Subgroups — C₂×C₅₃×#3228 #27268355

Order 13568.

2058 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3228	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#482	C₂	1	4	yes	maximal, complemented
9	C₂×C₅₃×#478	C₂	1	4	yes	maximal, complemented
10	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₅₃	—	2	1	no	—
12	C₅₃×#533	—	2	4	no	—
13	C₅₃×#533	C₂²	1	8	yes	complemented
14	C₅₃×#532	—	2	4	no	—
15	C₅₃×#532	C₂²	1	8	yes	—
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#125×C₅₃	C₂²	1	2	yes	—
21	C₂×#124×C₅₃	—	2	1	no	—
22	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
23	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
24	C₂×#121×C₅₃	—	2	1	no	—
25	C₂×#121×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
27	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
28	C₂×C₄×D₄×C₅₃	—	2	1	no	—
29	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
30	C₂×#117×C₅₃	C₂²	1	2	yes	—
31	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#500	C₂²	1	8	yes	complemented
33	C₅₃×#498	C₂²	1	8	yes	complemented
34	C₅₃×#486	C₂²	1	8	yes	complemented
35	C₅₃×#486	C₂²	1	8	yes	complemented
36	C₅₃×#482	C₂²	1	32	yes	complemented
37	C₅₃×#478	C₂²	1	32	yes	complemented
38	C₅₃×#431	C₂²	1	8	yes	complemented
39	C₂×#107×C₅₃	C₂²	1	4	yes	—
40	C₂×#106×C₅₃	C₂²	1	4	yes	—
41	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
42	C₂×#102×C₅₃	C₂²	1	4	yes	—
43	C₂×#102×C₅₃	C₂²	1	4	yes	—
44	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#41×C₅₃	—	2	8	no	—
48	C₂×#41×C₅₃	—	4	8	no	—
49	C₂²×Q₈×C₅₃	—	2	1	no	—
50	C₂²×Q₈×C₅₃	—	2	1	no	—
51	C₂²×Q₈×C₅₃	—	2	1	no	—
52	C₂²×D₄×C₅₃	—	2	2	no	—
53	C₂²×D₄×C₅₃	—	2	1	no	—
54	C₂²×D₄×C₅₃	—	2	1	no	—
55	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₅₃	—	2	2	no	—
57	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₅₃	—	2	2	no	—
63	C₄⋊Q₈×C₅₃	—	2	2	no	—
64	C₄⋊Q₈×C₅₃	—	2	2	no	—
65	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
66	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
67	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
68	#127×C₅₃	—	2	2	no	—
69	#127×C₅₃	C₂³	1	4	yes	—
70	#125×C₅₃	—	2	4	no	—
71	#125×C₅₃	C₂³	1	8	yes	—
72	#124×C₅₃	—	2	8	no	—
73	C₂²⋊Q₈×C₅₃	—	2	8	no	—
74	C₂²⋊Q₈×C₅₃	—	2	8	no	—
75	#121×C₅₃	—	2	8	no	—
76	#121×C₅₃	—	2	4	no	—
77	#121×C₅₃	C₂³	1	8	yes	—
78	C₄×Q₈×C₅₃	—	2	8	no	—
79	C₄×Q₈×C₅₃	—	2	8	no	—
80	C₄×D₄×C₅₃	—	2	8	no	—
81	C₄×D₄×C₅₃	—	2	4	no	—
82	C₄×D₄×C₅₃	C₂³	1	8	yes	—
83	#117×C₅₃	—	2	4	no	—
84	#117×C₅₃	C₂³	1	8	yes	—
85	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
96	C₂×C₄²×C₅₃	—	2	2	no	—
97	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	D₄	1	4	yes	—
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
103	#107×C₅₃	C₂³	1	16	yes	—
104	#106×C₅₃	C₂³	1	16	yes	—
105	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
106	#102×C₅₃	C₂³	1	16	yes	—
107	#102×C₅₃	C₂³	1	16	yes	—
108	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
109	C₄×C₈×C₅₃	C₂³	1	4	yes	—
110	C₄×C₈×C₅₃	C₂³	1	4	yes	—
111	C₂⁴×C₅₃	—	2	2	no	—
112	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
113	#41×C₅₃	—	4	32	no	—
114	C₂×Q₈×C₅₃	—	2	4	no	—
115	C₂×Q₈×C₅₃	—	2	4	no	—
116	C₂×Q₈×C₅₃	—	2	4	no	—
117	C₂×Q₈×C₅₃	—	4	4	no	—
118	C₂×Q₈×C₅₃	—	4	4	no	—
119	C₂×Q₈×C₅₃	—	4	4	no	—
120	C₂×D₄×C₅₃	—	2	8	no	—
121	C₂×D₄×C₅₃	—	2	2	no	—
122	C₂×D₄×C₅₃	—	2	4	no	—
123	C₂×D₄×C₅₃	—	2	4	no	—
124	C₂×D₄×C₅₃	—	2	8	no	—
125	C₂×D₄×C₅₃	—	4	8	no	—
126	C₂×D₄×C₅₃	—	4	4	no	—
127	C₂×D₄×C₅₃	—	4	4	no	—
128	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	2	4	no	—
131	C₂²×C₄×C₅₃	—	2	4	no	—
132	C₂²×C₄×C₅₃	—	2	2	no	—
133	C₂²×C₄×C₅₃	—	2	2	no	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	2	8	no	—
136	C₂²×C₄×C₅₃	—	4	8	no	—
137	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
147	C₂×C₈×C₅₃	—	2	4	no	—
148	C₂×C₈×C₅₃	—	2	4	no	—
149	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
150	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
151	C₄⋊C₄×C₅₃	—	2	4	no	—
152	C₄⋊C₄×C₅₃	—	2	8	no	—
153	C₄⋊C₄×C₅₃	—	2	8	no	—
154	C₄⋊C₄×C₅₃	—	2	4	no	—
155	C₄⋊C₄×C₅₃	—	2	4	no	—
156	C₄⋊C₄×C₅₃	—	2	2	no	—
157	C₄⋊C₄×C₅₃	—	2	2	no	—
158	C₄⋊C₄×C₅₃	—	2	2	no	—
159	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
160	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
161	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
162	C₂²⋊C₄×C₅₃	—	2	8	no	—
163	C₂²⋊C₄×C₅₃	—	2	8	no	—
164	C₂²⋊C₄×C₅₃	—	2	8	no	—
165	C₄²×C₅₃	—	2	8	no	—
166	C₄²×C₅₃	C₂×D₄	1	4	yes	—
167	C₄²×C₅₃	C₂×D₄	1	2	yes	—
168	C₄²×C₅₃	C₂×D₄	1	4	yes	—
169	C₄²×C₅₃	C₂×D₄	1	2	yes	—
170	C₄²×C₅₃	C₂⁴	1	2	yes	—
171	C₄²×C₅₃	C₂⁴	1	2	yes	—
172	C₂³×C₅₃	—	2	2	no	—
173	C₂³×C₅₃	—	2	2	no	—
174	C₂³×C₅₃	—	2	2	no	—
175	C₂³×C₅₃	—	2	4	no	—
176	C₂³×C₅₃	—	4	4	no	—
177	C₂³×C₅₃	—	4	4	no	—
178	C₂³×C₅₃	—	4	4	no	—
179	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
180	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₅₃	—	4	8	no	—
182	Q₈×C₅₃	—	4	8	no	—
183	Q₈×C₅₃	—	4	8	no	—
184	D₄×C₅₃	—	2	8	no	—
185	D₄×C₅₃	—	4	16	no	—
186	D₄×C₅₃	—	4	4	no	—
187	D₄×C₅₃	—	4	8	no	—
188	D₄×C₅₃	—	4	8	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	4	no	—
193	C₂×C₄×C₅₃	—	2	8	no	—
194	C₂×C₄×C₅₃	—	2	8	no	—
195	C₂×C₄×C₅₃	—	2	4	no	—
196	C₂×C₄×C₅₃	—	2	4	no	—
197	C₂×C₄×C₅₃	—	2	4	no	—
198	C₂×C₄×C₅₃	—	2	4	no	—
199	C₂×C₄×C₅₃	—	4	16	no	—
200	C₂×C₄×C₅₃	—	4	4	no	—
201	C₂×C₄×C₅₃	—	4	4	no	—
202	C₂×C₄×C₅₃	—	4	4	no	—
203	C₂×C₄×C₅₃	#135	1	4	yes	—
204	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
205	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
206	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
207	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
208	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
212	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
213	C₈×C₅₃	—	2	4	no	—
214	C₈×C₅₃	—	2	4	no	—
215	C₂×#3228	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂²×C₅₃	—	2	2	no	—
217	C₂²×C₅₃	—	2	2	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	4	4	no	—
220	C₂²×C₅₃	—	4	4	no	—
221	C₂²×C₅₃	—	4	4	no	—
222	C₂²×C₅₃	—	4	4	no	—
223	C₂²×C₅₃	—	8	8	no	—
224	C₂²×C₅₃	#536	1	2	yes	—
225	C₂²×C₅₃	C₂×#135	1	2	yes	—
226	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
227	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
228	C₄×C₅₃	—	2	2	no	—
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	2	2	no	—
231	C₄×C₅₃	—	4	4	no	—
232	C₄×C₅₃	—	4	4	no	—
233	C₄×C₅₃	—	4	4	no	—
234	C₄×C₅₃	#536	1	2	yes	—
235	C₄×C₅₃	C₂×#135	1	4	yes	—
236	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
237	C₄×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
238	C₂×#533	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
240	#3228	C₂×C₅₃	1	16	yes	direct product factor
241	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
242	C₂×#498	C₂×C₅₃	1	1	yes	characteristic, complemented
243	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
244	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
245	C₂×#482	C₂×C₅₃	1	4	yes	complemented
246	C₂×#478	C₂×C₅₃	1	4	yes	complemented
247	C₂×#431	C₂×C₅₃	1	1	yes	characteristic, complemented
248	C₂×C₅₃	—	4	2	no	—
249	C₂×C₅₃	—	8	4	no	—
250	C₂×C₅₃	#3228	1	4	yes	direct product factor
251	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
252	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
253	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₂²×#41	—	2	1	no	—
255	#533	—	2	4	no	—
256	#533	C₂²×C₅₃	1	8	yes	complemented
257	#532	—	2	4	no	—
258	#532	C₂²×C₅₃	1	8	yes	—
259	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
260	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×#125	C₂²×C₅₃	1	2	yes	—
264	C₂×#124	—	2	1	no	—
265	C₂×C₂²⋊Q₈	—	2	1	no	—
266	C₂×C₂²⋊Q₈	—	2	1	no	—
267	C₂×#121	—	2	1	no	—
268	C₂×#121	C₂²×C₅₃	1	2	yes	—
269	C₂×C₄×Q₈	—	2	1	no	—
270	C₂×C₄×Q₈	—	2	1	no	—
271	C₂×C₄×D₄	—	2	1	no	—
272	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
273	C₂×#117	C₂²×C₅₃	1	2	yes	—
274	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
275	#500	C₂²×C₅₃	1	8	yes	complemented
276	#498	C₂²×C₅₃	1	8	yes	complemented
277	#486	C₂²×C₅₃	1	8	yes	complemented
278	#486	C₂²×C₅₃	1	8	yes	complemented
279	#482	C₂²×C₅₃	1	32	yes	complemented
280	#478	C₂²×C₅₃	1	32	yes	complemented
281	#431	C₂²×C₅₃	1	8	yes	complemented
282	C₂×#107	C₂²×C₅₃	1	4	yes	—
283	C₂×#106	C₂²×C₅₃	1	4	yes	—
284	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
285	C₂×#102	C₂²×C₅₃	1	4	yes	—
286	C₂×#102	C₂²×C₅₃	1	4	yes	—
287	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
288	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
289	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
290	C₅₃	C₂×#3228	1	1	yes	characteristic, direct product factor, Sylow
291	C₂×#41	—	2	8	no	—
292	C₂×#41	—	4	8	no	—
293	C₂²×Q₈	—	2	1	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	—	2	1	no	—
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	—	2	1	no	—
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
300	C₂³×C₄	—	2	2	no	—
301	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
302	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
303	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
305	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
306	C₄⋊Q₈	—	2	2	no	—
307	C₄⋊Q₈	—	2	2	no	—
308	C₄⋊Q₈	—	2	2	no	—
309	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
310	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
311	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
312	#127	—	2	2	no	—
313	#127	C₂³×C₅₃	1	4	yes	—
314	#125	—	2	4	no	—
315	#125	C₂³×C₅₃	1	8	yes	—
316	#124	—	2	8	no	—
317	C₂²⋊Q₈	—	2	8	no	—
318	C₂²⋊Q₈	—	2	8	no	—
319	#121	—	2	8	no	—
320	#121	—	2	4	no	—
321	#121	C₂³×C₅₃	1	8	yes	—
322	C₄×Q₈	—	2	8	no	—
323	C₄×Q₈	—	2	8	no	—
324	C₄×D₄	—	2	8	no	—
325	C₄×D₄	—	2	4	no	—
326	C₄×D₄	C₂³×C₅₃	1	8	yes	—
327	#117	—	2	4	no	—
328	#117	C₂³×C₅₃	1	8	yes	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
342	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
343	C₂×C₄²	D₄×C₅₃	1	4	yes	—
344	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
347	#107	C₂³×C₅₃	1	16	yes	—
348	#106	C₂³×C₅₃	1	16	yes	—
349	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
350	#102	C₂³×C₅₃	1	16	yes	—
351	#102	C₂³×C₅₃	1	16	yes	—
352	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
353	C₄×C₈	C₂³×C₅₃	1	4	yes	—
354	C₄×C₈	C₂³×C₅₃	1	4	yes	—
355	C₂⁴	—	2	2	no	—
356	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
357	#41	—	4	32	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	—	4	4	no	—
363	C₂×Q₈	—	4	4	no	—
364	C₂×D₄	—	2	8	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	4	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	8	no	—
380	C₂²×C₄	—	4	8	no	—
381	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
383	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
386	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
387	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
391	C₂×C₈	—	2	4	no	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
394	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	8	no	—
397	C₄⋊C₄	—	2	8	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
404	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
405	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₂²⋊C₄	—	2	8	no	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₄²	—	2	8	no	—
410	C₄²	C₂×D₄×C₅₃	1	4	yes	—
411	C₄²	C₂×D₄×C₅₃	1	2	yes	—
412	C₄²	C₂×D₄×C₅₃	1	4	yes	—
413	C₄²	C₂×D₄×C₅₃	1	2	yes	—
414	C₄²	C₂⁴×C₅₃	1	2	yes	—
415	C₄²	C₂⁴×C₅₃	1	2	yes	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	4	4	no	—
421	C₂³	—	4	4	no	—
422	C₂³	—	4	4	no	—
423	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
424	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
425	Q₈	—	4	8	no	—
426	Q₈	—	4	8	no	—
427	Q₈	—	4	8	no	—
428	D₄	—	2	8	no	—
429	D₄	—	4	16	no	—
430	D₄	—	4	4	no	—
431	D₄	—	4	8	no	—
432	D₄	—	4	8	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	8	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	4	16	no	—
444	C₂×C₄	—	4	4	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	4	no	—
447	C₂×C₄	#135×C₅₃	1	4	yes	—
448	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
449	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
450	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
451	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
454	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
455	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
456	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
457	C₈	—	2	4	no	—
458	C₈	—	2	4	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	4	4	no	—
463	C₂²	—	4	4	no	—
464	C₂²	—	4	4	no	—
465	C₂²	—	4	4	no	—
466	C₂²	—	8	8	no	—
467	C₂²	C₅₃×#536	1	2	yes	—
468	C₂²	C₂×#135×C₅₃	1	2	yes	—
469	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
470	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	4	4	no	—
475	C₄	—	4	4	no	—
476	C₄	—	4	4	no	—
477	C₄	C₅₃×#536	1	2	yes	—
478	C₄	C₂×#135×C₅₃	1	4	yes	—
479	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	4	yes	—
480	C₄	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
481	C₂	—	4	2	no	—
482	C₂	—	8	4	no	—
483	C₂	C₅₃×#3228	1	4	yes	direct product factor
484	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
485	C₂	C₂²×#135×C₅₃	1	1	yes	characteristic
486	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#533	C₂	semidirect
3. C₅₃×#3228	C₂	direct (N × H)
4. C₂×C₅₃×#500	C₂	semidirect
5. C₂×C₅₃×#498	C₂	semidirect
6. C₂×C₅₃×#486	C₂	semidirect
7. C₂×C₅₃×#486	C₂	semidirect
8. C₂×C₅₃×#482	C₂	semidirect
9. C₂×C₅₃×#478	C₂	semidirect
10. C₂×C₅₃×#431	C₂	semidirect
13. C₅₃×#533	C₂²	semidirect
32. C₅₃×#500	C₂²	semidirect
33. C₅₃×#498	C₂²	semidirect
34. C₅₃×#486	C₂²	semidirect
35. C₅₃×#486	C₂²	semidirect
36. C₅₃×#482	C₂²	semidirect
37. C₅₃×#478	C₂²	semidirect
38. C₅₃×#431	C₂²	semidirect
215. C₂×#3228	C₅₃	direct (N × H)
238. C₂×#533	C₂×C₅₃	semidirect
240. #3228	C₂×C₅₃	direct (N × H)
241. C₂×#500	C₂×C₅₃	semidirect
242. C₂×#498	C₂×C₅₃	semidirect
243. C₂×#486	C₂×C₅₃	semidirect
244. C₂×#486	C₂×C₅₃	semidirect
245. C₂×#482	C₂×C₅₃	semidirect
246. C₂×#478	C₂×C₅₃	semidirect
247. C₂×#431	C₂×C₅₃	semidirect
250. C₂×C₅₃	#3228	direct (N × H)
256. #533	C₂²×C₅₃	semidirect
275. #500	C₂²×C₅₃	semidirect
276. #498	C₂²×C₅₃	semidirect
277. #486	C₂²×C₅₃	semidirect
278. #486	C₂²×C₅₃	semidirect
279. #482	C₂²×C₅₃	semidirect
280. #478	C₂²×C₅₃	semidirect
281. #431	C₂²×C₅₃	semidirect
290. C₅₃	C₂×#3228	direct (N × H)
483. C₂	C₅₃×#3228	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×#533	2
C₂×C₅₃×#532	2
C₅₃×#3228	2
C₂×C₅₃×#500	2
C₂×C₅₃×#498	2
C₂×C₅₃×#486	2
C₂×C₅₃×#482	2
C₂×C₅₃×#478	2
C₂×C₅₃×#431	2
C₂×#3228	53

Subgroups — C2×C53×#3228 #27268355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3228 #27268355