Subgroups — C₂×C₅₃×#1812 #27228182

Order 13568.

1346 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#427	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#410	C₂	1	2	yes	maximal
4	C₅₃×#1812	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#131×C₅₃	—	2	2	no	—
8	C₂×#131×C₅₃	—	2	2	no	—
9	C₂×#131×C₅₃	—	2	4	no	—
10	C₂×#131×C₅₃	C₄	1	4	yes	complemented
11	C₂×#131×C₅₃	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₅₃	C₄	1	2	yes	complemented
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₄	1	2	yes	complemented
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#427	—	2	4	no	—
18	C₅₃×#427	C₂²	1	8	yes	complemented
19	C₂×#107×C₅₃	—	2	2	no	—
20	C₂×#106×C₅₃	—	2	2	no	—
21	C₅₃×#410	—	2	4	no	—
22	C₅₃×#410	C₂²	1	8	yes	—
23	C₂×#99×C₅₃	—	2	4	no	—
24	C₅₃×H(3,ℤ/4)	C₂²	1	8	yes	complemented
25	C₂×#41×C₅₃	—	2	2	no	—
26	C₂×#41×C₅₃	—	2	2	no	—
27	C₂×#41×C₅₃	—	2	2	no	—
28	C₂×#41×C₅₃	—	2	2	no	—
29	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×#41×C₅₃	D₄	1	2	yes	—
31	C₂×#41×C₅₃	Q₈	1	2	yes	—
32	C₂×#41×C₅₃	C₂³	1	2	yes	—
33	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₂²×D₄×C₅₃	D₄	1	2	yes	—
35	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	#131×C₅₃	—	2	8	no	—
39	#131×C₅₃	—	2	8	no	—
40	#131×C₅₃	—	2	8	no	—
41	#131×C₅₃	—	2	8	no	—
42	C₂×M₁₆×C₅₃	—	2	4	no	—
43	C₂×M₁₆×C₅₃	—	2	4	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
48	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	—	2	2	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	—	2	4	no	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
57	C₂²×C₈×C₅₃	D₄	1	2	yes	—
58	C₂²×C₈×C₅₃	D₄	1	2	yes	—
59	C₂²×C₈×C₅₃	D₄	1	4	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
61	#117×C₅₃	—	2	4	no	—
62	#117×C₅₃	—	2	4	no	—
63	#117×C₅₃	—	2	4	no	—
64	#117×C₅₃	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
67	C₂×C₄²×C₅₃	—	2	4	no	—
68	#107×C₅₃	—	2	8	no	—
69	#106×C₅₃	—	2	8	no	—
70	#99×C₅₃	—	2	16	no	—
71	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
73	#41×C₅₃	—	2	4	no	—
74	#41×C₅₃	—	2	4	no	—
75	#41×C₅₃	—	2	4	no	—
76	#41×C₅₃	—	2	4	no	—
77	#41×C₅₃	—	2	4	no	—
78	#41×C₅₃	—	2	4	no	—
79	#41×C₅₃	—	2	4	no	—
80	#41×C₅₃	—	2	4	no	—
81	C₂×Q₈×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₂×Q₈×C₅₃	C₂×Q₈	1	2	yes	—
85	C₂×D₄×C₅₃	—	2	4	no	—
86	C₂×D₄×C₅₃	—	2	4	no	—
87	C₂×D₄×C₅₃	—	2	2	no	—
88	C₂×D₄×C₅₃	—	2	2	no	—
89	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
90	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
91	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
92	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	4	4	no	—
98	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	#41	1	2	yes	—
109	C₂²×C₄×C₅₃	#41	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
111	M₁₆×C₅₃	—	2	4	no	—
112	M₁₆×C₅₃	—	2	4	no	—
113	M₁₆×C₅₃	—	2	16	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	8	no	—
117	C₂×C₈×C₅₃	—	2	8	no	—
118	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
121	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
122	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
123	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
124	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
129	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
130	C₄⋊C₄×C₅₃	—	2	8	no	—
131	C₄⋊C₄×C₅₃	—	4	4	no	—
132	C₂²⋊C₄×C₅₃	—	2	8	no	—
133	C₂²⋊C₄×C₅₃	—	4	4	no	—
134	C₄²×C₅₃	—	2	8	no	—
135	C₄²×C₅₃	—	4	4	no	—
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	—	2	2	no	—
138	C₂³×C₅₃	—	4	2	no	—
139	C₂³×C₅₃	—	4	4	no	—
140	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
142	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂³×C₅₃	#121	1	2	yes	—
144	C₂³×C₅₃	#121	1	2	yes	—
145	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₅₃	—	2	4	no	—
147	Q₈×C₅₃	—	2	4	no	—
148	D₄×C₅₃	—	2	8	no	—
149	D₄×C₅₃	—	2	8	no	—
150	D₄×C₅₃	—	2	4	no	—
151	D₄×C₅₃	—	2	4	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	4	no	—
155	C₂×C₄×C₅₃	—	2	4	no	—
156	C₂×C₄×C₅₃	—	2	4	no	—
157	C₂×C₄×C₅₃	—	2	4	no	—
158	C₂×C₄×C₅₃	—	4	8	no	—
159	C₂×C₄×C₅₃	—	8	8	no	—
160	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
164	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
167	C₂×C₄×C₅₃	#117	1	2	yes	—
168	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
169	C₂×C₄×C₅₃	#121	1	2	yes	—
170	C₂×C₄×C₅₃	#121	1	2	yes	—
171	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
172	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
173	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
174	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
175	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
179	C₈×C₅₃	—	2	4	no	—
180	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
181	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
182	C₈×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
183	C₂×#1812	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂²×C₅₃	—	2	4	no	—
185	C₂²×C₅₃	—	2	2	no	—
186	C₂²×C₅₃	—	2	4	no	—
187	C₂²×C₅₃	—	4	1	no	—
188	C₂²×C₅₃	—	4	2	no	—
189	C₂²×C₅₃	—	4	1	no	—
190	C₂²×C₅₃	—	4	4	no	—
191	C₂²×C₅₃	—	4	4	no	—
192	C₂²×C₅₃	—	4	4	no	—
193	C₂²×C₅₃	#380	1	1	yes	characteristic
194	C₂²×C₅₃	#380	1	2	yes	—
195	C₂²×C₅₃	#380	1	1	yes	characteristic
196	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₅₃	C₂×#121	1	2	yes	—
198	C₄×C₅₃	—	2	4	no	—
199	C₄×C₅₃	—	8	8	no	—
200	C₄×C₅₃	#380	1	1	yes	characteristic
201	C₄×C₅₃	#380	1	1	yes	characteristic
202	C₄×C₅₃	#380	1	2	yes	—
203	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
204	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
205	C₄×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
206	C₂²×#131	C₂×C₅₃	1	1	yes	characteristic
207	C₂×#427	C₂×C₅₃	1	2	yes	complemented
208	C₂×#410	C₂×C₅₃	1	2	yes	—
209	#1812	C₂×C₅₃	1	8	yes	direct product factor
210	C₂×H(3,ℤ/4)	C₂×C₅₃	1	2	yes	complemented
211	C₂×C₅₃	—	2	2	no	—
212	C₂×C₅₃	—	2	1	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	—	2	2	no	—
215	C₂×C₅₃	—	4	4	no	—
216	C₂×C₅₃	#1812	1	2	yes	direct product factor
217	C₂×C₅₃	C₂×#380	1	1	yes	characteristic
218	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
219	C₂×#131	—	2	2	no	—
220	C₂×#131	—	2	2	no	—
221	C₂×#131	—	2	4	no	—
222	C₂×#131	C₄×C₅₃	1	4	yes	complemented
223	C₂×#131	C₂²×C₅₃	1	4	yes	—
224	C₂²×M₁₆	C₄×C₅₃	1	2	yes	complemented
225	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
226	C₂³×C₈	C₄×C₅₃	1	2	yes	complemented
227	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×#117	C₂²×C₅₃	1	4	yes	—
229	#427	—	2	4	no	—
230	#427	C₂²×C₅₃	1	8	yes	complemented
231	C₂×#107	—	2	2	no	—
232	C₂×#106	—	2	2	no	—
233	#410	—	2	4	no	—
234	#410	C₂²×C₅₃	1	8	yes	—
235	C₂×#99	—	2	4	no	—
236	H(3,ℤ/4)	C₂²×C₅₃	1	8	yes	complemented
237	C₅₃	C₂×#1812	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×#41	—	2	2	no	—
239	C₂×#41	—	2	2	no	—
240	C₂×#41	—	2	2	no	—
241	C₂×#41	—	2	2	no	—
242	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
243	C₂×#41	D₄×C₅₃	1	2	yes	—
244	C₂×#41	Q₈×C₅₃	1	2	yes	—
245	C₂×#41	C₂³×C₅₃	1	2	yes	—
246	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
247	C₂²×D₄	D₄×C₅₃	1	2	yes	—
248	C₂²×D₄	Q₈×C₅₃	1	1	yes	characteristic
249	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
250	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
251	#131	—	2	8	no	—
252	#131	—	2	8	no	—
253	#131	—	2	8	no	—
254	#131	—	2	8	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	4	no	—
258	C₂×M₁₆	—	2	4	no	—
259	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
260	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	complemented
261	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
262	C₂²×C₈	—	2	2	no	—
263	C₂²×C₈	—	2	2	no	—
264	C₂²×C₈	—	2	4	no	—
265	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
266	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
267	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
268	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
269	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	complemented
270	C₂²×C₈	D₄×C₅₃	1	2	yes	—
271	C₂²×C₈	D₄×C₅₃	1	2	yes	—
272	C₂²×C₈	D₄×C₅₃	1	4	yes	—
273	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
274	#117	—	2	4	no	—
275	#117	—	2	4	no	—
276	#117	—	2	4	no	—
277	#117	C₂³×C₅₃	1	8	yes	—
278	C₂×C₄⋊C₄	—	2	4	no	—
279	C₂×C₂²⋊C₄	—	2	4	no	—
280	C₂×C₄²	—	2	4	no	—
281	#107	—	2	8	no	—
282	#106	—	2	8	no	—
283	#99	—	2	16	no	—
284	C₂⁴	C₄⋊C₄×C₅₃	1	2	yes	—
285	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
286	#41	—	2	4	no	—
287	#41	—	2	4	no	—
288	#41	—	2	4	no	—
289	#41	—	2	4	no	—
290	#41	—	2	4	no	—
291	#41	—	2	4	no	—
292	#41	—	2	4	no	—
293	#41	—	2	4	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	C₄⋊C₄×C₅₃	1	2	yes	—
297	C₂×Q₈	C₂×Q₈×C₅₃	1	2	yes	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	4	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
303	C₂×D₄	C₄⋊C₄×C₅₃	1	2	yes	—
304	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
305	C₂×D₄	C₂×Q₈×C₅₃	1	2	yes	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	4	4	no	—
311	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
312	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
313	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
314	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
318	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
319	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
320	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
321	C₂²×C₄	#41×C₅₃	1	2	yes	—
322	C₂²×C₄	#41×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
324	M₁₆	—	2	4	no	—
325	M₁₆	—	2	4	no	—
326	M₁₆	—	2	16	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	8	no	—
330	C₂×C₈	—	2	8	no	—
331	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
332	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
333	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
334	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
335	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
336	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
337	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
338	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
339	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
340	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
342	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
343	C₄⋊C₄	—	2	8	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₂²⋊C₄	—	2	8	no	—
346	C₂²⋊C₄	—	4	4	no	—
347	C₄²	—	2	8	no	—
348	C₄²	—	4	4	no	—
349	C₂³	—	2	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	4	no	—
353	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
354	C₂³	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
355	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
356	C₂³	#121×C₅₃	1	2	yes	—
357	C₂³	#121×C₅₃	1	2	yes	—
358	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
359	Q₈	—	2	4	no	—
360	Q₈	—	2	4	no	—
361	D₄	—	2	8	no	—
362	D₄	—	2	8	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	4	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	4	8	no	—
372	C₂×C₄	—	8	8	no	—
373	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
376	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
377	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
379	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
380	C₂×C₄	#117×C₅₃	1	2	yes	—
381	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
382	C₂×C₄	#121×C₅₃	1	2	yes	—
383	C₂×C₄	#121×C₅₃	1	2	yes	—
384	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
385	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
386	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
387	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
388	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
389	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
390	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
392	C₈	—	2	4	no	—
393	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
394	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
395	C₈	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
396	C₂²	—	2	4	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	4	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	4	no	—
403	C₂²	—	4	4	no	—
404	C₂²	—	4	4	no	—
405	C₂²	C₅₃×#380	1	1	yes	characteristic
406	C₂²	C₅₃×#380	1	2	yes	—
407	C₂²	C₅₃×#380	1	1	yes	characteristic
408	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
409	C₂²	C₂×#121×C₅₃	1	2	yes	—
410	C₄	—	2	4	no	—
411	C₄	—	8	8	no	—
412	C₄	C₅₃×#380	1	1	yes	characteristic
413	C₄	C₅₃×#380	1	1	yes	characteristic
414	C₄	C₅₃×#380	1	2	yes	—
415	C₄	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
416	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
417	C₄	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
418	C₂	—	2	2	no	—
419	C₂	—	2	1	no	—
420	C₂	—	2	1	no	—
421	C₂	—	2	2	no	—
422	C₂	—	4	4	no	—
423	C₂	C₅₃×#1812	1	2	yes	direct product factor
424	C₂	C₂×C₅₃×#380	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	#380
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#427	C₂	semidirect
4. C₅₃×#1812	C₂	direct (N × H)
5. C₂×C₅₃×H(3,ℤ/4)	C₂	semidirect
10. C₂×#131×C₅₃	C₄	semidirect
12. C₂²×M₁₆×C₅₃	C₄	semidirect
14. C₂³×C₈×C₅₃	C₄	semidirect
18. C₅₃×#427	C₂²	semidirect
24. C₅₃×H(3,ℤ/4)	C₂²	semidirect
47. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
56. C₂²×C₈×C₅₃	C₂×C₄	semidirect
183. C₂×#1812	C₅₃	direct (N × H)
207. C₂×#427	C₂×C₅₃	semidirect
209. #1812	C₂×C₅₃	direct (N × H)
210. C₂×H(3,ℤ/4)	C₂×C₅₃	semidirect
216. C₂×C₅₃	#1812	direct (N × H)
222. C₂×#131	C₄×C₅₃	semidirect
224. C₂²×M₁₆	C₄×C₅₃	semidirect
226. C₂³×C₈	C₄×C₅₃	semidirect
230. #427	C₂²×C₅₃	semidirect
236. H(3,ℤ/4)	C₂²×C₅₃	semidirect
237. C₅₃	C₂×#1812	direct (N × H)
260. C₂×M₁₆	C₂×C₄×C₅₃	semidirect
269. C₂²×C₈	C₂×C₄×C₅₃	semidirect
423. C₂	C₅₃×#1812	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×#131×C₅₃	2
C₂×C₅₃×#427	2
C₂×C₅₃×#410	2
C₅₃×#1812	2
C₂×C₅₃×H(3,ℤ/4)	2
C₂×#1812	53

Subgroups — C2×C53×#1812 #27228182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1812 #27228182