Subgroups — C₂×C₅₃×#3127 #27268068

Order 13568.

1994 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#541	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#519	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3127	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#485	C₂	1	2	yes	maximal
5	C₂×C₅₃×#479	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#462	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#451	C₂	1	2	yes	maximal
8	C₂×C₅₃×#441	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₂×C₅₃×#433	C₂	1	1	yes	maximal, characteristic
11	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#541	—	2	4	no	—
15	C₅₃×#541	C₂²	1	8	yes	—
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#126×C₅₃	—	2	2	no	—
18	C₂×#125×C₅₃	—	2	2	no	—
19	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#519	—	2	4	no	—
22	C₅₃×#519	C₂²	1	8	yes	—
23	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
24	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
27	C₂×#117×C₅₃	—	2	2	no	—
28	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#485	C₂²	1	16	yes	—
31	C₅₃×#479	C₂²	1	16	yes	complemented
32	C₅₃×#462	C₂²	1	16	yes	complemented
33	C₅₃×#451	C₂²	1	16	yes	—
34	C₅₃×#441	C₂²	1	16	yes	complemented
35	C₄×Q₁₆×C₅₃	C₂²	1	16	yes	complemented
36	C₅₃×#433	C₂²	1	8	yes	—
37	C₂×#107×C₅₃	C₂²	1	2	yes	—
38	C₂×#106×C₅₃	C₂²	1	2	yes	—
39	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
40	C₂×#103×C₅₃	C₂²	1	2	yes	—
41	C₂×#103×C₅₃	C₂²	1	2	yes	—
42	C₂×#103×C₅₃	C₂²	1	2	yes	—
43	C₂×#103×C₅₃	C₂²	1	2	yes	—
44	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
45	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂²×Q₈×C₅₃	—	2	8	no	—
48	C₂²×Q₈×C₅₃	—	2	1	no	—
49	C₂²×Q₈×C₅₃	—	2	2	no	—
50	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
51	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
52	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₅₃	—	2	1	no	—
54	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×Q₁₆×C₅₃	—	2	8	no	—
56	C₂×Q₁₆×C₅₃	—	2	8	no	—
57	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
58	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
59	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₅₃	—	2	2	no	—
62	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
63	#126×C₅₃	—	2	16	no	—
64	#125×C₅₃	—	2	16	no	—
65	#125×C₅₃	—	2	2	no	—
66	#125×C₅₃	C₂³	1	4	yes	—
67	C₂²⋊Q₈×C₅₃	—	2	4	no	—
68	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
69	C₄×Q₈×C₅₃	—	2	16	no	—
70	C₄×Q₈×C₅₃	—	2	4	no	—
71	C₄×Q₈×C₅₃	—	2	4	no	—
72	C₄×Q₈×C₅₃	—	2	4	no	—
73	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
74	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
75	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
76	#117×C₅₃	—	2	16	no	—
77	#117×C₅₃	—	2	2	no	—
78	#117×C₅₃	D₄	1	4	yes	—
79	#117×C₅₃	C₂³	1	4	yes	—
80	#117×C₅₃	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
93	C₂×C₄²×C₅₃	—	2	2	no	—
94	C₂×C₄²×C₅₃	—	2	2	no	—
95	C₂×C₄²×C₅₃	—	2	1	no	—
96	C₂×C₄²×C₅₃	—	2	2	no	—
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	#107×C₅₃	C₂³	1	8	yes	—
100	#106×C₅₃	C₂³	1	8	yes	—
101	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
102	#103×C₅₃	C₂³	1	8	yes	—
103	#103×C₅₃	C₂³	1	8	yes	—
104	#103×C₅₃	C₂³	1	8	yes	—
105	#103×C₅₃	C₂³	1	8	yes	—
106	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
107	#97×C₅₃	C₂³	1	4	yes	—
108	C₄×C₈×C₅₃	C₂³	1	4	yes	—
109	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂×Q₈×C₅₃	—	2	16	no	—
111	C₂×Q₈×C₅₃	—	2	8	no	—
112	C₂×Q₈×C₅₃	—	2	16	no	—
113	C₂×Q₈×C₅₃	—	2	4	no	—
114	C₂×Q₈×C₅₃	—	2	2	no	—
115	C₂×Q₈×C₅₃	—	2	2	no	—
116	C₂×Q₈×C₅₃	—	2	4	no	—
117	C₂×Q₈×C₅₃	—	2	4	no	—
118	C₂×Q₈×C₅₃	—	2	4	no	—
119	C₂×Q₈×C₅₃	#41	1	8	yes	—
120	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
121	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
122	C₂²×C₄×C₅₃	—	2	8	no	—
123	C₂²×C₄×C₅₃	—	2	4	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	2	4	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	2	no	—
129	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
135	Q₁₆×C₅₃	—	2	16	no	—
136	Q₁₆×C₅₃	—	2	16	no	—
137	C₂×C₈×C₅₃	—	2	4	no	—
138	C₂×C₈×C₅₃	—	2	4	no	—
139	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
140	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
141	C₄⋊C₄×C₅₃	—	2	8	no	—
142	C₄⋊C₄×C₅₃	—	2	8	no	—
143	C₄⋊C₄×C₅₃	—	2	8	no	—
144	C₄⋊C₄×C₅₃	—	2	8	no	—
145	C₄⋊C₄×C₅₃	—	2	8	no	—
146	C₄⋊C₄×C₅₃	—	2	8	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
150	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
151	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²⋊C₄×C₅₃	—	2	4	no	—
153	C₂²⋊C₄×C₅₃	—	2	8	no	—
154	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
155	C₄²×C₅₃	—	2	8	no	—
156	C₄²×C₅₃	—	2	8	no	—
157	C₄²×C₅₃	—	2	4	no	—
158	C₄²×C₅₃	—	2	8	no	—
159	C₄²×C₅₃	C₂×D₄	1	2	yes	—
160	C₄²×C₅₃	C₂×D₄	1	2	yes	—
161	C₄²×C₅₃	C₂⁴	1	2	yes	—
162	C₄²×C₅₃	C₂⁴	1	2	yes	—
163	C₂³×C₅₃	—	2	2	no	—
164	C₂³×C₅₃	—	2	4	no	—
165	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
166	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
167	Q₈×C₅₃	—	2	16	no	—
168	Q₈×C₅₃	—	2	8	no	—
169	Q₈×C₅₃	—	2	4	no	—
170	Q₈×C₅₃	—	2	4	no	—
171	Q₈×C₅₃	—	2	4	no	—
172	Q₈×C₅₃	—	4	4	no	—
173	Q₈×C₅₃	C₂×#41	1	8	yes	—
174	C₂×C₄×C₅₃	—	2	16	no	—
175	C₂×C₄×C₅₃	—	2	8	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	4	no	—
178	C₂×C₄×C₅₃	—	2	4	no	—
179	C₂×C₄×C₅₃	—	2	4	no	—
180	C₂×C₄×C₅₃	—	2	4	no	—
181	C₂×C₄×C₅₃	—	2	4	no	—
182	C₂×C₄×C₅₃	—	2	4	no	—
183	C₂×C₄×C₅₃	—	2	4	no	—
184	C₂×C₄×C₅₃	—	2	4	no	—
185	C₂×C₄×C₅₃	—	4	4	no	—
186	C₂×C₄×C₅₃	—	4	4	no	—
187	C₂×C₄×C₅₃	—	4	4	no	—
188	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
190	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
191	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
193	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
194	C₈×C₅₃	—	2	4	no	—
195	C₈×C₅₃	—	2	4	no	—
196	C₂×#3127	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂²×C₅₃	—	2	2	no	—
198	C₂²×C₅₃	—	2	4	no	—
199	C₂²×C₅₃	—	4	4	no	—
200	C₂²×C₅₃	#525	1	2	yes	—
201	C₂²×C₅₃	#578	1	4	yes	—
202	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
203	C₄×C₅₃	—	2	8	no	—
204	C₄×C₅₃	—	2	2	no	—
205	C₄×C₅₃	—	2	4	no	—
206	C₄×C₅₃	—	2	4	no	—
207	C₄×C₅₃	—	4	4	no	—
208	C₄×C₅₃	—	4	4	no	—
209	C₄×C₅₃	—	4	4	no	—
210	C₄×C₅₃	#525	1	2	yes	—
211	C₄×C₅₃	C₂²×#41	1	2	yes	—
212	C₂×#541	C₂×C₅₃	1	1	yes	characteristic
213	C₂×#519	C₂×C₅₃	1	1	yes	characteristic
214	#3127	C₂×C₅₃	1	16	yes	direct product factor
215	C₂×#485	C₂×C₅₃	1	2	yes	—
216	C₂×#479	C₂×C₅₃	1	2	yes	complemented
217	C₂×#462	C₂×C₅₃	1	2	yes	complemented
218	C₂×#451	C₂×C₅₃	1	2	yes	—
219	C₂×#441	C₂×C₅₃	1	2	yes	complemented
220	C₂×C₄×Q₁₆	C₂×C₅₃	1	2	yes	complemented
221	C₂×#433	C₂×C₅₃	1	1	yes	characteristic
222	C₂×C₅₃	—	4	2	no	—
223	C₂×C₅₃	#3127	1	4	yes	direct product factor
224	C₂×C₅₃	C₂×#525	1	1	yes	characteristic
225	C₂×C₅₃	C₂×#578	1	2	yes	—
226	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
227	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
228	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
229	#541	—	2	4	no	—
230	#541	C₂²×C₅₃	1	8	yes	—
231	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
232	C₂×#126	—	2	2	no	—
233	C₂×#125	—	2	2	no	—
234	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
235	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
236	#519	—	2	4	no	—
237	#519	C₂²×C₅₃	1	8	yes	—
238	C₂×C₄×Q₈	—	2	2	no	—
239	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
240	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
241	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
242	C₂×#117	—	2	2	no	—
243	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
244	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
245	#485	C₂²×C₅₃	1	16	yes	—
246	#479	C₂²×C₅₃	1	16	yes	complemented
247	#462	C₂²×C₅₃	1	16	yes	complemented
248	#451	C₂²×C₅₃	1	16	yes	—
249	#441	C₂²×C₅₃	1	16	yes	complemented
250	C₄×Q₁₆	C₂²×C₅₃	1	16	yes	complemented
251	#433	C₂²×C₅₃	1	8	yes	—
252	C₂×#107	C₂²×C₅₃	1	2	yes	—
253	C₂×#106	C₂²×C₅₃	1	2	yes	—
254	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
255	C₂×#103	C₂²×C₅₃	1	2	yes	—
256	C₂×#103	C₂²×C₅₃	1	2	yes	—
257	C₂×#103	C₂²×C₅₃	1	2	yes	—
258	C₂×#103	C₂²×C₅₃	1	2	yes	—
259	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
260	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
261	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
262	C₅₃	C₂×#3127	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²×Q₈	—	2	8	no	—
264	C₂²×Q₈	—	2	1	no	—
265	C₂²×Q₈	—	2	2	no	—
266	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
267	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
268	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
269	C₂³×C₄	—	2	1	no	—
270	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
271	C₂×Q₁₆	—	2	8	no	—
272	C₂×Q₁₆	—	2	8	no	—
273	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
274	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
275	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
276	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
277	C₄⋊Q₈	—	2	2	no	—
278	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
279	#126	—	2	16	no	—
280	#125	—	2	16	no	—
281	#125	—	2	2	no	—
282	#125	C₂³×C₅₃	1	4	yes	—
283	C₂²⋊Q₈	—	2	4	no	—
284	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
285	C₄×Q₈	—	2	16	no	—
286	C₄×Q₈	—	2	4	no	—
287	C₄×Q₈	—	2	4	no	—
288	C₄×Q₈	—	2	4	no	—
289	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
290	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
291	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
292	#117	—	2	2	no	—
293	#117	—	2	16	no	—
294	#117	D₄×C₅₃	1	4	yes	—
295	#117	C₂³×C₅₃	1	4	yes	—
296	#117	C₂³×C₅₃	1	4	yes	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
305	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
309	C₂×C₄²	—	2	2	no	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	—	2	2	no	—
313	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
315	#107	C₂³×C₅₃	1	8	yes	—
316	#106	C₂³×C₅₃	1	8	yes	—
317	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
318	#103	C₂³×C₅₃	1	8	yes	—
319	#103	C₂³×C₅₃	1	8	yes	—
320	#103	C₂³×C₅₃	1	8	yes	—
321	#103	C₂³×C₅₃	1	8	yes	—
322	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
323	#97	C₂³×C₅₃	1	4	yes	—
324	C₄×C₈	C₂³×C₅₃	1	4	yes	—
325	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
326	C₂×Q₈	—	2	16	no	—
327	C₂×Q₈	—	2	8	no	—
328	C₂×Q₈	—	2	16	no	—
329	C₂×Q₈	—	2	4	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	4	no	—
333	C₂×Q₈	—	2	4	no	—
334	C₂×Q₈	—	2	4	no	—
335	C₂×Q₈	#41×C₅₃	1	8	yes	—
336	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
337	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
349	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
351	Q₁₆	—	2	16	no	—
352	Q₁₆	—	2	16	no	—
353	C₂×C₈	—	2	4	no	—
354	C₂×C₈	—	2	4	no	—
355	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
356	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
357	C₄⋊C₄	—	2	8	no	—
358	C₄⋊C₄	—	2	8	no	—
359	C₄⋊C₄	—	2	8	no	—
360	C₄⋊C₄	—	2	8	no	—
361	C₄⋊C₄	—	2	8	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
366	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
367	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
368	C₂²⋊C₄	—	2	4	no	—
369	C₂²⋊C₄	—	2	8	no	—
370	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
371	C₄²	—	2	8	no	—
372	C₄²	—	2	8	no	—
373	C₄²	—	2	4	no	—
374	C₄²	—	2	8	no	—
375	C₄²	C₂×D₄×C₅₃	1	2	yes	—
376	C₄²	C₂×D₄×C₅₃	1	2	yes	—
377	C₄²	C₂⁴×C₅₃	1	2	yes	—
378	C₄²	C₂⁴×C₅₃	1	2	yes	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	4	no	—
381	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
382	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
383	Q₈	—	2	16	no	—
384	Q₈	—	2	8	no	—
385	Q₈	—	2	4	no	—
386	Q₈	—	2	4	no	—
387	Q₈	—	2	4	no	—
388	Q₈	—	4	4	no	—
389	Q₈	C₂×#41×C₅₃	1	8	yes	—
390	C₂×C₄	—	2	16	no	—
391	C₂×C₄	—	2	8	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	4	4	no	—
402	C₂×C₄	—	4	4	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
405	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
406	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
407	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
409	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
410	C₈	—	2	4	no	—
411	C₈	—	2	4	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	2	2	no	—
414	C₂²	—	4	4	no	—
415	C₂²	C₅₃×#525	1	2	yes	—
416	C₂²	C₅₃×#578	1	4	yes	—
417	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
418	C₄	—	2	4	no	—
419	C₄	—	2	4	no	—
420	C₄	—	2	8	no	—
421	C₄	—	2	2	no	—
422	C₄	—	4	4	no	—
423	C₄	—	4	4	no	—
424	C₄	—	4	4	no	—
425	C₄	C₅₃×#525	1	2	yes	—
426	C₄	C₂²×#41×C₅₃	1	2	yes	—
427	C₂	—	4	2	no	—
428	C₂	C₅₃×#3127	1	4	yes	direct product factor
429	C₂	C₂×C₅₃×#525	1	1	yes	characteristic
430	C₂	C₂×C₅₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#3127	C₂	direct (N × H)
5. C₂×C₅₃×#479	C₂	semidirect
6. C₂×C₅₃×#462	C₂	semidirect
8. C₂×C₅₃×#441	C₂	semidirect
9. C₂×C₄×Q₁₆×C₅₃	C₂	semidirect
31. C₅₃×#479	C₂²	semidirect
32. C₅₃×#462	C₂²	semidirect
34. C₅₃×#441	C₂²	semidirect
35. C₄×Q₁₆×C₅₃	C₂²	semidirect
196. C₂×#3127	C₅₃	direct (N × H)
214. #3127	C₂×C₅₃	direct (N × H)
216. C₂×#479	C₂×C₅₃	semidirect
217. C₂×#462	C₂×C₅₃	semidirect
219. C₂×#441	C₂×C₅₃	semidirect
220. C₂×C₄×Q₁₆	C₂×C₅₃	semidirect
223. C₂×C₅₃	#3127	direct (N × H)
246. #479	C₂²×C₅₃	semidirect
247. #462	C₂²×C₅₃	semidirect
249. #441	C₂²×C₅₃	semidirect
250. C₄×Q₁₆	C₂²×C₅₃	semidirect
262. C₅₃	C₂×#3127	direct (N × H)
428. C₂	C₅₃×#3127	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×#541	2
C₂×C₅₃×#519	2
C₅₃×#3127	2
C₂×C₅₃×#485	2
C₂×C₅₃×#479	2
C₂×C₅₃×#462	2
C₂×C₅₃×#451	2
C₂×C₅₃×#441	2
C₂×C₄×Q₁₆×C₅₃	2
C₂×C₅₃×#433	2
C₂×#3127	53

Subgroups — C2×C53×#3127 #27268068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3127 #27268068