Subgroups — C₂×C₅₃×#2434 #27242196

Order 13568.

5286 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2434	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₅₃×#395	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#391	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₃×#385	C₂	1	4	yes	maximal
8	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₅₃	—	2	16	no	—
10	C₂³×Q₈×C₅₃	—	2	8	no	—
11	C₂³×Q₈×C₅₃	C₂²	1	8	yes	—
12	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
13	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₅₃	—	2	16	no	—
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	16	yes	complemented
19	C₂×C₂²⋊Q₈×C₅₃	—	2	8	no	—
20	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#395	C₂²	1	32	yes	complemented
28	C₅₃×#391	C₂²	1	16	yes	complemented
29	C₅₃×#385	C₂²	1	32	yes	—
30	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
31	C₂×#95×C₅₃	C₂²	1	2	yes	—
32	C₂×#95×C₅₃	C₂²	1	8	yes	—
33	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×Q₈×C₅₃	—	2	128	no	—
35	C₂²×Q₈×C₅₃	—	2	64	no	—
36	C₂²×Q₈×C₅₃	—	2	16	no	—
37	C₂²×Q₈×C₅₃	—	2	8	no	—
38	C₂²×Q₈×C₅₃	—	2	32	no	—
39	C₂²×Q₈×C₅₃	—	2	8	no	—
40	C₂²×Q₈×C₅₃	—	2	8	no	—
41	C₂²×Q₈×C₅₃	—	2	4	no	—
42	C₂²×Q₈×C₅₃	—	2	8	no	—
43	C₂²×Q₈×C₅₃	D₄	1	16	yes	—
44	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
45	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
46	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
47	C₂³×C₄×C₅₃	—	2	8	no	—
48	C₂³×C₄×C₅₃	—	2	16	no	—
49	C₂³×C₄×C₅₃	—	2	8	no	—
50	C₂³×C₄×C₅₃	D₄	1	8	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
54	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₅₃	—	2	64	no	—
60	C₂²⋊Q₈×C₅₃	—	2	32	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
63	C₂×C₄⋊C₄×C₅₃	—	2	32	no	—
64	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	32	yes	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
68	C₂×C₂²⋊C₄×C₅₃	Q₈	1	16	yes	complemented
69	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
70	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
71	C₂×C₄²×C₅₃	—	2	8	no	—
72	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
73	#95×C₅₃	C₂³	1	8	yes	—
74	#95×C₅₃	C₂³	1	32	yes	—
75	C₂⁴×C₅₃	—	2	4	no	—
76	C₂⁴×C₅₃	—	2	4	no	—
77	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
78	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
79	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
80	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂×Q₈×C₅₃	—	2	256	no	—
82	C₂×Q₈×C₅₃	—	2	64	no	—
83	C₂×Q₈×C₅₃	—	2	64	no	—
84	C₂×Q₈×C₅₃	—	2	32	no	—
85	C₂×Q₈×C₅₃	—	2	16	no	—
86	C₂×Q₈×C₅₃	—	2	16	no	—
87	C₂×Q₈×C₅₃	—	2	16	no	—
88	C₂×Q₈×C₅₃	—	2	8	no	—
89	C₂×Q₈×C₅₃	—	2	16	no	—
90	C₂×Q₈×C₅₃	—	4	16	no	—
91	C₂×Q₈×C₅₃	C₂×D₄	1	32	yes	—
92	C₂²×C₄×C₅₃	—	2	64	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	—	2	8	no	—
97	C₂²×C₄×C₅₃	—	2	4	no	—
98	C₂²×C₄×C₅₃	—	2	8	no	—
99	C₂²×C₄×C₅₃	—	2	32	no	—
100	C₂²×C₄×C₅₃	—	2	32	no	—
101	C₂²×C₄×C₅₃	—	2	16	no	—
102	C₂²×C₄×C₅₃	—	2	4	no	—
103	C₂²×C₄×C₅₃	—	2	4	no	—
104	C₂²×C₄×C₅₃	—	2	8	no	—
105	C₂²×C₄×C₅₃	—	2	8	no	—
106	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	16	yes	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
111	C₂²×C₄×C₅₃	C₂×Q₈	1	16	yes	complemented
112	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
114	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
115	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
118	C₄⋊C₄×C₅₃	—	2	16	no	—
119	C₄⋊C₄×C₅₃	—	2	8	no	—
120	C₄⋊C₄×C₅₃	—	2	64	no	—
121	C₂²⋊C₄×C₅₃	—	2	16	no	—
122	C₂²⋊C₄×C₅₃	C₂×Q₈	1	32	yes	complemented
123	C₄²×C₅₃	—	2	16	no	—
124	C₂³×C₅₃	—	2	8	no	—
125	C₂³×C₅₃	—	2	8	no	—
126	C₂³×C₅₃	—	2	32	no	—
127	C₂³×C₅₃	—	2	4	no	—
128	C₂³×C₅₃	—	2	4	no	—
129	C₂³×C₅₃	C₄⋊Q₈	1	16	yes	complemented
130	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
131	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
132	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
133	C₂³×C₅₃	C₂²×Q₈	1	8	yes	—
134	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
135	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
136	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
137	Q₈×C₅₃	—	2	64	no	—
138	Q₈×C₅₃	—	2	64	no	—
139	Q₈×C₅₃	—	4	16	no	—
140	C₂×C₄×C₅₃	—	2	64	no	—
141	C₂×C₄×C₅₃	—	2	32	no	—
142	C₂×C₄×C₅₃	—	2	8	no	—
143	C₂×C₄×C₅₃	—	2	32	no	—
144	C₂×C₄×C₅₃	—	2	8	no	—
145	C₂×C₄×C₅₃	—	2	8	no	—
146	C₂×C₄×C₅₃	—	2	4	no	—
147	C₂×C₄×C₅₃	—	2	8	no	—
148	C₂×C₄×C₅₃	—	2	8	no	—
149	C₂×C₄×C₅₃	—	2	32	no	—
150	C₂×C₄×C₅₃	—	2	4	no	—
151	C₂×C₄×C₅₃	—	2	8	no	—
152	C₂×C₄×C₅₃	—	2	4	no	—
153	C₂×C₄×C₅₃	—	2	8	no	—
154	C₂×C₄×C₅₃	—	4	8	no	—
155	C₂×C₄×C₅₃	—	4	8	no	—
156	C₂×C₄×C₅₃	#120	1	16	yes	—
157	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
158	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
159	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
160	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
161	C₂×C₄×C₅₃	C₂²×Q₈	1	16	yes	complemented
162	C₂×#2434	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂²×C₅₃	—	2	4	no	—
164	C₂²×C₅₃	—	2	4	no	—
165	C₂²×C₅₃	—	2	32	no	—
166	C₂²×C₅₃	—	2	16	no	—
167	C₂²×C₅₃	C₂×#120	1	4	yes	—
168	C₂²×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
169	C₂²×C₅₃	C₂×C₄⋊Q₈	1	8	yes	complemented
170	C₂²×C₅₃	#536	1	4	yes	—
171	C₂²×C₅₃	D₄×Q₈	1	16	yes	—
172	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
173	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
174	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
175	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
176	C₄×C₅₃	—	2	16	no	—
177	C₄×C₅₃	—	2	4	no	—
178	C₄×C₅₃	—	2	16	no	—
179	C₄×C₅₃	—	4	4	no	—
180	C₄×C₅₃	—	4	4	no	—
181	C₄×C₅₃	C₂×#120	1	8	yes	—
182	C₂⁴×Q₈	C₂×C₅₃	1	1	yes	characteristic
183	C₂²×C₄⋊Q₈	C₂×C₅₃	1	2	yes	complemented
184	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
185	#2434	C₂×C₅₃	1	16	yes	direct product factor
186	C₂×#395	C₂×C₅₃	1	4	yes	complemented
187	C₂×#391	C₂×C₅₃	1	2	yes	complemented
188	C₂×#385	C₂×C₅₃	1	4	yes	—
189	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
190	C₂×C₅₃	—	2	8	no	—
191	C₂×C₅₃	#2434	1	8	yes	direct product factor
192	C₂×C₅₃	C₂²×#120	1	1	yes	characteristic
193	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
194	C₂×C₅₃	C₂×#536	1	1	yes	characteristic
195	C₂×C₅₃	C₂×D₄×Q₈	1	4	yes	—
196	C₂³×Q₈	—	2	16	no	—
197	C₂³×Q₈	—	2	8	no	—
198	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
199	C₂³×Q₈	C₂²×C₅₃	1	8	yes	—
200	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
201	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
202	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
203	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
204	C₂×C₄⋊Q₈	—	2	16	no	—
205	C₂×C₄⋊Q₈	C₂²×C₅₃	1	16	yes	complemented
206	C₂×C₂²⋊Q₈	—	2	8	no	—
207	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	8	yes	—
208	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
209	C₂²×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
210	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
211	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
212	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
213	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
214	#395	C₂²×C₅₃	1	32	yes	complemented
215	#391	C₂²×C₅₃	1	16	yes	complemented
216	#385	C₂²×C₅₃	1	32	yes	—
217	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
218	C₂×#95	C₂²×C₅₃	1	2	yes	—
219	C₂×#95	C₂²×C₅₃	1	8	yes	—
220	C₅₃	C₂×#2434	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
222	C₂²×Q₈	—	2	128	no	—
223	C₂²×Q₈	—	2	64	no	—
224	C₂²×Q₈	—	2	16	no	—
225	C₂²×Q₈	—	2	8	no	—
226	C₂²×Q₈	—	2	32	no	—
227	C₂²×Q₈	—	2	8	no	—
228	C₂²×Q₈	—	2	8	no	—
229	C₂²×Q₈	—	2	4	no	—
230	C₂²×Q₈	—	2	8	no	—
231	C₂²×Q₈	D₄×C₅₃	1	16	yes	—
232	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
233	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
234	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
235	C₂³×C₄	—	2	8	no	—
236	C₂³×C₄	—	2	16	no	—
237	C₂³×C₄	—	2	8	no	—
238	C₂³×C₄	D₄×C₅₃	1	8	yes	—
239	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
240	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
241	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
242	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
243	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
244	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
245	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
246	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
247	C₄⋊Q₈	—	2	64	no	—
248	C₂²⋊Q₈	—	2	32	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	—	2	8	no	—
251	C₂×C₄⋊C₄	—	2	32	no	—
252	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
253	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
254	C₂×C₄⋊C₄	C₂³×C₅₃	1	32	yes	—
255	C₂×C₂²⋊C₄	—	2	8	no	—
256	C₂×C₂²⋊C₄	Q₈×C₅₃	1	16	yes	complemented
257	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
258	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
259	C₂×C₄²	—	2	8	no	—
260	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
261	#95	C₂³×C₅₃	1	8	yes	—
262	#95	C₂³×C₅₃	1	32	yes	—
263	C₂⁴	—	2	4	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	256	no	—
270	C₂×Q₈	—	2	64	no	—
271	C₂×Q₈	—	2	64	no	—
272	C₂×Q₈	—	2	32	no	—
273	C₂×Q₈	—	2	16	no	—
274	C₂×Q₈	—	2	16	no	—
275	C₂×Q₈	—	2	16	no	—
276	C₂×Q₈	—	2	8	no	—
277	C₂×Q₈	—	2	16	no	—
278	C₂×Q₈	—	4	16	no	—
279	C₂×Q₈	C₂×D₄×C₅₃	1	32	yes	—
280	C₂²×C₄	—	2	64	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	4	no	—
286	C₂²×C₄	—	2	8	no	—
287	C₂²×C₄	—	2	32	no	—
288	C₂²×C₄	—	2	32	no	—
289	C₂²×C₄	—	2	16	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	2	8	no	—
293	C₂²×C₄	—	2	8	no	—
294	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	—
295	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
296	C₂²×C₄	C₂×D₄×C₅₃	1	16	yes	—
297	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
298	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
299	C₂²×C₄	C₂×Q₈×C₅₃	1	16	yes	complemented
300	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
301	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
302	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
303	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
304	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
305	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
306	C₄⋊C₄	—	2	8	no	—
307	C₄⋊C₄	—	2	16	no	—
308	C₄⋊C₄	—	2	64	no	—
309	C₂²⋊C₄	—	2	16	no	—
310	C₂²⋊C₄	C₂×Q₈×C₅₃	1	32	yes	complemented
311	C₄²	—	2	16	no	—
312	C₂³	—	2	8	no	—
313	C₂³	—	2	8	no	—
314	C₂³	—	2	32	no	—
315	C₂³	—	2	4	no	—
316	C₂³	—	2	4	no	—
317	C₂³	C₄⋊Q₈×C₅₃	1	16	yes	complemented
318	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
319	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
320	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
321	C₂³	C₂²×Q₈×C₅₃	1	8	yes	—
322	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
323	C₂³	2₋¹⁺⁴×C₅₃	1	4	yes	—
324	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
325	Q₈	—	2	64	no	—
326	Q₈	—	2	64	no	—
327	Q₈	—	4	16	no	—
328	C₂×C₄	—	2	64	no	—
329	C₂×C₄	—	2	32	no	—
330	C₂×C₄	—	2	8	no	—
331	C₂×C₄	—	2	32	no	—
332	C₂×C₄	—	2	8	no	—
333	C₂×C₄	—	2	8	no	—
334	C₂×C₄	—	2	8	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	32	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	8	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	4	8	no	—
343	C₂×C₄	—	4	8	no	—
344	C₂×C₄	#120×C₅₃	1	16	yes	—
345	C₂×C₄	C₂²×D₄×C₅₃	1	16	yes	—
346	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
347	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
348	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
349	C₂×C₄	C₂²×Q₈×C₅₃	1	16	yes	complemented
350	C₂²	—	2	4	no	—
351	C₂²	—	2	32	no	—
352	C₂²	—	2	16	no	—
353	C₂²	—	2	4	no	—
354	C₂²	C₂×#120×C₅₃	1	4	yes	—
355	C₂²	C₂×C₄⋊Q₈×C₅₃	1	4	yes	—
356	C₂²	C₂×C₄⋊Q₈×C₅₃	1	8	yes	complemented
357	C₂²	C₅₃×#536	1	4	yes	—
358	C₂²	D₄×Q₈×C₅₃	1	16	yes	—
359	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
360	C₂²	C₂³×D₄×C₅₃	1	2	yes	—
361	C₂²	C₂³×Q₈×C₅₃	1	2	yes	—
362	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
363	C₄	—	2	16	no	—
364	C₄	—	2	16	no	—
365	C₄	—	2	4	no	—
366	C₄	—	4	4	no	—
367	C₄	—	4	4	no	—
368	C₄	C₂×#120×C₅₃	1	8	yes	—
369	C₂	—	2	8	no	—
370	C₂	C₅₃×#2434	1	8	yes	direct product factor
371	C₂	C₂²×#120×C₅₃	1	1	yes	characteristic
372	C₂	C₂²×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
373	C₂	C₂×C₅₃×#536	1	1	yes	characteristic
374	C₂	C₂×D₄×Q₈×C₅₃	1	4	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
2. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
4. C₅₃×#2434	C₂	direct (N × H)
5. C₂×C₅₃×#395	C₂	semidirect
6. C₂×C₅₃×#391	C₂	semidirect
18. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
27. C₅₃×#395	C₂²	semidirect
28. C₅₃×#391	C₂²	semidirect
68. C₂×C₂²⋊C₄×C₅₃	Q₈	semidirect
111. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
122. C₂²⋊C₄×C₅₃	C₂×Q₈	semidirect
129. C₂³×C₅₃	C₄⋊Q₈	semidirect
161. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
162. C₂×#2434	C₅₃	direct (N × H)
169. C₂²×C₅₃	C₂×C₄⋊Q₈	semidirect
183. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
185. #2434	C₂×C₅₃	direct (N × H)
186. C₂×#395	C₂×C₅₃	semidirect
187. C₂×#391	C₂×C₅₃	semidirect
191. C₂×C₅₃	#2434	direct (N × H)
205. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
214. #395	C₂²×C₅₃	semidirect
215. #391	C₂²×C₅₃	semidirect
220. C₅₃	C₂×#2434	direct (N × H)
256. C₂×C₂²⋊C₄	Q₈×C₅₃	semidirect
299. C₂²×C₄	C₂×Q₈×C₅₃	semidirect
310. C₂²⋊C₄	C₂×Q₈×C₅₃	semidirect
317. C₂³	C₄⋊Q₈×C₅₃	semidirect
349. C₂×C₄	C₂²×Q₈×C₅₃	semidirect
356. C₂²	C₂×C₄⋊Q₈×C₅₃	semidirect
370. C₂	C₅₃×#2434	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×Q₈×C₅₃	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₅₃×#2434	2
C₂×C₅₃×#395	2
C₂×C₅₃×#391	2
C₂×C₅₃×#385	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×#2434	53

Subgroups — C2×C53×#2434 #27242196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2434 #27242196