Subgroups — C₂×C₅₃×#1803 #27228155

Order 13568.

1598 conjugacy classes of subgroups.

The subgroup lattice has 418 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	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#103×C₅₃	C₂	1	2	yes	maximal
4	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1803	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₃×#340	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₅₃	—	2	4	no	—
10	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
11	C₂×C₄×Q₈×C₅₃	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
14	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₅₃	—	2	1	no	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₅₃	—	2	4	no	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
22	C₂×#103×C₅₃	—	2	8	no	—
23	C₂×#103×C₅₃	C₂²	1	8	yes	—
24	C₅₃×#384	—	2	2	no	—
25	C₅₃×#384	C₂²	1	4	yes	—
26	C₅₃×#340	C₂²	1	8	yes	—
27	C₂²×Q₈×C₅₃	—	2	2	no	—
28	C₂²×Q₈×C₅₃	—	2	1	no	—
29	C₂²×Q₈×C₅₃	—	2	2	no	—
30	C₂²×Q₈×C₅₃	—	2	2	no	—
31	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²×Q₈×C₅₃	D₄	1	4	yes	—
33	C₂²×Q₈×C₅₃	Q₈	1	4	yes	—
34	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
35	C₂³×C₄×C₅₃	—	2	1	no	—
36	C₂³×C₄×C₅₃	—	2	1	no	—
37	C₂³×C₄×C₅₃	—	2	2	no	—
38	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	—	2	4	no	—
42	C₂²×C₈×C₅₃	—	2	4	no	—
43	C₂²×C₈×C₅₃	—	2	4	no	—
44	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
45	C₄×Q₈×C₅₃	—	2	16	no	—
46	C₄×Q₈×C₅₃	—	2	8	no	—
47	C₄×Q₈×C₅₃	C₂×C₄	1	16	yes	complemented
48	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
58	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
59	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₅₃	—	2	8	no	—
63	C₂×C₄²×C₅₃	—	2	4	no	—
64	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
65	C₂×C₄²×C₅₃	D₄	1	4	yes	—
66	C₂×C₄²×C₅₃	D₄	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
68	C₄⋊C₈×C₅₃	—	2	8	no	—
69	#103×C₅₃	—	2	16	no	—
70	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₅₃	—	2	4	no	—
72	C₂×Q₈×C₅₃	—	2	8	no	—
73	C₂×Q₈×C₅₃	—	2	4	no	—
74	C₂×Q₈×C₅₃	—	2	8	no	—
75	C₂×Q₈×C₅₃	—	2	4	no	—
76	C₂×Q₈×C₅₃	—	2	8	no	—
77	C₂×Q₈×C₅₃	—	2	2	no	—
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	C₄⋊C₄	1	16	yes	complemented
81	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
82	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
83	C₂×Q₈×C₅₃	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₅₃	—	2	4	no	—
85	C₂²×C₄×C₅₃	—	2	4	no	—
86	C₂²×C₄×C₅₃	—	2	4	no	—
87	C₂²×C₄×C₅₃	—	2	2	no	—
88	C₂²×C₄×C₅₃	—	2	2	no	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	2	4	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	4	no	—
93	C₂²×C₄×C₅₃	—	4	2	no	—
94	C₂²×C₄×C₅₃	—	4	2	no	—
95	C₂²×C₄×C₅₃	—	4	4	no	—
96	C₂²×C₄×C₅₃	—	4	4	no	—
97	C₂²×C₄×C₅₃	—	4	4	no	—
98	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
99	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
100	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
101	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
102	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
109	C₂²×C₄×C₅₃	#41	1	2	yes	—
110	C₂²×C₄×C₅₃	#41	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₅₃	—	2	4	no	—
113	C₂×C₈×C₅₃	—	2	4	no	—
114	C₂×C₈×C₅₃	—	2	4	no	—
115	C₂×C₈×C₅₃	—	4	8	no	—
116	C₄⋊C₄×C₅₃	—	2	16	no	—
117	C₄⋊C₄×C₅₃	—	2	4	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	4	no	—
120	C₄⋊C₄×C₅₃	—	2	4	no	—
121	C₄⋊C₄×C₅₃	—	4	8	no	—
122	C₄⋊C₄×C₅₃	—	4	4	no	—
123	C₄⋊C₄×C₅₃	—	4	8	no	—
124	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
125	C₄²×C₅₃	—	2	16	no	—
126	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
127	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
128	C₄²×C₅₃	C₂×D₄	1	4	yes	—
129	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂³×C₅₃	#121	1	2	yes	—
131	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
132	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
133	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
134	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
135	C₂³×C₅₃	#137	1	2	yes	—
136	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	Q₈×C₅₃	—	2	4	no	—
138	Q₈×C₅₃	—	2	4	no	—
139	Q₈×C₅₃	—	2	4	no	—
140	Q₈×C₅₃	—	2	16	no	—
141	Q₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
142	C₂×C₄×C₅₃	—	2	4	no	—
143	C₂×C₄×C₅₃	—	2	4	no	—
144	C₂×C₄×C₅₃	—	2	4	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	16	no	—
148	C₂×C₄×C₅₃	—	2	8	no	—
149	C₂×C₄×C₅₃	—	2	4	no	—
150	C₂×C₄×C₅₃	—	4	2	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	—	4	8	no	—
153	C₂×C₄×C₅₃	—	4	4	no	—
154	C₂×C₄×C₅₃	—	4	4	no	—
155	C₂×C₄×C₅₃	—	4	4	no	—
156	C₂×C₄×C₅₃	—	8	8	no	—
157	C₂×C₄×C₅₃	#103	1	16	yes	complemented
158	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
159	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
160	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
162	C₂×C₄×C₅₃	#117	1	4	yes	—
163	C₂×C₄×C₅₃	#121	1	4	yes	—
164	C₂×C₄×C₅₃	#121	1	2	yes	—
165	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
166	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
167	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
168	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
169	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
175	C₈×C₅₃	—	4	4	no	—
176	C₂×#1803	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂²×C₅₃	#380	1	4	yes	—
178	C₂²×C₅₃	C₂×#103	1	4	yes	—
179	C₂²×C₅₃	#417	1	4	yes	—
180	C₂²×C₅₃	#460	1	4	yes	—
181	C₂²×C₅₃	#462	1	4	yes	—
182	C₂²×C₅₃	#475	1	4	yes	—
183	C₂²×C₅₃	#477	1	4	yes	—
184	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
186	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
188	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
191	C₄×C₅₃	—	2	4	no	—
192	C₄×C₅₃	—	2	8	no	—
193	C₄×C₅₃	—	4	4	no	—
194	C₄×C₅₃	—	8	4	no	—
195	C₄×C₅₃	#380	1	4	yes	—
196	C₄×C₅₃	C₂×#103	1	8	yes	complemented
197	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
198	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
199	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
200	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
201	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
202	C₂²×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
203	C₂²×#103	C₂×C₅₃	1	2	yes	—
204	C₂×#384	C₂×C₅₃	1	1	yes	characteristic
205	#1803	C₂×C₅₃	1	8	yes	direct product factor
206	C₂×#340	C₂×C₅₃	1	2	yes	—
207	C₂×C₅₃	#1803	1	8	yes	direct product factor
208	C₂×C₅₃	C₂×#380	1	1	yes	characteristic
209	C₂×C₅₃	C₂²×#103	1	1	yes	characteristic
210	C₂×C₅₃	C₂×#417	1	1	yes	characteristic
211	C₂×C₅₃	C₂×#460	1	1	yes	characteristic
212	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
213	C₂×C₅₃	C₂×#475	1	1	yes	characteristic
214	C₂×C₅₃	C₂×#477	1	1	yes	characteristic
215	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
216	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
217	C₂×C₄×Q₈	—	2	2	no	—
218	C₂×C₄×Q₈	—	2	4	no	—
219	C₂×C₄×Q₈	C₄×C₅₃	1	8	yes	complemented
220	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	—
221	C₂²×C₄⋊C₄	—	2	1	no	—
222	C₂²×C₄⋊C₄	—	2	1	no	—
223	C₂²×C₄⋊C₄	—	2	1	no	—
224	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
225	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
226	C₂²×C₄²	—	2	1	no	—
227	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×C₄⋊C₈	—	2	4	no	—
229	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
230	C₂×#103	—	2	8	no	—
231	C₂×#103	C₂²×C₅₃	1	8	yes	—
232	#384	—	2	2	no	—
233	#384	C₂²×C₅₃	1	4	yes	—
234	#340	C₂²×C₅₃	1	8	yes	—
235	C₅₃	C₂×#1803	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²×Q₈	—	2	2	no	—
237	C₂²×Q₈	—	2	1	no	—
238	C₂²×Q₈	—	2	2	no	—
239	C₂²×Q₈	—	2	2	no	—
240	C₂²×Q₈	C₂×C₄×C₅₃	1	4	yes	—
241	C₂²×Q₈	D₄×C₅₃	1	4	yes	—
242	C₂²×Q₈	Q₈×C₅₃	1	4	yes	—
243	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
244	C₂³×C₄	—	2	1	no	—
245	C₂³×C₄	—	2	1	no	—
246	C₂³×C₄	—	2	2	no	—
247	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
248	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
249	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
250	C₂²×C₈	—	2	4	no	—
251	C₂²×C₈	—	2	4	no	—
252	C₂²×C₈	—	2	4	no	—
253	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
254	C₄×Q₈	—	2	8	no	—
255	C₄×Q₈	—	2	16	no	—
256	C₄×Q₈	C₂×C₄×C₅₃	1	16	yes	complemented
257	C₂×C₄⋊C₄	—	2	4	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	8	no	—
261	C₂×C₄⋊C₄	—	2	4	no	—
262	C₂×C₄⋊C₄	—	2	4	no	—
263	C₂×C₄⋊C₄	—	2	4	no	—
264	C₂×C₄⋊C₄	—	2	4	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	—	4	4	no	—
267	C₂×C₄⋊C₄	—	4	4	no	—
268	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
269	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
270	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
271	C₂×C₄²	—	2	4	no	—
272	C₂×C₄²	—	2	8	no	—
273	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
274	C₂×C₄²	D₄×C₅₃	1	4	yes	—
275	C₂×C₄²	D₄×C₅₃	1	2	yes	—
276	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
277	C₄⋊C₈	—	2	8	no	—
278	#103	—	2	16	no	—
279	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
280	C₂×Q₈	—	2	4	no	—
281	C₂×Q₈	—	2	8	no	—
282	C₂×Q₈	—	2	4	no	—
283	C₂×Q₈	—	2	8	no	—
284	C₂×Q₈	—	2	4	no	—
285	C₂×Q₈	—	2	8	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	C₄⋊C₄×C₅₃	1	16	yes	complemented
290	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
291	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
292	C₂×Q₈	C₂×Q₈×C₅₃	1	4	yes	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	4	no	—
305	C₂²×C₄	—	4	4	no	—
306	C₂²×C₄	—	4	4	no	—
307	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
308	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
309	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
310	C₂²×C₄	Q₁₆×C₅₃	1	4	yes	—
311	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
317	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
318	C₂²×C₄	#41×C₅₃	1	2	yes	—
319	C₂²×C₄	#41×C₅₃	1	2	yes	—
320	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
321	C₂×C₈	—	2	4	no	—
322	C₂×C₈	—	2	4	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	—	4	8	no	—
325	C₄⋊C₄	—	2	16	no	—
326	C₄⋊C₄	—	2	4	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	4	8	no	—
331	C₄⋊C₄	—	4	4	no	—
332	C₄⋊C₄	—	4	8	no	—
333	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
334	C₄²	—	2	16	no	—
335	C₄²	C₂²⋊C₄×C₅₃	1	8	yes	—
336	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
337	C₄²	C₂×D₄×C₅₃	1	4	yes	—
338	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
339	C₂³	#121×C₅₃	1	2	yes	—
340	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
341	C₂³	C₂×SD₁₆×C₅₃	1	2	yes	—
342	C₂³	C₂×Q₁₆×C₅₃	1	2	yes	—
343	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
344	C₂³	#137×C₅₃	1	2	yes	—
345	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
346	Q₈	—	2	4	no	—
347	Q₈	—	2	4	no	—
348	Q₈	—	2	4	no	—
349	Q₈	—	2	16	no	—
350	Q₈	C₂×C₄⋊C₄×C₅₃	1	8	yes	complemented
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	16	no	—
357	C₂×C₄	—	2	8	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	8	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	—	8	8	no	—
366	C₂×C₄	#103×C₅₃	1	16	yes	complemented
367	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
369	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
370	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
371	C₂×C₄	#117×C₅₃	1	4	yes	—
372	C₂×C₄	#121×C₅₃	1	4	yes	—
373	C₂×C₄	#121×C₅₃	1	2	yes	—
374	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
375	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
376	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
377	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
378	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
384	C₈	—	4	4	no	—
385	C₂²	C₅₃×#380	1	4	yes	—
386	C₂²	C₂×#103×C₅₃	1	4	yes	—
387	C₂²	C₅₃×#417	1	4	yes	—
388	C₂²	C₅₃×#460	1	4	yes	—
389	C₂²	C₅₃×#462	1	4	yes	—
390	C₂²	C₅₃×#475	1	4	yes	—
391	C₂²	C₅₃×#477	1	4	yes	—
392	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
393	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
394	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
395	C₂²	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
396	C₂²	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
397	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
398	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
399	C₄	—	2	8	no	—
400	C₄	—	2	4	no	—
401	C₄	—	4	4	no	—
402	C₄	—	8	4	no	—
403	C₄	C₅₃×#380	1	4	yes	—
404	C₄	C₂×#103×C₅₃	1	8	yes	complemented
405	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
406	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
407	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
408	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
409	C₂	C₅₃×#1803	1	8	yes	direct product factor
410	C₂	C₂×C₅₃×#380	1	1	yes	characteristic
411	C₂	C₂²×#103×C₅₃	1	1	yes	characteristic
412	C₂	C₂×C₅₃×#417	1	1	yes	characteristic
413	C₂	C₂×C₅₃×#460	1	1	yes	characteristic
414	C₂	C₂×C₅₃×#462	1	1	yes	characteristic
415	C₂	C₂×C₅₃×#475	1	1	yes	characteristic
416	C₂	C₂×C₅₃×#477	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
5. C₅₃×#1803	C₂	direct (N × H)
11. C₂×C₄×Q₈×C₅₃	C₄	semidirect
47. C₄×Q₈×C₅₃	C₂×C₄	semidirect
80. C₂×Q₈×C₅₃	C₄⋊C₄	semidirect
141. Q₈×C₅₃	C₂×C₄⋊C₄	semidirect
157. C₂×C₄×C₅₃	#103	semidirect
176. C₂×#1803	C₅₃	direct (N × H)
196. C₄×C₅₃	C₂×#103	semidirect
205. #1803	C₂×C₅₃	direct (N × H)
207. C₂×C₅₃	#1803	direct (N × H)
219. C₂×C₄×Q₈	C₄×C₅₃	semidirect
235. C₅₃	C₂×#1803	direct (N × H)
256. C₄×Q₈	C₂×C₄×C₅₃	semidirect
289. C₂×Q₈	C₄⋊C₄×C₅₃	semidirect
350. Q₈	C₂×C₄⋊C₄×C₅₃	semidirect
366. C₂×C₄	#103×C₅₃	semidirect
404. C₄	C₂×#103×C₅₃	semidirect
409. C₂	C₅₃×#1803	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
C₂²×C₄⋊C₈×C₅₃	2
C₂²×#103×C₅₃	2
C₂×C₅₃×#384	2
C₅₃×#1803	2
C₂×C₅₃×#340	2
C₂×#1803	53

Subgroups — C2×C53×#1803 #27228155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1803 #27228155