Subgroups — #28290708

Order 13568.

11674 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)×D₅₃	C₂	1	1	yes	maximal, characteristic
2	H(5,𝔽₂)×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133263	C₂	1	6	yes	maximal, complemented
4	#20287921	C₂	1	6	yes	maximal, complemented
5	C₄×#13133262	C₂	1	9	yes	maximal, complemented
6	#20287919	C₂	1	9	yes	maximal, complemented
7	#20287915	C₂	1	6	yes	maximal, complemented
8	#20287914	C₂	1	6	yes	maximal, complemented
9	C₂×#16811095	C₂	1	9	yes	maximal, complemented
10	#518×D₅₃	C₂	1	9	yes	maximal, complemented
11	C₄×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133263	C₂²	1	6	yes	—
13	C₂×#13133262	C₂²	1	9	yes	—
14	H(5,𝔽₂)×D₅₃	C₄	1	32	yes	complemented
15	C₂×#41×D₅₃	C₂²	1	6	yes	—
16	C₂²×D₄×D₅₃	C₂²	1	9	yes	—
17	#16811106	C₂²	1	6	yes	complemented
18	#16811105	C₂²	1	6	yes	complemented
19	#16811105	C₂²	1	6	yes	complemented
20	#16811104	C₂²	1	18	yes	complemented
21	#16811103	C₂²	1	18	yes	complemented
22	#16811102	C₂²	1	18	yes	complemented
23	#16811102	C₂²	1	18	yes	complemented
24	#16811099	C₂²	1	2	yes	complemented
25	#16811099	C₂²	1	6	yes	complemented
26	#16811098	C₂²	1	9	yes	complemented
27	#16811097	C₂²	1	18	yes	complemented
28	#16811096	C₂²	1	36	yes	complemented
29	#16811095	C₂²	1	18	yes	complemented
30	#16811095	C₂²	1	36	yes	complemented
31	#16811095	C₂²	1	18	yes	complemented
32	#16811095	C₂²	1	36	yes	complemented
33	#16811095	C₂²	1	36	yes	complemented
34	#41×Dic₅₃	C₂²	1	6	yes	complemented
35	C₄×#11719949	C₂²	1	12	yes	complemented
36	C₄×#11719945	C₂²	1	18	yes	complemented
37	#16811091	C₂²	1	18	yes	complemented
38	C₄×#11719947	C₂²	1	36	yes	complemented
39	C₂×C₄×D₂₁₂	C₂²	1	9	yes	complemented
40	C₂×D₄×Dic₅₃	C₂²	1	9	yes	complemented
41	C₂×#13133179	C₂²	1	9	yes	complemented
42	C₂×C₄×C₂²⋊D₅₃	C₂²	1	18	yes	complemented
43	C₂×#13133177	C₂²	1	18	yes	complemented
44	C₄×D₄×D₅₃	C₂²	1	36	yes	complemented
45	C₄×D₄×D₅₃	C₂²	1	36	yes	complemented
46	C₂×#13133175	C₂²	1	9	yes	complemented
47	C₂×#13133174	C₂²	1	9	yes	complemented
48	#117×D₅₃	C₂²	1	18	yes	complemented
49	C₂²×#11719921	C₂²	1	6	yes	complemented
50	C₂×C₂²⋊C₄×D₅₃	C₂²	1	18	yes	complemented
51	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
52	C₅₃×#520	C₂²	1	6	yes	complemented
53	C₅₃×#518	C₂²	1	9	yes	complemented
54	C₄×#41×C₅₃	C₂²	1	6	yes	complemented
55	C₂×C₄×D₄×C₅₃	C₂²	1	9	yes	complemented
56	C₂⁴×D₅₃	C₂³	1	6	yes	—
57	#13133263	C₂×C₄	1	96	yes	complemented
58	#13133262	C₂×C₄	1	144	yes	complemented
59	C₂×#11719949	C₂³	1	6	yes	—
60	C₂×#11719945	C₂³	1	18	yes	—
61	C₂×#11719947	C₂³	1	18	yes	—
62	#41×D₅₃	C₂×C₄	1	96	yes	complemented
63	C₂×Q₈×D₅₃	C₂³	1	2	yes	—
64	C₂²×D₂₁₂	C₂³	1	9	yes	—
65	C₂²×C₂²⋊D₅₃	C₂³	1	18	yes	—
66	C₂×D₄×D₅₃	C₂×C₄	1	144	yes	complemented
67	C₂×D₄×D₅₃	C₂³	1	36	yes	—
68	C₂×D₄×D₅₃	C₂³	1	18	yes	—
69	C₂³×Dic₅₃	C₂³	1	6	yes	complemented
70	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
71	C₂²×C₄×D₅₃	C₂³	1	18	yes	complemented
72	C₂²×C₄×D₅₃	C₂³	1	18	yes	complemented
73	Q₈×Dic₅₃	C₂³	1	6	yes	complemented
74	Q₈×Dic₅₃	C₂³	1	2	yes	complemented
75	C₄×Dic₁₀₆	C₂³	1	18	yes	complemented
76	C₄×Dic₁₀₆	C₂³	1	6	yes	complemented
77	#13133182	C₂³	1	18	yes	complemented
78	#13133182	C₂³	1	6	yes	complemented
79	C₄×D₂₁₂	C₂³	1	18	yes	complemented
80	C₄×D₂₁₂	C₂³	1	36	yes	complemented
81	C₄×D₂₁₂	C₂³	1	18	yes	complemented
82	D₄×Dic₅₃	C₂³	1	18	yes	complemented
83	D₄×Dic₅₃	C₂³	1	36	yes	complemented
84	D₄×Dic₅₃	C₂³	1	18	yes	complemented
85	#13133179	C₂³	1	18	yes	complemented
86	#13133179	C₂³	1	36	yes	complemented
87	#13133179	C₂³	1	18	yes	complemented
88	C₄×C₂²⋊D₅₃	C₂³	1	36	yes	complemented
89	C₄×C₂²⋊D₅₃	C₂³	1	72	yes	complemented
90	C₄×C₂²⋊D₅₃	C₂³	1	36	yes	complemented
91	#13133177	C₂³	1	36	yes	complemented
92	#13133177	C₂³	1	72	yes	complemented
93	#13133177	C₂³	1	36	yes	complemented
94	#13133176	C₂³	1	18	yes	complemented
95	#13133175	C₂³	1	6	yes	—
96	#13133175	C₂³	1	18	yes	—
97	#13133175	C₂³	1	36	yes	complemented
98	#13133174	C₂³	1	18	yes	—
99	#13133174	C₂³	1	6	yes	—
100	#13133174	C₂³	1	18	yes	complemented
101	#13133174	C₂³	1	18	yes	complemented
102	#13133173	C₂³	1	36	yes	complemented
103	C₂×C₄⋊Dic₅₃	C₂³	1	9	yes	complemented
104	C₂×#11719919	C₂³	1	9	yes	complemented
105	C₂×#11719919	C₂³	1	9	yes	complemented
106	C₄⋊C₄×D₅₃	C₂³	1	18	yes	complemented
107	C₄⋊C₄×D₅₃	C₂³	1	18	yes	complemented
108	C₂×C₂²⋊Dic₅₃	C₂³	1	18	yes	complemented
109	C₂×#11719921	C₂³	1	9	yes	—
110	C₂×#11719921	C₂³	1	18	yes	—
111	C₂×#11719921	C₂³	1	9	yes	—
112	C₂×#11719921	C₂³	1	18	yes	complemented
113	C₂×#11719921	C₂³	1	18	yes	complemented
114	C₂²⋊C₄×D₅₃	C₂³	1	36	yes	—
115	C₂²⋊C₄×D₅₃	C₂³	1	72	yes	complemented
116	C₂×C₄×Dic₅₃	C₂³	1	9	yes	complemented
117	C₂×C₄×Dic₅₃	C₂³	1	18	yes	complemented
118	C₄²×D₅₃	C₂³	1	36	yes	complemented
119	H(5,𝔽₂)×C₅₃	C₂×C₄	1	16	yes	complemented
120	C₂×#41×C₅₃	C₂³	1	6	yes	—
121	C₂²×D₄×C₅₃	C₂³	1	9	yes	—
122	C₂³×C₄×C₅₃	C₂³	1	6	yes	complemented
123	C₄×Q₈×C₅₃	C₂³	1	2	yes	complemented
124	C₄×Q₈×C₅₃	C₂³	1	6	yes	complemented
125	C₄×D₄×C₅₃	C₂³	1	18	yes	complemented
126	C₄×D₄×C₅₃	C₂³	1	36	yes	complemented
127	C₄×D₄×C₅₃	C₂³	1	18	yes	complemented
128	#117×C₅₃	C₂³	1	18	yes	complemented
129	C₂×C₄⋊C₄×C₅₃	C₂³	1	9	yes	complemented
130	C₂×C₂²⋊C₄×C₅₃	C₂³	1	18	yes	complemented
131	C₂×C₄²×C₅₃	C₂³	1	9	yes	complemented
132	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
133	C₂³×D₅₃	—	8	6	no	—
134	C₂³×D₅₃	—	8	6	no	—
135	C₂³×D₅₃	C₂²×C₄	1	48	yes	complemented
136	C₂³×D₅₃	C₂⁴	1	9	yes	—
137	C₂³×D₅₃	C₂⁴	1	18	yes	—
138	#41×C₅₃	C₂²×C₄	1	48	yes	complemented
139	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
140	C₂×D₄×C₅₃	C₂²×C₄	1	72	yes	complemented
141	C₂×D₄×C₅₃	C₂⁴	1	18	yes	—
142	C₂²×C₄×C₅₃	—	8	6	no	—
143	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
147	C₂×C₂²⋊D₅₃	C₂²×C₄	1	144	yes	complemented
148	C₂×C₂²⋊D₅₃	C₂⁴	1	36	yes	—
149	C₂²×Dic₅₃	—	8	6	no	—
150	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
151	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
152	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
153	C₂²×Dic₅₃	C₂⁴	1	9	yes	—
154	#11719949	C₂²×C₄	1	48	yes	complemented
155	Q₈×D₅₃	C₂²×C₄	1	16	yes	complemented
156	#11719947	C₂²×C₄	1	144	yes	complemented
157	D₄×D₅₃	C₂²×C₄	1	144	yes	complemented
158	D₄×D₅₃	C₂²×C₄	1	288	yes	complemented
159	#11719945	C₂²×C₄	1	144	yes	complemented
160	C₂×D₂₁₂	C₂²×C₄	1	72	yes	complemented
161	C₂×D₂₁₂	C₂⁴	1	18	yes	—
162	C₂×C₄×D₅₃	—	8	18	no	—
163	C₂×C₄×D₅₃	—	8	18	no	—
164	C₂×C₄×D₅₃	C₂²×C₄	1	72	yes	complemented
165	C₂×C₄×D₅₃	C₂⁴	1	36	yes	—
166	C₂×C₄×D₅₃	C₂⁴	1	36	yes	—
167	C₂×C₄×D₅₃	C₂⁴	1	18	yes	—
168	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
169	C₂×Dic₁₀₆	C₂⁴	1	6	yes	—
170	C₄⋊C₄×C₅₃	C₂⁴	1	18	yes	—
171	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
172	C₂²⋊C₄×C₅₃	C₂⁴	1	36	yes	—
173	C₄²×C₅₃	C₂⁴	1	6	yes	—
174	C₄²×C₅₃	C₂⁴	1	18	yes	—
175	C₂²⋊Dic₅₃	C₂⁴	1	36	yes	—
176	#11719921	C₂⁴	1	6	yes	—
177	#11719921	C₂⁴	1	9	yes	—
178	#11719921	C₂⁴	1	18	yes	—
179	#11719921	C₂⁴	1	36	yes	—
180	#11719921	C₂⁴	1	1	yes	characteristic
181	#11719921	C₂⁴	1	9	yes	—
182	#11719921	C₂⁴	1	9	yes	—
183	C₄⋊Dic₅₃	C₂⁴	1	6	yes	—
184	C₄⋊Dic₅₃	C₂⁴	1	18	yes	—
185	#11719919	C₂⁴	1	18	yes	—
186	#11719919	C₂⁴	1	18	yes	—
187	#11719919	C₂⁴	1	6	yes	—
188	#11719919	C₂⁴	1	6	yes	—
189	C₄×Dic₅₃	C₂⁴	1	12	yes	—
190	C₄×Dic₅₃	C₂⁴	1	18	yes	—
191	C₄×Dic₅₃	C₂⁴	1	36	yes	—
192	C₄×Dic₅₃	C₂⁴	1	6	yes	—
193	C₂³×C₅₃	—	4	6	no	—
194	C₂³×C₅₃	—	4	6	no	—
195	C₂³×C₅₃	C₂³×C₄	1	24	yes	—
196	C₂³×C₅₃	C₂⁵	1	9	yes	—
197	C₂²×D₅₃	—	4	18	no	—
198	C₂²×D₅₃	—	4	18	no	—
199	C₂²×D₅₃	—	4	9	no	—
200	C₂²×D₅₃	—	4	9	no	—
201	C₂²×D₅₃	—	8	48	no	—
202	C₂²×D₅₃	C₂³×C₄	1	36	yes	—
203	C₂²×D₅₃	C₂³×C₄	1	72	yes	—
204	C₂²×D₅₃	C₂⁵	1	9	yes	—
205	C₂²×D₅₃	C₂⁵	1	1	yes	characteristic
206	Q₈×C₅₃	C₂³×C₄	1	8	yes	—
207	D₄×C₅₃	C₂³×C₄	1	72	yes	—
208	C₂×C₄×C₅₃	—	4	9	no	—
209	C₂×C₄×C₅₃	—	4	18	no	—
210	C₂×C₄×C₅₃	—	4	9	no	—
211	C₂×C₄×C₅₃	C₂³×C₄	1	36	yes	—
212	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
213	C₂×C₄×C₅₃	C₂⁵	1	9	yes	—
214	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
216	C₂²⋊D₅₃	C₂³×C₄	1	144	yes	—
217	C₂×Dic₅₃	—	4	9	no	—
218	C₂×Dic₅₃	—	4	18	no	—
219	C₂×Dic₅₃	—	4	9	no	—
220	C₂×Dic₅₃	C₂³×C₄	1	36	yes	—
221	C₂×Dic₅₃	C₂⁵	1	6	yes	—
222	C₂×Dic₅₃	C₂⁵	1	9	yes	—
223	C₂×Dic₅₃	C₂⁵	1	1	yes	characteristic
224	C₂×Dic₅₃	C₂⁵	1	6	yes	—
225	D₂₁₂	C₂³×C₄	1	72	yes	—
226	C₄×D₅₃	—	4	36	no	—
227	C₄×D₅₃	—	4	36	no	—
228	C₄×D₅₃	C₂³×C₄	1	24	yes	—
229	C₄×D₅₃	C₂³×C₄	1	72	yes	—
230	Dic₁₀₆	C₂³×C₄	1	24	yes	—
231	#62652	—	53	1	no	maximal, Sylow
232	C₂²×C₅₃	—	2	9	no	—
233	C₂²×C₅₃	—	2	9	no	—
234	C₂²×C₅₃	—	4	24	no	—
235	C₂²×C₅₃	C₂⁴×C₄	1	18	yes	—
236	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
237	C₂×D₅₃	—	2	9	no	—
238	C₂×D₅₃	—	2	9	no	—
239	C₂×D₅₃	—	2	1	no	—
240	C₂×D₅₃	—	2	1	no	—
241	C₂×D₅₃	—	4	36	no	—
242	C₂×D₅₃	—	4	72	no	—
243	C₂×D₅₃	C₂⁴×C₄	1	2	yes	—
244	C₂×D₅₃	C₂⁴×C₄	1	18	yes	—
245	C₄×C₅₃	—	2	6	no	—
246	C₄×C₅₃	—	2	9	no	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	C₂⁴×C₄	1	12	yes	—
249	Dic₅₃	—	2	6	no	—
250	Dic₅₃	—	2	9	no	—
251	Dic₅₃	—	2	1	no	—
252	Dic₅₃	C₂⁴×C₄	1	12	yes	—
253	C₂²×H(5,𝔽₂)	—	53	1	no	—
254	C₄×H(5,𝔽₂)	—	53	1	no	—
255	C₄×H(5,𝔽₂)	—	53	6	no	—
256	C₄×H(5,𝔽₂)	—	53	9	no	—
257	C₄×H(5,𝔽₂)	—	53	6	no	—
258	C₄×H(5,𝔽₂)	—	53	9	no	—
259	C₄×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic, complemented
260	#3428	—	53	6	no	—
261	#3428	—	53	6	no	—
262	C₂×#518	—	53	9	no	—
263	C₂×#518	—	53	9	no	—
264	C₂×C₅₃	—	2	18	no	—
265	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
266	C₂×C₅₃	H(7,𝔽₂)	1	1	yes	characteristic
267	C₂×C₅₃	H(7,𝔽₂)	1	1	yes	characteristic
268	D₅₃	—	2	2	no	—
269	D₅₃	—	2	18	no	—
270	C₂×H(5,𝔽₂)	—	53	32	no	—
271	C₂×H(5,𝔽₂)	—	53	6	no	—
272	C₂×H(5,𝔽₂)	—	53	9	no	—
273	C₂×H(5,𝔽₂)	C₂×D₅₃	1	1	yes	characteristic
274	C₂²×#41	—	53	6	no	—
275	C₂³×D₄	—	53	9	no	—
276	#520	—	53	6	no	—
277	#520	—	53	6	no	—
278	#520	—	53	6	no	—
279	#520	—	53	18	no	—
280	#520	—	53	18	no	—
281	#520	—	53	18	no	—
282	#520	—	53	18	no	—
283	#520	C₂×D₅₃	1	6	yes	complemented
284	#519	—	53	6	no	—
285	#519	—	53	2	no	—
286	#518	—	53	9	no	—
287	#518	—	53	18	no	—
288	#518	—	53	36	no	—
289	#518	—	53	18	no	—
290	#518	—	53	36	no	—
291	#518	—	53	36	no	—
292	#518	—	53	18	no	—
293	#518	—	53	36	no	—
294	#518	C₂×D₅₃	1	9	yes	complemented
295	C₄×#41	—	53	6	no	—
296	C₄×#41	—	53	12	no	—
297	C₄×#41	—	53	36	no	—
298	C₄×#41	—	53	18	no	—
299	C₄×#41	—	53	18	no	—
300	C₄×#41	C₂×D₅₃	1	6	yes	complemented
301	C₂×C₄×D₄	—	53	9	no	—
302	C₂×C₄×D₄	—	53	18	no	—
303	C₂×C₄×D₄	—	53	36	no	—
304	C₂×C₄×D₄	—	53	36	no	—
305	C₂×C₄×D₄	—	53	9	no	—
306	C₂×C₄×D₄	—	53	18	no	—
307	C₂×C₄×D₄	—	53	9	no	—
308	C₂×C₄×D₄	C₂×D₅₃	1	9	yes	complemented
309	C₂×#117	—	53	18	no	—
310	C₂×#117	—	53	9	no	—
311	C₂×#117	—	53	9	no	—
312	C₂²×C₂²⋊C₄	—	53	6	no	—
313	C₂²×C₂²⋊C₄	—	53	18	no	—
314	C₅₃	#62652	1	1	yes	characteristic, complemented, Sylow
315	C₂⁵	—	53	6	no	—
316	H(5,𝔽₂)	—	53	96	no	—
317	H(5,𝔽₂)	—	53	144	no	—
318	H(5,𝔽₂)	C₄×D₅₃	1	16	yes	complemented
319	C₂×#41	—	53	96	no	—
320	C₂×#41	—	53	6	no	—
321	C₂×#41	—	53	18	no	—
322	C₂×#41	—	53	18	no	—
323	C₂×#41	C₂²×D₅₃	1	6	yes	—
324	C₂²×Q₈	—	53	2	no	—
325	C₂²×D₄	—	53	18	no	—
326	C₂²×D₄	—	53	144	no	—
327	C₂²×D₄	—	53	36	no	—
328	C₂²×D₄	—	53	18	no	—
329	C₂²×D₄	—	53	9	no	—
330	C₂²×D₄	C₂²×D₅₃	1	9	yes	—
331	C₂³×C₄	—	53	9	no	—
332	C₂³×C₄	—	53	6	no	—
333	C₂³×C₄	—	53	18	no	—
334	C₂³×C₄	—	53	18	no	—
335	C₂³×C₄	C₂²×D₅₃	1	6	yes	complemented
336	C₄×Q₈	—	53	2	no	—
337	C₄×Q₈	—	53	6	no	—
338	C₄×Q₈	—	53	6	no	—
339	C₄×Q₈	—	53	18	no	—
340	C₄×Q₈	—	53	18	no	—
341	C₄×Q₈	—	53	6	no	—
342	C₄×Q₈	C₂²×D₅₃	1	6	yes	complemented
343	C₄×Q₈	C₂²×D₅₃	1	2	yes	complemented
344	C₄×D₄	—	53	18	no	—
345	C₄×D₄	—	53	36	no	—
346	C₄×D₄	—	53	18	no	—
347	C₄×D₄	—	53	72	no	—
348	C₄×D₄	—	53	36	no	—
349	C₄×D₄	—	53	36	no	—
350	C₄×D₄	—	53	18	no	—
351	C₄×D₄	—	53	36	no	—
352	C₄×D₄	—	53	18	no	—
353	C₄×D₄	—	53	36	no	—
354	C₄×D₄	—	53	72	no	—
355	C₄×D₄	—	53	36	no	—
356	C₄×D₄	—	53	18	no	—
357	C₄×D₄	—	53	36	no	—
358	C₄×D₄	—	53	18	no	—
359	C₄×D₄	C₂²×D₅₃	1	18	yes	complemented
360	C₄×D₄	C₂²×D₅₃	1	36	yes	complemented
361	C₄×D₄	C₂²×D₅₃	1	18	yes	complemented
362	#117	—	53	18	no	—
363	#117	—	53	6	no	—
364	#117	—	53	18	no	—
365	#117	—	53	36	no	—
366	#117	—	53	36	no	—
367	#117	—	53	18	no	—
368	#117	—	53	6	no	—
369	#117	—	53	18	no	—
370	#117	—	53	18	no	—
371	#117	C₂²×D₅₃	1	18	yes	complemented
372	C₂×C₄⋊C₄	—	53	9	no	—
373	C₂×C₄⋊C₄	—	53	9	no	—
374	C₂×C₄⋊C₄	—	53	9	no	—
375	C₂×C₄⋊C₄	—	53	18	no	—
376	C₂×C₄⋊C₄	—	53	18	no	—
377	C₂×C₄⋊C₄	C₂²×D₅₃	1	9	yes	complemented
378	C₂×C₂²⋊C₄	—	53	18	no	—
379	C₂×C₂²⋊C₄	—	53	18	no	—
380	C₂×C₂²⋊C₄	—	53	9	no	—
381	C₂×C₂²⋊C₄	—	53	9	no	—
382	C₂×C₂²⋊C₄	—	53	18	no	—
383	C₂×C₂²⋊C₄	—	53	18	no	—
384	C₂×C₂²⋊C₄	—	53	36	no	—
385	C₂×C₂²⋊C₄	—	53	72	no	—
386	C₂×C₂²⋊C₄	C₂²×D₅₃	1	18	yes	complemented
387	C₂×C₄²	—	53	18	no	—
388	C₂×C₄²	—	53	9	no	—
389	C₂×C₄²	—	53	36	no	—
390	C₂×C₄²	C₂²×D₅₃	1	9	yes	complemented
391	C₂⁴	—	53	48	no	—
392	C₂⁴	—	53	9	no	—
393	C₂⁴	—	53	18	no	—
394	C₂⁴	—	424	6	no	—
395	C₂⁴	—	424	6	no	—
396	C₂⁴	C₂³×D₅₃	1	6	yes	—
397	#41	—	53	48	no	—
398	#41	—	53	144	no	—
399	#41	—	53	144	no	—
400	#41	C₂×C₄×D₅₃	1	48	yes	complemented
401	C₂×Q₈	—	53	16	no	—
402	C₂×Q₈	—	53	6	no	—
403	C₂×Q₈	C₂³×D₅₃	1	2	yes	—
404	C₂×D₄	—	53	144	no	—
405	C₂×D₄	—	53	36	no	—
406	C₂×D₄	—	53	144	no	—
407	C₂×D₄	—	53	288	no	—
408	C₂×D₄	—	53	72	no	—
409	C₂×D₄	—	53	18	no	—
410	C₂×D₄	C₂×C₄×D₅₃	1	72	yes	complemented
411	C₂×D₄	C₂³×D₅₃	1	18	yes	—
412	C₂²×C₄	—	8	6	no	—
413	C₂²×C₄	—	53	9	no	—
414	C₂²×C₄	—	53	72	no	—
415	C₂²×C₄	—	53	18	no	—
416	C₂²×C₄	—	53	6	no	—
417	C₂²×C₄	—	53	9	no	—
418	C₂²×C₄	—	53	18	no	—
419	C₂²×C₄	—	53	9	no	—
420	C₂²×C₄	—	53	36	no	—
421	C₂²×C₄	—	53	36	no	—
422	C₂²×C₄	—	424	18	no	—
423	C₂²×C₄	—	424	18	no	—
424	C₂²×C₄	—	424	6	no	—
425	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
426	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
427	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
428	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
429	C₄⋊C₄	—	53	18	no	—
430	C₄⋊C₄	—	53	6	no	—
431	C₄⋊C₄	—	53	18	no	—
432	C₄⋊C₄	—	53	6	no	—
433	C₄⋊C₄	—	53	18	no	—
434	C₄⋊C₄	—	53	6	no	—
435	C₄⋊C₄	C₂³×D₅₃	1	6	yes	—
436	C₄⋊C₄	C₂³×D₅₃	1	18	yes	—
437	C₂²⋊C₄	—	53	36	no	—
438	C₂²⋊C₄	—	53	9	no	—
439	C₂²⋊C₄	—	53	6	no	—
440	C₂²⋊C₄	—	53	1	no	—
441	C₂²⋊C₄	—	53	9	no	—
442	C₂²⋊C₄	—	53	36	no	—
443	C₂²⋊C₄	—	53	18	no	—
444	C₂²⋊C₄	—	53	9	no	—
445	C₂²⋊C₄	C₂³×D₅₃	1	36	yes	—
446	C₄²	—	53	36	no	—
447	C₄²	—	53	12	no	—
448	C₄²	—	53	6	no	—
449	C₄²	—	53	18	no	—
450	C₄²	C₂³×D₅₃	1	18	yes	—
451	C₄²	C₂³×D₅₃	1	6	yes	—
452	C₂³	—	4	6	no	—
453	C₂³	—	4	6	no	—
454	C₂³	—	53	1	no	—
455	C₂³	—	53	9	no	—
456	C₂³	—	53	36	no	—
457	C₂³	—	53	72	no	—
458	C₂³	—	424	48	no	—
459	C₂³	—	212	18	no	—
460	C₂³	—	212	18	no	—
461	C₂³	—	212	9	no	—
462	C₂³	—	212	9	no	—
463	C₂³	C₂²×C₄×D₅₃	1	24	yes	—
464	C₂³	C₂⁴×D₅₃	1	9	yes	—
465	Q₈	—	53	24	no	—
466	Q₈	C₂²×C₄×D₅₃	1	8	yes	—
467	D₄	—	53	144	no	—
468	D₄	—	53	72	no	—
469	D₄	C₂²×C₄×D₅₃	1	72	yes	—
470	C₂×C₄	—	4	9	no	—
471	C₂×C₄	—	4	18	no	—
472	C₂×C₄	—	4	9	no	—
473	C₂×C₄	—	53	36	no	—
474	C₂×C₄	—	53	6	no	—
475	C₂×C₄	—	53	24	no	—
476	C₂×C₄	—	53	72	no	—
477	C₂×C₄	—	53	6	no	—
478	C₂×C₄	—	53	9	no	—
479	C₂×C₄	—	53	1	no	—
480	C₂×C₄	—	212	36	no	—
481	C₂×C₄	—	212	36	no	—
482	C₂×C₄	—	212	9	no	—
483	C₂×C₄	—	212	18	no	—
484	C₂×C₄	—	212	9	no	—
485	C₂×C₄	C₂²×C₄×D₅₃	1	36	yes	—
486	C₂×C₄	C₂⁴×D₅₃	1	9	yes	—
487	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
488	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
489	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
490	C₂²	—	2	9	no	—
491	C₂²	—	2	9	no	—
492	C₂²	—	4	24	no	—
493	C₂²	—	53	2	no	—
494	C₂²	—	53	18	no	—
495	C₂²	—	106	9	no	—
496	C₂²	—	106	1	no	—
497	C₂²	—	106	9	no	—
498	C₂²	—	106	1	no	—
499	C₂²	—	212	36	no	—
500	C₂²	—	212	72	no	—
501	C₂²	C₂³×C₄×D₅₃	1	18	yes	—
502	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
503	C₄	—	2	1	no	—
504	C₄	—	2	9	no	—
505	C₄	—	2	6	no	—
506	C₄	—	53	12	no	—
507	C₄	—	106	6	no	—
508	C₄	—	106	9	no	—
509	C₄	—	106	1	no	—
510	C₄	C₂³×C₄×D₅₃	1	12	yes	—
511	C₂	—	2	18	no	—
512	C₂	—	106	2	no	—
513	C₂	—	106	18	no	—
514	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
515	C₂	#20289525	1	1	yes	characteristic
516	C₂	#20289526	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Fitting	C₄×H(5,𝔽₂)×C₅₃	C₂
socle, cosocle	C₂²×C₅₃	C₂⁶
prime radical	C₂×H(5,𝔽₂)×D₅₃	C₂
hypercenter	C₄×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	#62652
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. H(5,𝔽₂)×Dic₅₃	C₂	semidirect
3. C₄×#13133263	C₂	semidirect
4. #20287921	C₂	semidirect
5. C₄×#13133262	C₂	semidirect
6. #20287919	C₂	semidirect
7. #20287915	C₂	semidirect
8. #20287914	C₂	semidirect
9. C₂×#16811095	C₂	semidirect
10. #518×D₅₃	C₂	semidirect
11. C₄×H(5,𝔽₂)×C₅₃	C₂	semidirect
14. H(5,𝔽₂)×D₅₃	C₄	semidirect
17. #16811106	C₂²	semidirect
18. #16811105	C₂²	semidirect
19. #16811105	C₂²	semidirect
20. #16811104	C₂²	semidirect
21. #16811103	C₂²	semidirect
22. #16811102	C₂²	semidirect
23. #16811102	C₂²	semidirect
24. #16811099	C₂²	semidirect
25. #16811099	C₂²	semidirect
26. #16811098	C₂²	semidirect
27. #16811097	C₂²	semidirect
28. #16811096	C₂²	semidirect
29. #16811095	C₂²	semidirect
30. #16811095	C₂²	semidirect
31. #16811095	C₂²	semidirect
32. #16811095	C₂²	semidirect
33. #16811095	C₂²	semidirect
34. #41×Dic₅₃	C₂²	semidirect
35. C₄×#11719949	C₂²	semidirect
36. C₄×#11719945	C₂²	semidirect
37. #16811091	C₂²	semidirect
38. C₄×#11719947	C₂²	semidirect
39. C₂×C₄×D₂₁₂	C₂²	semidirect
40. C₂×D₄×Dic₅₃	C₂²	semidirect
41. C₂×#13133179	C₂²	semidirect
42. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
43. C₂×#13133177	C₂²	semidirect
44. C₄×D₄×D₅₃	C₂²	semidirect
45. C₄×D₄×D₅₃	C₂²	semidirect
46. C₂×#13133175	C₂²	semidirect
47. C₂×#13133174	C₂²	semidirect
48. #117×D₅₃	C₂²	semidirect
49. C₂²×#11719921	C₂²	semidirect
50. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
52. C₅₃×#520	C₂²	semidirect
53. C₅₃×#518	C₂²	semidirect
54. C₄×#41×C₅₃	C₂²	semidirect
55. C₂×C₄×D₄×C₅₃	C₂²	semidirect
57. #13133263	C₂×C₄	semidirect
58. #13133262	C₂×C₄	semidirect
62. #41×D₅₃	C₂×C₄	semidirect
66. C₂×D₄×D₅₃	C₂×C₄	semidirect
69. C₂³×Dic₅₃	C₂³	semidirect
71. C₂²×C₄×D₅₃	C₂³	semidirect
72. C₂²×C₄×D₅₃	C₂³	semidirect
73. Q₈×Dic₅₃	C₂³	semidirect
74. Q₈×Dic₅₃	C₂³	semidirect
75. C₄×Dic₁₀₆	C₂³	semidirect
76. C₄×Dic₁₀₆	C₂³	semidirect
77. #13133182	C₂³	semidirect
78. #13133182	C₂³	semidirect
79. C₄×D₂₁₂	C₂³	semidirect
80. C₄×D₂₁₂	C₂³	semidirect
81. C₄×D₂₁₂	C₂³	semidirect
82. D₄×Dic₅₃	C₂³	semidirect
83. D₄×Dic₅₃	C₂³	semidirect
84. D₄×Dic₅₃	C₂³	semidirect
85. #13133179	C₂³	semidirect
86. #13133179	C₂³	semidirect
87. #13133179	C₂³	semidirect
88. C₄×C₂²⋊D₅₃	C₂³	semidirect
89. C₄×C₂²⋊D₅₃	C₂³	semidirect
90. C₄×C₂²⋊D₅₃	C₂³	semidirect
91. #13133177	C₂³	semidirect
92. #13133177	C₂³	semidirect
93. #13133177	C₂³	semidirect
94. #13133176	C₂³	semidirect
97. #13133175	C₂³	semidirect
100. #13133174	C₂³	semidirect
101. #13133174	C₂³	semidirect
102. #13133173	C₂³	semidirect
103. C₂×C₄⋊Dic₅₃	C₂³	semidirect
104. C₂×#11719919	C₂³	semidirect
105. C₂×#11719919	C₂³	semidirect
106. C₄⋊C₄×D₅₃	C₂³	semidirect
107. C₄⋊C₄×D₅₃	C₂³	semidirect
108. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
112. C₂×#11719921	C₂³	semidirect
113. C₂×#11719921	C₂³	semidirect
115. C₂²⋊C₄×D₅₃	C₂³	semidirect
116. C₂×C₄×Dic₅₃	C₂³	semidirect
117. C₂×C₄×Dic₅₃	C₂³	semidirect
118. C₄²×D₅₃	C₂³	semidirect
119. H(5,𝔽₂)×C₅₃	C₂×C₄	semidirect
122. C₂³×C₄×C₅₃	C₂³	semidirect
123. C₄×Q₈×C₅₃	C₂³	semidirect
124. C₄×Q₈×C₅₃	C₂³	semidirect
125. C₄×D₄×C₅₃	C₂³	semidirect
126. C₄×D₄×C₅₃	C₂³	semidirect
127. C₄×D₄×C₅₃	C₂³	semidirect
128. #117×C₅₃	C₂³	semidirect
129. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
130. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
131. C₂×C₄²×C₅₃	C₂³	semidirect
135. C₂³×D₅₃	C₂²×C₄	semidirect
138. #41×C₅₃	C₂²×C₄	semidirect
140. C₂×D₄×C₅₃	C₂²×C₄	semidirect
147. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
154. #11719949	C₂²×C₄	semidirect
155. Q₈×D₅₃	C₂²×C₄	semidirect
156. #11719947	C₂²×C₄	semidirect
157. D₄×D₅₃	C₂²×C₄	semidirect
158. D₄×D₅₃	C₂²×C₄	semidirect
159. #11719945	C₂²×C₄	semidirect
160. C₂×D₂₁₂	C₂²×C₄	semidirect
164. C₂×C₄×D₅₃	C₂²×C₄	semidirect
259. C₄×H(5,𝔽₂)	D₅₃	semidirect
283. #520	C₂×D₅₃	semidirect
294. #518	C₂×D₅₃	semidirect
300. C₄×#41	C₂×D₅₃	semidirect
308. C₂×C₄×D₄	C₂×D₅₃	semidirect
314. C₅₃	#62652	semidirect
318. H(5,𝔽₂)	C₄×D₅₃	semidirect
335. C₂³×C₄	C₂²×D₅₃	semidirect
342. C₄×Q₈	C₂²×D₅₃	semidirect
343. C₄×Q₈	C₂²×D₅₃	semidirect
359. C₄×D₄	C₂²×D₅₃	semidirect
360. C₄×D₄	C₂²×D₅₃	semidirect
361. C₄×D₄	C₂²×D₅₃	semidirect
371. #117	C₂²×D₅₃	semidirect
377. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
386. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
390. C₂×C₄²	C₂²×D₅₃	semidirect
400. #41	C₂×C₄×D₅₃	semidirect
410. C₂×D₄	C₂×C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)×D₅₃	2
H(5,𝔽₂)×Dic₅₃	2
C₄×#13133263	2
#20287921	2
C₄×#13133262	2
#20287919	2
#20287915	2
#20287914	2
C₂×#16811095	2
#518×D₅₃	2
C₄×H(5,𝔽₂)×C₅₃	2
#62652	53