Subgroups — C₂×C₅₃×#2857 #27244135

Order 13568.

3790 conjugacy classes of subgroups.

The subgroup lattice has 384 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, complemented
2	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal
3	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2857	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅₃×#391	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#389	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#380	C₂	1	4	yes	maximal
9	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₅₃	—	2	8	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	8	yes	complemented
14	C₂×C₂²⋊Q₈×C₅₃	—	2	16	no	—
15	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	16	yes	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
20	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₄²×C₅₃	—	2	8	no	—
22	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₅₃	C₂²	1	8	yes	—
25	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#391	C₂²	1	16	yes	complemented
27	C₅₃×#389	C₂²	1	32	yes	complemented
28	C₅₃×#384	C₂²	1	8	yes	complemented
29	C₅₃×#380	C₂²	1	32	yes	—
30	C₂×#95×C₅₃	C₂²	1	8	yes	—
31	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×Q₈×C₅₃	—	2	8	no	—
33	C₂²×Q₈×C₅₃	—	2	8	no	—
34	C₂²×Q₈×C₅₃	—	2	8	no	—
35	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
36	C₂³×C₄×C₅₃	—	2	4	no	—
37	C₂³×C₄×C₅₃	—	2	8	no	—
38	C₂³×C₄×C₅₃	—	2	4	no	—
39	C₂³×C₄×C₅₃	—	2	8	no	—
40	C₂³×C₄×C₅₃	D₄	1	8	yes	—
41	C₂³×C₄×C₅₃	Q₈	1	16	yes	complemented
42	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₅₃	—	2	32	no	—
51	C₂²⋊Q₈×C₅₃	—	2	64	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
56	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
57	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
61	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
62	C₂×C₄²×C₅₃	—	2	32	no	—
63	C₂×C₄²×C₅₃	—	2	4	no	—
64	C₂×C₄²×C₅₃	—	2	8	no	—
65	C₂×C₄²×C₅₃	D₄	1	8	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
69	#95×C₅₃	C₂³	1	32	yes	—
70	C₂⁴×C₅₃	—	2	2	no	—
71	C₂⁴×C₅₃	—	2	2	no	—
72	C₂⁴×C₅₃	—	2	4	no	—
73	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
74	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
75	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
76	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂×Q₈×C₅₃	—	2	16	no	—
78	C₂×Q₈×C₅₃	—	2	16	no	—
79	C₂×Q₈×C₅₃	—	2	16	no	—
80	C₂×Q₈×C₅₃	—	4	32	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	4	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	2	4	no	—
85	C₂²×C₄×C₅₃	—	2	32	no	—
86	C₂²×C₄×C₅₃	—	2	64	no	—
87	C₂²×C₄×C₅₃	—	2	8	no	—
88	C₂²×C₄×C₅₃	—	2	16	no	—
89	C₂²×C₄×C₅₃	—	2	8	no	—
90	C₂²×C₄×C₅₃	—	2	16	no	—
91	C₂²×C₄×C₅₃	—	2	8	no	—
92	C₂²×C₄×C₅₃	—	2	8	no	—
93	C₂²×C₄×C₅₃	—	2	8	no	—
94	C₂²×C₄×C₅₃	—	2	16	no	—
95	C₂²×C₄×C₅₃	—	2	8	no	—
96	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
97	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	—
98	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
101	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
102	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
103	C₂²×C₄×C₅₃	C₂×Q₈	1	32	yes	complemented
104	C₂²×C₄×C₅₃	#41	1	4	yes	—
105	C₂²×C₄×C₅₃	#41	1	8	yes	—
106	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
107	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
108	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
112	C₄⋊C₄×C₅₃	—	2	32	no	—
113	C₄⋊C₄×C₅₃	—	2	16	no	—
114	C₄⋊C₄×C₅₃	—	2	32	no	—
115	C₄⋊C₄×C₅₃	—	2	32	no	—
116	C₂²⋊C₄×C₅₃	—	2	32	no	—
117	C₄²×C₅₃	—	2	8	no	—
118	C₄²×C₅₃	—	2	16	no	—
119	C₄²×C₅₃	C₂×D₄	1	16	yes	—
120	C₂³×C₅₃	—	2	8	no	—
121	C₂³×C₅₃	—	2	16	no	—
122	C₂³×C₅₃	—	2	8	no	—
123	C₂³×C₅₃	—	2	16	no	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	—	2	4	no	—
127	C₂³×C₅₃	C₄⋊Q₈	1	16	yes	complemented
128	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
129	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
130	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
131	C₂³×C₅₃	C₂²×Q₈	1	8	yes	—
132	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
133	C₂³×C₅₃	C₂×#41	1	2	yes	—
134	C₂³×C₅₃	C₂×#41	1	4	yes	—
135	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
136	Q₈×C₅₃	—	4	32	no	—
137	C₂×C₄×C₅₃	—	2	8	no	—
138	C₂×C₄×C₅₃	—	2	16	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	4	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	4	no	—
143	C₂×C₄×C₅₃	—	2	32	no	—
144	C₂×C₄×C₅₃	—	2	64	no	—
145	C₂×C₄×C₅₃	—	2	8	no	—
146	C₂×C₄×C₅₃	—	2	8	no	—
147	C₂×C₄×C₅₃	—	2	8	no	—
148	C₂×C₄×C₅₃	—	2	8	no	—
149	C₂×C₄×C₅₃	—	2	16	no	—
150	C₂×C₄×C₅₃	—	4	16	no	—
151	C₂×C₄×C₅₃	—	4	16	no	—
152	C₂×C₄×C₅₃	#121	1	16	yes	—
153	C₂×C₄×C₅₃	C₂²⋊Q₈	1	32	yes	complemented
154	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
155	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
156	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
157	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
158	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
159	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
160	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
161	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
162	C₂×#2857	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂²×C₅₃	—	2	32	no	—
164	C₂²×C₅₃	—	2	4	no	—
165	C₂²×C₅₃	—	2	8	no	—
166	C₂²×C₅₃	—	2	4	no	—
167	C₂²×C₅₃	—	2	8	no	—
168	C₂²×C₅₃	C₂×#121	1	4	yes	—
169	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	8	yes	—
170	C₂²×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
171	C₂²×C₅₃	C₂×C₄⋊Q₈	1	8	yes	complemented
172	C₂²×C₅₃	#532	1	4	yes	—
173	C₂²×C₅₃	#533	1	8	yes	—
174	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
175	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
176	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
177	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
178	C₂²×C₅₃	C₂²×#41	1	2	yes	—
179	C₄×C₅₃	—	2	4	no	—
180	C₄×C₅₃	—	2	8	no	—
181	C₄×C₅₃	—	4	8	no	—
182	C₄×C₅₃	—	4	8	no	—
183	C₄×C₅₃	C₂×#121	1	8	yes	—
184	C₄×C₅₃	C₂×C₂²⋊Q₈	1	16	yes	complemented
185	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
186	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	—
187	C₂³×C₄²	C₂×C₅₃	1	1	yes	characteristic
188	#2857	C₂×C₅₃	1	16	yes	direct product factor
189	C₂×#391	C₂×C₅₃	1	2	yes	complemented
190	C₂×#389	C₂×C₅₃	1	4	yes	complemented
191	C₂×#384	C₂×C₅₃	1	1	yes	characteristic, complemented
192	C₂×#380	C₂×C₅₃	1	4	yes	—
193	C₂×C₅₃	—	2	8	no	—
194	C₂×C₅₃	#2857	1	8	yes	direct product factor
195	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
196	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	2	yes	—
197	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
198	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#533	1	2	yes	—
200	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
201	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
202	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
203	C₂×C₄⋊Q₈	—	2	8	no	—
204	C₂×C₄⋊Q₈	C₂²×C₅₃	1	8	yes	complemented
205	C₂×C₂²⋊Q₈	—	2	16	no	—
206	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	16	yes	—
207	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
208	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
209	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
210	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
211	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
212	C₂²×C₄²	—	2	8	no	—
213	C₂²×C₄²	C₂²×C₅₃	1	8	yes	—
214	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
215	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
216	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
217	#391	C₂²×C₅₃	1	16	yes	complemented
218	#389	C₂²×C₅₃	1	32	yes	complemented
219	#384	C₂²×C₅₃	1	8	yes	complemented
220	#380	C₂²×C₅₃	1	32	yes	—
221	C₂×#95	C₂²×C₅₃	1	8	yes	—
222	C₅₃	C₂×#2857	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
224	C₂²×Q₈	—	2	8	no	—
225	C₂²×Q₈	—	2	8	no	—
226	C₂²×Q₈	—	2	8	no	—
227	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
228	C₂³×C₄	—	2	4	no	—
229	C₂³×C₄	—	2	8	no	—
230	C₂³×C₄	—	2	4	no	—
231	C₂³×C₄	—	2	8	no	—
232	C₂³×C₄	D₄×C₅₃	1	8	yes	—
233	C₂³×C₄	Q₈×C₅₃	1	16	yes	complemented
234	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
235	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
236	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
237	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
238	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
239	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
240	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
241	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
242	C₄⋊Q₈	—	2	32	no	—
243	C₂²⋊Q₈	—	2	64	no	—
244	C₂×C₄⋊C₄	—	2	8	no	—
245	C₂×C₄⋊C₄	—	2	16	no	—
246	C₂×C₄⋊C₄	—	2	16	no	—
247	C₂×C₄⋊C₄	—	2	16	no	—
248	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
249	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
250	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
251	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
252	C₂×C₂²⋊C₄	—	2	16	no	—
253	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
254	C₂×C₄²	—	2	4	no	—
255	C₂×C₄²	—	2	8	no	—
256	C₂×C₄²	—	2	32	no	—
257	C₂×C₄²	D₄×C₅₃	1	8	yes	—
258	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
259	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
260	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
261	#95	C₂³×C₅₃	1	32	yes	—
262	C₂⁴	—	2	2	no	—
263	C₂⁴	—	2	2	no	—
264	C₂⁴	—	2	4	no	—
265	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
266	C₂⁴	C₂×Q₈×C₅₃	1	8	yes	—
267	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
268	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
269	C₂×Q₈	—	2	16	no	—
270	C₂×Q₈	—	2	16	no	—
271	C₂×Q₈	—	2	16	no	—
272	C₂×Q₈	—	4	32	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	2	32	no	—
278	C₂²×C₄	—	2	64	no	—
279	C₂²×C₄	—	2	8	no	—
280	C₂²×C₄	—	2	16	no	—
281	C₂²×C₄	—	2	8	no	—
282	C₂²×C₄	—	2	16	no	—
283	C₂²×C₄	—	2	8	no	—
284	C₂²×C₄	—	2	8	no	—
285	C₂²×C₄	—	2	8	no	—
286	C₂²×C₄	—	2	16	no	—
287	C₂²×C₄	—	2	8	no	—
288	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
289	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	—
290	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
291	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
292	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
293	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
294	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
295	C₂²×C₄	C₂×Q₈×C₅₃	1	32	yes	complemented
296	C₂²×C₄	#41×C₅₃	1	4	yes	—
297	C₂²×C₄	#41×C₅₃	1	8	yes	—
298	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
299	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
300	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
301	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
302	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
303	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
304	C₄⋊C₄	—	2	16	no	—
305	C₄⋊C₄	—	2	32	no	—
306	C₄⋊C₄	—	2	32	no	—
307	C₄⋊C₄	—	2	32	no	—
308	C₂²⋊C₄	—	2	32	no	—
309	C₄²	—	2	8	no	—
310	C₄²	—	2	16	no	—
311	C₄²	C₂×D₄×C₅₃	1	16	yes	—
312	C₂³	—	2	8	no	—
313	C₂³	—	2	16	no	—
314	C₂³	—	2	8	no	—
315	C₂³	—	2	16	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	2	2	no	—
318	C₂³	—	2	4	no	—
319	C₂³	C₄⋊Q₈×C₅₃	1	16	yes	complemented
320	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
321	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
322	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
323	C₂³	C₂²×Q₈×C₅₃	1	8	yes	—
324	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
325	C₂³	C₂×#41×C₅₃	1	2	yes	—
326	C₂³	C₂×#41×C₅₃	1	4	yes	—
327	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
328	Q₈	—	4	32	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	2	16	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	2	32	no	—
336	C₂×C₄	—	2	64	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	8	no	—
339	C₂×C₄	—	2	8	no	—
340	C₂×C₄	—	2	16	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	4	16	no	—
343	C₂×C₄	—	4	16	no	—
344	C₂×C₄	#121×C₅₃	1	16	yes	—
345	C₂×C₄	C₂²⋊Q₈×C₅₃	1	32	yes	complemented
346	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
347	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
348	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
349	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
350	C₂×C₄	C₂²×Q₈×C₅₃	1	8	yes	—
351	C₂×C₄	C₂²×Q₈×C₅₃	1	8	yes	—
352	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
353	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
354	C₂²	—	2	32	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	2	8	no	—
357	C₂²	—	2	4	no	—
358	C₂²	—	2	8	no	—
359	C₂²	C₂×#121×C₅₃	1	4	yes	—
360	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	8	yes	—
361	C₂²	C₂×C₄⋊Q₈×C₅₃	1	4	yes	—
362	C₂²	C₂×C₄⋊Q₈×C₅₃	1	8	yes	complemented
363	C₂²	C₅₃×#532	1	4	yes	—
364	C₂²	C₅₃×#533	1	8	yes	—
365	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
366	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
367	C₂²	C₂³×Q₈×C₅₃	1	2	yes	—
368	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
369	C₂²	C₂²×#41×C₅₃	1	2	yes	—
370	C₄	—	2	4	no	—
371	C₄	—	2	8	no	—
372	C₄	—	4	8	no	—
373	C₄	—	4	8	no	—
374	C₄	C₂×#121×C₅₃	1	8	yes	—
375	C₄	C₂×C₂²⋊Q₈×C₅₃	1	16	yes	complemented
376	C₂	—	2	8	no	—
377	C₂	C₅₃×#2857	1	8	yes	direct product factor
378	C₂	C₂²×#121×C₅₃	1	1	yes	characteristic
379	C₂	C₂²×C₂²⋊Q₈×C₅₃	1	2	yes	—
380	C₂	C₂²×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
381	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
382	C₂	C₂×C₅₃×#533	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
4. C₅₃×#2857	C₂	direct (N × H)
5. C₂×C₅₃×#391	C₂	semidirect
6. C₂×C₅₃×#389	C₂	semidirect
7. C₂×C₅₃×#384	C₂	semidirect
13. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
26. C₅₃×#391	C₂²	semidirect
27. C₅₃×#389	C₂²	semidirect
28. C₅₃×#384	C₂²	semidirect
41. C₂³×C₄×C₅₃	Q₈	semidirect
103. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
127. C₂³×C₅₃	C₄⋊Q₈	semidirect
153. C₂×C₄×C₅₃	C₂²⋊Q₈	semidirect
162. C₂×#2857	C₅₃	direct (N × H)
171. C₂²×C₅₃	C₂×C₄⋊Q₈	semidirect
184. C₄×C₅₃	C₂×C₂²⋊Q₈	semidirect
185. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
188. #2857	C₂×C₅₃	direct (N × H)
189. C₂×#391	C₂×C₅₃	semidirect
190. C₂×#389	C₂×C₅₃	semidirect
191. C₂×#384	C₂×C₅₃	semidirect
194. C₂×C₅₃	#2857	direct (N × H)
204. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
217. #391	C₂²×C₅₃	semidirect
218. #389	C₂²×C₅₃	semidirect
219. #384	C₂²×C₅₃	semidirect
222. C₅₃	C₂×#2857	direct (N × H)
233. C₂³×C₄	Q₈×C₅₃	semidirect
295. C₂²×C₄	C₂×Q₈×C₅₃	semidirect
319. C₂³	C₄⋊Q₈×C₅₃	semidirect
345. C₂×C₄	C₂²⋊Q₈×C₅₃	semidirect
362. C₂²	C₂×C₄⋊Q₈×C₅₃	semidirect
375. C₄	C₂×C₂²⋊Q₈×C₅₃	semidirect
377. C₂	C₅₃×#2857	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×C₂²⋊Q₈×C₅₃	2
C₂³×C₄²×C₅₃	2
C₅₃×#2857	2
C₂×C₅₃×#391	2
C₂×C₅₃×#389	2
C₂×C₅₃×#384	2
C₂×C₅₃×#380	2
C₂×#2857	53

Subgroups — C2×C53×#2857 #27244135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2857 #27244135