Subgroups — C₅₃×#3514 #20272451

Order 6784.

1142 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#555	C₂	1	2	yes	maximal, complemented
4	C₅₃×#551	C₂	1	4	yes	maximal, complemented
5	D₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#548	C₂	1	2	yes	maximal, complemented
7	C₅₃×#546	C₂	1	2	yes	maximal, complemented
8	C₅₃×#546	C₂	1	2	yes	maximal, complemented
9	D₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#542	C₂	1	2	yes	maximal, complemented
11	C₅₃×#538	C₂	1	4	yes	maximal, complemented
12	C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#529	C₂	1	2	yes	maximal, complemented
15	C₂×#124×C₅₃	C₂	1	2	yes	maximal, complemented
16	C₅₃×#518	C₂	1	2	yes	maximal, complemented
17	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
20	C₂²×Q₈×C₅₃	C₂²	1	2	yes	complemented
21	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
22	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
23	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
25	#127×C₅₃	C₂²	1	1	yes	characteristic
26	#126×C₅₃	C₂²	1	4	yes	complemented
27	#126×C₅₃	C₂²	1	2	yes	complemented
28	#126×C₅₃	C₂²	1	4	yes	complemented
29	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	#124×C₅₃	C₂²	1	1	yes	characteristic
31	#124×C₅₃	C₂²	1	2	yes	—
32	#124×C₅₃	C₂²	1	2	yes	complemented
33	#124×C₅₃	C₂²	1	4	yes	complemented
34	#124×C₅₃	C₂²	1	2	yes	complemented
35	#124×C₅₃	C₂²	1	4	yes	complemented
36	#124×C₅₃	C₂²	1	4	yes	complemented
37	#124×C₅₃	C₂²	1	2	yes	complemented
38	#123×C₅₃	C₂²	1	2	yes	complemented
39	#123×C₅₃	C₂²	1	4	yes	complemented
40	#123×C₅₃	C₂²	1	4	yes	complemented
41	#123×C₅₃	C₂²	1	4	yes	complemented
42	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
43	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
45	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
46	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
47	#121×C₅₃	C₂²	1	2	yes	—
48	#121×C₅₃	C₂²	1	1	yes	characteristic
49	#121×C₅₃	C₂²	1	2	yes	—
50	#121×C₅₃	C₂²	1	2	yes	complemented
51	#121×C₅₃	C₂²	1	2	yes	complemented
52	#121×C₅₃	C₂²	1	4	yes	complemented
53	#121×C₅₃	C₂²	1	4	yes	complemented
54	#120×C₅₃	C₂²	1	4	yes	—
55	#120×C₅₃	C₂²	1	4	yes	complemented
56	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
57	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
59	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
60	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
61	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
62	C₄×D₄×C₅₃	C₂²	1	2	yes	—
63	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
64	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
65	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
66	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
67	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
68	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
69	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
70	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
71	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
72	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
73	#117×C₅₃	C₂²	1	2	yes	complemented
74	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
75	#117×C₅₃	C₂²	1	2	yes	complemented
76	#117×C₅₃	C₂²	1	2	yes	complemented
77	#117×C₅₃	C₂²	1	4	yes	complemented
78	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
79	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
80	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
81	C₂×C₄²×C₅₃	C₂²	1	2	yes	complemented
82	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
83	C₂⁴×C₅₃	C₂³	1	4	yes	—
84	#41×C₅₃	—	2	4	no	—
85	#41×C₅₃	—	4	4	no	—
86	C₂×Q₈×C₅₃	—	4	4	no	—
87	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
90	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
91	C₂×D₄×C₅₃	—	2	8	no	—
92	C₂×D₄×C₅₃	—	4	4	no	—
93	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₅₃	C₂³	1	4	yes	—
95	C₂×D₄×C₅₃	C₂³	1	2	yes	—
96	C₂×D₄×C₅₃	C₂³	1	2	yes	—
97	C₂×D₄×C₅₃	C₂³	1	2	yes	—
98	C₂×D₄×C₅₃	C₂³	1	2	yes	—
99	C₂×D₄×C₅₃	C₂³	1	2	yes	—
100	C₂×D₄×C₅₃	C₂³	1	4	yes	—
101	C₂×D₄×C₅₃	C₂³	1	4	yes	—
102	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
106	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
111	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
112	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
113	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
114	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
115	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
116	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
117	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
118	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
119	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
120	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
121	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
122	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
123	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
124	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
125	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
126	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
128	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
130	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
132	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
133	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
134	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
135	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
137	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
138	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
139	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
140	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
141	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
142	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
143	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
144	C₄²×C₅₃	C₂³	1	1	yes	characteristic
145	C₄²×C₅₃	C₂³	1	2	yes	—
146	C₄²×C₅₃	C₂³	1	1	yes	characteristic
147	C₄²×C₅₃	C₂³	1	1	yes	characteristic
148	C₄²×C₅₃	C₂³	1	1	yes	characteristic
149	C₄²×C₅₃	C₂³	1	2	yes	—
150	C₄²×C₅₃	C₂³	1	2	yes	—
151	C₄²×C₅₃	C₂³	1	4	yes	—
152	C₄²×C₅₃	C₂³	1	2	yes	—
153	C₂³×C₅₃	—	2	8	no	—
154	C₂³×C₅₃	—	4	4	no	—
155	C₂³×C₅₃	—	4	4	no	—
156	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
157	C₂³×C₅₃	C₂⁴	1	4	yes	—
158	C₂³×C₅₃	C₂⁴	1	2	yes	—
159	C₂³×C₅₃	C₂⁴	1	2	yes	—
160	Q₈×C₅₃	—	2	2	no	—
161	Q₈×C₅₃	—	2	2	no	—
162	Q₈×C₅₃	—	2	4	no	—
163	Q₈×C₅₃	—	4	4	no	—
164	D₄×C₅₃	—	2	8	no	—
165	D₄×C₅₃	—	2	4	no	—
166	D₄×C₅₃	—	2	4	no	—
167	D₄×C₅₃	—	2	2	no	—
168	D₄×C₅₃	—	2	4	no	—
169	D₄×C₅₃	—	4	4	no	—
170	D₄×C₅₃	—	4	2	no	—
171	D₄×C₅₃	—	4	2	no	—
172	D₄×C₅₃	—	4	4	no	—
173	D₄×C₅₃	#41	1	4	yes	complemented
174	C₂×C₄×C₅₃	—	2	4	no	—
175	C₂×C₄×C₅₃	—	2	4	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	4	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	4	2	no	—
180	C₂×C₄×C₅₃	—	4	4	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	4	no	—
183	C₂×C₄×C₅₃	—	4	4	no	—
184	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
188	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
189	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
190	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
191	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
192	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
193	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
195	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
197	C₂²×C₅₃	—	2	2	no	—
198	C₂²×C₅₃	—	2	2	no	—
199	C₂²×C₅₃	—	2	1	no	—
200	C₂²×C₅₃	—	2	2	no	—
201	C₂²×C₅₃	—	2	4	no	—
202	C₂²×C₅₃	—	2	1	no	—
203	C₂²×C₅₃	—	2	4	no	—
204	C₂²×C₅₃	—	2	2	no	—
205	C₂²×C₅₃	—	2	1	no	—
206	C₂²×C₅₃	—	2	4	no	—
207	C₂²×C₅₃	—	2	2	no	—
208	C₂²×C₅₃	—	8	8	no	—
209	C₂²×C₅₃	C₂×#41	1	4	yes	—
210	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
211	C₄×C₅₃	—	2	1	no	—
212	C₄×C₅₃	—	2	2	no	—
213	C₄×C₅₃	—	2	2	no	—
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	—	2	2	no	—
216	C₄×C₅₃	—	2	4	no	—
217	C₄×C₅₃	—	2	2	no	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	2	2	no	—
220	C₄×C₅₃	—	2	2	no	—
221	C₄×C₅₃	—	2	1	no	—
222	C₄×C₅₃	C₂×#41	1	2	yes	—
223	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
224	#3514	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂×C₅₃	—	2	4	no	—
226	C₂×C₅₃	—	4	1	no	—
227	C₂×C₅₃	—	4	4	no	—
228	C₂×C₅₃	—	4	2	no	—
229	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
230	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
231	C₂×C₅₃	#585	1	1	yes	characteristic
232	#559	C₂×C₅₃	1	1	yes	characteristic, complemented
233	#556	C₂×C₅₃	1	1	yes	characteristic, complemented
234	#555	C₂×C₅₃	1	2	yes	complemented
235	#551	C₂×C₅₃	1	4	yes	complemented
236	D₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
237	#548	C₂×C₅₃	1	2	yes	complemented
238	#546	C₂×C₅₃	1	2	yes	complemented
239	#546	C₂×C₅₃	1	2	yes	complemented
240	D₄²	C₂×C₅₃	1	1	yes	characteristic, complemented
241	#542	C₂×C₅₃	1	2	yes	complemented
242	#538	C₂×C₅₃	1	4	yes	complemented
243	#532	C₂×C₅₃	1	1	yes	characteristic, complemented
244	#529	C₂×C₅₃	1	1	yes	characteristic, complemented
245	#529	C₂×C₅₃	1	2	yes	complemented
246	C₂×#124	C₂×C₅₃	1	2	yes	complemented
247	#518	C₂×C₅₃	1	2	yes	complemented
248	C₄×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
249	C₅₃	#3514	1	1	yes	characteristic, direct product factor, Sylow
250	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
251	C₂×#41	C₂²×C₅₃	1	2	yes	complemented
252	C₂²×Q₈	C₂²×C₅₃	1	2	yes	complemented
253	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
254	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
255	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
256	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
257	#127	C₂²×C₅₃	1	1	yes	characteristic
258	#126	C₂²×C₅₃	1	4	yes	complemented
259	#126	C₂²×C₅₃	1	2	yes	complemented
260	#126	C₂²×C₅₃	1	4	yes	complemented
261	#125	C₂²×C₅₃	1	1	yes	characteristic, complemented
262	#124	C₂²×C₅₃	1	1	yes	characteristic
263	#124	C₂²×C₅₃	1	2	yes	—
264	#124	C₂²×C₅₃	1	2	yes	complemented
265	#124	C₂²×C₅₃	1	4	yes	complemented
266	#124	C₂²×C₅₃	1	2	yes	complemented
267	#124	C₂²×C₅₃	1	4	yes	complemented
268	#124	C₂²×C₅₃	1	4	yes	complemented
269	#124	C₂²×C₅₃	1	2	yes	complemented
270	#123	C₂²×C₅₃	1	2	yes	complemented
271	#123	C₂²×C₅₃	1	4	yes	complemented
272	#123	C₂²×C₅₃	1	4	yes	complemented
273	#123	C₂²×C₅₃	1	4	yes	complemented
274	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
275	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
276	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
277	C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
278	C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	complemented
279	#121	C₂²×C₅₃	1	2	yes	—
280	#121	C₂²×C₅₃	1	1	yes	characteristic
281	#121	C₂²×C₅₃	1	2	yes	—
282	#121	C₂²×C₅₃	1	2	yes	complemented
283	#121	C₂²×C₅₃	1	2	yes	complemented
284	#121	C₂²×C₅₃	1	4	yes	complemented
285	#121	C₂²×C₅₃	1	4	yes	complemented
286	#120	C₂²×C₅₃	1	4	yes	—
287	#120	C₂²×C₅₃	1	4	yes	complemented
288	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
289	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
290	C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
291	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
292	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
293	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
294	C₄×D₄	C₂²×C₅₃	1	2	yes	—
295	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
296	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
297	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
298	C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
299	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
300	C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
301	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
302	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
303	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
304	C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
305	#117	C₂²×C₅₃	1	2	yes	complemented
306	#117	C₂²×C₅₃	1	1	yes	characteristic, complemented
307	#117	C₂²×C₅₃	1	2	yes	complemented
308	#117	C₂²×C₅₃	1	2	yes	complemented
309	#117	C₂²×C₅₃	1	4	yes	complemented
310	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
311	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
312	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
313	C₂×C₄²	C₂²×C₅₃	1	2	yes	complemented
314	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic, complemented
315	C₂⁴	C₂³×C₅₃	1	4	yes	—
316	#41	—	2	4	no	—
317	#41	—	4	4	no	—
318	C₂×Q₈	—	4	4	no	—
319	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
320	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
321	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
322	C₂×Q₈	C₂³×C₅₃	1	4	yes	—
323	C₂×D₄	—	2	8	no	—
324	C₂×D₄	—	4	4	no	—
325	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
326	C₂×D₄	C₂³×C₅₃	1	4	yes	—
327	C₂×D₄	C₂³×C₅₃	1	2	yes	—
328	C₂×D₄	C₂³×C₅₃	1	2	yes	—
329	C₂×D₄	C₂³×C₅₃	1	2	yes	—
330	C₂×D₄	C₂³×C₅₃	1	2	yes	—
331	C₂×D₄	C₂³×C₅₃	1	2	yes	—
332	C₂×D₄	C₂³×C₅₃	1	4	yes	—
333	C₂×D₄	C₂³×C₅₃	1	4	yes	—
334	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
335	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
336	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
337	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
338	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
340	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
341	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
342	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
343	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
344	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
345	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
346	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
347	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
348	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
349	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
350	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
351	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
352	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
353	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
354	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
355	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
356	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
357	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
358	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
359	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
360	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
361	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
362	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
363	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
364	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
365	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
366	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
367	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
368	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
369	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
370	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
371	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
372	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
373	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
374	C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
375	C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
376	C₄²	C₂³×C₅₃	1	1	yes	characteristic
377	C₄²	C₂³×C₅₃	1	2	yes	—
378	C₄²	C₂³×C₅₃	1	1	yes	characteristic
379	C₄²	C₂³×C₅₃	1	1	yes	characteristic
380	C₄²	C₂³×C₅₃	1	1	yes	characteristic
381	C₄²	C₂³×C₅₃	1	2	yes	—
382	C₄²	C₂³×C₅₃	1	2	yes	—
383	C₄²	C₂³×C₅₃	1	4	yes	—
384	C₄²	C₂³×C₅₃	1	2	yes	—
385	C₂³	—	2	8	no	—
386	C₂³	—	4	4	no	—
387	C₂³	—	4	4	no	—
388	C₂³	C₂⁴×C₅₃	1	2	yes	—
389	C₂³	C₂⁴×C₅₃	1	4	yes	—
390	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
391	C₂³	C₂⁴×C₅₃	1	2	yes	—
392	Q₈	—	2	2	no	—
393	Q₈	—	2	2	no	—
394	Q₈	—	2	4	no	—
395	Q₈	—	4	4	no	—
396	D₄	—	2	8	no	—
397	D₄	—	2	4	no	—
398	D₄	—	2	4	no	—
399	D₄	—	2	2	no	—
400	D₄	—	2	4	no	—
401	D₄	—	4	4	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	4	no	—
405	D₄	#41×C₅₃	1	4	yes	complemented
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	4	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	4	no	—
416	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
417	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
418	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
419	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
420	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
421	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
422	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
424	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
425	C₂×C₄	C₂⁴×C₅₃	1	2	yes	—
426	C₂×C₄	C₂⁴×C₅₃	1	4	yes	—
427	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
428	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
429	C₂²	—	2	2	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	1	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	4	no	—
434	C₂²	—	2	1	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	2	4	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	8	8	no	—
441	C₂²	C₂×#41×C₅₃	1	4	yes	—
442	C₂²	C₂⁵×C₅₃	1	1	yes	characteristic
443	C₄	—	2	1	no	—
444	C₄	—	2	2	no	—
445	C₄	—	2	2	no	—
446	C₄	—	2	1	no	—
447	C₄	—	2	2	no	—
448	C₄	—	2	4	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	1	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	2	1	no	—
454	C₄	C₂×#41×C₅₃	1	2	yes	—
455	C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
456	C₂	—	2	4	no	—
457	C₂	—	4	2	no	—
458	C₂	—	4	4	no	—
459	C₂	—	4	1	no	—
460	C₂	C₂²×#41×C₅₃	1	1	yes	characteristic
461	C₂	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
462	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. C₅₃×#559	C₂	semidirect
2. C₅₃×#556	C₂	semidirect
3. C₅₃×#555	C₂	semidirect
4. C₅₃×#551	C₂	semidirect
5. D₄×Q₈×C₅₃	C₂	semidirect
6. C₅₃×#548	C₂	semidirect
7. C₅₃×#546	C₂	semidirect
8. C₅₃×#546	C₂	semidirect
9. D₄²×C₅₃	C₂	semidirect
10. C₅₃×#542	C₂	semidirect
11. C₅₃×#538	C₂	semidirect
12. C₅₃×#532	C₂	semidirect
13. C₅₃×#529	C₂	semidirect
14. C₅₃×#529	C₂	semidirect
15. C₂×#124×C₅₃	C₂	semidirect
16. C₅₃×#518	C₂	semidirect
17. C₄×#41×C₅₃	C₂	semidirect
19. C₂×#41×C₅₃	C₂²	semidirect
20. C₂²×Q₈×C₅₃	C₂²	semidirect
23. C₄⋊Q₈×C₅₃	C₂²	semidirect
24. C₄⋊Q₈×C₅₃	C₂²	semidirect
26. #126×C₅₃	C₂²	semidirect
27. #126×C₅₃	C₂²	semidirect
28. #126×C₅₃	C₂²	semidirect
29. #125×C₅₃	C₂²	semidirect
32. #124×C₅₃	C₂²	semidirect
33. #124×C₅₃	C₂²	semidirect
34. #124×C₅₃	C₂²	semidirect
35. #124×C₅₃	C₂²	semidirect
36. #124×C₅₃	C₂²	semidirect
37. #124×C₅₃	C₂²	semidirect
38. #123×C₅₃	C₂²	semidirect
39. #123×C₅₃	C₂²	semidirect
40. #123×C₅₃	C₂²	semidirect
41. #123×C₅₃	C₂²	semidirect
42. C₂²⋊Q₈×C₅₃	C₂²	semidirect
43. C₂²⋊Q₈×C₅₃	C₂²	semidirect
44. C₂²⋊Q₈×C₅₃	C₂²	semidirect
45. C₂²⋊Q₈×C₅₃	C₂²	semidirect
46. C₂²⋊Q₈×C₅₃	C₂²	semidirect
50. #121×C₅₃	C₂²	semidirect
51. #121×C₅₃	C₂²	semidirect
52. #121×C₅₃	C₂²	semidirect
53. #121×C₅₃	C₂²	semidirect
55. #120×C₅₃	C₂²	semidirect
56. C₄×Q₈×C₅₃	C₂²	semidirect
57. C₄×Q₈×C₅₃	C₂²	semidirect
58. C₄×Q₈×C₅₃	C₂²	semidirect
59. C₄×Q₈×C₅₃	C₂²	semidirect
64. C₄×D₄×C₅₃	C₂²	semidirect
65. C₄×D₄×C₅₃	C₂²	semidirect
66. C₄×D₄×C₅₃	C₂²	semidirect
67. C₄×D₄×C₅₃	C₂²	semidirect
68. C₄×D₄×C₅₃	C₂²	semidirect
69. C₄×D₄×C₅₃	C₂²	semidirect
70. C₄×D₄×C₅₃	C₂²	semidirect
71. C₄×D₄×C₅₃	C₂²	semidirect
72. C₄×D₄×C₅₃	C₂²	semidirect
73. #117×C₅₃	C₂²	semidirect
74. #117×C₅₃	C₂²	semidirect
75. #117×C₅₃	C₂²	semidirect
76. #117×C₅₃	C₂²	semidirect
77. #117×C₅₃	C₂²	semidirect
78. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
79. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
80. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
81. C₂×C₄²×C₅₃	C₂²	semidirect
82. C₂×C₄²×C₅₃	C₂²	semidirect
173. D₄×C₅₃	#41	semidirect
224. #3514	C₅₃	direct (N × H)
232. #559	C₂×C₅₃	semidirect
233. #556	C₂×C₅₃	semidirect
234. #555	C₂×C₅₃	semidirect
235. #551	C₂×C₅₃	semidirect
236. D₄×Q₈	C₂×C₅₃	semidirect
237. #548	C₂×C₅₃	semidirect
238. #546	C₂×C₅₃	semidirect
239. #546	C₂×C₅₃	semidirect
240. D₄²	C₂×C₅₃	semidirect
241. #542	C₂×C₅₃	semidirect
242. #538	C₂×C₅₃	semidirect
243. #532	C₂×C₅₃	semidirect
244. #529	C₂×C₅₃	semidirect
245. #529	C₂×C₅₃	semidirect
246. C₂×#124	C₂×C₅₃	semidirect
247. #518	C₂×C₅₃	semidirect
248. C₄×#41	C₂×C₅₃	semidirect
249. C₅₃	#3514	direct (N × H)
251. C₂×#41	C₂²×C₅₃	semidirect
252. C₂²×Q₈	C₂²×C₅₃	semidirect
255. C₄⋊Q₈	C₂²×C₅₃	semidirect
256. C₄⋊Q₈	C₂²×C₅₃	semidirect
258. #126	C₂²×C₅₃	semidirect
259. #126	C₂²×C₅₃	semidirect
260. #126	C₂²×C₅₃	semidirect
261. #125	C₂²×C₅₃	semidirect
264. #124	C₂²×C₅₃	semidirect
265. #124	C₂²×C₅₃	semidirect
266. #124	C₂²×C₅₃	semidirect
267. #124	C₂²×C₅₃	semidirect
268. #124	C₂²×C₅₃	semidirect
269. #124	C₂²×C₅₃	semidirect
270. #123	C₂²×C₅₃	semidirect
271. #123	C₂²×C₅₃	semidirect
272. #123	C₂²×C₅₃	semidirect
273. #123	C₂²×C₅₃	semidirect
274. C₂²⋊Q₈	C₂²×C₅₃	semidirect
275. C₂²⋊Q₈	C₂²×C₅₃	semidirect
276. C₂²⋊Q₈	C₂²×C₅₃	semidirect
277. C₂²⋊Q₈	C₂²×C₅₃	semidirect
278. C₂²⋊Q₈	C₂²×C₅₃	semidirect
282. #121	C₂²×C₅₃	semidirect
283. #121	C₂²×C₅₃	semidirect
284. #121	C₂²×C₅₃	semidirect
285. #121	C₂²×C₅₃	semidirect
287. #120	C₂²×C₅₃	semidirect
288. C₄×Q₈	C₂²×C₅₃	semidirect
289. C₄×Q₈	C₂²×C₅₃	semidirect
290. C₄×Q₈	C₂²×C₅₃	semidirect
291. C₄×Q₈	C₂²×C₅₃	semidirect
296. C₄×D₄	C₂²×C₅₃	semidirect
297. C₄×D₄	C₂²×C₅₃	semidirect
298. C₄×D₄	C₂²×C₅₃	semidirect
299. C₄×D₄	C₂²×C₅₃	semidirect
300. C₄×D₄	C₂²×C₅₃	semidirect
301. C₄×D₄	C₂²×C₅₃	semidirect
302. C₄×D₄	C₂²×C₅₃	semidirect
303. C₄×D₄	C₂²×C₅₃	semidirect
304. C₄×D₄	C₂²×C₅₃	semidirect
305. #117	C₂²×C₅₃	semidirect
306. #117	C₂²×C₅₃	semidirect
307. #117	C₂²×C₅₃	semidirect
308. #117	C₂²×C₅₃	semidirect
309. #117	C₂²×C₅₃	semidirect
310. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
311. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
312. C₂×C₂²⋊C₄	C₂²×C₅₃	semidirect
313. C₂×C₄²	C₂²×C₅₃	semidirect
314. C₂×C₄²	C₂²×C₅₃	semidirect
405. D₄	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#559	2
C₅₃×#556	2
C₅₃×#555	2
C₅₃×#551	2
D₄×Q₈×C₅₃	2
C₅₃×#548	2
C₅₃×#546	2
D₄²×C₅₃	2
C₅₃×#542	2
C₅₃×#538	2
C₅₃×#532	2
C₅₃×#529	2
C₂×#124×C₅₃	2
C₅₃×#518	2
C₄×#41×C₅₃	2
#3514	53

Subgroups — C53×#3514 #20272451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3514 #20272451