Subgroups — C₂×C₅₃×#3156 #27268154

Order 13568.

3150 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂²×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#536	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3156	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₅₃×#497	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#495	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#494	C₂	1	2	yes	maximal, complemented
9	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₅₃	C₂²	1	2	yes	—
11	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#137×C₅₃	—	2	16	no	—
13	C₂×#137×C₅₃	C₂²	1	16	yes	complemented
14	C₂×#135×C₅₃	—	2	8	no	—
15	C₂×#135×C₅₃	C₂²	1	8	yes	—
16	C₂²×Q₁₆×C₅₃	—	2	8	no	—
17	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆×C₅₃	C₂²	1	4	yes	—
21	C₂²×Q₁₆×C₅₃	C₂²	1	8	yes	complemented
22	C₂²×SD₁₆×C₅₃	C₂²	1	2	yes	—
23	C₂²×SD₁₆×C₅₃	C₂²	1	4	yes	—
24	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#536	—	2	8	no	—
28	C₅₃×#536	C₂²	1	16	yes	complemented
29	C₂×C₄⋊Q₈×C₅₃	C₂²	1	4	yes	—
30	C₂×#124×C₅₃	C₂²	1	4	yes	—
31	C₂×#123×C₅₃	—	2	4	no	—
32	C₂×C₂²⋊Q₈×C₅₃	—	2	4	no	—
33	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#497	C₂²	1	32	yes	complemented
35	C₅₃×#495	C₂²	1	16	yes	complemented
36	C₅₃×#494	C₂²	1	16	yes	complemented
37	C₅₃×#405	C₂²	1	8	yes	complemented
38	C₂×#97×C₅₃	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
40	C₂×#41×C₅₃	—	2	4	no	—
41	C₂×#41×C₅₃	—	2	8	no	—
42	C₂×#41×C₅₃	—	2	8	no	—
43	C₂×#41×C₅₃	C₂³	1	8	yes	—
44	C₂²×Q₈×C₅₃	—	2	2	no	—
45	C₂²×Q₈×C₅₃	—	2	4	no	—
46	C₂²×Q₈×C₅₃	—	2	8	no	—
47	C₂²×Q₈×C₅₃	—	2	8	no	—
48	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
49	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
50	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
51	C₂²×D₄×C₅₃	—	2	2	no	—
52	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₅₃	—	2	2	no	—
54	C₂³×C₄×C₅₃	—	2	2	no	—
55	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	#137×C₅₃	—	2	64	no	—
57	#135×C₅₃	—	2	32	no	—
58	C₂×Q₁₆×C₅₃	—	2	8	no	—
59	C₂×Q₁₆×C₅₃	—	2	8	no	—
60	C₂×Q₁₆×C₅₃	—	2	32	no	—
61	C₂×Q₁₆×C₅₃	—	2	4	no	—
62	C₂×Q₁₆×C₅₃	—	2	16	no	—
63	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
64	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
65	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	—
66	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
67	C₂×SD₁₆×C₅₃	—	2	8	no	—
68	C₂×SD₁₆×C₅₃	—	2	16	no	—
69	C₂×SD₁₆×C₅₃	C₂³	1	16	yes	—
70	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
71	C₂×D₈×C₅₃	—	2	4	no	—
72	C₂×D₈×C₅₃	C₂³	1	4	yes	—
73	C₂×M₁₆×C₅₃	D₄	1	8	yes	—
74	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
75	C₂²×C₈×C₅₃	D₄	1	8	yes	—
76	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
79	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₅₃	—	2	8	no	—
81	C₄⋊Q₈×C₅₃	C₂³	1	16	yes	—
82	#124×C₅₃	—	2	8	no	—
83	#124×C₅₃	C₂³	1	16	yes	—
84	#123×C₅₃	—	2	32	no	—
85	C₂²⋊Q₈×C₅₃	—	2	32	no	—
86	#117×C₅₃	D₄	1	4	yes	—
87	#117×C₅₃	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
89	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
91	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
93	#97×C₅₃	C₂³	1	8	yes	—
94	C₄×C₈×C₅₃	C₂³	1	8	yes	—
95	C₂⁴×C₅₃	—	2	2	no	—
96	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
97	#41×C₅₃	—	2	16	no	—
98	#41×C₅₃	—	2	16	no	—
99	#41×C₅₃	—	4	16	no	—
100	C₂×Q₈×C₅₃	—	2	8	no	—
101	C₂×Q₈×C₅₃	—	2	4	no	—
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	—	2	8	no	—
104	C₂×Q₈×C₅₃	—	2	4	no	—
105	C₂×Q₈×C₅₃	—	2	8	no	—
106	C₂×Q₈×C₅₃	—	2	16	no	—
107	C₂×Q₈×C₅₃	—	2	4	no	—
108	C₂×Q₈×C₅₃	—	2	16	no	—
109	C₂×Q₈×C₅₃	—	4	8	no	—
110	C₂×Q₈×C₅₃	—	4	16	no	—
111	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
112	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
113	C₂×D₄×C₅₃	—	2	8	no	—
114	C₂×D₄×C₅₃	—	2	4	no	—
115	C₂×D₄×C₅₃	—	2	4	no	—
116	C₂×D₄×C₅₃	—	2	2	no	—
117	C₂×D₄×C₅₃	—	4	8	no	—
118	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₅₃	—	2	8	no	—
120	C₂²×C₄×C₅₃	—	2	4	no	—
121	C₂²×C₄×C₅₃	—	2	2	no	—
122	C₂²×C₄×C₅₃	—	2	8	no	—
123	C₂²×C₄×C₅₃	—	4	8	no	—
124	C₂²×C₄×C₅₃	—	4	8	no	—
125	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₅₃	—	2	16	no	—
133	Q₁₆×C₅₃	—	2	16	no	—
134	Q₁₆×C₅₃	—	2	8	no	—
135	Q₁₆×C₅₃	—	2	32	no	—
136	SD₁₆×C₅₃	—	2	16	no	—
137	SD₁₆×C₅₃	—	2	32	no	—
138	D₈×C₅₃	—	2	8	no	—
139	M₁₆×C₅₃	C₂×D₄	1	16	yes	—
140	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
141	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
142	C₂×C₈×C₅₃	C₂×D₄	1	16	yes	—
143	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
144	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
145	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
146	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
147	C₄⋊C₄×C₅₃	—	2	32	no	—
148	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
149	C₂²⋊C₄×C₅₃	—	2	32	no	—
150	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
151	C₄²×C₅₃	C₂×D₄	1	4	yes	—
152	C₄²×C₅₃	C₂⁴	1	4	yes	—
153	C₂³×C₅₃	—	2	4	no	—
154	C₂³×C₅₃	—	2	4	no	—
155	C₂³×C₅₃	—	4	4	no	—
156	C₂³×C₅₃	—	4	8	no	—
157	C₂³×C₅₃	#127	1	4	yes	—
158	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
159	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	Q₈×C₅₃	—	2	8	no	—
161	Q₈×C₅₃	—	2	8	no	—
162	Q₈×C₅₃	—	2	4	no	—
163	Q₈×C₅₃	—	2	4	no	—
164	Q₈×C₅₃	—	4	16	no	—
165	Q₈×C₅₃	—	4	8	no	—
166	Q₈×C₅₃	—	4	4	no	—
167	D₄×C₅₃	—	2	4	no	—
168	D₄×C₅₃	—	2	4	no	—
169	D₄×C₅₃	—	4	16	no	—
170	D₄×C₅₃	—	4	4	no	—
171	C₂×C₄×C₅₃	—	2	8	no	—
172	C₂×C₄×C₅₃	—	2	4	no	—
173	C₂×C₄×C₅₃	—	2	8	no	—
174	C₂×C₄×C₅₃	—	4	4	no	—
175	C₂×C₄×C₅₃	—	4	16	no	—
176	C₂×C₄×C₅₃	—	4	8	no	—
177	C₂×C₄×C₅₃	—	4	16	no	—
178	C₂×C₄×C₅₃	#127	1	4	yes	—
179	C₂×C₄×C₅₃	#127	1	8	yes	—
180	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
181	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
182	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
184	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
185	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
186	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
187	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
188	C₈×C₅₃	C₂²×D₄	1	4	yes	—
189	C₈×C₅₃	C₂²×D₄	1	4	yes	—
190	C₈×C₅₃	C₂²×D₄	1	8	yes	—
191	C₂×#3156	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂²×C₅₃	—	2	8	no	—
193	C₂²×C₅₃	—	2	4	no	—
194	C₂²×C₅₃	—	4	4	no	—
195	C₂²×C₅₃	—	4	8	no	—
196	C₂²×C₅₃	—	8	8	no	—
197	C₂²×C₅₃	C₂×#127	1	4	yes	—
198	C₂²×C₅₃	C₂×#127	1	2	yes	—
199	C₂²×C₅₃	#578	1	4	yes	—
200	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
201	C₄×C₅₃	—	2	8	no	—
202	C₄×C₅₃	—	4	4	no	—
203	C₄×C₅₃	—	4	8	no	—
204	C₄×C₅₃	C₂×#127	1	4	yes	—
205	C₄×C₅₃	C₂×#127	1	2	yes	—
206	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
207	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
208	C₂²×#137	C₂×C₅₃	1	2	yes	complemented
209	C₂²×#135	C₂×C₅₃	1	1	yes	characteristic
210	C₂³×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
211	C₂×#536	C₂×C₅₃	1	2	yes	complemented
212	#3156	C₂×C₅₃	1	16	yes	direct product factor
213	C₂×#497	C₂×C₅₃	1	4	yes	complemented
214	C₂×#495	C₂×C₅₃	1	2	yes	complemented
215	C₂×#494	C₂×C₅₃	1	2	yes	complemented
216	C₂×#405	C₂×C₅₃	1	1	yes	characteristic, complemented
217	C₂×C₅₃	—	2	4	no	—
218	C₂×C₅₃	—	8	4	no	—
219	C₂×C₅₃	#3156	1	4	yes	direct product factor
220	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
221	C₂×C₅₃	C₂×#578	1	2	yes	—
222	C₂²×#41	C₂²×C₅₃	1	2	yes	—
223	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
224	C₂×#137	—	2	16	no	—
225	C₂×#137	C₂²×C₅₃	1	16	yes	complemented
226	C₂×#135	—	2	8	no	—
227	C₂×#135	C₂²×C₅₃	1	8	yes	—
228	C₂²×Q₁₆	—	2	8	no	—
229	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
230	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
231	C₂²×Q₁₆	C₂²×C₅₃	1	4	yes	—
232	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
233	C₂²×Q₁₆	C₂²×C₅₃	1	8	yes	complemented
234	C₂²×SD₁₆	C₂²×C₅₃	1	4	yes	—
235	C₂²×SD₁₆	C₂²×C₅₃	1	2	yes	—
236	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
237	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
238	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
239	#536	—	2	8	no	—
240	#536	C₂²×C₅₃	1	16	yes	complemented
241	C₂×C₄⋊Q₈	C₂²×C₅₃	1	4	yes	—
242	C₂×#124	C₂²×C₅₃	1	4	yes	—
243	C₂×#123	—	2	4	no	—
244	C₂×C₂²⋊Q₈	—	2	4	no	—
245	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
246	#497	C₂²×C₅₃	1	32	yes	complemented
247	#495	C₂²×C₅₃	1	16	yes	complemented
248	#494	C₂²×C₅₃	1	16	yes	complemented
249	#405	C₂²×C₅₃	1	8	yes	complemented
250	C₂×#97	C₂²×C₅₃	1	2	yes	—
251	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
252	C₅₃	C₂×#3156	1	1	yes	characteristic, direct product factor, Sylow
253	C₂×#41	—	2	4	no	—
254	C₂×#41	—	2	8	no	—
255	C₂×#41	—	2	8	no	—
256	C₂×#41	C₂³×C₅₃	1	8	yes	—
257	C₂²×Q₈	—	2	2	no	—
258	C₂²×Q₈	—	2	4	no	—
259	C₂²×Q₈	—	2	8	no	—
260	C₂²×Q₈	—	2	8	no	—
261	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
262	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
263	C₂²×Q₈	C₂³×C₅₃	1	8	yes	—
264	C₂²×D₄	—	2	2	no	—
265	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
266	C₂³×C₄	—	2	2	no	—
267	C₂³×C₄	—	2	2	no	—
268	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
269	#137	—	2	64	no	—
270	#135	—	2	32	no	—
271	C₂×Q₁₆	—	2	8	no	—
272	C₂×Q₁₆	—	2	8	no	—
273	C₂×Q₁₆	—	2	32	no	—
274	C₂×Q₁₆	—	2	4	no	—
275	C₂×Q₁₆	—	2	16	no	—
276	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
277	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
278	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
279	C₂×Q₁₆	C₂³×C₅₃	1	16	yes	—
280	C₂×SD₁₆	—	2	8	no	—
281	C₂×SD₁₆	—	2	16	no	—
282	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
283	C₂×SD₁₆	C₂³×C₅₃	1	16	yes	—
284	C₂×D₈	—	2	4	no	—
285	C₂×D₈	C₂³×C₅₃	1	4	yes	—
286	C₂×M₁₆	D₄×C₅₃	1	8	yes	—
287	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
288	C₂²×C₈	D₄×C₅₃	1	8	yes	—
289	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
292	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
293	C₄⋊Q₈	—	2	8	no	—
294	C₄⋊Q₈	C₂³×C₅₃	1	16	yes	—
295	#124	—	2	8	no	—
296	#124	C₂³×C₅₃	1	16	yes	—
297	#123	—	2	32	no	—
298	C₂²⋊Q₈	—	2	32	no	—
299	#117	D₄×C₅₃	1	4	yes	—
300	#117	C₂³×C₅₃	1	4	yes	—
301	C₂×C₄⋊C₄	—	2	8	no	—
302	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
303	C₂×C₂²⋊C₄	—	2	8	no	—
304	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
305	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
306	#97	C₂³×C₅₃	1	8	yes	—
307	C₄×C₈	C₂³×C₅₃	1	8	yes	—
308	C₂⁴	—	2	2	no	—
309	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
310	#41	—	2	16	no	—
311	#41	—	2	16	no	—
312	#41	—	4	16	no	—
313	C₂×Q₈	—	2	8	no	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	8	no	—
317	C₂×Q₈	—	2	4	no	—
318	C₂×Q₈	—	2	8	no	—
319	C₂×Q₈	—	2	16	no	—
320	C₂×Q₈	—	2	4	no	—
321	C₂×Q₈	—	2	16	no	—
322	C₂×Q₈	—	4	8	no	—
323	C₂×Q₈	—	4	16	no	—
324	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
325	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
326	C₂×D₄	—	2	8	no	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	4	8	no	—
331	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	8	no	—
335	C₂²×C₄	—	2	8	no	—
336	C₂²×C₄	—	4	8	no	—
337	C₂²×C₄	—	4	8	no	—
338	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
339	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
342	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
343	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
344	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
345	Q₁₆	—	2	16	no	—
346	Q₁₆	—	2	16	no	—
347	Q₁₆	—	2	8	no	—
348	Q₁₆	—	2	32	no	—
349	SD₁₆	—	2	16	no	—
350	SD₁₆	—	2	32	no	—
351	D₈	—	2	8	no	—
352	M₁₆	C₂×D₄×C₅₃	1	16	yes	—
353	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
354	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
355	C₂×C₈	C₂×D₄×C₅₃	1	16	yes	—
356	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
357	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
358	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
359	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
360	C₄⋊C₄	—	2	32	no	—
361	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
362	C₂²⋊C₄	—	2	32	no	—
363	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
364	C₄²	C₂×D₄×C₅₃	1	4	yes	—
365	C₄²	C₂⁴×C₅₃	1	4	yes	—
366	C₂³	—	2	4	no	—
367	C₂³	—	2	4	no	—
368	C₂³	—	4	4	no	—
369	C₂³	—	4	8	no	—
370	C₂³	#127×C₅₃	1	4	yes	—
371	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
372	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
373	Q₈	—	2	8	no	—
374	Q₈	—	2	4	no	—
375	Q₈	—	2	8	no	—
376	Q₈	—	2	4	no	—
377	Q₈	—	4	16	no	—
378	Q₈	—	4	8	no	—
379	Q₈	—	4	4	no	—
380	D₄	—	2	4	no	—
381	D₄	—	2	4	no	—
382	D₄	—	4	16	no	—
383	D₄	—	4	4	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	2	8	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	16	no	—
389	C₂×C₄	—	4	8	no	—
390	C₂×C₄	—	4	16	no	—
391	C₂×C₄	#127×C₅₃	1	4	yes	—
392	C₂×C₄	#127×C₅₃	1	8	yes	—
393	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
394	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
396	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
397	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
398	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
399	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
400	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
401	C₈	C₂²×D₄×C₅₃	1	4	yes	—
402	C₈	C₂²×D₄×C₅₃	1	4	yes	—
403	C₈	C₂²×D₄×C₅₃	1	8	yes	—
404	C₂²	—	2	4	no	—
405	C₂²	—	2	8	no	—
406	C₂²	—	4	8	no	—
407	C₂²	—	4	4	no	—
408	C₂²	—	8	8	no	—
409	C₂²	C₂×#127×C₅₃	1	4	yes	—
410	C₂²	C₂×#127×C₅₃	1	2	yes	—
411	C₂²	C₅₃×#578	1	4	yes	—
412	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
413	C₄	—	2	8	no	—
414	C₄	—	4	4	no	—
415	C₄	—	4	8	no	—
416	C₄	C₂×#127×C₅₃	1	4	yes	—
417	C₄	C₂×#127×C₅₃	1	2	yes	—
418	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
419	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
420	C₂	—	2	4	no	—
421	C₂	—	8	4	no	—
422	C₂	C₅₃×#3156	1	4	yes	direct product factor
423	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
424	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	C₂²×#135×C₅₃	C₂
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
1. C₂²×#137×C₅₃	C₂	semidirect
3. C₂³×Q₁₆×C₅₃	C₂	semidirect
4. C₂×C₅₃×#536	C₂	semidirect
5. C₅₃×#3156	C₂	direct (N × H)
6. C₂×C₅₃×#497	C₂	semidirect
7. C₂×C₅₃×#495	C₂	semidirect
8. C₂×C₅₃×#494	C₂	semidirect
9. C₂×C₅₃×#405	C₂	semidirect
13. C₂×#137×C₅₃	C₂²	semidirect
21. C₂²×Q₁₆×C₅₃	C₂²	semidirect
28. C₅₃×#536	C₂²	semidirect
34. C₅₃×#497	C₂²	semidirect
35. C₅₃×#495	C₂²	semidirect
36. C₅₃×#494	C₂²	semidirect
37. C₅₃×#405	C₂²	semidirect
191. C₂×#3156	C₅₃	direct (N × H)
208. C₂²×#137	C₂×C₅₃	semidirect
210. C₂³×Q₁₆	C₂×C₅₃	semidirect
211. C₂×#536	C₂×C₅₃	semidirect
212. #3156	C₂×C₅₃	direct (N × H)
213. C₂×#497	C₂×C₅₃	semidirect
214. C₂×#495	C₂×C₅₃	semidirect
215. C₂×#494	C₂×C₅₃	semidirect
216. C₂×#405	C₂×C₅₃	semidirect
219. C₂×C₅₃	#3156	direct (N × H)
225. C₂×#137	C₂²×C₅₃	semidirect
233. C₂²×Q₁₆	C₂²×C₅₃	semidirect
240. #536	C₂²×C₅₃	semidirect
246. #497	C₂²×C₅₃	semidirect
247. #495	C₂²×C₅₃	semidirect
248. #494	C₂²×C₅₃	semidirect
249. #405	C₂²×C₅₃	semidirect
252. C₅₃	C₂×#3156	direct (N × H)
422. C₂	C₅₃×#3156	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×#137×C₅₃	2
C₂²×#135×C₅₃	2
C₂³×Q₁₆×C₅₃	2
C₂×C₅₃×#536	2
C₅₃×#3156	2
C₂×C₅₃×#497	2
C₂×C₅₃×#495	2
C₂×C₅₃×#494	2
C₂×C₅₃×#405	2
C₂×#3156	53

Subgroups — C2×C53×#3156 #27268154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3156 #27268154