Subgroups — #28293202

Order 13568.

11618 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133263	C₂	1	1	yes	maximal, characteristic, complemented
2	#20288509	C₂	1	8	yes	maximal, complemented
3	#20288489	C₂	1	8	yes	maximal, complemented
4	#20288384	C₂	1	16	yes	maximal, complemented
5	#20288370	C₂	1	4	yes	maximal, complemented
6	#20288346	C₂	1	12	yes	maximal, complemented
7	#20288044	C₂	1	6	yes	maximal, complemented
8	C₂×#16811187	C₂	1	6	yes	maximal, complemented
9	#20287918	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13133263	—	2	16	no	—
12	C₂×#13133263	C₂²	1	8	yes	complemented
13	C₂×#13133263	C₂²	1	8	yes	complemented
14	C₂²×#11719949	C₂²	1	2	yes	complemented
15	C₂²×#11719945	C₂²	1	3	yes	complemented
16	C₂³×D₂₁₂	C₂²	1	3	yes	complemented
17	C₂²×D₄×D₅₃	C₂²	1	6	yes	complemented
18	#16811292	C₂²	1	32	yes	complemented
19	#16811284	C₂²	1	32	yes	complemented
20	#16811261	C₂²	1	96	yes	complemented
21	D₄×C₂²⋊D₅₃	C₂²	1	96	yes	complemented
22	#16811201	C₂²	1	24	yes	complemented
23	#16811197	C₂²	1	12	yes	complemented
24	#16811194	C₂²	1	4	yes	complemented
25	#16811190	C₂²	1	12	yes	complemented
26	#16811187	C₂²	1	12	yes	complemented
27	#16811187	C₂²	1	24	yes	complemented
28	#16811187	C₂²	1	24	yes	complemented
29	#16811187	C₂²	1	24	yes	complemented
30	#16811182	C₂²	1	24	yes	complemented
31	#16811169	C₂²	1	24	yes	complemented
32	C₂×#13133213	C₂²	1	12	yes	complemented
33	C₂×#13133206	C₂²	1	12	yes	complemented
34	#16811134	C₂²	1	24	yes	complemented
35	#16811130	C₂²	1	24	yes	complemented
36	C₂×#13133191	C₂²	1	24	yes	complemented
37	C₂×#13133185	C₂²	1	24	yes	complemented
38	#16811106	C₂²	1	8	yes	complemented
39	#16811101	C₂²	1	2	yes	complemented
40	#16811098	C₂²	1	6	yes	complemented
41	#41×Dic₅₃	C₂²	1	8	yes	complemented
42	C₂×#13133176	C₂²	1	6	yes	complemented
43	C₅₃×#585	—	2	16	no	—
44	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	2	yes	complemented
45	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	4	yes	complemented
46	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	6	yes	complemented
47	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	4	yes	complemented
48	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
49	C₂²×#41×C₅₃	C₂²	1	6	yes	complemented
50	C₂²×#41×C₅₃	C₂²	1	8	yes	complemented
51	C₂⁴×D₅₃	C₂³	1	6	yes	—
52	#13133263	—	2	64	no	—
53	#13133263	—	2	64	no	—
54	C₂×#11719949	—	2	16	no	—
55	C₂×#11719949	C₂³	1	16	yes	—
56	C₂×#11719945	—	2	24	no	—
57	C₂×#11719945	C₂³	1	12	yes	—
58	C₂×#11719945	C₂³	1	12	yes	—
59	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
60	C₂²×D₂₁₂	—	2	24	no	—
61	C₂²×D₂₁₂	C₂³	1	12	yes	—
62	C₂²×D₂₁₂	C₂³	1	12	yes	—
63	C₂²×D₂₁₂	C₂³	1	6	yes	—
64	C₂²×D₂₁₂	C₂³	1	3	yes	—
65	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
66	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
67	C₂×D₄×D₅₃	—	2	48	no	—
68	C₂×D₄×D₅₃	C₂³	1	48	yes	—
69	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
70	#13133215	C₂³	1	8	yes	complemented
71	#13133213	C₂³	1	48	yes	complemented
72	#13133213	C₂³	1	24	yes	complemented
73	#13133206	C₂³	1	48	yes	complemented
74	#13133206	C₂³	1	24	yes	complemented
75	#13133204	C₂³	1	8	yes	complemented
76	#13133200	C₂³	1	48	yes	complemented
77	#13133198	C₂³	1	16	yes	—
78	#13133196	C₂³	1	16	yes	complemented
79	#13133193	C₂³	1	48	yes	complemented
80	#13133191	C₂³	1	96	yes	complemented
81	#13133191	C₂³	1	48	yes	complemented
82	#13133189	C₂³	1	96	yes	complemented
83	#13133188	C₂³	1	48	yes	complemented
84	#13133186	C₂³	1	48	yes	complemented
85	#13133185	C₂³	1	48	yes	—
86	#13133185	C₂³	1	96	yes	complemented
87	Q₈×Dic₅₃	C₂³	1	16	yes	complemented
88	D₄×Dic₅₃	C₂³	1	48	yes	complemented
89	C₄×C₂²⋊D₅₃	C₂³	1	96	yes	complemented
90	#13133176	C₂³	1	4	yes	—
91	#13133176	C₂³	1	12	yes	—
92	#13133176	C₂³	1	12	yes	complemented
93	#13133176	C₂³	1	12	yes	complemented
94	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
95	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
96	C₂×#11719921	C₂³	1	48	yes	complemented
97	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
98	2₋¹⁺⁴×C₅₃	—	2	32	no	—
99	2₋¹⁺⁴×C₅₃	—	2	16	no	—
100	H(5,𝔽₂)×C₅₃	—	2	32	no	—
101	H(5,𝔽₂)×C₅₃	—	2	48	no	—
102	C₂×#41×C₅₃	—	2	64	no	—
103	C₂×#41×C₅₃	—	2	48	no	—
104	C₂×#41×C₅₃	D₄	1	16	yes	complemented
105	C₂×#41×C₅₃	C₂³	1	4	yes	—
106	C₂×#41×C₅₃	C₂³	1	12	yes	—
107	C₂×#41×C₅₃	C₂³	1	4	yes	—
108	C₂×#41×C₅₃	C₂³	1	4	yes	—
109	C₂×#41×C₅₃	C₂³	1	8	yes	complemented
110	C₂×#41×C₅₃	C₂³	1	12	yes	complemented
111	C₂×#41×C₅₃	C₂³	1	12	yes	complemented
112	C₂×#41×C₅₃	C₂³	1	24	yes	complemented
113	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₂²×Q₈×C₅₃	C₂³	1	8	yes	complemented
115	C₂²×Q₈×C₅₃	C₂³	1	6	yes	complemented
116	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
117	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
118	C₂²×D₄×C₅₃	C₂³	1	6	yes	complemented
119	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
120	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
121	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
122	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
123	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
124	C₂³×D₅₃	—	2	24	no	—
125	C₂³×D₅₃	—	8	12	no	—
126	C₂³×D₅₃	C₂⁴	1	24	yes	—
127	C₂³×D₅₃	C₂⁴	1	6	yes	—
128	#41×C₅₃	—	2	32	no	—
129	#41×C₅₃	—	2	96	no	—
130	#41×C₅₃	—	2	16	no	—
131	#41×C₅₃	—	2	48	no	—
132	#41×C₅₃	—	2	48	no	—
133	#41×C₅₃	—	2	48	no	—
134	#41×C₅₃	C₂×D₄	1	32	yes	complemented
135	#41×C₅₃	C₂×D₄	1	32	yes	complemented
136	C₂×Q₈×C₅₃	—	2	32	no	—
137	C₂×Q₈×C₅₃	—	2	24	no	—
138	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
139	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
140	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
141	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
142	C₂×D₄×C₅₃	—	2	48	no	—
143	C₂×D₄×C₅₃	—	2	96	no	—
144	C₂×D₄×C₅₃	—	2	24	no	—
145	C₂×D₄×C₅₃	C₂×D₄	1	24	yes	complemented
146	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
147	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
148	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
149	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
150	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
151	C₂²×C₄×C₅₃	—	2	48	no	—
152	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	complemented
153	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
160	C₂×C₂²⋊D₅₃	—	2	24	no	—
161	C₂×C₂²⋊D₅₃	—	8	48	no	—
162	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
163	C₂×C₂²⋊D₅₃	C₂⁴	1	96	yes	—
164	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
165	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
166	#11719949	—	2	64	no	—
167	D₄×D₅₃	—	2	192	no	—
168	#11719945	—	2	48	no	—
169	#11719945	—	2	48	no	—
170	C₂×D₂₁₂	—	2	24	no	—
171	C₂×D₂₁₂	—	2	12	no	—
172	C₂×D₂₁₂	—	2	48	no	—
173	C₂×D₂₁₂	—	2	48	no	—
174	C₂×D₂₁₂	C₂⁴	1	6	yes	—
175	C₂×D₂₁₂	C₂⁴	1	24	yes	—
176	C₂×D₂₁₂	C₂⁴	1	6	yes	—
177	C₂×C₄×D₅₃	—	2	24	no	—
178	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
179	C₂×Dic₁₀₆	—	2	4	no	—
180	C₂×Dic₁₀₆	C₂⁴	1	2	yes	—
181	C₂×Dic₁₀₆	C₂⁴	1	2	yes	—
182	C₂²⋊Dic₅₃	C₂⁴	1	8	yes	—
183	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
184	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
185	#11719921	C₂⁴	1	96	yes	—
186	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
187	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
188	#11719919	C₂⁴	1	32	yes	—
189	C₄×Dic₅₃	C₂⁴	1	8	yes	—
190	C₄×Dic₅₃	C₂⁴	1	24	yes	—
191	C₂³×C₅₃	—	2	24	no	—
192	C₂³×C₅₃	—	4	6	no	—
193	C₂³×C₅₃	—	4	24	no	—
194	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
195	C₂³×C₅₃	C₂⁵	1	8	yes	—
196	C₂³×C₅₃	C₂⁵	1	6	yes	—
197	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
198	C₂²×D₅₃	—	2	12	no	—
199	C₂²×D₅₃	—	2	48	no	—
200	C₂²×D₅₃	—	4	12	no	—
201	C₂²×D₅₃	—	4	48	no	—
202	C₂²×D₅₃	—	8	48	no	—
203	C₂²×D₅₃	C₂⁵	1	12	yes	—
204	Q₈×C₅₃	—	2	8	no	—
205	Q₈×C₅₃	—	2	24	no	—
206	Q₈×C₅₃	C₂²×D₄	1	16	yes	—
207	D₄×C₅₃	—	2	16	no	—
208	D₄×C₅₃	—	2	48	no	—
209	D₄×C₅₃	—	2	8	no	—
210	D₄×C₅₃	—	2	24	no	—
211	D₄×C₅₃	C₂²×D₄	1	48	yes	—
212	C₂×C₄×C₅₃	—	2	24	no	—
213	C₂×C₄×C₅₃	—	2	16	no	—
214	C₂×C₄×C₅₃	—	2	48	no	—
215	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
216	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
217	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
218	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
219	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
220	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
221	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
222	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
223	C₂²⋊D₅₃	—	2	48	no	—
224	C₂²⋊D₅₃	—	4	96	no	—
225	C₂×Dic₅₃	—	2	4	no	—
226	C₂×Dic₅₃	—	4	24	no	—
227	C₂×Dic₅₃	C₂⁵	1	4	yes	—
228	C₂×Dic₅₃	C₂⁵	1	16	yes	—
229	D₂₁₂	—	2	12	no	—
230	D₂₁₂	—	2	12	no	—
231	D₂₁₂	—	2	48	no	—
232	C₄×D₅₃	—	2	48	no	—
233	Dic₁₀₆	—	2	4	no	—
234	Dic₁₀₆	—	2	4	no	—
235	#62852	—	53	1	no	maximal, Sylow
236	C₂²×C₅₃	—	2	12	no	—
237	C₂²×C₅₃	—	2	2	no	—
238	C₂²×C₅₃	—	2	8	no	—
239	C₂²×C₅₃	—	2	16	no	—
240	C₂²×C₅₃	—	4	12	no	—
241	C₂²×C₅₃	—	8	24	no	—
242	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
243	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
244	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
245	C₂×D₅₃	—	2	12	no	—
246	C₂×D₅₃	—	2	24	no	—
247	C₂×D₅₃	—	4	24	no	—
248	C₂×D₅₃	—	8	48	no	—
249	C₄×C₅₃	—	2	8	no	—
250	C₄×C₅₃	C₂³×D₄	1	2	yes	—
251	C₄×C₅₃	C₂³×D₄	1	2	yes	—
252	C₄×C₅₃	C₂³×D₄	1	12	yes	—
253	Dic₅₃	—	2	4	no	—
254	Dic₅₃	—	2	16	no	—
255	C₂×#585	D₅₃	1	1	yes	characteristic, complemented
256	C₂²×H(5,𝔽₂)	—	53	1	no	—
257	#3500	—	53	8	no	—
258	#3498	—	53	8	no	—
259	#3488	—	53	16	no	—
260	#3487	—	53	4	no	—
261	#3484	—	53	12	no	—
262	#3449	—	53	6	no	—
263	C₂×#535	—	53	6	no	—
264	#3428	—	53	1	no	—
265	C₂×C₅₃	—	2	2	no	—
266	C₂×C₅₃	—	2	12	no	—
267	C₂×C₅₃	—	4	8	no	—
268	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
269	C₂×C₅₃	H(7,𝔽₂)	1	2	yes	—
270	D₅₃	—	4	12	no	—
271	#585	—	2	16	no	—
272	C₂×2₋¹⁺⁴	C₂×D₅₃	1	2	yes	complemented
273	C₂×2₋¹⁺⁴	C₂×D₅₃	1	4	yes	complemented
274	C₂×H(5,𝔽₂)	—	53	8	no	—
275	C₂×H(5,𝔽₂)	—	53	8	no	—
276	C₂×H(5,𝔽₂)	—	106	16	no	—
277	C₂×H(5,𝔽₂)	C₂×D₅₃	1	6	yes	complemented
278	C₂×H(5,𝔽₂)	C₂×D₅₃	1	4	yes	complemented
279	C₂²×#41	—	53	2	no	—
280	C₂²×#41	—	53	3	no	—
281	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
282	C₂²×#41	C₂×D₅₃	1	6	yes	complemented
283	C₂²×#41	C₂×D₅₃	1	8	yes	complemented
284	C₂³×D₄	—	53	6	no	—
285	C₂³×D₄	—	53	3	no	—
286	#550	—	53	32	no	—
287	#548	—	53	32	no	—
288	#546	—	53	96	no	—
289	D₄²	—	53	96	no	—
290	#537	—	53	24	no	—
291	#536	—	53	4	no	—
292	#536	—	53	12	no	—
293	#535	—	53	12	no	—
294	#535	—	53	24	no	—
295	#535	—	53	24	no	—
296	#535	—	53	24	no	—
297	#535	—	53	12	no	—
298	#534	—	53	24	no	—
299	#532	—	53	24	no	—
300	C₂×#127	—	53	12	no	—
301	C₂×#124	—	53	12	no	—
302	#525	—	53	24	no	—
303	#525	—	53	24	no	—
304	C₂×#121	—	53	24	no	—
305	C₂×#120	—	53	24	no	—
306	#520	—	53	8	no	—
307	#519	—	53	2	no	—
308	#518	—	53	6	no	—
309	C₄×#41	—	53	8	no	—
310	C₂×#117	—	53	6	no	—
311	C₅₃	#62852	1	1	yes	characteristic, complemented, Sylow
312	C₂⁵	—	53	6	no	—
313	2₋¹⁺⁴	—	2	32	no	—
314	2₋¹⁺⁴	—	2	16	no	—
315	H(5,𝔽₂)	—	2	32	no	—
316	H(5,𝔽₂)	—	2	48	no	—
317	H(5,𝔽₂)	—	106	64	no	—
318	H(5,𝔽₂)	—	106	64	no	—
319	C₂×#41	—	2	64	no	—
320	C₂×#41	—	2	48	no	—
321	C₂×#41	—	53	16	no	—
322	C₂×#41	—	53	12	no	—
323	C₂×#41	—	53	12	no	—
324	C₂×#41	—	106	16	no	—
325	C₂×#41	—	106	24	no	—
326	C₂×#41	C₂²⋊D₅₃	1	16	yes	complemented
327	C₂×#41	C₂²×D₅₃	1	12	yes	—
328	C₂×#41	C₂²×D₅₃	1	4	yes	—
329	C₂×#41	C₂²×D₅₃	1	4	yes	—
330	C₂×#41	C₂²×D₅₃	1	4	yes	—
331	C₂×#41	C₂²×D₅₃	1	12	yes	complemented
332	C₂×#41	C₂²×D₅₃	1	12	yes	complemented
333	C₂×#41	C₂²×D₅₃	1	24	yes	complemented
334	C₂×#41	C₂²×D₅₃	1	8	yes	complemented
335	C₂²×Q₈	—	53	1	no	—
336	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
337	C₂²×Q₈	C₂²×D₅₃	1	6	yes	complemented
338	C₂²×Q₈	C₂²×D₅₃	1	8	yes	complemented
339	C₂²×D₄	—	53	6	no	—
340	C₂²×D₄	—	53	48	no	—
341	C₂²×D₄	—	53	48	no	—
342	C₂²×D₄	—	53	12	no	—
343	C₂²×D₄	—	53	12	no	—
344	C₂²×D₄	—	53	6	no	—
345	C₂²×D₄	—	53	3	no	—
346	C₂²×D₄	—	106	48	no	—
347	C₂²×D₄	—	106	24	no	—
348	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
349	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
350	C₂²×D₄	C₂²×D₅₃	1	6	yes	complemented
351	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
352	C₂³×C₄	—	53	6	no	—
353	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
354	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
355	C₄⋊Q₈	—	53	8	no	—
356	#127	—	53	24	no	—
357	#127	—	53	48	no	—
358	#124	—	53	24	no	—
359	#124	—	53	48	no	—
360	#124	—	53	8	no	—
361	#123	—	53	16	no	—
362	#123	—	53	48	no	—
363	C₂²⋊Q₈	—	53	48	no	—
364	C₂²⋊Q₈	—	53	16	no	—
365	#121	—	53	48	no	—
366	#121	—	53	96	no	—
367	#121	—	53	48	no	—
368	#121	—	53	96	no	—
369	#120	—	53	48	no	—
370	#120	—	53	48	no	—
371	#120	—	53	96	no	—
372	C₄×Q₈	—	53	16	no	—
373	C₄×D₄	—	53	48	no	—
374	C₄×D₄	—	53	96	no	—
375	#117	—	53	4	no	—
376	#117	—	53	12	no	—
377	#117	—	53	12	no	—
378	#117	—	53	12	no	—
379	C₂×C₄⋊C₄	—	53	12	no	—
380	C₂×C₂²⋊C₄	—	53	12	no	—
381	C₂×C₂²⋊C₄	—	53	48	no	—
382	C₂×C₄²	—	53	12	no	—
383	C₂⁴	—	53	24	no	—
384	C₂⁴	—	53	6	no	—
385	C₂⁴	—	106	24	no	—
386	C₂⁴	—	424	12	no	—
387	C₂⁴	C₂³×D₅₃	1	12	yes	—
388	C₂⁴	C₂³×D₅₃	1	3	yes	—
389	#41	—	2	32	no	—
390	#41	—	2	96	no	—
391	#41	—	2	16	no	—
392	#41	—	2	48	no	—
393	#41	—	2	48	no	—
394	#41	—	2	48	no	—
395	#41	—	106	64	no	—
396	#41	—	106	48	no	—
397	#41	—	106	48	no	—
398	#41	C₂×C₂²⋊D₅₃	1	32	yes	complemented
399	#41	C₂×C₂²⋊D₅₃	1	32	yes	complemented
400	C₂×Q₈	—	2	32	no	—
401	C₂×Q₈	—	2	24	no	—
402	C₂×Q₈	—	53	2	no	—
403	C₂×Q₈	—	53	2	no	—
404	C₂×Q₈	—	106	4	no	—
405	C₂×Q₈	C₂×C₂²⋊D₅₃	1	8	yes	complemented
406	C₂×Q₈	C₂³×D₅₃	1	12	yes	—
407	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
408	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
409	C₂×D₄	—	2	48	no	—
410	C₂×D₄	—	2	96	no	—
411	C₂×D₄	—	2	24	no	—
412	C₂×D₄	—	53	24	no	—
413	C₂×D₄	—	53	96	no	—
414	C₂×D₄	—	53	6	no	—
415	C₂×D₄	—	53	24	no	—
416	C₂×D₄	—	53	6	no	—
417	C₂×D₄	—	106	24	no	—
418	C₂×D₄	—	106	192	no	—
419	C₂×D₄	—	106	24	no	—
420	C₂×D₄	—	106	12	no	—
421	C₂×D₄	—	106	48	no	—
422	C₂×D₄	—	106	48	no	—
423	C₂×D₄	—	424	48	no	—
424	C₂×D₄	C₂×C₂²⋊D₅₃	1	24	yes	complemented
425	C₂×D₄	C₂³×D₅₃	1	8	yes	—
426	C₂×D₄	C₂³×D₅₃	1	24	yes	—
427	C₂×D₄	C₂³×D₅₃	1	12	yes	—
428	C₂×D₄	C₂³×D₅₃	1	4	yes	—
429	C₂×D₄	C₂³×D₅₃	1	12	yes	—
430	C₂²×C₄	—	2	48	no	—
431	C₂²×C₄	—	53	2	no	—
432	C₂²×C₄	—	53	24	no	—
433	C₂²×C₄	—	53	24	no	—
434	C₂²×C₄	—	106	24	no	—
435	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	complemented
436	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
437	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
438	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
439	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
440	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
441	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
442	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
443	C₄⋊C₄	—	53	8	no	—
444	C₄⋊C₄	—	53	24	no	—
445	C₄⋊C₄	—	53	32	no	—
446	C₂²⋊C₄	—	53	8	no	—
447	C₂²⋊C₄	—	53	24	no	—
448	C₂²⋊C₄	—	53	32	no	—
449	C₂²⋊C₄	—	53	96	no	—
450	C₄²	—	53	8	no	—
451	C₄²	—	53	24	no	—
452	C₂³	—	2	24	no	—
453	C₂³	—	4	6	no	—
454	C₂³	—	4	24	no	—
455	C₂³	—	53	12	no	—
456	C₂³	—	106	48	no	—
457	C₂³	—	106	12	no	—
458	C₂³	—	424	48	no	—
459	C₂³	—	212	48	no	—
460	C₂³	—	212	12	no	—
461	C₂³	C₂²×C₂²⋊D₅₃	1	12	yes	—
462	C₂³	C₂⁴×D₅₃	1	8	yes	—
463	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
464	C₂³	C₂⁴×D₅₃	1	6	yes	—
465	Q₈	—	2	24	no	—
466	Q₈	—	2	8	no	—
467	Q₈	—	106	4	no	—
468	Q₈	—	106	4	no	—
469	Q₈	C₂²×C₂²⋊D₅₃	1	16	yes	—
470	D₄	—	2	48	no	—
471	D₄	—	2	16	no	—
472	D₄	—	2	24	no	—
473	D₄	—	2	8	no	—
474	D₄	—	106	48	no	—
475	D₄	—	106	12	no	—
476	D₄	—	106	48	no	—
477	D₄	—	106	12	no	—
478	D₄	—	212	96	no	—
479	D₄	C₂²×C₂²⋊D₅₃	1	48	yes	—
480	C₂×C₄	—	2	24	no	—
481	C₂×C₄	—	2	16	no	—
482	C₂×C₄	—	2	48	no	—
483	C₂×C₄	—	53	16	no	—
484	C₂×C₄	—	53	4	no	—
485	C₂×C₄	—	106	4	no	—
486	C₂×C₄	—	106	48	no	—
487	C₂×C₄	—	212	24	no	—
488	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
489	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
490	C₂×C₄	C₂²×C₂²⋊D₅₃	1	4	yes	—
491	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
492	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
493	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
494	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
495	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
496	C₂²	—	2	12	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	8	no	—
499	C₂²	—	2	16	no	—
500	C₂²	—	4	12	no	—
501	C₂²	—	8	24	no	—
502	C₂²	—	106	12	no	—
503	C₂²	—	106	24	no	—
504	C₂²	—	424	48	no	—
505	C₂²	—	212	24	no	—
506	C₂²	C₂³×C₂²⋊D₅₃	1	12	yes	—
507	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
508	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
509	C₄	—	2	8	no	—
510	C₄	—	106	4	no	—
511	C₄	—	106	16	no	—
512	C₄	C₂³×C₂²⋊D₅₃	1	2	yes	—
513	C₄	C₂³×C₂²⋊D₅₃	1	12	yes	—
514	C₄	C₂³×C₂²⋊D₅₃	1	2	yes	—
515	C₂	—	2	2	no	—
516	C₂	—	2	12	no	—
517	C₂	—	4	8	no	—
518	C₂	—	212	12	no	—
519	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
520	C₂	#20289526	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₂×C₅₃×#585	C₂
hypercenter	C₂×#585	D₅₃
hypocenter	C₅₃	#62852
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133263	C₂	semidirect
2. #20288509	C₂	semidirect
3. #20288489	C₂	semidirect
4. #20288384	C₂	semidirect
5. #20288370	C₂	semidirect
6. #20288346	C₂	semidirect
7. #20288044	C₂	semidirect
8. C₂×#16811187	C₂	semidirect
9. #20287918	C₂	semidirect
10. C₂×C₅₃×#585	C₂	semidirect
12. C₂×#13133263	C₂²	semidirect
13. C₂×#13133263	C₂²	semidirect
14. C₂²×#11719949	C₂²	semidirect
15. C₂²×#11719945	C₂²	semidirect
16. C₂³×D₂₁₂	C₂²	semidirect
17. C₂²×D₄×D₅₃	C₂²	semidirect
18. #16811292	C₂²	semidirect
19. #16811284	C₂²	semidirect
20. #16811261	C₂²	semidirect
21. D₄×C₂²⋊D₅₃	C₂²	semidirect
22. #16811201	C₂²	semidirect
23. #16811197	C₂²	semidirect
24. #16811194	C₂²	semidirect
25. #16811190	C₂²	semidirect
26. #16811187	C₂²	semidirect
27. #16811187	C₂²	semidirect
28. #16811187	C₂²	semidirect
29. #16811187	C₂²	semidirect
30. #16811182	C₂²	semidirect
31. #16811169	C₂²	semidirect
32. C₂×#13133213	C₂²	semidirect
33. C₂×#13133206	C₂²	semidirect
34. #16811134	C₂²	semidirect
35. #16811130	C₂²	semidirect
36. C₂×#13133191	C₂²	semidirect
37. C₂×#13133185	C₂²	semidirect
38. #16811106	C₂²	semidirect
39. #16811101	C₂²	semidirect
40. #16811098	C₂²	semidirect
41. #41×Dic₅₃	C₂²	semidirect
42. C₂×#13133176	C₂²	semidirect
44. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
45. C₂×2₋¹⁺⁴×C₅₃	C₂²	semidirect
46. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
47. C₂×H(5,𝔽₂)×C₅₃	C₂²	semidirect
49. C₂²×#41×C₅₃	C₂²	semidirect
50. C₂²×#41×C₅₃	C₂²	semidirect
66. C₂²×C₂²⋊D₅₃	C₂³	semidirect
70. #13133215	C₂³	semidirect
71. #13133213	C₂³	semidirect
72. #13133213	C₂³	semidirect
73. #13133206	C₂³	semidirect
74. #13133206	C₂³	semidirect
75. #13133204	C₂³	semidirect
76. #13133200	C₂³	semidirect
78. #13133196	C₂³	semidirect
79. #13133193	C₂³	semidirect
80. #13133191	C₂³	semidirect
81. #13133191	C₂³	semidirect
82. #13133189	C₂³	semidirect
83. #13133188	C₂³	semidirect
84. #13133186	C₂³	semidirect
86. #13133185	C₂³	semidirect
87. Q₈×Dic₅₃	C₂³	semidirect
88. D₄×Dic₅₃	C₂³	semidirect
89. C₄×C₂²⋊D₅₃	C₂³	semidirect
92. #13133176	C₂³	semidirect
93. #13133176	C₂³	semidirect
94. C₂×C₄⋊Dic₅₃	C₂³	semidirect
96. C₂×#11719921	C₂³	semidirect
97. C₂×C₄×Dic₅₃	C₂³	semidirect
104. C₂×#41×C₅₃	D₄	semidirect
109. C₂×#41×C₅₃	C₂³	semidirect
110. C₂×#41×C₅₃	C₂³	semidirect
111. C₂×#41×C₅₃	C₂³	semidirect
112. C₂×#41×C₅₃	C₂³	semidirect
114. C₂²×Q₈×C₅₃	C₂³	semidirect
115. C₂²×Q₈×C₅₃	C₂³	semidirect
118. C₂²×D₄×C₅₃	C₂³	semidirect
119. C₂²×D₄×C₅₃	C₂³	semidirect
121. C₂³×C₄×C₅₃	C₂³	semidirect
134. #41×C₅₃	C₂×D₄	semidirect
135. #41×C₅₃	C₂×D₄	semidirect
138. C₂×Q₈×C₅₃	C₂×D₄	semidirect
145. C₂×D₄×C₅₃	C₂×D₄	semidirect
152. C₂²×C₄×C₅₃	C₂×D₄	semidirect
255. C₂×#585	D₅₃	semidirect
272. C₂×2₋¹⁺⁴	C₂×D₅₃	semidirect
273. C₂×2₋¹⁺⁴	C₂×D₅₃	semidirect
277. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
278. C₂×H(5,𝔽₂)	C₂×D₅₃	semidirect
282. C₂²×#41	C₂×D₅₃	semidirect
283. C₂²×#41	C₂×D₅₃	semidirect
311. C₅₃	#62852	semidirect
326. C₂×#41	C₂²⋊D₅₃	semidirect
331. C₂×#41	C₂²×D₅₃	semidirect
332. C₂×#41	C₂²×D₅₃	semidirect
333. C₂×#41	C₂²×D₅₃	semidirect
334. C₂×#41	C₂²×D₅₃	semidirect
337. C₂²×Q₈	C₂²×D₅₃	semidirect
338. C₂²×Q₈	C₂²×D₅₃	semidirect
350. C₂²×D₄	C₂²×D₅₃	semidirect
351. C₂²×D₄	C₂²×D₅₃	semidirect
354. C₂³×C₄	C₂²×D₅₃	semidirect
398. #41	C₂×C₂²⋊D₅₃	semidirect
399. #41	C₂×C₂²⋊D₅₃	semidirect
405. C₂×Q₈	C₂×C₂²⋊D₅₃	semidirect
424. C₂×D₄	C₂×C₂²⋊D₅₃	semidirect
435. 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×#13133263	2
#20288509	2
#20288489	2
#20288384	2
#20288370	2
#20288346	2
#20288044	2
C₂×#16811187	2
#20287918	2
C₂×C₅₃×#585	2
#62852	53