Subgroups — C₂×C₅₃×#1487 #27224953

Order 13568.

1282 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1487	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×Q₈⋊C₈	C₂	1	4	yes	maximal, complemented
6	C₂³×Q₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×M₁₆×C₅₃	—	2	1	no	—
8	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
9	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
10	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
11	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
12	C₂×C₄×Q₈×C₅₃	C₄	1	4	yes	—
13	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
16	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×C₄²×C₅₃	—	2	1	no	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#431	—	2	2	no	—
21	C₅₃×#431	C₂²	1	4	yes	—
22	C₅₃×#423	—	2	2	no	—
23	C₅₃×#423	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
25	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
29	C₅₃×Q₈⋊C₈	C₂²	1	16	yes	complemented
30	C₂²×Q₈×C₅₃	—	2	8	no	—
31	C₂²×Q₈×C₅₃	—	2	2	no	—
32	C₂²×Q₈×C₅₃	—	2	1	no	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₅₃	—	2	1	no	—
36	C₂³×C₄×C₅₃	—	2	1	no	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	—	2	4	no	—
41	C₂×M₁₆×C₅₃	—	4	4	no	—
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	—	2	2	no	—
44	C₄×Q₈×C₅₃	—	2	4	no	—
45	C₄×Q₈×C₅₃	—	2	4	no	—
46	C₄×Q₈×C₅₃	—	2	4	no	—
47	C₄×Q₈×C₅₃	—	2	2	no	—
48	C₄×Q₈×C₅₃	—	2	4	no	—
49	C₄×Q₈×C₅₃	—	2	2	no	—
50	C₄×Q₈×C₅₃	—	2	4	no	—
51	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
52	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
53	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₅₃	—	2	4	no	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	—	2	4	no	—
65	C₂×C₄²×C₅₃	—	2	2	no	—
66	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	D₄	1	2	yes	—
70	C₂×C₄²×C₅₃	D₄	1	2	yes	—
71	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	—	2	4	no	—
75	C₄⋊C₈×C₅₃	—	2	4	no	—
76	C₄⋊C₈×C₅₃	—	2	2	no	—
77	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
78	C₂²⋊C₈×C₅₃	—	2	4	no	—
79	C₄×C₈×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	C₂³	1	4	yes	—
81	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	16	no	—
83	C₂×Q₈×C₅₃	—	2	8	no	—
84	C₂×Q₈×C₅₃	—	2	16	no	—
85	C₂×Q₈×C₅₃	—	2	4	no	—
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	—	2	4	no	—
88	C₂×Q₈×C₅₃	M₁₆	1	8	yes	complemented
89	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₅₃	—	2	8	no	—
91	C₂²×C₄×C₅₃	—	2	4	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	4	no	—
94	C₂²×C₄×C₅₃	—	2	4	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	4	2	no	—
97	C₂²×C₄×C₅₃	—	4	4	no	—
98	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
101	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
102	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
104	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
105	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
106	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	M₁₆×C₅₃	—	4	8	no	—
113	C₂×C₈×C₅₃	—	2	4	no	—
114	C₂×C₈×C₅₃	—	2	4	no	—
115	C₂×C₈×C₅₃	—	4	4	no	—
116	C₂×C₈×C₅₃	—	4	4	no	—
117	C₄⋊C₄×C₅₃	—	2	8	no	—
118	C₄⋊C₄×C₅₃	—	2	4	no	—
119	C₄⋊C₄×C₅₃	—	2	4	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	4	no	—
122	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
123	C₄²×C₅₃	—	2	8	no	—
124	C₄²×C₅₃	—	2	4	no	—
125	C₄²×C₅₃	—	2	4	no	—
126	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
127	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
128	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
129	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
130	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	C₂×D₄	1	2	yes	—
133	C₄²×C₅₃	C₂×D₄	1	2	yes	—
134	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	—	2	2	no	—
138	C₂³×C₅₃	#103	1	4	yes	—
139	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	Q₈×C₅₃	—	2	16	no	—
142	Q₈×C₅₃	—	2	4	no	—
143	Q₈×C₅₃	—	2	8	no	—
144	Q₈×C₅₃	C₂×M₁₆	1	8	yes	complemented
145	C₂×C₄×C₅₃	—	2	16	no	—
146	C₂×C₄×C₅₃	—	2	4	no	—
147	C₂×C₄×C₅₃	—	2	8	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	4	no	—
152	C₂×C₄×C₅₃	—	2	4	no	—
153	C₂×C₄×C₅₃	—	2	4	no	—
154	C₂×C₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	4	2	no	—
156	C₂×C₄×C₅₃	—	4	8	no	—
157	C₂×C₄×C₅₃	—	4	4	no	—
158	C₂×C₄×C₅₃	#103	1	8	yes	—
159	C₂×C₄×C₅₃	#103	1	4	yes	—
160	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
161	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
167	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
168	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
169	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
170	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
171	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₅₃	—	4	4	no	—
175	C₈×C₅₃	—	4	4	no	—
176	C₂×#1487	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂²×C₅₃	—	2	2	no	—
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	2	8	no	—
180	C₂²×C₅₃	#407	1	2	yes	—
181	C₂²×C₅₃	C₂×#103	1	4	yes	—
182	C₂²×C₅₃	C₂×#103	1	2	yes	—
183	C₂²×C₅₃	#421	1	2	yes	—
184	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₄×C₅₃	—	2	8	no	—
186	C₄×C₅₃	—	2	2	no	—
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	4	4	no	—
191	C₄×C₅₃	#407	1	2	yes	—
192	C₄×C₅₃	C₂×#103	1	4	yes	—
193	C₄×C₅₃	C₂×#103	1	2	yes	—
194	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
195	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
196	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
197	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
198	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
199	C₂×#423	C₂×C₅₃	1	1	yes	characteristic
200	#1487	C₂×C₅₃	1	8	yes	direct product factor
201	C₂×Q₈⋊C₈	C₂×C₅₃	1	4	yes	complemented
202	C₂×C₅₃	—	2	4	no	—
203	C₂×C₅₃	#1487	1	4	yes	direct product factor
204	C₂×C₅₃	C₂×#407	1	1	yes	characteristic
205	C₂×C₅₃	C₂²×#103	1	1	yes	characteristic
206	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
207	C₂³×Q₈	C₄×C₅₃	1	1	yes	characteristic
208	C₂²×M₁₆	—	2	1	no	—
209	C₂×C₄×Q₈	—	2	1	no	—
210	C₂×C₄×Q₈	—	2	2	no	—
211	C₂×C₄×Q₈	—	2	1	no	—
212	C₂×C₄×Q₈	—	2	2	no	—
213	C₂×C₄×Q₈	C₄×C₅₃	1	4	yes	—
214	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
215	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
216	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	—
217	C₂²×C₄⋊C₄	—	2	1	no	—
218	C₂²×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄²	—	2	1	no	—
220	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
221	#431	—	2	2	no	—
222	#431	C₂²×C₅₃	1	4	yes	—
223	#423	—	2	2	no	—
224	#423	C₂²×C₅₃	1	4	yes	—
225	C₂×C₄⋊C₈	—	2	1	no	—
226	C₂×C₄⋊C₈	—	2	1	no	—
227	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
228	C₂×C₂²⋊C₈	—	2	1	no	—
229	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
230	Q₈⋊C₈	C₂²×C₅₃	1	16	yes	complemented
231	C₅₃	C₂×#1487	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²×Q₈	—	2	8	no	—
233	C₂²×Q₈	—	2	2	no	—
234	C₂²×Q₈	—	2	1	no	—
235	C₂²×Q₈	C₂×C₄×C₅₃	1	4	yes	—
236	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
237	C₂³×C₄	—	2	1	no	—
238	C₂³×C₄	—	2	1	no	—
239	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
240	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
241	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
242	C₂×M₁₆	—	2	4	no	—
243	C₂×M₁₆	—	4	4	no	—
244	C₂²×C₈	—	2	2	no	—
245	C₂²×C₈	—	2	2	no	—
246	C₄×Q₈	—	2	2	no	—
247	C₄×Q₈	—	2	4	no	—
248	C₄×Q₈	—	2	2	no	—
249	C₄×Q₈	—	2	4	no	—
250	C₄×Q₈	—	2	4	no	—
251	C₄×Q₈	—	2	4	no	—
252	C₄×Q₈	—	2	4	no	—
253	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
254	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	—
255	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	4	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	4	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
263	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
264	C₂×C₄²	—	2	4	no	—
265	C₂×C₄²	—	2	2	no	—
266	C₂×C₄²	—	2	4	no	—
267	C₂×C₄²	—	2	2	no	—
268	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
269	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
270	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄²	D₄×C₅₃	1	2	yes	—
273	C₂×C₄²	D₄×C₅₃	1	2	yes	—
274	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
276	C₄⋊C₈	—	2	4	no	—
277	C₄⋊C₈	—	2	4	no	—
278	C₄⋊C₈	—	2	2	no	—
279	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
280	C₂²⋊C₈	—	2	4	no	—
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	C₂³×C₅₃	1	4	yes	—
283	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
284	C₂×Q₈	—	2	16	no	—
285	C₂×Q₈	—	2	8	no	—
286	C₂×Q₈	—	2	16	no	—
287	C₂×Q₈	—	2	4	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	4	no	—
290	C₂×Q₈	M₁₆×C₅₃	1	8	yes	complemented
291	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
292	C₂²×C₄	—	2	8	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	4	no	—
300	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
303	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
304	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
305	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
306	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
307	C₂²×C₄	Q₁₆×C₅₃	1	4	yes	—
308	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
314	M₁₆	—	4	8	no	—
315	C₂×C₈	—	2	4	no	—
316	C₂×C₈	—	2	4	no	—
317	C₂×C₈	—	4	4	no	—
318	C₂×C₈	—	4	4	no	—
319	C₄⋊C₄	—	2	4	no	—
320	C₄⋊C₄	—	2	8	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
325	C₄²	—	2	8	no	—
326	C₄²	—	2	4	no	—
327	C₄²	—	2	4	no	—
328	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
329	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
330	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
331	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
332	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
333	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
334	C₄²	C₂×D₄×C₅₃	1	2	yes	—
335	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₄²	C₂×D₄×C₅₃	1	2	yes	—
337	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
338	C₂³	—	2	2	no	—
339	C₂³	—	2	2	no	—
340	C₂³	#103×C₅₃	1	4	yes	—
341	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
342	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
343	Q₈	—	2	16	no	—
344	Q₈	—	2	4	no	—
345	Q₈	—	2	8	no	—
346	Q₈	C₂×M₁₆×C₅₃	1	8	yes	complemented
347	C₂×C₄	—	2	16	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	8	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	8	no	—
359	C₂×C₄	—	4	4	no	—
360	C₂×C₄	#103×C₅₃	1	8	yes	—
361	C₂×C₄	#103×C₅₃	1	4	yes	—
362	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
363	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
364	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
365	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
367	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
369	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
370	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
371	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
372	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
373	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
376	C₈	—	4	4	no	—
377	C₈	—	4	4	no	—
378	C₂²	—	2	8	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	C₅₃×#407	1	2	yes	—
382	C₂²	C₂×#103×C₅₃	1	4	yes	—
383	C₂²	C₂×#103×C₅₃	1	2	yes	—
384	C₂²	C₅₃×#421	1	2	yes	—
385	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
386	C₄	—	2	8	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	4	no	—
392	C₄	C₅₃×#407	1	2	yes	—
393	C₄	C₂×#103×C₅₃	1	2	yes	—
394	C₄	C₂×#103×C₅₃	1	4	yes	—
395	C₄	C₂²×M₁₆×C₅₃	1	2	yes	—
396	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
397	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
398	C₂	—	2	4	no	—
399	C₂	C₅₃×#1487	1	4	yes	direct product factor
400	C₂	C₂×C₅₃×#407	1	1	yes	characteristic
401	C₂	C₂²×#103×C₅₃	1	1	yes	characteristic
402	C₂	C₂×C₅₃×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1487	C₂	direct (N × H)
5. C₂×C₅₃×Q₈⋊C₈	C₂	semidirect
29. C₅₃×Q₈⋊C₈	C₂²	semidirect
88. C₂×Q₈×C₅₃	M₁₆	semidirect
144. Q₈×C₅₃	C₂×M₁₆	semidirect
176. C₂×#1487	C₅₃	direct (N × H)
200. #1487	C₂×C₅₃	direct (N × H)
201. C₂×Q₈⋊C₈	C₂×C₅₃	semidirect
203. C₂×C₅₃	#1487	direct (N × H)
230. Q₈⋊C₈	C₂²×C₅₃	semidirect
231. C₅₃	C₂×#1487	direct (N × H)
290. C₂×Q₈	M₁₆×C₅₃	semidirect
346. Q₈	C₂×M₁₆×C₅₃	semidirect
399. C₂	C₅₃×#1487	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×C₄×Q₈×C₅₃	2
C₂×C₅₃×#431	2
C₂×C₅₃×#423	2
C₅₃×#1487	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×#1487	53

Subgroups — C2×C53×#1487 #27224953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1487 #27224953