Subgroups — #28292628

Order 13568.

11534 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20288388	C₂	1	8	yes	maximal, complemented
3	#20288377	C₂	1	8	yes	maximal, complemented
4	#20288370	C₂	1	2	yes	maximal, complemented
5	#20288363	C₂	1	2	yes	maximal, complemented
6	#20288355	C₂	1	6	yes	maximal, complemented
7	#20288346	C₂	1	6	yes	maximal, complemented
8	#41×C₂²⋊D₅₃	C₂	1	16	yes	maximal, complemented
9	#20288044	C₂	1	6	yes	maximal, complemented
10	C₂×#16811130	C₂	1	6	yes	maximal, complemented
11	#20287918	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#11719949	C₂²	1	1	yes	characteristic, complemented
14	C₂²×#11719945	C₂²	1	3	yes	complemented
15	C₂²×#11719947	C₂²	1	3	yes	complemented
16	C₂×#41×D₅₃	—	2	16	no	—
17	C₂×#41×D₅₃	C₂²	1	8	yes	complemented
18	C₂×#41×D₅₃	C₂²	1	8	yes	complemented
19	C₂²×Q₈×D₅₃	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄×D₅₃	C₂²	1	3	yes	complemented
21	C₂³×C₄×D₅₃	C₂²	1	3	yes	complemented
22	#16811292	C₂²	1	16	yes	complemented
23	Q₈×C₂²⋊D₅₃	C₂²	1	16	yes	complemented
24	#16811284	C₂²	1	16	yes	complemented
25	#16811280	C₂²	1	16	yes	complemented
26	#16811272	C₂²	1	48	yes	complemented
27	#16811267	C₂²	1	48	yes	complemented
28	#16811261	C₂²	1	48	yes	complemented
29	D₄×C₂²⋊D₅₃	C₂²	1	48	yes	complemented
30	#16811201	C₂²	1	24	yes	complemented
31	#16811197	C₂²	1	12	yes	complemented
32	#16811194	C₂²	1	12	yes	complemented
33	#16811190	C₂²	1	12	yes	complemented
34	#16811187	C₂²	1	12	yes	complemented
35	#16811182	C₂²	1	24	yes	complemented
36	#16811169	C₂²	1	24	yes	complemented
37	#16811134	C₂²	1	24	yes	complemented
38	#16811130	C₂²	1	12	yes	complemented
39	#16811130	C₂²	1	4	yes	complemented
40	#16811130	C₂²	1	12	yes	complemented
41	#16811130	C₂²	1	12	yes	complemented
42	#16811130	C₂²	1	24	yes	complemented
43	#16811130	C₂²	1	24	yes	complemented
44	C₂×#13133198	C₂²	1	12	yes	complemented
45	C₂×#13133193	C₂²	1	12	yes	complemented
46	C₂×#13133188	C₂²	1	12	yes	complemented
47	C₂×#13133185	C₂²	1	12	yes	complemented
48	#16811106	C₂²	1	8	yes	complemented
49	#16811101	C₂²	1	2	yes	complemented
50	#16811098	C₂²	1	6	yes	complemented
51	#41×Dic₅₃	C₂²	1	8	yes	complemented
52	C₂×C₄×C₂²⋊D₅₃	C₂²	1	24	yes	complemented
53	C₂×#13133176	C₂²	1	6	yes	complemented
54	C₅₃×#585	—	2	16	no	—
55	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	complemented
56	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	4	yes	complemented
57	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	6	yes	complemented
58	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	4	yes	complemented
59	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
60	C₂²×#41×C₅₃	C₂²	1	6	yes	complemented
61	C₂²×#41×C₅₃	C₂²	1	8	yes	complemented
62	C₂⁴×D₅₃	C₂³	1	3	yes	—
63	C₂×#11719949	—	2	8	no	—
64	C₂×#11719949	C₂³	1	8	yes	—
65	C₂×#11719945	—	2	24	no	—
66	C₂×#11719945	C₂³	1	12	yes	—
67	C₂×#11719945	C₂³	1	12	yes	—
68	C₂×#11719947	—	2	24	no	—
69	C₂×#11719947	C₂³	1	24	yes	—
70	#41×D₅₃	—	2	64	no	—
71	#41×D₅₃	—	2	64	no	—
72	C₂²×Dic₁₀₆	C₂³	1	3	yes	—
73	C₂×Q₈×D₅₃	—	2	8	no	—
74	C₂×Q₈×D₅₃	C₂³	1	8	yes	—
75	C₂²×D₂₁₂	C₂³	1	3	yes	—
76	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
77	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
78	C₂×D₄×D₅₃	—	2	24	no	—
79	C₂×D₄×D₅₃	C₂³	1	24	yes	—
80	C₂³×Dic₅₃	C₂³	1	3	yes	—
81	C₂²×C₄×D₅₃	—	2	24	no	—
82	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
83	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
84	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
85	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
86	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
87	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
88	#13133215	C₂³	1	24	yes	complemented
89	#13133213	C₂³	1	24	yes	complemented
90	#13133206	C₂³	1	24	yes	complemented
91	#13133204	C₂³	1	24	yes	complemented
92	#13133200	C₂³	1	48	yes	complemented
93	#13133198	C₂³	1	8	yes	—
94	#13133198	C₂³	1	24	yes	—
95	#13133198	C₂³	1	48	yes	complemented
96	#13133196	C₂³	1	48	yes	complemented
97	#13133193	C₂³	1	24	yes	—
98	#13133193	C₂³	1	8	yes	—
99	#13133193	C₂³	1	24	yes	complemented
100	#13133193	C₂³	1	24	yes	complemented
101	#13133191	C₂³	1	48	yes	complemented
102	#13133189	C₂³	1	96	yes	complemented
103	#13133188	C₂³	1	24	yes	—
104	#13133188	C₂³	1	8	yes	—
105	#13133188	C₂³	1	24	yes	complemented
106	#13133188	C₂³	1	24	yes	complemented
107	#13133186	C₂³	1	48	yes	complemented
108	#13133185	C₂³	1	8	yes	—
109	#13133185	C₂³	1	24	yes	—
110	#13133185	C₂³	1	48	yes	—
111	Q₈×Dic₅₃	C₂³	1	16	yes	complemented
112	D₄×Dic₅₃	C₂³	1	48	yes	complemented
113	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	—
114	C₄×C₂²⋊D₅₃	C₂³	1	16	yes	—
115	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
116	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
117	#13133176	C₂³	1	4	yes	—
118	#13133176	C₂³	1	12	yes	—
119	#13133176	C₂³	1	12	yes	complemented
120	#13133176	C₂³	1	12	yes	complemented
121	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
122	C₂×#11719919	C₂³	1	24	yes	complemented
123	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
124	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
125	C₂×#11719921	C₂³	1	24	yes	—
126	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
127	2₋¹⁺⁴×C₅₃	—	2	32	no	—
128	2₋¹⁺⁴×C₅₃	—	2	16	no	—
129	H(5,𝔽₂)×C₅₃	—	2	32	no	—
130	H(5,𝔽₂)×C₅₃	—	2	48	no	—
131	C₂×#41×C₅₃	—	2	64	no	—
132	C₂×#41×C₅₃	—	2	48	no	—
133	C₂×#41×C₅₃	D₄	1	16	yes	complemented
134	C₂×#41×C₅₃	C₂³	1	4	yes	—
135	C₂×#41×C₅₃	C₂³	1	12	yes	—
136	C₂×#41×C₅₃	C₂³	1	4	yes	—
137	C₂×#41×C₅₃	C₂³	1	4	yes	—
138	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
139	C₂×#41×C₅₃	C₂³	1	8	yes	complemented
140	C₂×#41×C₅₃	C₂³	1	12	yes	complemented
141	C₂×#41×C₅₃	C₂³	1	12	yes	complemented
142	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
143	C₂²×Q₈×C₅₃	C₂³	1	8	yes	complemented
144	C₂²×Q₈×C₅₃	C₂³	1	6	yes	complemented
145	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
146	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
147	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
148	C₂²×D₄×C₅₃	C₂³	1	6	yes	complemented
149	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
150	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
151	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
152	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
153	C₂³×D₅₃	—	2	12	no	—
154	C₂³×D₅₃	—	4	12	no	—
155	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
156	C₂³×D₅₃	C₂⁴	1	12	yes	—
157	C₂³×D₅₃	C₂⁴	1	3	yes	—
158	#41×C₅₃	—	2	32	no	—
159	#41×C₅₃	—	2	96	no	—
160	#41×C₅₃	—	2	16	no	—
161	#41×C₅₃	—	2	48	no	—
162	#41×C₅₃	—	2	48	no	—
163	#41×C₅₃	—	2	48	no	—
164	#41×C₅₃	C₂×D₄	1	32	yes	complemented
165	#41×C₅₃	C₂×D₄	1	32	yes	complemented
166	C₂×Q₈×C₅₃	—	2	32	no	—
167	C₂×Q₈×C₅₃	—	2	24	no	—
168	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
169	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
170	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
171	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
172	C₂×D₄×C₅₃	—	2	48	no	—
173	C₂×D₄×C₅₃	—	2	96	no	—
174	C₂×D₄×C₅₃	—	2	24	no	—
175	C₂×D₄×C₅₃	C₂×D₄	1	24	yes	complemented
176	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
177	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
178	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
179	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
180	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
181	C₂²×C₄×C₅₃	—	2	48	no	—
182	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	complemented
183	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
184	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
185	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
186	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
188	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
189	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
190	C₂×C₂²⋊D₅₃	—	2	24	no	—
191	C₂×C₂²⋊D₅₃	—	4	48	no	—
192	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
193	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
194	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
195	C₂²×Dic₅₃	—	2	12	no	—
196	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
197	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
198	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
199	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
200	#11719949	—	2	32	no	—
201	Q₈×D₅₃	—	2	32	no	—
202	#11719947	—	2	96	no	—
203	D₄×D₅₃	—	2	96	no	—
204	#11719945	—	2	48	no	—
205	#11719945	—	2	48	no	—
206	C₂×D₂₁₂	—	2	12	no	—
207	C₂×D₂₁₂	C₂⁴	1	12	yes	—
208	C₂×C₄×D₅₃	—	2	4	no	—
209	C₂×C₄×D₅₃	—	2	12	no	—
210	C₂×C₄×D₅₃	—	2	48	no	—
211	C₂×C₄×D₅₃	—	2	48	no	—
212	C₂×C₄×D₅₃	—	2	12	no	—
213	C₂×C₄×D₅₃	—	2	12	no	—
214	C₂×C₄×D₅₃	C₂⁴	1	2	yes	—
215	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
216	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
217	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
218	C₂×C₄×D₅₃	C₂⁴	1	2	yes	—
219	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
220	C₂×Dic₁₀₆	—	2	12	no	—
221	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
222	C₂²⋊Dic₅₃	C₂⁴	1	8	yes	—
223	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
224	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
225	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
226	#11719921	C₂⁴	1	16	yes	—
227	#11719921	C₂⁴	1	48	yes	—
228	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
229	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
230	#11719919	C₂⁴	1	16	yes	—
231	#11719919	C₂⁴	1	48	yes	—
232	C₄×Dic₅₃	C₂⁴	1	8	yes	—
233	C₄×Dic₅₃	C₂⁴	1	24	yes	—
234	C₂³×C₅₃	—	2	24	no	—
235	C₂³×C₅₃	—	4	6	no	—
236	C₂³×C₅₃	—	4	24	no	—
237	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
238	C₂³×C₅₃	C₂⁵	1	8	yes	—
239	C₂³×C₅₃	C₂⁵	1	6	yes	—
240	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
241	C₂²×D₅₃	—	2	2	no	—
242	C₂²×D₅₃	—	2	4	no	—
243	C₂²×D₅₃	—	2	6	no	—
244	C₂²×D₅₃	—	2	24	no	—
245	C₂²×D₅₃	—	2	24	no	—
246	C₂²×D₅₃	—	4	12	no	—
247	C₂²×D₅₃	—	4	6	no	—
248	C₂²×D₅₃	—	4	48	no	—
249	C₂²×D₅₃	C₂⁵	1	6	yes	—
250	C₂²×D₅₃	C₂⁵	1	2	yes	—
251	Q₈×C₅₃	—	2	24	no	—
252	Q₈×C₅₃	—	2	8	no	—
253	Q₈×C₅₃	C₂²×D₄	1	16	yes	—
254	D₄×C₅₃	—	2	16	no	—
255	D₄×C₅₃	—	2	48	no	—
256	D₄×C₅₃	—	2	24	no	—
257	D₄×C₅₃	—	2	8	no	—
258	D₄×C₅₃	C₂²×D₄	1	48	yes	—
259	C₂×C₄×C₅₃	—	2	24	no	—
260	C₂×C₄×C₅₃	—	2	16	no	—
261	C₂×C₄×C₅₃	—	2	48	no	—
262	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
263	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
265	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
266	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
267	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
268	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
269	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
270	C₂²⋊D₅₃	—	2	48	no	—
271	C₂²⋊D₅₃	—	2	32	no	—
272	C₂²⋊D₅₃	—	4	48	no	—
273	C₂×Dic₅₃	—	2	6	no	—
274	C₂×Dic₅₃	—	2	2	no	—
275	C₂×Dic₅₃	—	2	24	no	—
276	C₂×Dic₅₃	—	4	24	no	—
277	C₂×Dic₅₃	C₂⁵	1	6	yes	—
278	C₂×Dic₅₃	C₂⁵	1	2	yes	—
279	C₂×Dic₅₃	C₂⁵	1	16	yes	—
280	D₂₁₂	—	2	24	no	—
281	C₄×D₅₃	—	2	4	no	—
282	C₄×D₅₃	—	2	4	no	—
283	C₄×D₅₃	—	2	12	no	—
284	C₄×D₅₃	—	2	12	no	—
285	C₄×D₅₃	—	2	24	no	—
286	C₄×D₅₃	—	2	24	no	—
287	Dic₁₀₆	—	2	24	no	—
288	#62811	—	53	1	no	maximal, Sylow
289	C₂²×C₅₃	—	2	12	no	—
290	C₂²×C₅₃	—	2	2	no	—
291	C₂²×C₅₃	—	2	8	no	—
292	C₂²×C₅₃	—	2	16	no	—
293	C₂²×C₅₃	—	4	12	no	—
294	C₂²×C₅₃	—	8	24	no	—
295	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
296	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
297	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
298	C₂×D₅₃	—	2	8	no	—
299	C₂×D₅₃	—	2	2	no	—
300	C₂×D₅₃	—	2	2	no	—
301	C₂×D₅₃	—	2	6	no	—
302	C₂×D₅₃	—	2	12	no	—
303	C₂×D₅₃	—	4	48	no	—
304	C₂×D₅₃	—	4	12	no	—
305	C₂×D₅₃	#585	1	4	yes	complemented
306	C₄×C₅₃	—	2	8	no	—
307	C₄×C₅₃	C₂³×D₄	1	12	yes	—
308	C₄×C₅₃	C₂³×D₄	1	2	yes	—
309	C₄×C₅₃	C₂³×D₄	1	2	yes	—
310	Dic₅₃	—	2	6	no	—
311	Dic₅₃	—	2	2	no	—
312	Dic₅₃	—	2	16	no	—
313	C₂×#585	D₅₃	1	1	yes	characteristic, complemented
314	C₂³×#41	—	53	1	no	—
315	#3489	—	53	8	no	—
316	#3488	—	53	8	no	—
317	#3487	—	53	2	no	—
318	#3486	—	53	2	no	—
319	#3485	—	53	6	no	—
320	#3484	—	53	6	no	—
321	D₄×#41	—	53	16	no	—
322	#3449	—	53	6	no	—
323	C₂×#525	—	53	6	no	—
324	#3428	—	53	1	no	—
325	C₂×C₅₃	—	2	2	no	—
326	C₂×C₅₃	—	2	12	no	—
327	C₂×C₅₃	—	4	8	no	—
328	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
329	C₂×C₅₃	C₂×#585	1	2	yes	—
330	D₅₃	—	2	4	no	—
331	D₅₃	—	4	6	no	—
332	#585	—	2	16	no	—
333	C₂×2₋¹⁺⁴	C₂×D₅₃	1	2	yes	complemented
334	C₂×2₋¹⁺⁴	C₂×D₅₃	1	4	yes	complemented
335	C₂×H(5,𝔽₂)	C₂×D₅₃	1	6	yes	complemented
336	C₂×H(5,𝔽₂)	C₂×D₅₃	1	4	yes	complemented
337	C₂²×#41	—	53	8	no	—
338	C₂²×#41	—	53	8	no	—
339	C₂²×#41	—	53	1	no	—
340	C₂²×#41	—	53	3	no	—
341	C₂²×#41	—	53	3	no	—
342	C₂²×#41	—	106	16	no	—
343	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
344	C₂²×#41	C₂×D₅₃	1	6	yes	complemented
345	C₂²×#41	C₂×D₅₃	1	8	yes	complemented
346	C₂³×Q₈	—	53	1	no	—
347	C₂³×D₄	—	53	3	no	—
348	C₂⁴×C₄	—	53	3	no	—
349	#550	—	53	16	no	—
350	D₄×Q₈	—	53	16	no	—
351	#548	—	53	16	no	—
352	#548	—	53	16	no	—
353	#547	—	53	48	no	—
354	#546	—	53	48	no	—
355	#546	—	53	48	no	—
356	D₄²	—	53	48	no	—
357	#537	—	53	24	no	—
358	#536	—	53	12	no	—
359	#536	—	53	12	no	—
360	#535	—	53	12	no	—
361	#535	—	53	12	no	—
362	#534	—	53	24	no	—
363	#532	—	53	24	no	—
364	#525	—	53	4	no	—
365	#525	—	53	12	no	—
366	#525	—	53	12	no	—
367	#525	—	53	12	no	—
368	#525	—	53	24	no	—
369	#525	—	53	24	no	—
370	#525	—	53	24	no	—
371	C₂×#123	—	53	12	no	—
372	C₂×C₂²⋊Q₈	—	53	12	no	—
373	C₂×#121	—	53	12	no	—
374	C₂×#120	—	53	12	no	—
375	#520	—	53	8	no	—
376	#519	—	53	2	no	—
377	#518	—	53	6	no	—
378	C₄×#41	—	53	8	no	—
379	C₂×C₄×D₄	—	53	24	no	—
380	C₂×#117	—	53	6	no	—
381	C₅₃	#62811	1	1	yes	characteristic, complemented, Sylow
382	C₂⁵	—	53	3	no	—
383	2₋¹⁺⁴	—	2	32	no	—
384	2₋¹⁺⁴	—	2	16	no	—
385	H(5,𝔽₂)	—	2	32	no	—
386	H(5,𝔽₂)	—	2	48	no	—
387	C₂×#41	—	2	64	no	—
388	C₂×#41	—	2	48	no	—
389	C₂×#41	—	53	8	no	—
390	C₂×#41	—	53	24	no	—
391	C₂×#41	—	53	12	no	—
392	C₂×#41	—	53	12	no	—
393	C₂×#41	—	106	64	no	—
394	C₂×#41	—	106	64	no	—
395	C₂×#41	—	106	8	no	—
396	C₂×#41	—	106	24	no	—
397	C₂×#41	—	106	24	no	—
398	C₂×#41	C₂²⋊D₅₃	1	16	yes	complemented
399	C₂×#41	C₂²×D₅₃	1	4	yes	—
400	C₂×#41	C₂²×D₅₃	1	12	yes	—
401	C₂×#41	C₂²×D₅₃	1	4	yes	—
402	C₂×#41	C₂²×D₅₃	1	4	yes	—
403	C₂×#41	C₂²×D₅₃	1	12	yes	complemented
404	C₂×#41	C₂²×D₅₃	1	12	yes	complemented
405	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
406	C₂×#41	C₂²×D₅₃	1	8	yes	complemented
407	C₂²×Q₈	—	53	8	no	—
408	C₂²×Q₈	—	53	3	no	—
409	C₂²×Q₈	—	106	8	no	—
410	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
411	C₂²×Q₈	C₂²×D₅₃	1	6	yes	complemented
412	C₂²×Q₈	C₂²×D₅₃	1	8	yes	complemented
413	C₂²×D₄	—	53	6	no	—
414	C₂²×D₄	—	53	24	no	—
415	C₂²×D₄	—	53	24	no	—
416	C₂²×D₄	—	53	3	no	—
417	C₂²×D₄	—	106	24	no	—
418	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
419	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
420	C₂²×D₄	C₂²×D₅₃	1	6	yes	complemented
421	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
422	C₂³×C₄	—	53	3	no	—
423	C₂³×C₄	—	53	12	no	—
424	C₂³×C₄	—	53	12	no	—
425	C₂³×C₄	—	53	1	no	—
426	C₂³×C₄	—	53	3	no	—
427	C₂³×C₄	—	53	3	no	—
428	C₂³×C₄	—	53	3	no	—
429	C₂³×C₄	—	106	24	no	—
430	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
431	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
432	C₄⋊Q₈	—	53	24	no	—
433	#127	—	53	24	no	—
434	#124	—	53	24	no	—
435	#124	—	53	24	no	—
436	#123	—	53	8	no	—
437	#123	—	53	24	no	—
438	#123	—	53	48	no	—
439	#123	—	53	48	no	—
440	C₂²⋊Q₈	—	53	8	no	—
441	C₂²⋊Q₈	—	53	24	no	—
442	C₂²⋊Q₈	—	53	24	no	—
443	C₂²⋊Q₈	—	53	24	no	—
444	C₂²⋊Q₈	—	53	48	no	—
445	#121	—	53	8	no	—
446	#121	—	53	24	no	—
447	#121	—	53	24	no	—
448	#121	—	53	24	no	—
449	#121	—	53	96	no	—
450	#121	—	53	48	no	—
451	#120	—	53	48	no	—
452	#120	—	53	8	no	—
453	#120	—	53	24	no	—
454	#120	—	53	48	no	—
455	C₄×Q₈	—	53	16	no	—
456	C₄×D₄	—	53	48	no	—
457	C₄×D₄	—	53	16	no	—
458	C₄×D₄	—	53	48	no	—
459	C₄×D₄	—	53	48	no	—
460	C₄×D₄	—	53	48	no	—
461	#117	—	53	4	no	—
462	#117	—	53	12	no	—
463	#117	—	53	12	no	—
464	#117	—	53	12	no	—
465	C₂×C₄⋊C₄	—	53	12	no	—
466	C₂×C₄⋊C₄	—	53	24	no	—
467	C₂×C₂²⋊C₄	—	53	12	no	—
468	C₂×C₂²⋊C₄	—	53	24	no	—
469	C₂×C₂²⋊C₄	—	53	24	no	—
470	C₂×C₄²	—	53	12	no	—
471	C₂⁴	—	53	1	no	—
472	C₂⁴	—	53	12	no	—
473	C₂⁴	—	53	3	no	—
474	C₂⁴	—	106	12	no	—
475	C₂⁴	—	212	12	no	—
476	C₂⁴	C₂³×D₅₃	1	12	yes	—
477	C₂⁴	C₂³×D₅₃	1	3	yes	—
478	#41	—	2	32	no	—
479	#41	—	2	96	no	—
480	#41	—	2	16	no	—
481	#41	—	2	48	no	—
482	#41	—	2	48	no	—
483	#41	—	2	48	no	—
484	#41	—	106	32	no	—
485	#41	—	106	96	no	—
486	#41	—	106	48	no	—
487	#41	—	106	48	no	—
488	#41	C₂×C₂²⋊D₅₃	1	32	yes	complemented
489	#41	C₂×C₂²⋊D₅₃	1	32	yes	complemented
490	C₂×Q₈	—	2	32	no	—
491	C₂×Q₈	—	2	24	no	—
492	C₂×Q₈	—	53	12	no	—
493	C₂×Q₈	—	106	32	no	—
494	C₂×Q₈	—	106	12	no	—
495	C₂×Q₈	C₂×C₂²⋊D₅₃	1	8	yes	complemented
496	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
497	C₂×Q₈	C₂³×D₅₃	1	12	yes	—
498	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
499	C₂×D₄	—	2	48	no	—
500	C₂×D₄	—	2	96	no	—
501	C₂×D₄	—	2	24	no	—
502	C₂×D₄	—	53	16	no	—
503	C₂×D₄	—	53	24	no	—
504	C₂×D₄	—	53	48	no	—
505	C₂×D₄	—	53	12	no	—
506	C₂×D₄	—	106	24	no	—
507	C₂×D₄	—	106	96	no	—
508	C₂×D₄	—	106	12	no	—
509	C₂×D₄	—	212	48	no	—
510	C₂×D₄	C₂×C₂²⋊D₅₃	1	24	yes	complemented
511	C₂×D₄	C₂³×D₅₃	1	8	yes	—
512	C₂×D₄	C₂³×D₅₃	1	24	yes	—
513	C₂×D₄	C₂³×D₅₃	1	4	yes	—
514	C₂×D₄	C₂³×D₅₃	1	12	yes	—
515	C₂×D₄	C₂³×D₅₃	1	12	yes	—
516	C₂²×C₄	—	2	48	no	—
517	C₂²×C₄	—	53	12	no	—
518	C₂²×C₄	—	53	1	no	—
519	C₂²×C₄	—	53	3	no	—
520	C₂²×C₄	—	53	24	no	—
521	C₂²×C₄	—	53	12	no	—
522	C₂²×C₄	—	53	12	no	—
523	C₂²×C₄	—	53	2	no	—
524	C₂²×C₄	—	53	6	no	—
525	C₂²×C₄	—	53	2	no	—
526	C₂²×C₄	—	53	6	no	—
527	C₂²×C₄	—	106	12	no	—
528	C₂²×C₄	—	106	4	no	—
529	C₂²×C₄	—	106	12	no	—
530	C₂²×C₄	—	106	12	no	—
531	C₂²×C₄	—	106	12	no	—
532	C₂²×C₄	—	106	48	no	—
533	C₂²×C₄	—	106	48	no	—
534	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	complemented
535	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
536	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
537	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
538	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
539	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
540	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
541	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
542	C₄⋊C₄	—	53	8	no	—
543	C₄⋊C₄	—	53	16	no	—
544	C₄⋊C₄	—	53	24	no	—
545	C₄⋊C₄	—	53	48	no	—
546	C₂²⋊C₄	—	53	8	no	—
547	C₂²⋊C₄	—	53	16	no	—
548	C₂²⋊C₄	—	53	24	no	—
549	C₂²⋊C₄	—	53	48	no	—
550	C₂²⋊C₄	—	53	16	no	—
551	C₂²⋊C₄	—	53	48	no	—
552	C₄²	—	53	24	no	—
553	C₄²	—	53	8	no	—
554	C₂³	—	2	24	no	—
555	C₂³	—	4	6	no	—
556	C₂³	—	4	24	no	—
557	C₂³	—	53	2	no	—
558	C₂³	—	53	6	no	—
559	C₂³	—	106	4	no	—
560	C₂³	—	106	24	no	—
561	C₂³	—	106	2	no	—
562	C₂³	—	106	24	no	—
563	C₂³	—	106	6	no	—
564	C₂³	—	212	12	no	—
565	C₂³	—	212	6	no	—
566	C₂³	—	212	48	no	—
567	C₂³	C₂²×C₂²⋊D₅₃	1	12	yes	—
568	C₂³	C₂⁴×D₅₃	1	8	yes	—
569	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
570	C₂³	C₂⁴×D₅₃	1	6	yes	—
571	Q₈	—	2	24	no	—
572	Q₈	—	2	8	no	—
573	Q₈	—	106	24	no	—
574	Q₈	C₂²×C₂²⋊D₅₃	1	16	yes	—
575	D₄	—	2	48	no	—
576	D₄	—	2	16	no	—
577	D₄	—	2	24	no	—
578	D₄	—	2	8	no	—
579	D₄	—	106	32	no	—
580	D₄	—	106	48	no	—
581	D₄	—	106	24	no	—
582	D₄	—	212	48	no	—
583	D₄	C₂²×C₂²⋊D₅₃	1	48	yes	—
584	C₂×C₄	—	2	24	no	—
585	C₂×C₄	—	2	16	no	—
586	C₂×C₄	—	2	48	no	—
587	C₂×C₄	—	53	16	no	—
588	C₂×C₄	—	53	2	no	—
589	C₂×C₄	—	53	6	no	—
590	C₂×C₄	—	106	2	no	—
591	C₂×C₄	—	106	6	no	—
592	C₂×C₄	—	106	24	no	—
593	C₂×C₄	—	106	4	no	—
594	C₂×C₄	—	106	4	no	—
595	C₂×C₄	—	106	24	no	—
596	C₂×C₄	—	106	24	no	—
597	C₂×C₄	—	106	12	no	—
598	C₂×C₄	—	106	12	no	—
599	C₂×C₄	—	212	24	no	—
600	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
601	C₂×C₄	C₂²×C₂²⋊D₅₃	1	4	yes	—
602	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
603	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
604	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
605	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
606	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
607	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
608	C₂²	—	2	12	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	8	no	—
611	C₂²	—	2	16	no	—
612	C₂²	—	4	12	no	—
613	C₂²	—	8	24	no	—
614	C₂²	—	53	4	no	—
615	C₂²	—	106	8	no	—
616	C₂²	—	106	2	no	—
617	C₂²	—	106	2	no	—
618	C₂²	—	106	6	no	—
619	C₂²	—	106	12	no	—
620	C₂²	—	212	48	no	—
621	C₂²	—	212	12	no	—
622	C₂²	C₂³×C₂²⋊D₅₃	1	12	yes	—
623	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
624	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
625	C₄	—	2	8	no	—
626	C₄	—	106	2	no	—
627	C₄	—	106	6	no	—
628	C₄	—	106	16	no	—
629	C₄	C₂³×C₂²⋊D₅₃	1	2	yes	—
630	C₄	C₂³×C₂²⋊D₅₃	1	12	yes	—
631	C₄	C₂³×C₂²⋊D₅₃	1	2	yes	—
632	C₂	—	2	2	no	—
633	C₂	—	2	12	no	—
634	C₂	—	4	8	no	—
635	C₂	—	106	4	no	—
636	C₂	—	212	6	no	—
637	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
638	C₂	#585×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#585	C₂
hypercenter	C₂×#585	D₅₃
hypocenter	C₅₃	#62811
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×D₅₃	C₂	semidirect
2. #20288388	C₂	semidirect
3. #20288377	C₂	semidirect
4. #20288370	C₂	semidirect
5. #20288363	C₂	semidirect
6. #20288355	C₂	semidirect
7. #20288346	C₂	semidirect
8. #41×C₂²⋊D₅₃	C₂	semidirect
9. #20288044	C₂	semidirect
10. C₂×#16811130	C₂	semidirect
11. #20287918	C₂	semidirect
12. C₂×C₅₃×#585	C₂	semidirect
13. C₂²×#11719949	C₂²	semidirect
14. C₂²×#11719945	C₂²	semidirect
15. C₂²×#11719947	C₂²	semidirect
17. C₂×#41×D₅₃	C₂²	semidirect
18. C₂×#41×D₅₃	C₂²	semidirect
19. C₂²×Q₈×D₅₃	C₂²	semidirect
20. C₂²×D₄×D₅₃	C₂²	semidirect
21. C₂³×C₄×D₅₃	C₂²	semidirect
22. #16811292	C₂²	semidirect
23. Q₈×C₂²⋊D₅₃	C₂²	semidirect
24. #16811284	C₂²	semidirect
25. #16811280	C₂²	semidirect
26. #16811272	C₂²	semidirect
27. #16811267	C₂²	semidirect
28. #16811261	C₂²	semidirect
29. D₄×C₂²⋊D₅₃	C₂²	semidirect
30. #16811201	C₂²	semidirect
31. #16811197	C₂²	semidirect
32. #16811194	C₂²	semidirect
33. #16811190	C₂²	semidirect
34. #16811187	C₂²	semidirect
35. #16811182	C₂²	semidirect
36. #16811169	C₂²	semidirect
37. #16811134	C₂²	semidirect
38. #16811130	C₂²	semidirect
39. #16811130	C₂²	semidirect
40. #16811130	C₂²	semidirect
41. #16811130	C₂²	semidirect
42. #16811130	C₂²	semidirect
43. #16811130	C₂²	semidirect
44. C₂×#13133198	C₂²	semidirect
45. C₂×#13133193	C₂²	semidirect
46. C₂×#13133188	C₂²	semidirect
47. C₂×#13133185	C₂²	semidirect
48. #16811106	C₂²	semidirect
49. #16811101	C₂²	semidirect
50. #16811098	C₂²	semidirect
51. #41×Dic₅₃	C₂²	semidirect
52. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
53. C₂×#13133176	C₂²	semidirect
55. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
56. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
57. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
58. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
60. C₂²×#41×C₅₃	C₂²	semidirect
61. C₂²×#41×C₅₃	C₂²	semidirect
88. #13133215	C₂³	semidirect
89. #13133213	C₂³	semidirect
90. #13133206	C₂³	semidirect
91. #13133204	C₂³	semidirect
92. #13133200	C₂³	semidirect
95. #13133198	C₂³	semidirect
96. #13133196	C₂³	semidirect
99. #13133193	C₂³	semidirect
100. #13133193	C₂³	semidirect
101. #13133191	C₂³	semidirect
102. #13133189	C₂³	semidirect
105. #13133188	C₂³	semidirect
106. #13133188	C₂³	semidirect
107. #13133186	C₂³	semidirect
111. Q₈×Dic₅₃	C₂³	semidirect
112. D₄×Dic₅₃	C₂³	semidirect
115. C₄×C₂²⋊D₅₃	C₂³	semidirect
116. C₄×C₂²⋊D₅₃	C₂³	semidirect
119. #13133176	C₂³	semidirect
120. #13133176	C₂³	semidirect
121. C₂×C₄⋊Dic₅₃	C₂³	semidirect
122. C₂×#11719919	C₂³	semidirect
124. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
126. C₂×C₄×Dic₅₃	C₂³	semidirect
133. C₂×#41×C₅₃	D₄	semidirect
138. C₂×#41×C₅₃	C₂³	semidirect
139. C₂×#41×C₅₃	C₂³	semidirect
140. C₂×#41×C₅₃	C₂³	semidirect
141. C₂×#41×C₅₃	C₂³	semidirect
143. C₂²×Q₈×C₅₃	C₂³	semidirect
144. C₂²×Q₈×C₅₃	C₂³	semidirect
147. C₂²×D₄×C₅₃	C₂³	semidirect
148. C₂²×D₄×C₅₃	C₂³	semidirect
150. C₂³×C₄×C₅₃	C₂³	semidirect
164. #41×C₅₃	C₂×D₄	semidirect
165. #41×C₅₃	C₂×D₄	semidirect
168. C₂×Q₈×C₅₃	C₂×D₄	semidirect
175. C₂×D₄×C₅₃	C₂×D₄	semidirect
182. C₂²×C₄×C₅₃	C₂×D₄	semidirect
305. C₂×D₅₃	#585	semidirect
313. C₂×#585	D₅₃	semidirect
333. C₂×2₋¹⁺⁴	C₂×D₅₃	semidirect
334. C₂×2₋¹⁺⁴	C₂×D₅₃	semidirect
335. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
336. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
344. C₂²×#41	C₂×D₅₃	semidirect
345. C₂²×#41	C₂×D₅₃	semidirect
381. C₅₃	#62811	semidirect
398. C₂×#41	C₂²⋊D₅₃	semidirect
403. C₂×#41	C₂²×D₅₃	semidirect
404. C₂×#41	C₂²×D₅₃	semidirect
405. C₂×#41	C₂²×D₅₃	semidirect
406. C₂×#41	C₂²×D₅₃	semidirect
411. C₂²×Q₈	C₂²×D₅₃	semidirect
412. C₂²×Q₈	C₂²×D₅₃	semidirect
420. C₂²×D₄	C₂²×D₅₃	semidirect
421. C₂²×D₄	C₂²×D₅₃	semidirect
431. C₂³×C₄	C₂²×D₅₃	semidirect
488. #41	C₂×C₂²⋊D₅₃	semidirect
489. #41	C₂×C₂²⋊D₅₃	semidirect
495. C₂×Q₈	C₂×C₂²⋊D₅₃	semidirect
510. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
534. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×D₅₃	2
#20288388	2
#20288377	2
#20288370	2
#20288363	2
#20288355	2
#20288346	2
#41×C₂²⋊D₅₃	2
#20288044	2
C₂×#16811130	2
#20287918	2
C₂×C₅₃×#585	2
#62811	53