Subgroups — C₂×C₅₃×#3340 #27268743

Order 13568.

1918 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#555	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3340	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#485	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#482	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#465	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅₃×#427	C₂	1	2	yes	maximal, complemented
13	C₂²×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#556	—	2	4	no	—
16	C₅₃×#556	C₂²	1	8	yes	complemented
17	C₅₃×#555	—	2	4	no	—
18	C₅₃×#555	C₂²	1	8	yes	—
19	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#126×C₅₃	—	2	2	no	—
22	C₂×#124×C₅₃	—	2	1	no	—
23	C₂×#124×C₅₃	—	2	2	no	—
24	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
25	C₂×#121×C₅₃	—	2	1	no	—
26	C₂×#121×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
28	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×#117×C₅₃	—	2	1	no	—
32	C₂×#117×C₅₃	C₂²	1	2	yes	—
33	C₂×#117×C₅₃	C₂²	1	2	yes	—
34	C₅₃×#498	C₂²	1	8	yes	complemented
35	C₅₃×#485	C₂²	1	16	yes	complemented
36	C₅₃×#482	C₂²	1	16	yes	complemented
37	C₅₃×#477	C₂²	1	8	yes	complemented
38	C₅₃×#474	C₂²	1	8	yes	complemented
39	C₅₃×#465	C₂²	1	16	yes	complemented
40	C₄×SD₁₆×C₅₃	C₂²	1	8	yes	complemented
41	C₈×D₄×C₅₃	C₂²	1	8	yes	complemented
42	C₅₃×#427	C₂²	1	16	yes	complemented
43	C₂×#107×C₅₃	C₂²	1	2	yes	—
44	C₂×#107×C₅₃	C₂²	1	2	yes	—
45	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×#106×C₅₃	C₂²	1	2	yes	—
47	C₂×#106×C₅₃	C₂²	1	2	yes	—
48	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#103×C₅₃	C₂²	1	1	yes	characteristic
50	C₂×#103×C₅₃	C₂²	1	2	yes	—
51	C₂×#102×C₅₃	C₂²	1	1	yes	characteristic
52	C₂×#102×C₅₃	C₂²	1	2	yes	—
53	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₅₃	—	2	1	no	—
56	C₂²×Q₈×C₅₃	—	2	1	no	—
57	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₅₃	—	2	2	no	—
59	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	—	2	1	no	—
62	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
63	C₂×SD₁₆×C₅₃	—	2	4	no	—
64	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
65	C₂²×C₈×C₅₃	—	2	8	no	—
66	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₅₃	—	2	2	no	—
71	C₄⋊Q₈×C₅₃	—	2	2	no	—
72	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
73	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
74	#126×C₅₃	—	2	16	no	—
75	#124×C₅₃	—	2	16	no	—
76	#124×C₅₃	—	2	8	no	—
77	C₂²⋊Q₈×C₅₃	—	2	4	no	—
78	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
79	#121×C₅₃	—	2	8	no	—
80	#121×C₅₃	—	2	4	no	—
81	#121×C₅₃	C₂³	1	8	yes	—
82	C₄×Q₈×C₅₃	—	2	8	no	—
83	C₄×Q₈×C₅₃	—	2	2	no	—
84	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
85	C₄×D₄×C₅₃	—	2	2	no	—
86	C₄×D₄×C₅₃	D₄	1	4	yes	—
87	C₄×D₄×C₅₃	C₂³	1	4	yes	—
88	C₄×D₄×C₅₃	C₂³	1	4	yes	—
89	#117×C₅₃	—	2	8	no	—
90	#117×C₅₃	—	2	4	no	—
91	#117×C₅₃	—	2	4	no	—
92	#117×C₅₃	C₂³	1	8	yes	—
93	#117×C₅₃	C₂³	1	8	yes	—
94	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
104	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
109	C₂×C₄²×C₅₃	—	2	2	no	—
110	C₂×C₄²×C₅₃	—	2	2	no	—
111	C₂×C₄²×C₅₃	—	2	1	no	—
112	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
113	#107×C₅₃	C₂³	1	8	yes	—
114	#107×C₅₃	C₂³	1	8	yes	—
115	#106×C₅₃	C₂³	1	8	yes	—
116	#106×C₅₃	C₂³	1	8	yes	—
117	#106×C₅₃	C₂³	1	4	yes	—
118	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
119	#103×C₅₃	C₂³	1	4	yes	—
120	#103×C₅₃	C₂³	1	8	yes	—
121	#102×C₅₃	C₂³	1	4	yes	—
122	#102×C₅₃	C₂³	1	8	yes	—
123	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
124	C₄×C₈×C₅₃	C₂³	1	4	yes	—
125	C₂⁴×C₅₃	—	2	1	no	—
126	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
127	C₂×Q₈×C₅₃	—	2	1	no	—
128	C₂×Q₈×C₅₃	—	2	4	no	—
129	C₂×Q₈×C₅₃	—	2	4	no	—
130	C₂×Q₈×C₅₃	—	2	2	no	—
131	C₂×Q₈×C₅₃	—	2	2	no	—
132	C₂×Q₈×C₅₃	—	4	4	no	—
133	C₂×Q₈×C₅₃	—	8	2	no	—
134	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
135	C₂×D₄×C₅₃	—	2	8	no	—
136	C₂×D₄×C₅₃	—	2	4	no	—
137	C₂×D₄×C₅₃	—	2	1	no	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	—	2	2	no	—
140	C₂×D₄×C₅₃	—	8	4	no	—
141	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
142	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₅₃	—	2	4	no	—
144	C₂²×C₄×C₅₃	—	2	4	no	—
145	C₂²×C₄×C₅₃	—	2	1	no	—
146	C₂²×C₄×C₅₃	—	2	1	no	—
147	C₂²×C₄×C₅₃	—	2	1	no	—
148	C₂²×C₄×C₅₃	—	2	2	no	—
149	C₂²×C₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	2	4	no	—
151	C₂²×C₄×C₅₃	—	4	4	no	—
152	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
153	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	SD₁₆×C₅₃	—	2	8	no	—
159	C₂×C₈×C₅₃	—	2	4	no	—
160	C₂×C₈×C₅₃	—	2	2	no	—
161	C₂×C₈×C₅₃	—	2	16	no	—
162	C₂×C₈×C₅₃	#41	1	4	yes	—
163	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
164	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
165	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
166	C₄⋊C₄×C₅₃	—	2	4	no	—
167	C₄⋊C₄×C₅₃	—	2	8	no	—
168	C₄⋊C₄×C₅₃	—	2	4	no	—
169	C₄⋊C₄×C₅₃	—	2	4	no	—
170	C₄⋊C₄×C₅₃	—	2	1	no	—
171	C₄⋊C₄×C₅₃	—	2	1	no	—
172	C₄⋊C₄×C₅₃	—	2	2	no	—
173	C₄⋊C₄×C₅₃	—	2	2	no	—
174	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
175	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
176	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
177	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
178	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
179	C₂²⋊C₄×C₅₃	—	2	8	no	—
180	C₂²⋊C₄×C₅₃	—	2	8	no	—
181	C₂²⋊C₄×C₅₃	—	2	8	no	—
182	C₂²⋊C₄×C₅₃	—	2	8	no	—
183	C₂²⋊C₄×C₅₃	—	2	4	no	—
184	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
185	C₄²×C₅₃	—	2	8	no	—
186	C₄²×C₅₃	—	2	8	no	—
187	C₄²×C₅₃	—	2	4	no	—
188	C₄²×C₅₃	C₂×D₄	1	2	yes	—
189	C₄²×C₅₃	C₂⁴	1	2	yes	—
190	C₂³×C₅₃	—	2	4	no	—
191	C₂³×C₅₃	—	2	4	no	—
192	C₂³×C₅₃	—	2	4	no	—
193	C₂³×C₅₃	—	2	2	no	—
194	C₂³×C₅₃	—	4	2	no	—
195	C₂³×C₅₃	—	4	2	no	—
196	C₂³×C₅₃	—	4	2	no	—
197	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
198	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₅₃	—	2	2	no	—
200	Q₈×C₅₃	—	2	2	no	—
201	Q₈×C₅₃	—	4	2	no	—
202	Q₈×C₅₃	—	4	8	no	—
203	Q₈×C₅₃	—	8	4	no	—
204	D₄×C₅₃	—	2	8	no	—
205	D₄×C₅₃	—	2	2	no	—
206	D₄×C₅₃	—	2	2	no	—
207	D₄×C₅₃	—	4	2	no	—
208	D₄×C₅₃	—	8	8	no	—
209	C₂×C₄×C₅₃	—	2	4	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	8	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	2	2	no	—
218	C₂×C₄×C₅₃	—	2	2	no	—
219	C₂×C₄×C₅₃	—	4	2	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	4	no	—
222	C₂×C₄×C₅₃	—	4	2	no	—
223	C₂×C₄×C₅₃	—	4	4	no	—
224	C₂×C₄×C₅₃	—	4	8	no	—
225	C₂×C₄×C₅₃	#135	1	4	yes	—
226	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
231	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
232	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
233	C₈×C₅₃	—	2	4	no	—
234	C₈×C₅₃	—	2	2	no	—
235	C₈×C₅₃	C₂×#41	1	4	yes	—
236	C₂×#3340	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂²×C₅₃	—	2	4	no	—
238	C₂²×C₅₃	—	2	4	no	—
239	C₂²×C₅₃	—	2	4	no	—
240	C₂²×C₅₃	—	4	8	no	—
241	C₂²×C₅₃	—	4	2	no	—
242	C₂²×C₅₃	—	4	2	no	—
243	C₂²×C₅₃	—	4	2	no	—
244	C₂²×C₅₃	—	8	4	no	—
245	C₂²×C₅₃	#547	1	2	yes	—
246	C₂²×C₅₃	C₂×#135	1	2	yes	—
247	C₂²×C₅₃	#577	1	2	yes	—
248	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
249	C₄×C₅₃	—	2	4	no	—
250	C₄×C₅₃	—	2	2	no	—
251	C₄×C₅₃	—	4	2	no	—
252	C₄×C₅₃	—	4	2	no	—
253	C₄×C₅₃	—	4	4	no	—
254	C₄×C₅₃	—	4	2	no	—
255	C₄×C₅₃	—	4	4	no	—
256	C₄×C₅₃	#547	1	2	yes	—
257	C₄×C₅₃	C₂×#135	1	4	yes	—
258	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
259	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
260	C₂×#556	C₂×C₅₃	1	1	yes	characteristic, complemented
261	C₂×#555	C₂×C₅₃	1	1	yes	characteristic
262	#3340	C₂×C₅₃	1	16	yes	direct product factor
263	C₂×#498	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₂×#485	C₂×C₅₃	1	2	yes	complemented
265	C₂×#482	C₂×C₅₃	1	2	yes	complemented
266	C₂×#477	C₂×C₅₃	1	1	yes	characteristic, complemented
267	C₂×#474	C₂×C₅₃	1	1	yes	characteristic, complemented
268	C₂×#465	C₂×C₅₃	1	2	yes	complemented
269	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
270	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
271	C₂×#427	C₂×C₅₃	1	2	yes	complemented
272	C₂×C₅₃	—	4	4	no	—
273	C₂×C₅₃	—	8	2	no	—
274	C₂×C₅₃	#3340	1	4	yes	direct product factor
275	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
276	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
277	C₂×C₅₃	C₂×#577	1	1	yes	characteristic
278	C₂²×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
279	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
280	#556	—	2	4	no	—
281	#556	C₂²×C₅₃	1	8	yes	complemented
282	#555	—	2	4	no	—
283	#555	C₂²×C₅₃	1	8	yes	—
284	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
286	C₂×#126	—	2	2	no	—
287	C₂×#124	—	2	1	no	—
288	C₂×#124	—	2	2	no	—
289	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
290	C₂×#121	—	2	1	no	—
291	C₂×#121	C₂²×C₅₃	1	2	yes	—
292	C₂×C₄×Q₈	—	2	1	no	—
293	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
294	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
295	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
296	C₂×#117	—	2	1	no	—
297	C₂×#117	C₂²×C₅₃	1	2	yes	—
298	C₂×#117	C₂²×C₅₃	1	2	yes	—
299	#498	C₂²×C₅₃	1	8	yes	complemented
300	#485	C₂²×C₅₃	1	16	yes	complemented
301	#482	C₂²×C₅₃	1	16	yes	complemented
302	#477	C₂²×C₅₃	1	8	yes	complemented
303	#474	C₂²×C₅₃	1	8	yes	complemented
304	#465	C₂²×C₅₃	1	16	yes	complemented
305	C₄×SD₁₆	C₂²×C₅₃	1	8	yes	complemented
306	C₈×D₄	C₂²×C₅₃	1	8	yes	complemented
307	#427	C₂²×C₅₃	1	16	yes	complemented
308	C₂×#107	C₂²×C₅₃	1	2	yes	—
309	C₂×#107	C₂²×C₅₃	1	2	yes	—
310	C₂×#106	C₂²×C₅₃	1	2	yes	—
311	C₂×#106	C₂²×C₅₃	1	2	yes	—
312	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
313	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
314	C₂×#103	C₂²×C₅₃	1	2	yes	—
315	C₂×#103	C₂²×C₅₃	1	1	yes	characteristic
316	C₂×#102	C₂²×C₅₃	1	1	yes	characteristic
317	C₂×#102	C₂²×C₅₃	1	2	yes	—
318	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
319	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
320	C₅₃	C₂×#3340	1	1	yes	characteristic, direct product factor, Sylow
321	C₂²×Q₈	—	2	1	no	—
322	C₂²×Q₈	—	2	1	no	—
323	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
324	C₂²×D₄	—	2	2	no	—
325	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
326	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
327	C₂³×C₄	—	2	1	no	—
328	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
329	C₂×SD₁₆	—	2	4	no	—
330	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
331	C₂²×C₈	—	2	8	no	—
332	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
333	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
334	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
336	C₄⋊Q₈	—	2	2	no	—
337	C₄⋊Q₈	—	2	2	no	—
338	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
339	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
340	#126	—	2	16	no	—
341	#124	—	2	8	no	—
342	#124	—	2	16	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
345	#121	—	2	8	no	—
346	#121	—	2	4	no	—
347	#121	C₂³×C₅₃	1	8	yes	—
348	C₄×Q₈	—	2	8	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
351	C₄×D₄	—	2	2	no	—
352	C₄×D₄	D₄×C₅₃	1	4	yes	—
353	C₄×D₄	C₂³×C₅₃	1	4	yes	—
354	C₄×D₄	C₂³×C₅₃	1	4	yes	—
355	#117	—	2	4	no	—
356	#117	—	2	4	no	—
357	#117	—	2	8	no	—
358	#117	C₂³×C₅₃	1	8	yes	—
359	#117	C₂³×C₅₃	1	8	yes	—
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	2	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
365	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
375	C₂×C₄²	—	2	2	no	—
376	C₂×C₄²	—	2	2	no	—
377	C₂×C₄²	—	2	1	no	—
378	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
379	#107	C₂³×C₅₃	1	8	yes	—
380	#107	C₂³×C₅₃	1	8	yes	—
381	#106	C₂³×C₅₃	1	4	yes	—
382	#106	C₂³×C₅₃	1	8	yes	—
383	#106	C₂³×C₅₃	1	8	yes	—
384	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
385	#103	C₂³×C₅₃	1	4	yes	—
386	#103	C₂³×C₅₃	1	8	yes	—
387	#102	C₂³×C₅₃	1	4	yes	—
388	#102	C₂³×C₅₃	1	8	yes	—
389	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
390	C₄×C₈	C₂³×C₅₃	1	4	yes	—
391	C₂⁴	—	2	1	no	—
392	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	—	2	4	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	4	4	no	—
399	C₂×Q₈	—	8	2	no	—
400	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	8	4	no	—
407	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
408	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	4	4	no	—
418	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
419	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
421	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
422	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
423	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
424	SD₁₆	—	2	8	no	—
425	C₂×C₈	—	2	4	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	16	no	—
428	C₂×C₈	#41×C₅₃	1	4	yes	—
429	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
430	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
431	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
441	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
442	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
443	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
444	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
445	C₂²⋊C₄	—	2	8	no	—
446	C₂²⋊C₄	—	2	8	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
451	C₄²	—	2	8	no	—
452	C₄²	—	2	8	no	—
453	C₄²	—	2	4	no	—
454	C₄²	C₂×D₄×C₅₃	1	2	yes	—
455	C₄²	C₂⁴×C₅₃	1	2	yes	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	4	2	no	—
462	C₂³	—	4	2	no	—
463	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
464	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
465	Q₈	—	2	2	no	—
466	Q₈	—	2	2	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	4	8	no	—
469	Q₈	—	8	4	no	—
470	D₄	—	2	8	no	—
471	D₄	—	2	2	no	—
472	D₄	—	2	2	no	—
473	D₄	—	4	2	no	—
474	D₄	—	8	8	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	—	4	8	no	—
491	C₂×C₄	#135×C₅₃	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
493	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
495	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
496	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
497	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
498	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
499	C₈	—	2	4	no	—
500	C₈	—	2	2	no	—
501	C₈	C₂×#41×C₅₃	1	4	yes	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	4	8	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	8	4	no	—
510	C₂²	C₅₃×#547	1	2	yes	—
511	C₂²	C₂×#135×C₅₃	1	2	yes	—
512	C₂²	C₅₃×#577	1	2	yes	—
513	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
514	C₄	—	2	4	no	—
515	C₄	—	2	2	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	4	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	2	no	—
521	C₄	C₅₃×#547	1	2	yes	—
522	C₄	C₂×#135×C₅₃	1	4	yes	—
523	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
524	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
525	C₂	—	4	4	no	—
526	C₂	—	8	2	no	—
527	C₂	C₅₃×#3340	1	4	yes	direct product factor
528	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
529	C₂	C₂²×#135×C₅₃	1	1	yes	characteristic
530	C₂	C₂×C₅₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#556	C₂	semidirect
3. C₅₃×#3340	C₂	direct (N × H)
4. C₂×C₅₃×#498	C₂	semidirect
5. C₂×C₅₃×#485	C₂	semidirect
6. C₂×C₅₃×#482	C₂	semidirect
7. C₂×C₅₃×#477	C₂	semidirect
8. C₂×C₅₃×#474	C₂	semidirect
9. C₂×C₅₃×#465	C₂	semidirect
10. C₂×C₄×SD₁₆×C₅₃	C₂	semidirect
11. C₂×C₈×D₄×C₅₃	C₂	semidirect
12. C₂×C₅₃×#427	C₂	semidirect
16. C₅₃×#556	C₂²	semidirect
34. C₅₃×#498	C₂²	semidirect
35. C₅₃×#485	C₂²	semidirect
36. C₅₃×#482	C₂²	semidirect
37. C₅₃×#477	C₂²	semidirect
38. C₅₃×#474	C₂²	semidirect
39. C₅₃×#465	C₂²	semidirect
40. C₄×SD₁₆×C₅₃	C₂²	semidirect
41. C₈×D₄×C₅₃	C₂²	semidirect
42. C₅₃×#427	C₂²	semidirect
236. C₂×#3340	C₅₃	direct (N × H)
260. C₂×#556	C₂×C₅₃	semidirect
262. #3340	C₂×C₅₃	direct (N × H)
263. C₂×#498	C₂×C₅₃	semidirect
264. C₂×#485	C₂×C₅₃	semidirect
265. C₂×#482	C₂×C₅₃	semidirect
266. C₂×#477	C₂×C₅₃	semidirect
267. C₂×#474	C₂×C₅₃	semidirect
268. C₂×#465	C₂×C₅₃	semidirect
269. C₂×C₄×SD₁₆	C₂×C₅₃	semidirect
270. C₂×C₈×D₄	C₂×C₅₃	semidirect
271. C₂×#427	C₂×C₅₃	semidirect
274. C₂×C₅₃	#3340	direct (N × H)
281. #556	C₂²×C₅₃	semidirect
299. #498	C₂²×C₅₃	semidirect
300. #485	C₂²×C₅₃	semidirect
301. #482	C₂²×C₅₃	semidirect
302. #477	C₂²×C₅₃	semidirect
303. #474	C₂²×C₅₃	semidirect
304. #465	C₂²×C₅₃	semidirect
305. C₄×SD₁₆	C₂²×C₅₃	semidirect
306. C₈×D₄	C₂²×C₅₃	semidirect
307. #427	C₂²×C₅₃	semidirect
320. C₅₃	C₂×#3340	direct (N × H)
527. C₂	C₅₃×#3340	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×#556	2
C₂×C₅₃×#555	2
C₅₃×#3340	2
C₂×C₅₃×#498	2
C₂×C₅₃×#485	2
C₂×C₅₃×#482	2
C₂×C₅₃×#477	2
C₂×C₅₃×#474	2
C₂×C₅₃×#465	2
C₂×C₄×SD₁₆×C₅₃	2
C₂×C₈×D₄×C₅₃	2
C₂×C₅₃×#427	2
C₂×#3340	53

Subgroups — C2×C53×#3340 #27268743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3340 #27268743