121	C₂×C₄×C₅₃	—	2	12	no	—
122	C₂×C₄×C₅₃	—	4	6	no	—
123	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	complemented
124	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
126	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
127	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
128	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
129	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂⁴	1	3	yes	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	3	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	3	no	—
135	C₂²×C₅₃	—	2	3	no	—
136	C₂²×C₅₃	—	2	6	no	—
137	C₂²×C₅₃	—	2	6	no	—
138	C₂²×C₅₃	—	2	3	no	—
139	C₂²×C₅₃	—	2	3	no	—
140	C₂²×C₅₃	—	2	6	no	—
141	C₂²×C₅₃	—	4	12	no	—
142	C₂²×C₅₃	—	8	12	no	—
143	C₂²×C₅₃	C₂²×D₄	1	6	yes	—
144	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	complemented
145	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	3	no	—
149	C₄×C₅₃	—	2	6	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	C₂²×D₄	1	2	yes	—
152	C₄×C₅₃	C₂²×D₄	1	6	yes	—
153	#3488	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	—	2	6	no	—
156	C₂×C₅₃	—	4	3	no	—
157	C₂×C₅₃	—	4	6	no	—
158	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
159	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₅₃	#585	1	1	yes	characteristic
161	C₂×H(5,𝔽₂)	C₂×C₅₃	1	1	yes	characteristic, complemented
162	C₂²×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
163	#550	C₂×C₅₃	1	2	yes	complemented
164	#548	C₂×C₅₃	1	2	yes	complemented
165	#546	C₂×C₅₃	1	6	yes	complemented
166	D₄²	C₂×C₅₃	1	6	yes	complemented
167	#537	C₂×C₅₃	1	3	yes	complemented
168	#535	C₂×C₅₃	1	3	yes	complemented
169	#525	C₂×C₅₃	1	3	yes	complemented
170	C₂×#121	C₂×C₅₃	1	3	yes	complemented
171	#520	C₂×C₅₃	1	1	yes	characteristic, complemented
172	C₅₃	#3488	1	1	yes	characteristic, direct product factor, Sylow
173	H(5,𝔽₂)	—	2	8	no	—
174	C₂×#41	—	2	8	no	—
175	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
176	C₂×#41	C₂²×C₅₃	1	3	yes	complemented
177	C₂×#41	C₂²×C₅₃	1	3	yes	complemented
178	C₂×#41	C₂²×C₅₃	1	3	yes	complemented
179	C₂×#41	C₂²×C₅₃	1	2	yes	complemented
180	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic, complemented
181	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
182	C₂²×D₄	C₂²×C₅₃	1	3	yes	complemented
183	C₂²×D₄	C₂²×C₅₃	1	3	yes	complemented
184	C₂²×D₄	C₂²×C₅₃	1	6	yes	complemented
185	C₂²×D₄	C₂²×C₅₃	1	6	yes	complemented
186	C₂³×C₄	C₂²×C₅₃	1	3	yes	complemented
187	#127	C₂²×C₅₃	1	6	yes	complemented
188	#124	C₂²×C₅₃	1	6	yes	complemented
189	#123	C₂²×C₅₃	1	6	yes	complemented
190	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
191	C₂²⋊Q₈	C₂²×C₅₃	1	3	yes	complemented
192	C₂²⋊Q₈	C₂²×C₅₃	1	6	yes	complemented
193	#121	C₂²×C₅₃	1	3	yes	complemented
194	#121	C₂²×C₅₃	1	6	yes	complemented
195	#121	C₂²×C₅₃	1	6	yes	complemented
196	#121	C₂²×C₅₃	1	6	yes	complemented
197	#121	C₂²×C₅₃	1	3	yes	complemented
198	#121	C₂²×C₅₃	1	12	yes	complemented
199	#121	C₂²×C₅₃	1	6	yes	complemented
200	#120	C₂²×C₅₃	1	6	yes	complemented
201	#120	C₂²×C₅₃	1	12	yes	complemented
202	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
203	C₄×D₄	C₂²×C₅₃	1	6	yes	complemented
204	C₄×D₄	C₂²×C₅₃	1	6	yes	complemented
205	C₄×D₄	C₂²×C₅₃	1	6	yes	complemented
206	#117	C₂²×C₅₃	1	3	yes	complemented
207	C₂×C₄⋊C₄	C₂²×C₅₃	1	3	yes	complemented
208	C₂×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	complemented
209	C₂⁴	C₂³×C₅₃	1	3	yes	—
210	C₂⁴	C₂³×C₅₃	1	6	yes	—
211	#41	—	2	4	no	—
212	#41	—	2	8	no	—
213	#41	—	2	12	no	—
214	#41	—	2	12	no	—
215	#41	—	2	12	no	—
216	#41	D₄×C₅₃	1	8	yes	complemented
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
219	C₂×Q₈	C₂³×C₅₃	1	3	yes	—
220	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
221	C₂×D₄	—	2	12	no	—
222	C₂×D₄	—	2	24	no	—
223	C₂×D₄	—	2	12	no	—
224	C₂×D₄	—	4	12	no	—
225	C₂×D₄	C₂³×C₅₃	1	6	yes	—
226	C₂×D₄	C₂³×C₅₃	1	6	yes	—
227	C₂×D₄	C₂³×C₅₃	1	6	yes	—
228	C₂×D₄	C₂³×C₅₃	1	3	yes	—
229	C₂×D₄	C₂³×C₅₃	1	6	yes	—
230	C₂×D₄	C₂³×C₅₃	1	3	yes	—
231	C₂×D₄	C₂³×C₅₃	1	6	yes	—
232	C₂×D₄	C₂³×C₅₃	1	12	yes	—
233	C₂×D₄	C₂³×C₅₃	1	3	yes	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
236	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
237	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
238	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
239	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
241	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
242	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
243	C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
244	C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
245	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
246	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
247	C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
248	C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
249	C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
250	C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
251	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
252	C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
253	C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
254	C₄²	C₂³×C₅₃	1	6	yes	—
255	C₂³	—	2	6	no	—
256	C₂³	—	2	6	no	—
257	C₂³	—	2	12	no	—
258	C₂³	—	4	3	no	—
259	C₂³	—	4	6	no	—
260	C₂³	—	4	6	no	—
261	C₂³	C₂⁴×C₅₃	1	6	yes	—
262	C₂³	C₂⁴×C₅₃	1	3	yes	—
263	C₂³	C₂⁴×C₅₃	1	3	yes	—
264	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
265	Q₈	—	2	6	no	—
266	Q₈	—	2	2	no	—
267	Q₈	C₂×D₄×C₅₃	1	4	yes	complemented
268	D₄	—	2	12	no	—
269	D₄	—	2	12	no	—
270	D₄	—	2	12	no	—
271	D₄	—	2	6	no	—
272	D₄	—	2	12	no	—
273	D₄	—	2	6	no	—
274	D₄	—	4	12	no	—
275	D₄	—	4	6	no	—
276	D₄	C₂×D₄×C₅₃	1	12	yes	complemented
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	2	6	no	—
281	C₂×C₄	—	2	6	no	—
282	C₂×C₄	—	2	12	no	—
283	C₂×C₄	—	4	6	no	—
284	C₂×C₄	C₂×D₄×C₅₃	1	12	yes	complemented
285	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
286	C₂×C₄	C₂⁴×C₅₃	1	6	yes	—
287	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂⁴×C₅₃	1	3	yes	—
289	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
290	C₂×C₄	C₂⁴×C₅₃	1	3	yes	—
291	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	3	no	—
296	C₂²	—	2	3	no	—
297	C₂²	—	2	6	no	—
298	C₂²	—	2	6	no	—
299	C₂²	—	2	3	no	—
300	C₂²	—	2	3	no	—
301	C₂²	—	2	6	no	—
302	C₂²	—	4	12	no	—
303	C₂²	—	8	12	no	—
304	C₂²	C₂²×D₄×C₅₃	1	6	yes	—
305	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	complemented
306	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	3	no	—
310	C₄	—	2	6	no	—
311	C₄	—	2	2	no	—
312	C₄	C₂²×D₄×C₅₃	1	2	yes	—
313	C₄	C₂²×D₄×C₅₃	1	6	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	2	6	no	—
316	C₂	—	4	3	no	—
317	C₂	—	4	6	no	—
318	C₂	C₂³×D₄×C₅₃	1	1	yes	characteristic
319	C₂	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
320	C₂	C₅₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×H(5,𝔽₂)×C₅₃	C₂	semidirect
2. C₂²×#41×C₅₃	C₂	semidirect
3. C₅₃×#550	C₂	semidirect
4. C₅₃×#548	C₂	semidirect
5. C₅₃×#546	C₂	semidirect
6. D₄²×C₅₃	C₂	semidirect
7. C₅₃×#537	C₂	semidirect
8. C₅₃×#535	C₂	semidirect
9. C₅₃×#525	C₂	semidirect
10. C₂×#121×C₅₃	C₂	semidirect
11. C₅₃×#520	C₂	semidirect
15. C₂×#41×C₅₃	C₂²	semidirect
16. C₂×#41×C₅₃	C₂²	semidirect
17. C₂×#41×C₅₃	C₂²	semidirect
18. C₂×#41×C₅₃	C₂²	semidirect
19. C₂×#41×C₅₃	C₂²	semidirect
20. C₂²×Q₈×C₅₃	C₂²	semidirect
21. C₂²×D₄×C₅₃	C₂²	semidirect
22. C₂²×D₄×C₅₃	C₂²	semidirect
23. C₂²×D₄×C₅₃	C₂²	semidirect
24. C₂²×D₄×C₅₃	C₂²	semidirect
25. C₂³×C₄×C₅₃	C₂²	semidirect
26. #127×C₅₃	C₂²	semidirect
27. #124×C₅₃	C₂²	semidirect
28. #123×C₅₃	C₂²	semidirect
29. C₂²⋊Q₈×C₅₃	C₂²	semidirect
30. C₂²⋊Q₈×C₅₃	C₂²	semidirect
31. C₂²⋊Q₈×C₅₃	C₂²	semidirect
32. #121×C₅₃	C₂²	semidirect
33. #121×C₅₃	C₂²	semidirect
34. #121×C₅₃	C₂²	semidirect
35. #121×C₅₃	C₂²	semidirect
36. #121×C₅₃	C₂²	semidirect
37. #121×C₅₃	C₂²	semidirect
38. #121×C₅₃	C₂²	semidirect
39. #120×C₅₃	C₂²	semidirect
40. #120×C₅₃	C₂²	semidirect
41. C₄×Q₈×C₅₃	C₂²	semidirect
42. C₄×D₄×C₅₃	C₂²	semidirect
43. C₄×D₄×C₅₃	C₂²	semidirect
44. C₄×D₄×C₅₃	C₂²	semidirect
45. #117×C₅₃	C₂²	semidirect
46. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
47. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
55. #41×C₅₃	D₄	semidirect
106. Q₈×C₅₃	C₂×D₄	semidirect
115. D₄×C₅₃	C₂×D₄	semidirect
123. C₂×C₄×C₅₃	C₂×D₄	semidirect
144. C₂²×C₅₃	H(5,𝔽₂)	semidirect
153. #3488	C₅₃	direct (N × H)
161. C₂×H(5,𝔽₂)	C₂×C₅₃	semidirect
162. C₂²×#41	C₂×C₅₃	semidirect
163. #550	C₂×C₅₃	semidirect
164. #548	C₂×C₅₃	semidirect
165. #546	C₂×C₅₃	semidirect
166. D₄²	C₂×C₅₃	semidirect
167. #537	C₂×C₅₃	semidirect
168. #535	C₂×C₅₃	semidirect
169. #525	C₂×C₅₃	semidirect
170. C₂×#121	C₂×C₅₃	semidirect
171. #520	C₂×C₅₃	semidirect
172. C₅₃	#3488	direct (N × H)
176. C₂×#41	C₂²×C₅₃	semidirect
177. C₂×#41	C₂²×C₅₃	semidirect
178. C₂×#41	C₂²×C₅₃	semidirect
179. C₂×#41	C₂²×C₅₃	semidirect
180. C₂×#41	C₂²×C₅₃	semidirect
181. C₂²×Q₈	C₂²×C₅₃	semidirect
182. C₂²×D₄	C₂²×C₅₃	semidirect
183. C₂²×D₄	C₂²×C₅₃	semidirect
184. C₂²×D₄	C₂²×C₅₃	semidirect
185. C₂²×D₄	C₂²×C₅₃	semidirect
186. C₂³×C₄	C₂²×C₅₃	semidirect
187. #127	C₂²×C₅₃	semidirect
188. #124	C₂²×C₅₃	semidirect
189. #123	C₂²×C₅₃	semidirect
190. C₂²⋊Q₈	C₂²×C₅₃	semidirect
191. C₂²⋊Q₈	C₂²×C₅₃	semidirect
192. C₂²⋊Q₈	C₂²×C₅₃	semidirect
193. #121	C₂²×C₅₃	semidirect
194. #121	C₂²×C₅₃	semidirect
195. #121	C₂²×C₅₃	semidirect
196. #121	C₂²×C₅₃	semidirect
197. #121	C₂²×C₅₃	semidirect
198. #121	C₂²×C₅₃	semidirect
199. #121	C₂²×C₅₃	semidirect
200. #120	C₂²×C₅₃	semidirect
201. #120	C₂²×C₅₃	semidirect
202. C₄×Q₈	C₂²×C₅₃	semidirect
203. C₄×D₄	C₂²×C₅₃	semidirect
204. C₄×D₄	C₂²×C₅₃	semidirect
205. C₄×D₄	C₂²×C₅₃	semidirect
206. #117	C₂²×C₅₃	semidirect
207. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
208. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
216. #41	D₄×C₅₃	semidirect
267. Q₈	C₂×D₄×C₅₃	semidirect
276. D₄	C₂×D₄×C₅₃	semidirect
284. C₂×C₄	C₂×D₄×C₅₃	semidirect
305. C₂²	H(5,𝔽₂)×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)×C₅₃	2
C₂²×#41×C₅₃	2
C₅₃×#550	2
C₅₃×#548	2
C₅₃×#546	2
D₄²×C₅₃	2
C₅₃×#537	2
C₅₃×#535	2
C₅₃×#525	2
C₂×#121×C₅₃	2
C₅₃×#520	2
#3488	53

Subgroups — C53×#3488 #20272425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3488 #20272425