Subgroups — C₅₃×#36702 #27244214

Order 13568.

4154 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#120×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2846	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2684	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2669	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2668	C₂	1	8	yes	maximal, complemented
7	C₅₃×#2622	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2482	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2410	C₂	1	4	yes	maximal, complemented
10	C₅₃×#2404	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2308	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
14	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
15	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#535	—	2	8	no	—
17	C₂×#127×C₅₃	C₂²	1	2	yes	complemented
18	C₂×#124×C₅₃	C₂²	1	2	yes	complemented
19	C₂×#121×C₅₃	C₂²	1	4	yes	complemented
20	C₂×#121×C₅₃	C₂²	1	8	yes	complemented
21	C₂×#120×C₅₃	—	2	8	no	—
22	C₂×#120×C₅₃	C₂²	1	4	yes	—
23	C₂×#120×C₅₃	C₂²	1	2	yes	—
24	C₂×#120×C₅₃	C₂²	1	4	yes	—
25	C₂×#120×C₅₃	C₂²	1	4	yes	—
26	C₂×#120×C₅₃	C₂²	1	8	yes	complemented
27	C₂×#120×C₅₃	C₂²	1	2	yes	complemented
28	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
32	C₅₃×#399	C₂²	1	4	yes	complemented
33	C₅₃×#397	C₂²	1	8	yes	complemented
34	C₅₃×#397	C₂²	1	4	yes	complemented
35	C₅₃×#394	C₂²	1	8	yes	complemented
36	C₅₃×#394	C₂²	1	8	yes	complemented
37	C₅₃×#393	C₂²	1	4	yes	complemented
38	C₅₃×#392	C₂²	1	4	yes	complemented
39	C₅₃×#392	C₂²	1	8	yes	complemented
40	C₅₃×#388	C₂²	1	8	yes	complemented
41	C₅₃×#386	C₂²	1	4	yes	complemented
42	C₅₃×#386	C₂²	1	8	yes	complemented
43	C₅₃×#386	C₂²	1	4	yes	complemented
44	C₅₃×#386	C₂²	1	8	yes	complemented
45	C₅₃×#386	C₂²	1	4	yes	complemented
46	C₅₃×#385	C₂²	1	4	yes	complemented
47	C₅₃×#381	C₂²	1	2	yes	complemented
48	C₅₃×#379	C₂²	1	4	yes	—
49	C₅₃×#379	C₂²	1	2	yes	—
50	C₅₃×#379	C₂²	1	8	yes	complemented
51	C₂⁵×C₅₃	—	2	4	no	—
52	C₂⁵×C₅₃	—	4	8	no	—
53	C₂⁵×C₅₃	—	4	4	no	—
54	C₂⁵×C₅₃	C₂³	1	2	yes	—
55	C₂⁵×C₅₃	C₂³	1	4	yes	—
56	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
57	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×#41×C₅₃	—	2	4	no	—
59	C₂×#41×C₅₃	—	4	4	no	—
60	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic, complemented
61	C₂²×D₄×C₅₃	—	2	8	no	—
62	C₂²×D₄×C₅₃	—	4	4	no	—
63	C₂²×D₄×C₅₃	—	4	8	no	—
64	C₂²×D₄×C₅₃	D₄	1	8	yes	complemented
65	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
66	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
67	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
68	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
69	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
70	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
71	C₂²×D₄×C₅₃	C₂³	1	4	yes	complemented
72	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic, complemented
73	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
75	C₂³×C₄×C₅₃	C₂³	1	2	yes	complemented
76	C₂³×C₄×C₅₃	C₂³	1	4	yes	complemented
77	#127×C₅₃	—	2	8	no	—
78	#124×C₅₃	—	2	8	no	—
79	#121×C₅₃	—	2	16	no	—
80	#121×C₅₃	—	4	16	no	—
81	#120×C₅₃	—	2	8	no	—
82	#120×C₅₃	—	2	8	no	—
83	#120×C₅₃	—	2	16	no	—
84	#120×C₅₃	—	2	16	no	—
85	#120×C₅₃	—	4	8	no	—
86	#120×C₅₃	—	4	16	no	—
87	#117×C₅₃	—	2	4	no	—
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	complemented
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
90	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
92	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	complemented
104	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	complemented
105	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	complemented
106	C₂×C₄²×C₅₃	C₂³	1	2	yes	complemented
107	#95×C₅₃	C₂³	1	4	yes	complemented
108	#95×C₅₃	C₂³	1	4	yes	complemented
109	#95×C₅₃	C₂³	1	8	yes	complemented
110	#95×C₅₃	C₂³	1	8	yes	complemented
111	C₂⁴×C₅₃	—	2	4	no	—
112	C₂⁴×C₅₃	—	2	8	no	—
113	C₂⁴×C₅₃	—	2	4	no	—
114	C₂⁴×C₅₃	—	2	4	no	—
115	C₂⁴×C₅₃	—	2	8	no	—
116	C₂⁴×C₅₃	—	2	16	no	—
117	C₂⁴×C₅₃	—	2	8	no	—
118	C₂⁴×C₅₃	—	4	4	no	—
119	C₂⁴×C₅₃	—	4	4	no	—
120	C₂⁴×C₅₃	—	4	4	no	—
121	C₂⁴×C₅₃	—	4	8	no	—
122	C₂⁴×C₅₃	—	4	4	no	—
123	C₂⁴×C₅₃	—	4	4	no	—
124	C₂⁴×C₅₃	—	4	32	no	—
125	C₂⁴×C₅₃	—	4	16	no	—
126	C₂⁴×C₅₃	—	4	64	no	—
127	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
128	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
129	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
130	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
131	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
132	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
133	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
134	#41×C₅₃	—	4	16	no	—
135	C₂×Q₈×C₅₃	—	2	2	no	—
136	C₂×Q₈×C₅₃	—	4	2	no	—
137	C₂×D₄×C₅₃	—	2	16	no	—
138	C₂×D₄×C₅₃	—	2	4	no	—
139	C₂×D₄×C₅₃	—	2	8	no	—
140	C₂×D₄×C₅₃	—	2	16	no	—
141	C₂×D₄×C₅₃	—	2	8	no	—
142	C₂×D₄×C₅₃	—	2	8	no	—
143	C₂×D₄×C₅₃	—	2	16	no	—
144	C₂×D₄×C₅₃	—	2	2	no	—
145	C₂×D₄×C₅₃	—	4	16	no	—
146	C₂×D₄×C₅₃	—	4	2	no	—
147	C₂×D₄×C₅₃	—	4	2	no	—
148	C₂×D₄×C₅₃	—	4	2	no	—
149	C₂×D₄×C₅₃	—	4	2	no	—
150	C₂×D₄×C₅₃	—	4	8	no	—
151	C₂×D₄×C₅₃	—	4	2	no	—
152	C₂×D₄×C₅₃	—	4	8	no	—
153	C₂×D₄×C₅₃	—	4	8	no	—
154	C₂×D₄×C₅₃	—	4	8	no	—
155	C₂×D₄×C₅₃	—	4	8	no	—
156	C₂×D₄×C₅₃	—	4	32	no	—
157	C₂×D₄×C₅₃	C₂×D₄	1	8	yes	—
158	C₂×D₄×C₅₃	C₂×D₄	1	8	yes	—
159	C₂²×C₄×C₅₃	—	2	4	no	—
160	C₂²×C₄×C₅₃	—	2	4	no	—
161	C₂²×C₄×C₅₃	—	2	4	no	—
162	C₂²×C₄×C₅₃	—	4	4	no	—
163	C₂²×C₄×C₅₃	—	4	4	no	—
164	C₂²×C₄×C₅₃	—	4	4	no	—
165	C₂²×C₄×C₅₃	—	4	8	no	—
166	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
173	C₄⋊C₄×C₅₃	—	2	4	no	—
174	C₄⋊C₄×C₅₃	—	4	8	no	—
175	C₂²⋊C₄×C₅₃	—	2	8	no	—
176	C₂²⋊C₄×C₅₃	—	2	4	no	—
177	C₂²⋊C₄×C₅₃	—	2	8	no	—
178	C₂²⋊C₄×C₅₃	—	2	8	no	—
179	C₂²⋊C₄×C₅₃	—	2	8	no	—
180	C₂²⋊C₄×C₅₃	—	2	16	no	—
181	C₂²⋊C₄×C₅₃	—	2	8	no	—
182	C₂²⋊C₄×C₅₃	—	2	8	no	—
183	C₂²⋊C₄×C₅₃	—	4	4	no	—
184	C₂²⋊C₄×C₅₃	—	4	8	no	—
185	C₂²⋊C₄×C₅₃	—	4	8	no	—
186	C₂²⋊C₄×C₅₃	—	4	8	no	—
187	C₂²⋊C₄×C₅₃	—	4	8	no	—
188	C₄²×C₅₃	—	2	4	no	—
189	C₂³×C₅₃	—	2	4	no	—
190	C₂³×C₅₃	—	2	16	no	—
191	C₂³×C₅₃	—	2	8	no	—
192	C₂³×C₅₃	—	2	4	no	—
193	C₂³×C₅₃	—	2	2	no	—
194	C₂³×C₅₃	—	2	8	no	—
195	C₂³×C₅₃	—	2	2	no	—
196	C₂³×C₅₃	—	2	2	no	—
197	C₂³×C₅₃	—	2	4	no	—
198	C₂³×C₅₃	—	2	2	no	—
199	C₂³×C₅₃	—	2	8	no	—
200	C₂³×C₅₃	—	2	8	no	—
201	C₂³×C₅₃	—	2	8	no	—
202	C₂³×C₅₃	—	2	8	no	—
203	C₂³×C₅₃	—	2	2	no	—
204	C₂³×C₅₃	—	2	4	no	—
205	C₂³×C₅₃	—	2	8	no	—
206	C₂³×C₅₃	—	4	8	no	—
207	C₂³×C₅₃	—	4	8	no	—
208	C₂³×C₅₃	—	4	32	no	—
209	C₂³×C₅₃	—	4	32	no	—
210	C₂³×C₅₃	—	4	16	no	—
211	C₂³×C₅₃	—	4	8	no	—
212	C₂³×C₅₃	—	4	16	no	—
213	C₂³×C₅₃	—	4	16	no	—
214	C₂³×C₅₃	—	4	32	no	—
215	C₂³×C₅₃	—	4	128	no	—
216	C₂³×C₅₃	—	4	16	no	—
217	C₂³×C₅₃	—	4	8	no	—
218	C₂³×C₅₃	—	8	8	no	—
219	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
220	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
221	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
222	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
223	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
224	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
225	Q₈×C₅₃	—	4	4	no	—
226	D₄×C₅₃	—	4	8	no	—
227	D₄×C₅₃	—	4	8	no	—
228	D₄×C₅₃	—	4	4	no	—
229	D₄×C₅₃	—	4	32	no	—
230	D₄×C₅₃	—	4	8	no	—
231	D₄×C₅₃	—	8	8	no	—
232	D₄×C₅₃	—	8	8	no	—
233	D₄×C₅₃	—	8	16	no	—
234	C₂×C₄×C₅₃	—	2	8	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	2	8	no	—
241	C₂×C₄×C₅₃	—	2	4	no	—
242	C₂×C₄×C₅₃	—	2	8	no	—
243	C₂×C₄×C₅₃	—	2	8	no	—
244	C₂×C₄×C₅₃	—	2	8	no	—
245	C₂×C₄×C₅₃	—	2	8	no	—
246	C₂×C₄×C₅₃	—	4	8	no	—
247	C₂×C₄×C₅₃	—	4	8	no	—
248	C₂×C₄×C₅₃	—	8	8	no	—
249	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
251	#36702	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂²×C₅₃	—	2	4	no	—
253	C₂²×C₅₃	—	2	8	no	—
254	C₂²×C₅₃	—	2	4	no	—
255	C₂²×C₅₃	—	2	4	no	—
256	C₂²×C₅₃	—	2	8	no	—
257	C₂²×C₅₃	—	2	8	no	—
258	C₂²×C₅₃	—	2	8	no	—
259	C₂²×C₅₃	—	2	16	no	—
260	C₂²×C₅₃	—	4	32	no	—
261	C₂²×C₅₃	—	4	16	no	—
262	C₂²×C₅₃	—	4	64	no	—
263	C₂²×C₅₃	—	4	4	no	—
264	C₂²×C₅₃	—	4	4	no	—
265	C₂²×C₅₃	—	4	4	no	—
266	C₂²×C₅₃	—	4	4	no	—
267	C₂²×C₅₃	—	4	8	no	—
268	C₂²×C₅₃	—	4	4	no	—
269	C₂²×C₅₃	—	4	4	no	—
270	C₂²×C₅₃	—	4	2	no	—
271	C₂²×C₅₃	—	4	4	no	—
272	C₂²×C₅₃	—	4	8	no	—
273	C₂²×C₅₃	—	8	8	no	—
274	C₂²×C₅₃	#534	1	4	yes	—
275	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
276	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
277	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
278	C₄×C₅₃	—	2	2	no	—
279	C₄×C₅₃	—	2	2	no	—
280	C₄×C₅₃	—	4	2	no	—
281	C₄×C₅₃	—	4	4	no	—
282	C₄×C₅₃	—	4	4	no	—
283	C₄×C₅₃	—	4	8	no	—
284	C₂×#535	C₂×C₅₃	1	1	yes	characteristic, complemented
285	C₂²×#120	C₂×C₅₃	1	1	yes	characteristic, complemented
286	#2846	C₂×C₅₃	1	2	yes	complemented
287	#2684	C₂×C₅₃	1	4	yes	complemented
288	#2669	C₂×C₅₃	1	4	yes	complemented
289	#2668	C₂×C₅₃	1	8	yes	complemented
290	#2622	C₂×C₅₃	1	2	yes	complemented
291	#2482	C₂×C₅₃	1	2	yes	complemented
292	#2410	C₂×C₅₃	1	4	yes	complemented
293	#2404	C₂×C₅₃	1	2	yes	complemented
294	#2308	C₂×C₅₃	1	1	yes	characteristic, complemented
295	C₂×C₅₃	—	2	4	no	—
296	C₂×C₅₃	—	4	4	no	—
297	C₂×C₅₃	—	4	8	no	—
298	C₂×C₅₃	—	4	4	no	—
299	C₂×C₅₃	—	8	2	no	—
300	C₂×C₅₃	C₂×#534	1	1	yes	characteristic
301	C₂×C₅₃	#3484	1	2	yes	—
302	C₂×C₅₃	#3532	1	4	yes	—
303	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
304	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic, complemented
305	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
306	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
307	#535	—	2	8	no	—
308	C₂×#127	C₂²×C₅₃	1	2	yes	complemented
309	C₂×#124	C₂²×C₅₃	1	2	yes	complemented
310	C₂×#121	C₂²×C₅₃	1	4	yes	complemented
311	C₂×#121	C₂²×C₅₃	1	8	yes	complemented
312	C₂×#120	—	2	8	no	—
313	C₂×#120	C₂²×C₅₃	1	2	yes	—
314	C₂×#120	C₂²×C₅₃	1	4	yes	—
315	C₂×#120	C₂²×C₅₃	1	4	yes	—
316	C₂×#120	C₂²×C₅₃	1	4	yes	—
317	C₂×#120	C₂²×C₅₃	1	8	yes	complemented
318	C₂×#120	C₂²×C₅₃	1	2	yes	complemented
319	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic, complemented
320	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
321	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
322	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
323	#399	C₂²×C₅₃	1	4	yes	complemented
324	#397	C₂²×C₅₃	1	8	yes	complemented
325	#397	C₂²×C₅₃	1	4	yes	complemented
326	#394	C₂²×C₅₃	1	8	yes	complemented
327	#394	C₂²×C₅₃	1	8	yes	complemented
328	#393	C₂²×C₅₃	1	4	yes	complemented
329	#392	C₂²×C₅₃	1	4	yes	complemented
330	#392	C₂²×C₅₃	1	8	yes	complemented
331	#388	C₂²×C₅₃	1	8	yes	complemented
332	#386	C₂²×C₅₃	1	4	yes	complemented
333	#386	C₂²×C₅₃	1	4	yes	complemented
334	#386	C₂²×C₅₃	1	8	yes	complemented
335	#386	C₂²×C₅₃	1	8	yes	complemented
336	#386	C₂²×C₅₃	1	4	yes	complemented
337	#385	C₂²×C₅₃	1	4	yes	complemented
338	#381	C₂²×C₅₃	1	2	yes	complemented
339	#379	C₂²×C₅₃	1	2	yes	—
340	#379	C₂²×C₅₃	1	4	yes	—
341	#379	C₂²×C₅₃	1	8	yes	complemented
342	C₅₃	#36702	1	1	yes	characteristic, direct product factor, Sylow
343	C₂⁵	—	2	4	no	—
344	C₂⁵	—	4	8	no	—
345	C₂⁵	—	4	4	no	—
346	C₂⁵	C₂³×C₅₃	1	2	yes	—
347	C₂⁵	C₂³×C₅₃	1	4	yes	—
348	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
349	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
350	C₂×#41	—	2	4	no	—
351	C₂×#41	—	4	4	no	—
352	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic, complemented
353	C₂²×D₄	—	2	8	no	—
354	C₂²×D₄	—	4	4	no	—
355	C₂²×D₄	—	4	8	no	—
356	C₂²×D₄	D₄×C₅₃	1	8	yes	complemented
357	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
358	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
359	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
360	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
361	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
362	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
363	C₂²×D₄	C₂³×C₅₃	1	4	yes	complemented
364	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic, complemented
365	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
367	C₂³×C₄	C₂³×C₅₃	1	2	yes	complemented
368	C₂³×C₄	C₂³×C₅₃	1	4	yes	complemented
369	#127	—	2	8	no	—
370	#124	—	2	8	no	—
371	#121	—	2	16	no	—
372	#121	—	4	16	no	—
373	#120	—	2	8	no	—
374	#120	—	2	8	no	—
375	#120	—	2	16	no	—
376	#120	—	2	16	no	—
377	#120	—	4	8	no	—
378	#120	—	4	16	no	—
379	#117	—	2	4	no	—
380	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	complemented
381	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	complemented
382	C₂×C₂²⋊C₄	—	2	4	no	—
383	C₂×C₂²⋊C₄	—	2	8	no	—
384	C₂×C₂²⋊C₄	—	4	4	no	—
385	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	complemented
396	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	complemented
397	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	complemented
398	C₂×C₄²	C₂³×C₅₃	1	2	yes	complemented
399	#95	C₂³×C₅₃	1	4	yes	complemented
400	#95	C₂³×C₅₃	1	4	yes	complemented
401	#95	C₂³×C₅₃	1	8	yes	complemented
402	#95	C₂³×C₅₃	1	8	yes	complemented
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	2	8	no	—
405	C₂⁴	—	2	4	no	—
406	C₂⁴	—	2	4	no	—
407	C₂⁴	—	2	8	no	—
408	C₂⁴	—	2	16	no	—
409	C₂⁴	—	2	8	no	—
410	C₂⁴	—	4	4	no	—
411	C₂⁴	—	4	4	no	—
412	C₂⁴	—	4	8	no	—
413	C₂⁴	—	4	4	no	—
414	C₂⁴	—	4	4	no	—
415	C₂⁴	—	4	4	no	—
416	C₂⁴	—	4	32	no	—
417	C₂⁴	—	4	16	no	—
418	C₂⁴	—	4	64	no	—
419	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
420	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
421	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
422	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
423	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
424	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
425	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
426	#41	—	4	16	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	—	4	2	no	—
429	C₂×D₄	—	2	16	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	8	no	—
432	C₂×D₄	—	2	16	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	8	no	—
435	C₂×D₄	—	2	16	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	4	16	no	—
438	C₂×D₄	—	4	2	no	—
439	C₂×D₄	—	4	2	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	—	4	2	no	—
442	C₂×D₄	—	4	8	no	—
443	C₂×D₄	—	4	2	no	—
444	C₂×D₄	—	4	8	no	—
445	C₂×D₄	—	4	8	no	—
446	C₂×D₄	—	4	8	no	—
447	C₂×D₄	—	4	8	no	—
448	C₂×D₄	—	4	32	no	—
449	C₂×D₄	C₂×D₄×C₅₃	1	8	yes	—
450	C₂×D₄	C₂×D₄×C₅₃	1	8	yes	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	4	no	—
457	C₂²×C₄	—	4	8	no	—
458	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
460	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
462	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
463	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	4	8	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	—	2	16	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	4	4	no	—
476	C₂²⋊C₄	—	4	8	no	—
477	C₂²⋊C₄	—	4	8	no	—
478	C₂²⋊C₄	—	4	8	no	—
479	C₂²⋊C₄	—	4	8	no	—
480	C₄²	—	2	4	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	16	no	—
483	C₂³	—	2	8	no	—
484	C₂³	—	2	4	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	8	no	—
487	C₂³	—	2	8	no	—
488	C₂³	—	2	8	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	8	no	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	8	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	8	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	4	8	no	—
499	C₂³	—	4	8	no	—
500	C₂³	—	4	32	no	—
501	C₂³	—	4	32	no	—
502	C₂³	—	4	16	no	—
503	C₂³	—	4	8	no	—
504	C₂³	—	4	16	no	—
505	C₂³	—	4	32	no	—
506	C₂³	—	4	16	no	—
507	C₂³	—	4	128	no	—
508	C₂³	—	4	16	no	—
509	C₂³	—	4	8	no	—
510	C₂³	—	8	8	no	—
511	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
512	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
513	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
514	C₂³	H(5,𝔽₂)×C₅₃	1	8	yes	—
515	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
516	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
517	Q₈	—	4	4	no	—
518	D₄	—	4	8	no	—
519	D₄	—	4	8	no	—
520	D₄	—	4	4	no	—
521	D₄	—	4	32	no	—
522	D₄	—	4	8	no	—
523	D₄	—	8	8	no	—
524	D₄	—	8	8	no	—
525	D₄	—	8	16	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	8	no	—
533	C₂×C₄	—	2	8	no	—
534	C₂×C₄	—	2	8	no	—
535	C₂×C₄	—	2	8	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	—	8	8	no	—
541	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
542	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
543	C₂²	—	2	4	no	—
544	C₂²	—	2	8	no	—
545	C₂²	—	2	8	no	—
546	C₂²	—	2	8	no	—
547	C₂²	—	2	8	no	—
548	C₂²	—	2	16	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	4	32	no	—
552	C₂²	—	4	16	no	—
553	C₂²	—	4	64	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	4	8	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	4	no	—
562	C₂²	—	4	8	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	8	8	no	—
565	C₂²	C₅₃×#534	1	4	yes	—
566	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
567	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
568	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
569	C₄	—	2	2	no	—
570	C₄	—	2	2	no	—
571	C₄	—	4	4	no	—
572	C₄	—	4	4	no	—
573	C₄	—	4	8	no	—
574	C₄	—	4	2	no	—
575	C₂	—	2	4	no	—
576	C₂	—	4	8	no	—
577	C₂	—	4	4	no	—
578	C₂	—	4	4	no	—
579	C₂	—	8	2	no	—
580	C₂	C₂×C₅₃×#534	1	1	yes	characteristic
581	C₂	C₅₃×#3484	1	2	yes	—
582	C₂	C₅₃×#3532	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#535	C₂	semidirect
2. C₂²×#120×C₅₃	C₂	semidirect
3. C₅₃×#2846	C₂	semidirect
4. C₅₃×#2684	C₂	semidirect
5. C₅₃×#2669	C₂	semidirect
6. C₅₃×#2668	C₂	semidirect
7. C₅₃×#2622	C₂	semidirect
8. C₅₃×#2482	C₂	semidirect
9. C₅₃×#2410	C₂	semidirect
10. C₅₃×#2404	C₂	semidirect
11. C₅₃×#2308	C₂	semidirect
13. C₂²×#41×C₅₃	C₂²	semidirect
17. C₂×#127×C₅₃	C₂²	semidirect
18. C₂×#124×C₅₃	C₂²	semidirect
19. C₂×#121×C₅₃	C₂²	semidirect
20. C₂×#121×C₅₃	C₂²	semidirect
26. C₂×#120×C₅₃	C₂²	semidirect
27. C₂×#120×C₅₃	C₂²	semidirect
28. C₂×#117×C₅₃	C₂²	semidirect
31. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
32. C₅₃×#399	C₂²	semidirect
33. C₅₃×#397	C₂²	semidirect
34. C₅₃×#397	C₂²	semidirect
35. C₅₃×#394	C₂²	semidirect
36. C₅₃×#394	C₂²	semidirect
37. C₅₃×#393	C₂²	semidirect
38. C₅₃×#392	C₂²	semidirect
39. C₅₃×#392	C₂²	semidirect
40. C₅₃×#388	C₂²	semidirect
41. C₅₃×#386	C₂²	semidirect
42. C₅₃×#386	C₂²	semidirect
43. C₅₃×#386	C₂²	semidirect
44. C₅₃×#386	C₂²	semidirect
45. C₅₃×#386	C₂²	semidirect
46. C₅₃×#385	C₂²	semidirect
47. C₅₃×#381	C₂²	semidirect
50. C₅₃×#379	C₂²	semidirect
60. C₂²×Q₈×C₅₃	C₂³	semidirect
64. C₂²×D₄×C₅₃	D₄	semidirect
71. C₂²×D₄×C₅₃	C₂³	semidirect
72. C₂²×D₄×C₅₃	C₂³	semidirect
75. C₂³×C₄×C₅₃	C₂³	semidirect
76. C₂³×C₄×C₅₃	C₂³	semidirect
88. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
89. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
103. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
104. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
105. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
106. C₂×C₄²×C₅₃	C₂³	semidirect
107. #95×C₅₃	C₂³	semidirect
108. #95×C₅₃	C₂³	semidirect
109. #95×C₅₃	C₂³	semidirect
110. #95×C₅₃	C₂³	semidirect
251. #36702	C₅₃	direct (N × H)
284. C₂×#535	C₂×C₅₃	semidirect
285. C₂²×#120	C₂×C₅₃	semidirect
286. #2846	C₂×C₅₃	semidirect
287. #2684	C₂×C₅₃	semidirect
288. #2669	C₂×C₅₃	semidirect
289. #2668	C₂×C₅₃	semidirect
290. #2622	C₂×C₅₃	semidirect
291. #2482	C₂×C₅₃	semidirect
292. #2410	C₂×C₅₃	semidirect
293. #2404	C₂×C₅₃	semidirect
294. #2308	C₂×C₅₃	semidirect
304. C₂²×#41	C₂²×C₅₃	semidirect
308. C₂×#127	C₂²×C₅₃	semidirect
309. C₂×#124	C₂²×C₅₃	semidirect
310. C₂×#121	C₂²×C₅₃	semidirect
311. C₂×#121	C₂²×C₅₃	semidirect
317. C₂×#120	C₂²×C₅₃	semidirect
318. C₂×#120	C₂²×C₅₃	semidirect
319. C₂×#117	C₂²×C₅₃	semidirect
322. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
323. #399	C₂²×C₅₃	semidirect
324. #397	C₂²×C₅₃	semidirect
325. #397	C₂²×C₅₃	semidirect
326. #394	C₂²×C₅₃	semidirect
327. #394	C₂²×C₅₃	semidirect
328. #393	C₂²×C₅₃	semidirect
329. #392	C₂²×C₅₃	semidirect
330. #392	C₂²×C₅₃	semidirect
331. #388	C₂²×C₅₃	semidirect
332. #386	C₂²×C₅₃	semidirect
333. #386	C₂²×C₅₃	semidirect
334. #386	C₂²×C₅₃	semidirect
335. #386	C₂²×C₅₃	semidirect
336. #386	C₂²×C₅₃	semidirect
337. #385	C₂²×C₅₃	semidirect
338. #381	C₂²×C₅₃	semidirect
341. #379	C₂²×C₅₃	semidirect
342. C₅₃	#36702	direct (N × H)
352. C₂²×Q₈	C₂³×C₅₃	semidirect
356. C₂²×D₄	D₄×C₅₃	semidirect
363. C₂²×D₄	C₂³×C₅₃	semidirect
364. C₂²×D₄	C₂³×C₅₃	semidirect
367. C₂³×C₄	C₂³×C₅₃	semidirect
368. C₂³×C₄	C₂³×C₅₃	semidirect
380. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
381. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
395. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
396. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
397. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
398. C₂×C₄²	C₂³×C₅₃	semidirect
399. #95	C₂³×C₅₃	semidirect
400. #95	C₂³×C₅₃	semidirect
401. #95	C₂³×C₅₃	semidirect
402. #95	C₂³×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#535	2
C₂²×#120×C₅₃	2
C₅₃×#2846	2
C₅₃×#2684	2
C₅₃×#2669	2
C₅₃×#2668	2
C₅₃×#2622	2
C₅₃×#2482	2
C₅₃×#2410	2
C₅₃×#2404	2
C₅₃×#2308	2
#36702	53

Subgroups — C53×#36702 #27244214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36702 #27244214