Subgroups — C₂×C₅₃×#1632 #27225989

Order 13568.

1054 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#476	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1632	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₅₃	—	2	2	no	—
10	C₅₃×#554	—	2	2	no	—
11	C₅₃×#554	—	2	2	no	—
12	C₅₃×#554	—	2	2	no	—
13	C₅₃×#554	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#125×C₅₃	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	—	2	1	no	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₅₃×#494	—	2	2	no	—
24	C₅₃×#494	C₂²	1	4	yes	complemented
25	C₅₃×#477	—	2	2	no	—
26	C₅₃×#477	C₂²	1	4	yes	complemented
27	C₅₃×#476	—	2	2	no	—
28	C₅₃×#476	C₂²	1	4	yes	—
29	C₂×#107×C₅₃	—	2	1	no	—
30	C₂×#106×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#103×C₅₃	—	2	1	no	—
34	C₂×#102×C₅₃	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#333	C₂²	1	4	yes	complemented
37	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
38	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
39	C₂²×Q₈×C₅₃	—	2	1	no	—
40	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
42	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	—	2	2	no	—
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂×Q₁₆×C₅₃	—	2	8	no	—
46	C₂×Q₁₆×C₅₃	—	4	8	no	—
47	C₂²×C₈×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	—	2	1	no	—
49	C₂²×C₈×C₅₃	—	2	1	no	—
50	C₄⋊Q₈×C₅₃	—	2	1	no	—
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
58	#125×C₅₃	—	2	4	no	—
59	#125×C₅₃	—	2	4	no	—
60	C₂²⋊Q₈×C₅₃	—	2	8	no	—
61	C₂²⋊Q₈×C₅₃	—	2	4	no	—
62	C₂²⋊Q₈×C₅₃	—	2	8	no	—
63	C₂²⋊Q₈×C₅₃	—	2	4	no	—
64	C₄×Q₈×C₅₃	—	2	1	no	—
65	C₄×Q₈×C₅₃	—	2	1	no	—
66	C₄×Q₈×C₅₃	D₄	1	2	yes	—
67	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
68	C₄×D₄×C₅₃	—	2	4	no	—
69	C₄×D₄×C₅₃	—	2	1	no	—
70	C₄×D₄×C₅₃	—	2	1	no	—
71	C₄×D₄×C₅₃	—	2	4	no	—
72	C₄×D₄×C₅₃	D₄	1	2	yes	—
73	C₄×D₄×C₅₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
76	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
87	C₂×C₄²×C₅₃	—	2	1	no	—
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	#107×C₅₃	—	2	4	no	—
90	#106×C₅₃	—	2	4	no	—
91	C₄⋊C₈×C₅₃	—	2	1	no	—
92	C₄⋊C₈×C₅₃	—	2	1	no	—
93	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
94	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
95	#103×C₅₃	—	2	4	no	—
96	#102×C₅₃	—	2	4	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	C₂³	1	2	yes	—
99	C₂⁴×C₅₃	—	2	1	no	—
100	C₂×Q₈×C₅₃	—	2	1	no	—
101	C₂×Q₈×C₅₃	—	2	2	no	—
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	—	4	4	no	—
106	C₂×Q₈×C₅₃	—	4	4	no	—
107	C₂×Q₈×C₅₃	—	4	1	no	—
108	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
109	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	Q₁₆	1	4	yes	complemented
113	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	2	no	—
118	C₂²×C₄×C₅₃	—	2	8	no	—
119	C₂²×C₄×C₅₃	—	2	2	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	—	4	2	no	—
122	C₂²×C₄×C₅₃	—	4	2	no	—
123	C₂²×C₄×C₅₃	—	4	1	no	—
124	C₂²×C₄×C₅₃	—	8	4	no	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	Q₁₆×C₅₃	—	4	16	no	—
129	C₂×C₈×C₅₃	—	2	4	no	—
130	C₂×C₈×C₅₃	—	2	2	no	—
131	C₂×C₈×C₅₃	—	2	2	no	—
132	C₂×C₈×C₅₃	—	2	2	no	—
133	C₂×C₈×C₅₃	—	4	2	no	—
134	C₂×C₈×C₅₃	—	4	2	no	—
135	C₄⋊C₄×C₅₃	—	2	8	no	—
136	C₄⋊C₄×C₅₃	—	2	8	no	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	—	2	2	no	—
139	C₄⋊C₄×C₅₃	—	2	2	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	1	no	—
142	C₄⋊C₄×C₅₃	—	2	2	no	—
143	C₄⋊C₄×C₅₃	—	2	2	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	—	2	2	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	4	1	no	—
148	C₄⋊C₄×C₅₃	—	4	1	no	—
149	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₅₃	—	2	8	no	—
152	C₂²⋊C₄×C₅₃	—	2	2	no	—
153	C₂²⋊C₄×C₅₃	—	2	2	no	—
154	C₄²×C₅₃	—	2	2	no	—
155	C₄²×C₅₃	—	2	2	no	—
156	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
160	C₂³×C₅₃	—	2	2	no	—
161	C₂³×C₅₃	—	2	4	no	—
162	C₂³×C₅₃	—	4	2	no	—
163	C₂³×C₅₃	—	4	2	no	—
164	C₂³×C₅₃	#120	1	1	yes	characteristic
165	Q₈×C₅₃	—	2	2	no	—
166	Q₈×C₅₃	—	2	2	no	—
167	Q₈×C₅₃	—	4	4	no	—
168	Q₈×C₅₃	—	4	4	no	—
169	Q₈×C₅₃	—	4	2	no	—
170	Q₈×C₅₃	—	8	2	no	—
171	Q₈×C₅₃	—	8	2	no	—
172	D₄×C₅₃	—	2	2	no	—
173	D₄×C₅₃	—	2	4	no	—
174	D₄×C₅₃	C₂×Q₁₆	1	4	yes	complemented
175	C₂×C₄×C₅₃	—	2	2	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	—	2	2	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	2	no	—
182	C₂×C₄×C₅₃	—	4	2	no	—
183	C₂×C₄×C₅₃	—	4	4	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	—	4	2	no	—
186	C₂×C₄×C₅₃	—	4	4	no	—
187	C₂×C₄×C₅₃	—	4	2	no	—
188	C₂×C₄×C₅₃	—	8	8	no	—
189	C₂×C₄×C₅₃	—	8	2	no	—
190	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
191	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
192	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
194	C₂×C₄×C₅₃	#135	1	2	yes	—
195	C₂×C₄×C₅₃	#137	1	2	yes	—
196	C₂×C₄×C₅₃	#137	1	2	yes	—
197	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
200	C₈×C₅₃	—	2	4	no	—
201	C₈×C₅₃	—	4	2	no	—
202	C₈×C₅₃	—	4	2	no	—
203	C₂×#1632	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂²×C₅₃	—	2	4	no	—
205	C₂²×C₅₃	—	4	8	no	—
206	C₂²×C₅₃	—	4	2	no	—
207	C₂²×C₅₃	—	4	2	no	—
208	C₂²×C₅₃	#451	1	2	yes	—
209	C₂²×C₅₃	#452	1	2	yes	—
210	C₂²×C₅₃	#453	1	2	yes	—
211	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
212	C₄×C₅₃	—	2	1	no	—
213	C₄×C₅₃	—	2	1	no	—
214	C₄×C₅₃	—	4	2	no	—
215	C₄×C₅₃	—	4	2	no	—
216	C₄×C₅₃	—	4	2	no	—
217	C₄×C₅₃	—	4	4	no	—
218	C₄×C₅₃	—	8	2	no	—
219	C₄×C₅₃	#451	1	2	yes	—
220	C₄×C₅₃	#452	1	2	yes	—
221	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
222	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
223	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
224	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
225	C₂×#554	C₂×C₅₃	1	1	yes	characteristic
226	C₂×#494	C₂×C₅₃	1	1	yes	characteristic, complemented
227	C₂×#477	C₂×C₅₃	1	1	yes	characteristic, complemented
228	C₂×#476	C₂×C₅₃	1	1	yes	characteristic
229	#1632	C₂×C₅₃	1	8	yes	direct product factor
230	C₂×#333	C₂×C₅₃	1	1	yes	characteristic, complemented
231	C₂×Q₈⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
232	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
233	C₂×C₅₃	—	4	4	no	—
234	C₂×C₅₃	#1632	1	4	yes	direct product factor
235	C₂×C₅₃	C₂×#451	1	1	yes	characteristic
236	C₂×C₅₃	C₂×#452	1	1	yes	characteristic
237	C₂×C₅₃	C₂×#453	1	1	yes	characteristic
238	C₂²×Q₁₆	—	2	2	no	—
239	#554	—	2	2	no	—
240	#554	—	2	2	no	—
241	#554	—	2	2	no	—
242	#554	C₂²×C₅₃	1	4	yes	—
243	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×#125	—	2	1	no	—
246	C₂×C₂²⋊Q₈	—	2	1	no	—
247	C₂×C₂²⋊Q₈	—	2	2	no	—
248	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
249	C₂×C₄×D₄	—	2	1	no	—
250	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
251	C₂²×C₄⋊C₄	—	2	1	no	—
252	#494	—	2	2	no	—
253	#494	C₂²×C₅₃	1	4	yes	complemented
254	#477	—	2	2	no	—
255	#477	C₂²×C₅₃	1	4	yes	complemented
256	#476	—	2	2	no	—
257	#476	C₂²×C₅₃	1	4	yes	—
258	C₂×#107	—	2	1	no	—
259	C₂×#106	—	2	1	no	—
260	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×#103	—	2	1	no	—
263	C₂×#102	—	2	1	no	—
264	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
265	#333	C₂²×C₅₃	1	4	yes	complemented
266	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	complemented
267	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
268	C₅₃	C₂×#1632	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²×Q₈	—	2	1	no	—
270	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
271	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
272	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
273	C₂³×C₄	—	2	2	no	—
274	C₂³×C₄	—	2	1	no	—
275	C₂×Q₁₆	—	2	8	no	—
276	C₂×Q₁₆	—	4	8	no	—
277	C₂²×C₈	—	2	1	no	—
278	C₂²×C₈	—	2	1	no	—
279	C₂²×C₈	—	2	1	no	—
280	C₄⋊Q₈	—	2	1	no	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	—	2	1	no	—
283	C₄⋊Q₈	—	2	1	no	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
286	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
287	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
288	#125	—	2	4	no	—
289	#125	—	2	4	no	—
290	C₂²⋊Q₈	—	2	4	no	—
291	C₂²⋊Q₈	—	2	8	no	—
292	C₂²⋊Q₈	—	2	4	no	—
293	C₂²⋊Q₈	—	2	8	no	—
294	C₄×Q₈	—	2	1	no	—
295	C₄×Q₈	—	2	1	no	—
296	C₄×Q₈	D₄×C₅₃	1	2	yes	—
297	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
298	C₄×D₄	—	2	4	no	—
299	C₄×D₄	—	2	4	no	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	—	2	1	no	—
302	C₄×D₄	D₄×C₅₃	1	2	yes	—
303	C₄×D₄	C₂³×C₅₃	1	2	yes	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	4	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	4	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
314	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	1	no	—
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
319	#107	—	2	4	no	—
320	#106	—	2	4	no	—
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
324	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
325	#103	—	2	4	no	—
326	#102	—	2	4	no	—
327	C₄×C₈	—	2	1	no	—
328	C₄×C₈	C₂³×C₅₃	1	2	yes	—
329	C₂⁴	—	2	1	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	4	4	no	—
336	C₂×Q₈	—	4	4	no	—
337	C₂×Q₈	—	4	1	no	—
338	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
339	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	Q₁₆×C₅₃	1	4	yes	complemented
343	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	4	2	no	—
352	C₂²×C₄	—	4	2	no	—
353	C₂²×C₄	—	4	1	no	—
354	C₂²×C₄	—	8	4	no	—
355	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
358	Q₁₆	—	4	16	no	—
359	C₂×C₈	—	2	4	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	4	2	no	—
364	C₂×C₈	—	4	2	no	—
365	C₄⋊C₄	—	2	8	no	—
366	C₄⋊C₄	—	2	8	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	4	1	no	—
378	C₄⋊C₄	—	4	1	no	—
379	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
380	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
381	C₂²⋊C₄	—	2	8	no	—
382	C₂²⋊C₄	—	2	2	no	—
383	C₂²⋊C₄	—	2	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
387	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
388	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
389	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
390	C₂³	—	2	2	no	—
391	C₂³	—	2	4	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	#120×C₅₃	1	1	yes	characteristic
395	Q₈	—	2	2	no	—
396	Q₈	—	2	2	no	—
397	Q₈	—	4	4	no	—
398	Q₈	—	4	4	no	—
399	Q₈	—	4	2	no	—
400	Q₈	—	8	2	no	—
401	Q₈	—	8	2	no	—
402	D₄	—	2	2	no	—
403	D₄	—	2	4	no	—
404	D₄	C₂×Q₁₆×C₅₃	1	4	yes	complemented
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	4	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	4	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	8	8	no	—
419	C₂×C₄	—	8	2	no	—
420	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
421	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
422	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
424	C₂×C₄	#135×C₅₃	1	2	yes	—
425	C₂×C₄	#137×C₅₃	1	2	yes	—
426	C₂×C₄	#137×C₅₃	1	2	yes	—
427	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
430	C₈	—	2	4	no	—
431	C₈	—	4	2	no	—
432	C₈	—	4	2	no	—
433	C₂²	—	2	4	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	8	no	—
437	C₂²	C₅₃×#451	1	2	yes	—
438	C₂²	C₅₃×#452	1	2	yes	—
439	C₂²	C₅₃×#453	1	2	yes	—
440	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
441	C₄	—	2	1	no	—
442	C₄	—	2	1	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	4	no	—
447	C₄	—	8	2	no	—
448	C₄	C₅₃×#451	1	2	yes	—
449	C₄	C₅₃×#452	1	2	yes	—
450	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
451	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
452	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
453	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
454	C₂	—	4	4	no	—
455	C₂	C₅₃×#1632	1	4	yes	direct product factor
456	C₂	C₂×C₅₃×#451	1	1	yes	characteristic
457	C₂	C₂×C₅₃×#452	1	1	yes	characteristic
458	C₂	C₂×C₅₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#494	C₂	semidirect
3. C₂×C₅₃×#477	C₂	semidirect
5. C₅₃×#1632	C₂	direct (N × H)
6. C₂×C₅₃×#333	C₂	semidirect
7. C₂×C₅₃×Q₈⋊C₈	C₂	semidirect
24. C₅₃×#494	C₂²	semidirect
26. C₅₃×#477	C₂²	semidirect
36. C₅₃×#333	C₂²	semidirect
37. C₅₃×Q₈⋊C₈	C₂²	semidirect
112. C₂×D₄×C₅₃	Q₁₆	semidirect
174. D₄×C₅₃	C₂×Q₁₆	semidirect
203. C₂×#1632	C₅₃	direct (N × H)
226. C₂×#494	C₂×C₅₃	semidirect
227. C₂×#477	C₂×C₅₃	semidirect
229. #1632	C₂×C₅₃	direct (N × H)
230. C₂×#333	C₂×C₅₃	semidirect
231. C₂×Q₈⋊C₈	C₂×C₅₃	semidirect
234. C₂×C₅₃	#1632	direct (N × H)
253. #494	C₂²×C₅₃	semidirect
255. #477	C₂²×C₅₃	semidirect
265. #333	C₂²×C₅₃	semidirect
266. Q₈⋊C₈	C₂²×C₅₃	semidirect
268. C₅₃	C₂×#1632	direct (N × H)
342. C₂×D₄	Q₁₆×C₅₃	semidirect
404. D₄	C₂×Q₁₆×C₅₃	semidirect
455. C₂	C₅₃×#1632	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#554	2
C₂×C₅₃×#494	2
C₂×C₅₃×#477	2
C₂×C₅₃×#476	2
C₅₃×#1632	2
C₂×C₅₃×#333	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×C₅₃×D₄⋊C₈	2
C₂×#1632	53

Subgroups — C2×C53×#1632 #27225989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1632 #27225989