Subgroups — #28292633

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	C₂×#16811459	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#41×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20288396	C₂	1	8	yes	maximal, complemented
4	#20288385	C₂	1	8	yes	maximal, complemented
5	#20288374	C₂	1	2	yes	maximal, complemented
6	#20288367	C₂	1	2	yes	maximal, complemented
7	#20288360	C₂	1	6	yes	maximal, complemented
8	#20288351	C₂	1	6	yes	maximal, complemented
9	#41×D₂₁₂	C₂	1	16	yes	maximal, complemented
10	#20288048	C₂	1	6	yes	maximal, complemented
11	C₂×#16811135	C₂	1	6	yes	maximal, complemented
12	C₅₃×#3428	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811459	—	2	16	no	—
14	C₂×#13133265	C₂²	1	2	yes	complemented
15	C₂×#13133264	C₂²	1	4	yes	complemented
16	C₂×#13133263	C₂²	1	4	yes	complemented
17	C₂×#13133262	C₂²	1	6	yes	complemented
18	C₂²×#11719949	C₂²	1	1	yes	characteristic, complemented
19	C₂²×#11719945	C₂²	1	3	yes	complemented
20	C₂²×#11719945	C₂²	1	6	yes	complemented
21	C₂²×#11719947	C₂²	1	3	yes	complemented
22	C₂×#41×D₅₃	—	2	16	no	—
23	C₂×#41×D₅₃	C₂²	1	8	yes	complemented
24	C₂×#41×D₅₃	C₂²	1	8	yes	complemented
25	C₂×#41×D₅₃	C₂²	1	8	yes	complemented
26	C₂²×Q₈×D₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₄×D₅₃	C₂²	1	3	yes	complemented
28	C₂³×C₄×D₅₃	C₂²	1	3	yes	complemented
29	#16811295	C₂²	1	16	yes	complemented
30	Q₈×D₂₁₂	C₂²	1	16	yes	complemented
31	#16811285	C₂²	1	16	yes	complemented
32	#16811285	C₂²	1	16	yes	complemented
33	#16811277	C₂²	1	48	yes	complemented
34	#16811268	C₂²	1	48	yes	complemented
35	#16811268	C₂²	1	48	yes	complemented
36	D₄×D₂₁₂	C₂²	1	48	yes	complemented
37	#16811204	C₂²	1	24	yes	complemented
38	#16811199	C₂²	1	12	yes	complemented
39	#16811199	C₂²	1	12	yes	complemented
40	#16811192	C₂²	1	12	yes	complemented
41	#16811192	C₂²	1	12	yes	complemented
42	#16811185	C₂²	1	24	yes	complemented
43	#16811173	C₂²	1	24	yes	complemented
44	#16811135	C₂²	1	12	yes	complemented
45	#16811135	C₂²	1	4	yes	complemented
46	#16811135	C₂²	1	12	yes	complemented
47	#16811135	C₂²	1	12	yes	complemented
48	#16811135	C₂²	1	24	yes	complemented
49	#16811135	C₂²	1	24	yes	complemented
50	#16811135	C₂²	1	24	yes	complemented
51	C₂×#13133201	C₂²	1	12	yes	complemented
52	C₂×#13133197	C₂²	1	12	yes	complemented
53	C₂×#13133192	C₂²	1	12	yes	complemented
54	C₂×#13133187	C₂²	1	12	yes	complemented
55	C₂×C₄×D₂₁₂	C₂²	1	24	yes	complemented
56	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
57	C₅₃×#520	C₂²	1	8	yes	complemented
58	C₅₃×#519	C₂²	1	2	yes	complemented
59	C₅₃×#518	C₂²	1	6	yes	complemented
60	C₄×#41×C₅₃	C₂²	1	8	yes	complemented
61	C₂×#117×C₅₃	C₂²	1	6	yes	complemented
62	C₂⁴×D₅₃	C₂³	1	3	yes	—
63	#13133265	—	2	16	no	—
64	#13133264	—	2	32	no	—
65	#13133263	—	2	32	no	—
66	#13133262	—	2	48	no	—
67	C₂×#11719949	—	2	8	no	—
68	C₂×#11719949	C₂³	1	8	yes	—
69	C₂×#11719949	C₂³	1	8	yes	complemented
70	C₂×#11719945	—	2	24	no	—
71	C₂×#11719945	—	2	48	no	—
72	C₂×#11719945	C₂³	1	4	yes	—
73	C₂×#11719945	C₂³	1	12	yes	—
74	C₂×#11719945	C₂³	1	12	yes	—
75	C₂×#11719945	C₂³	1	12	yes	—
76	C₂×#11719945	C₂³	1	12	yes	complemented
77	C₂×#11719945	C₂³	1	12	yes	complemented
78	C₂×#11719947	—	2	24	no	—
79	C₂×#11719947	C₂³	1	24	yes	—
80	C₂×#11719947	C₂³	1	24	yes	complemented
81	#41×D₅₃	—	2	64	no	—
82	#41×D₅₃	—	2	64	no	—
83	#41×D₅₃	—	2	64	no	—
84	C₂²×Dic₁₀₆	C₂³	1	3	yes	—
85	C₂²×Dic₁₀₆	C₂³	1	6	yes	complemented
86	C₂×Q₈×D₅₃	—	2	8	no	—
87	C₂×Q₈×D₅₃	C₂³	1	8	yes	—
88	C₂×Q₈×D₅₃	C₂³	1	8	yes	complemented
89	C₂²×D₂₁₂	C₂³	1	3	yes	—
90	C₂²×D₂₁₂	C₂³	1	24	yes	—
91	C₂²×D₂₁₂	C₂³	1	6	yes	complemented
92	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
93	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
94	C₂×D₄×D₅₃	—	2	24	no	—
95	C₂×D₄×D₅₃	C₂³	1	24	yes	—
96	C₂×D₄×D₅₃	C₂³	1	24	yes	complemented
97	C₂³×Dic₅₃	C₂³	1	3	yes	—
98	C₂²×C₄×D₅₃	—	2	24	no	—
99	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
100	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
101	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
102	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
103	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
104	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
105	C₂²×C₄×D₅₃	C₂³	1	12	yes	complemented
106	#13133217	C₂³	1	24	yes	complemented
107	#13133214	C₂³	1	24	yes	complemented
108	#13133205	C₂³	1	24	yes	complemented
109	#13133205	C₂³	1	24	yes	complemented
110	#13133201	C₂³	1	8	yes	—
111	#13133201	C₂³	1	24	yes	—
112	#13133201	C₂³	1	48	yes	complemented
113	#13133201	C₂³	1	48	yes	complemented
114	#13133197	C₂³	1	24	yes	—
115	#13133197	C₂³	1	8	yes	—
116	#13133197	C₂³	1	48	yes	complemented
117	#13133197	C₂³	1	24	yes	complemented
118	#13133197	C₂³	1	24	yes	complemented
119	#13133192	C₂³	1	24	yes	—
120	#13133192	C₂³	1	8	yes	—
121	#13133192	C₂³	1	24	yes	complemented
122	#13133192	C₂³	1	48	yes	complemented
123	#13133192	C₂³	1	24	yes	complemented
124	#13133191	C₂³	1	96	yes	complemented
125	#13133187	C₂³	1	48	yes	—
126	#13133187	C₂³	1	8	yes	—
127	#13133187	C₂³	1	24	yes	—
128	#13133187	C₂³	1	48	yes	complemented
129	C₄×D₂₁₂	C₂³	1	48	yes	—
130	C₄×D₂₁₂	C₂³	1	16	yes	—
131	C₄×D₂₁₂	C₂³	1	48	yes	complemented
132	C₄×D₂₁₂	C₂³	1	48	yes	complemented
133	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
134	C₂×#11719921	C₂³	1	24	yes	—
135	C₂×#11719921	C₂³	1	24	yes	complemented
136	C₂×#41×C₅₃	D₄	1	16	yes	complemented
137	C₂×#41×C₅₃	C₂³	1	4	yes	—
138	C₂×#41×C₅₃	C₂³	1	4	yes	—
139	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
140	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
141	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
142	C₄×Q₈×C₅₃	C₂³	1	16	yes	complemented
143	C₄×D₄×C₅₃	C₂³	1	48	yes	complemented
144	#117×C₅₃	C₂³	1	4	yes	—
145	#117×C₅₃	C₂³	1	12	yes	—
146	#117×C₅₃	C₂³	1	12	yes	complemented
147	#117×C₅₃	C₂³	1	12	yes	complemented
148	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
149	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
150	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
151	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
152	C₂³×D₅₃	—	2	12	no	—
153	C₂³×D₅₃	—	4	12	no	—
154	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
155	C₂³×D₅₃	C₂⁴	1	12	yes	—
156	C₂³×D₅₃	C₂⁴	1	12	yes	—
157	C₂³×D₅₃	C₂⁴	1	3	yes	—
158	#41×C₅₃	C₂×D₄	1	32	yes	complemented
159	#41×C₅₃	C₂×D₄	1	32	yes	complemented
160	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
161	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
162	C₂×D₄×C₅₃	C₂×D₄	1	24	yes	complemented
163	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
164	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	complemented
165	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
166	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
170	C₂×C₂²⋊D₅₃	—	2	24	no	—
171	C₂×C₂²⋊D₅₃	—	2	48	no	—
172	C₂×C₂²⋊D₅₃	C₂⁴	1	8	yes	—
173	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
174	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
175	C₂²×Dic₅₃	—	2	12	no	—
176	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
177	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
178	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
179	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
180	#11719949	—	2	32	no	—
181	#11719949	—	2	32	no	—
182	Q₈×D₅₃	—	2	32	no	—
183	Q₈×D₅₃	—	2	32	no	—
184	#11719947	—	2	96	no	—
185	#11719947	—	2	96	no	—
186	D₄×D₅₃	—	2	96	no	—
187	D₄×D₅₃	—	2	96	no	—
188	#11719945	—	2	48	no	—
189	#11719945	—	2	48	no	—
190	#11719945	—	2	48	no	—
191	#11719945	—	2	16	no	—
192	#11719945	—	2	48	no	—
193	#11719945	—	2	48	no	—
194	C₂×D₂₁₂	—	2	12	no	—
195	C₂×D₂₁₂	—	2	24	no	—
196	C₂×D₂₁₂	—	4	48	no	—
197	C₂×D₂₁₂	C₂⁴	1	16	yes	—
198	C₂×D₂₁₂	C₂⁴	1	4	yes	—
199	C₂×D₂₁₂	C₂⁴	1	12	yes	—
200	C₂×D₂₁₂	C₂⁴	1	12	yes	—
201	C₂×D₂₁₂	C₂⁴	1	48	yes	—
202	C₂×C₄×D₅₃	—	2	4	no	—
203	C₂×C₄×D₅₃	—	2	12	no	—
204	C₂×C₄×D₅₃	—	2	48	no	—
205	C₂×C₄×D₅₃	—	2	48	no	—
206	C₂×C₄×D₅₃	—	2	12	no	—
207	C₂×C₄×D₅₃	—	2	12	no	—
208	C₂×C₄×D₅₃	—	2	48	no	—
209	C₂×C₄×D₅₃	C₂⁴	1	2	yes	—
210	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
211	C₂×C₄×D₅₃	C₂⁴	1	8	yes	—
212	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
213	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
214	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
215	C₂×C₄×D₅₃	C₂⁴	1	2	yes	—
216	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
217	C₂×Dic₁₀₆	—	2	12	no	—
218	C₂×Dic₁₀₆	—	2	24	no	—
219	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
220	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
221	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
222	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
223	C₄⋊C₄×C₅₃	C₂⁴	1	24	yes	—
224	C₂²⋊C₄×C₅₃	C₂⁴	1	8	yes	—
225	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
226	C₄²×C₅₃	C₂⁴	1	8	yes	—
227	C₄²×C₅₃	C₂⁴	1	24	yes	—
228	#11719921	C₂⁴	1	16	yes	—
229	#11719921	C₂⁴	1	48	yes	—
230	#11719921	C₂⁴	1	16	yes	—
231	#11719921	C₂⁴	1	48	yes	—
232	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
233	C₄⋊Dic₅₃	C₂⁴	1	48	yes	—
234	C₂³×C₅₃	—	4	6	no	—
235	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
236	C₂³×C₅₃	C₂⁵	1	6	yes	—
237	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
238	C₂²×D₅₃	—	2	2	no	—
239	C₂²×D₅₃	—	2	4	no	—
240	C₂²×D₅₃	—	2	6	no	—
241	C₂²×D₅₃	—	2	24	no	—
242	C₂²×D₅₃	—	2	24	no	—
243	C₂²×D₅₃	—	2	24	no	—
244	C₂²×D₅₃	—	4	12	no	—
245	C₂²×D₅₃	—	4	6	no	—
246	C₂²×D₅₃	—	4	24	no	—
247	C₂²×D₅₃	—	4	48	no	—
248	C₂²×D₅₃	C₂⁵	1	8	yes	—
249	C₂²×D₅₃	C₂⁵	1	6	yes	—
250	C₂²×D₅₃	C₂⁵	1	2	yes	—
251	Q₈×C₅₃	C₂²×D₄	1	16	yes	—
252	D₄×C₅₃	C₂²×D₄	1	48	yes	—
253	C₂×C₄×C₅₃	—	4	24	no	—
254	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
255	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
256	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
257	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
258	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
259	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
261	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
262	C₂²⋊D₅₃	—	2	16	no	—
263	C₂²⋊D₅₃	—	2	48	no	—
264	C₂²⋊D₅₃	—	2	48	no	—
265	C₂×Dic₅₃	—	2	6	no	—
266	C₂×Dic₅₃	—	2	24	no	—
267	C₂×Dic₅₃	—	2	2	no	—
268	C₂×Dic₅₃	—	2	24	no	—
269	C₂×Dic₅₃	C₂⁵	1	8	yes	—
270	C₂×Dic₅₃	C₂⁵	1	6	yes	—
271	C₂×Dic₅₃	C₂⁵	1	2	yes	—
272	D₂₁₂	—	2	32	no	—
273	D₂₁₂	—	2	8	no	—
274	D₂₁₂	—	2	24	no	—
275	D₂₁₂	—	2	24	no	—
276	D₂₁₂	—	4	48	no	—
277	C₄×D₅₃	—	2	4	no	—
278	C₄×D₅₃	—	2	4	no	—
279	C₄×D₅₃	—	2	12	no	—
280	C₄×D₅₃	—	2	12	no	—
281	C₄×D₅₃	—	2	16	no	—
282	C₄×D₅₃	—	2	24	no	—
283	C₄×D₅₃	—	2	24	no	—
284	C₄×D₅₃	—	2	48	no	—
285	Dic₁₀₆	—	2	8	no	—
286	Dic₁₀₆	—	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	—	4	12	no	—
292	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
293	C₂²×C₅₃	C₂³×D₄	1	12	yes	—
294	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
295	C₂×D₅₃	—	2	8	no	—
296	C₂×D₅₃	—	2	2	no	—
297	C₂×D₅₃	—	2	2	no	—
298	C₂×D₅₃	—	2	6	no	—
299	C₂×D₅₃	—	2	8	no	—
300	C₂×D₅₃	—	2	16	no	—
301	C₂×D₅₃	—	2	12	no	—
302	C₂×D₅₃	—	4	48	no	—
303	C₂×D₅₃	—	4	12	no	—
304	C₂×D₅₃	—	8	24	no	—
305	C₂×D₅₃	#585	1	4	yes	complemented
306	C₄×C₅₃	—	2	16	no	—
307	C₄×C₅₃	C₂³×D₄	1	2	yes	—
308	C₄×C₅₃	C₂³×D₄	1	12	yes	—
309	C₄×C₅₃	C₂³×D₄	1	2	yes	—
310	Dic₅₃	—	2	8	no	—
311	Dic₅₃	—	2	6	no	—
312	Dic₅₃	—	2	2	no	—
313	C₂×#585	—	53	1	no	—
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	D₅₃	1	1	yes	characteristic, complemented
325	C₂×C₅₃	—	2	2	no	—
326	C₂×C₅₃	—	2	12	no	—
327	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
328	C₂×C₅₃	C₂×#585	1	2	yes	—
329	D₅₃	—	2	4	no	—
330	D₅₃	—	4	6	no	—
331	D₅₃	—	4	8	no	—
332	#585	—	106	16	no	—
333	C₂×2₋¹⁺⁴	—	53	4	no	—
334	C₂×2₋¹⁺⁴	—	53	2	no	—
335	C₂×H(5,𝔽₂)	—	53	4	no	—
336	C₂×H(5,𝔽₂)	—	53	6	no	—
337	C₂²×#41	—	53	8	no	—
338	C₂²×#41	—	53	8	no	—
339	C₂²×#41	—	53	8	no	—
340	C₂²×#41	—	53	1	no	—
341	C₂²×#41	—	53	3	no	—
342	C₂²×#41	—	53	3	no	—
343	C₂²×#41	—	53	6	no	—
344	C₂²×#41	—	106	16	no	—
345	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
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	C₂×D₅₃	1	8	yes	complemented
376	#519	C₂×D₅₃	1	2	yes	complemented
377	#518	C₂×D₅₃	1	6	yes	complemented
378	C₄×#41	C₂×D₅₃	1	8	yes	complemented
379	C₂×C₄×D₄	—	53	24	no	—
380	C₂×#117	C₂×D₅₃	1	6	yes	complemented
381	C₅₃	#62811	1	1	yes	characteristic, complemented, Sylow
382	C₂⁵	—	53	3	no	—
383	2₋¹⁺⁴	—	106	32	no	—
384	2₋¹⁺⁴	—	106	16	no	—
385	H(5,𝔽₂)	—	106	32	no	—
386	H(5,𝔽₂)	—	106	48	no	—
387	C₂×#41	—	53	8	no	—
388	C₂×#41	—	53	8	no	—
389	C₂×#41	—	53	24	no	—
390	C₂×#41	—	53	24	no	—
391	C₂×#41	—	53	4	no	—
392	C₂×#41	—	53	12	no	—
393	C₂×#41	—	53	12	no	—
394	C₂×#41	—	53	12	no	—
395	C₂×#41	—	53	12	no	—
396	C₂×#41	—	53	12	no	—
397	C₂×#41	—	106	64	no	—
398	C₂×#41	—	106	64	no	—
399	C₂×#41	—	106	64	no	—
400	C₂×#41	—	106	8	no	—
401	C₂×#41	—	106	24	no	—
402	C₂×#41	—	106	24	no	—
403	C₂×#41	—	106	48	no	—
404	C₂×#41	D₂₁₂	1	16	yes	complemented
405	C₂×#41	C₂²×D₅₃	1	4	yes	—
406	C₂×#41	C₂²×D₅₃	1	4	yes	—
407	C₂²×Q₈	—	53	8	no	—
408	C₂²×Q₈	—	53	8	no	—
409	C₂²×Q₈	—	53	6	no	—
410	C₂²×Q₈	—	53	3	no	—
411	C₂²×Q₈	—	106	8	no	—
412	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
413	C₂²×D₄	—	53	12	no	—
414	C₂²×D₄	—	53	6	no	—
415	C₂²×D₄	—	53	24	no	—
416	C₂²×D₄	—	53	24	no	—
417	C₂²×D₄	—	53	6	no	—
418	C₂²×D₄	—	53	3	no	—
419	C₂²×D₄	—	53	24	no	—
420	C₂²×D₄	—	106	24	no	—
421	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
422	C₂³×C₄	—	53	3	no	—
423	C₂³×C₄	—	53	12	no	—
424	C₂³×C₄	—	53	12	no	—
425	C₂³×C₄	—	53	12	no	—
426	C₂³×C₄	—	53	1	no	—
427	C₂³×C₄	—	53	3	no	—
428	C₂³×C₄	—	53	3	no	—
429	C₂³×C₄	—	53	3	no	—
430	C₂³×C₄	—	106	24	no	—
431	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
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	96	no	—
446	#121	—	53	8	no	—
447	#121	—	53	24	no	—
448	#121	—	53	24	no	—
449	#121	—	53	24	no	—
450	#121	—	53	48	no	—
451	#120	—	53	8	no	—
452	#120	—	53	24	no	—
453	#120	—	53	48	no	—
454	#120	—	53	48	no	—
455	C₄×Q₈	C₂²×D₅₃	1	16	yes	complemented
456	C₄×D₄	—	53	16	no	—
457	C₄×D₄	—	53	48	no	—
458	C₄×D₄	—	53	48	no	—
459	C₄×D₄	—	53	48	no	—
460	C₄×D₄	C₂²×D₅₃	1	48	yes	complemented
461	#117	C₂²×D₅₃	1	4	yes	—
462	#117	C₂²×D₅₃	1	12	yes	—
463	#117	C₂²×D₅₃	1	12	yes	complemented
464	#117	C₂²×D₅₃	1	12	yes	complemented
465	C₂×C₄⋊C₄	—	53	24	no	—
466	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
467	C₂×C₂²⋊C₄	—	53	24	no	—
468	C₂×C₂²⋊C₄	—	53	24	no	—
469	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	—
470	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
471	C₂⁴	—	53	1	no	—
472	C₂⁴	—	53	12	no	—
473	C₂⁴	—	53	12	no	—
474	C₂⁴	—	53	3	no	—
475	C₂⁴	—	106	12	no	—
476	C₂⁴	—	212	12	no	—
477	C₂⁴	C₂³×D₅₃	1	3	yes	—
478	#41	—	106	32	no	—
479	#41	—	106	32	no	—
480	#41	—	106	96	no	—
481	#41	—	106	96	no	—
482	#41	—	106	16	no	—
483	#41	—	106	48	no	—
484	#41	—	106	48	no	—
485	#41	—	106	48	no	—
486	#41	—	106	48	no	—
487	#41	—	106	48	no	—
488	#41	C₂×D₂₁₂	1	32	yes	complemented
489	#41	C₂×D₂₁₂	1	32	yes	complemented
490	C₂×Q₈	—	53	4	no	—
491	C₂×Q₈	—	53	12	no	—
492	C₂×Q₈	—	53	12	no	—
493	C₂×Q₈	—	106	32	no	—
494	C₂×Q₈	—	106	32	no	—
495	C₂×Q₈	—	106	12	no	—
496	C₂×Q₈	—	106	24	no	—
497	C₂×Q₈	C₂×D₂₁₂	1	8	yes	complemented
498	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
499	C₂×D₄	—	53	8	no	—
500	C₂×D₄	—	53	24	no	—
501	C₂×D₄	—	53	24	no	—
502	C₂×D₄	—	53	16	no	—
503	C₂×D₄	—	53	4	no	—
504	C₂×D₄	—	53	12	no	—
505	C₂×D₄	—	53	12	no	—
506	C₂×D₄	—	53	48	no	—
507	C₂×D₄	—	106	24	no	—
508	C₂×D₄	—	106	48	no	—
509	C₂×D₄	—	106	96	no	—
510	C₂×D₄	—	106	96	no	—
511	C₂×D₄	—	106	12	no	—
512	C₂×D₄	—	106	24	no	—
513	C₂×D₄	—	212	48	no	—
514	C₂×D₄	C₂×D₂₁₂	1	24	yes	complemented
515	C₂×D₄	C₂³×D₅₃	1	12	yes	—
516	C₂²×C₄	—	53	12	no	—
517	C₂²×C₄	—	53	12	no	—
518	C₂²×C₄	—	53	1	no	—
519	C₂²×C₄	—	53	3	no	—
520	C₂²×C₄	—	53	12	no	—
521	C₂²×C₄	—	53	12	no	—
522	C₂²×C₄	—	53	24	no	—
523	C₂²×C₄	—	53	2	no	—
524	C₂²×C₄	—	53	6	no	—
525	C₂²×C₄	—	53	2	no	—
526	C₂²×C₄	—	53	8	no	—
527	C₂²×C₄	—	53	6	no	—
528	C₂²×C₄	—	106	12	no	—
529	C₂²×C₄	—	106	4	no	—
530	C₂²×C₄	—	106	12	no	—
531	C₂²×C₄	—	106	12	no	—
532	C₂²×C₄	—	106	12	no	—
533	C₂²×C₄	—	106	48	no	—
534	C₂²×C₄	—	106	48	no	—
535	C₂²×C₄	—	106	48	no	—
536	C₂²×C₄	C₂×D₂₁₂	1	24	yes	complemented
537	C₂²×C₄	C₂³×D₅₃	1	24	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	16	no	—
543	C₄⋊C₄	—	53	48	no	—
544	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
545	C₄⋊C₄	C₂³×D₅₃	1	24	yes	—
546	C₂²⋊C₄	—	53	16	no	—
547	C₂²⋊C₄	—	53	16	no	—
548	C₂²⋊C₄	—	53	48	no	—
549	C₂²⋊C₄	—	53	48	no	—
550	C₂²⋊C₄	C₂³×D₅₃	1	8	yes	—
551	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
552	C₄²	C₂³×D₅₃	1	24	yes	—
553	C₄²	C₂³×D₅₃	1	8	yes	—
554	C₂³	—	4	6	no	—
555	C₂³	—	53	2	no	—
556	C₂³	—	53	8	no	—
557	C₂³	—	53	6	no	—
558	C₂³	—	106	4	no	—
559	C₂³	—	106	24	no	—
560	C₂³	—	106	2	no	—
561	C₂³	—	106	24	no	—
562	C₂³	—	106	24	no	—
563	C₂³	—	106	6	no	—
564	C₂³	—	212	24	no	—
565	C₂³	—	212	12	no	—
566	C₂³	—	212	6	no	—
567	C₂³	—	212	48	no	—
568	C₂³	C₂²×D₂₁₂	1	12	yes	—
569	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
570	C₂³	C₂⁴×D₅₃	1	6	yes	—
571	Q₈	—	106	8	no	—
572	Q₈	—	106	24	no	—
573	Q₈	—	106	24	no	—
574	Q₈	C₂²×D₂₁₂	1	16	yes	—
575	D₄	—	106	32	no	—
576	D₄	—	106	16	no	—
577	D₄	—	106	48	no	—
578	D₄	—	106	48	no	—
579	D₄	—	106	8	no	—
580	D₄	—	106	24	no	—
581	D₄	—	106	24	no	—
582	D₄	—	212	48	no	—
583	D₄	C₂²×D₂₁₂	1	48	yes	—
584	C₂×C₄	—	4	24	no	—
585	C₂×C₄	—	53	2	no	—
586	C₂×C₄	—	53	8	no	—
587	C₂×C₄	—	53	6	no	—
588	C₂×C₄	—	106	2	no	—
589	C₂×C₄	—	106	6	no	—
590	C₂×C₄	—	106	24	no	—
591	C₂×C₄	—	106	24	no	—
592	C₂×C₄	—	106	4	no	—
593	C₂×C₄	—	106	4	no	—
594	C₂×C₄	—	106	24	no	—
595	C₂×C₄	—	106	24	no	—
596	C₂×C₄	—	106	12	no	—
597	C₂×C₄	—	106	16	no	—
598	C₂×C₄	—	106	12	no	—
599	C₂×C₄	—	106	48	no	—
600	C₂×C₄	C₂²×D₂₁₂	1	4	yes	—
601	C₂×C₄	C₂²×D₂₁₂	1	24	yes	—
602	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
603	C₂×C₄	C₂²×D₂₁₂	1	24	yes	—
604	C₂×C₄	C₂⁴×D₅₃	1	16	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	—	4	12	no	—
611	C₂²	—	53	4	no	—
612	C₂²	—	106	8	no	—
613	C₂²	—	106	2	no	—
614	C₂²	—	106	2	no	—
615	C₂²	—	106	8	no	—
616	C₂²	—	106	6	no	—
617	C₂²	—	106	16	no	—
618	C₂²	—	106	12	no	—
619	C₂²	—	424	24	no	—
620	C₂²	—	212	48	no	—
621	C₂²	—	212	12	no	—
622	C₂²	C₂³×D₂₁₂	1	2	yes	—
623	C₂²	C₂³×D₂₁₂	1	12	yes	—
624	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
625	C₄	—	2	16	no	—
626	C₄	—	106	2	no	—
627	C₄	—	106	6	no	—
628	C₄	—	106	8	no	—
629	C₄	C₂³×D₂₁₂	1	2	yes	—
630	C₄	C₂³×D₂₁₂	1	2	yes	—
631	C₄	C₂³×D₂₁₂	1	12	yes	—
632	C₂	—	2	2	no	—
633	C₂	—	2	12	no	—
634	C₂	—	106	4	no	—
635	C₂	—	212	8	no	—
636	C₂	—	212	6	no	—
637	C₂	C₂⁴×D₂₁₂	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	C₅₃×#3428	C₂
hypercenter	#3428	D₅₃
hypocenter	C₅₃	#62811
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #3428

Complements

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

N	H ≅ G/N	type
1. C₂×#16811459	C₂	semidirect
2. C₂²×#41×D₅₃	C₂	semidirect
3. #20288396	C₂	semidirect
4. #20288385	C₂	semidirect
5. #20288374	C₂	semidirect
6. #20288367	C₂	semidirect
7. #20288360	C₂	semidirect
8. #20288351	C₂	semidirect
9. #41×D₂₁₂	C₂	semidirect
10. #20288048	C₂	semidirect
11. C₂×#16811135	C₂	semidirect
12. C₅₃×#3428	C₂	semidirect
14. C₂×#13133265	C₂²	semidirect
15. C₂×#13133264	C₂²	semidirect
16. C₂×#13133263	C₂²	semidirect
17. C₂×#13133262	C₂²	semidirect
18. C₂²×#11719949	C₂²	semidirect
19. C₂²×#11719945	C₂²	semidirect
20. C₂²×#11719945	C₂²	semidirect
21. C₂²×#11719947	C₂²	semidirect
23. C₂×#41×D₅₃	C₂²	semidirect
24. C₂×#41×D₅₃	C₂²	semidirect
25. C₂×#41×D₅₃	C₂²	semidirect
26. C₂²×Q₈×D₅₃	C₂²	semidirect
27. C₂²×D₄×D₅₃	C₂²	semidirect
28. C₂³×C₄×D₅₃	C₂²	semidirect
29. #16811295	C₂²	semidirect
30. Q₈×D₂₁₂	C₂²	semidirect
31. #16811285	C₂²	semidirect
32. #16811285	C₂²	semidirect
33. #16811277	C₂²	semidirect
34. #16811268	C₂²	semidirect
35. #16811268	C₂²	semidirect
36. D₄×D₂₁₂	C₂²	semidirect
37. #16811204	C₂²	semidirect
38. #16811199	C₂²	semidirect
39. #16811199	C₂²	semidirect
40. #16811192	C₂²	semidirect
41. #16811192	C₂²	semidirect
42. #16811185	C₂²	semidirect
43. #16811173	C₂²	semidirect
44. #16811135	C₂²	semidirect
45. #16811135	C₂²	semidirect
46. #16811135	C₂²	semidirect
47. #16811135	C₂²	semidirect
48. #16811135	C₂²	semidirect
49. #16811135	C₂²	semidirect
50. #16811135	C₂²	semidirect
51. C₂×#13133201	C₂²	semidirect
52. C₂×#13133197	C₂²	semidirect
53. C₂×#13133192	C₂²	semidirect
54. C₂×#13133187	C₂²	semidirect
55. C₂×C₄×D₂₁₂	C₂²	semidirect
57. C₅₃×#520	C₂²	semidirect
58. C₅₃×#519	C₂²	semidirect
59. C₅₃×#518	C₂²	semidirect
60. C₄×#41×C₅₃	C₂²	semidirect
61. C₂×#117×C₅₃	C₂²	semidirect
69. C₂×#11719949	C₂³	semidirect
76. C₂×#11719945	C₂³	semidirect
77. C₂×#11719945	C₂³	semidirect
80. C₂×#11719947	C₂³	semidirect
85. C₂²×Dic₁₀₆	C₂³	semidirect
88. C₂×Q₈×D₅₃	C₂³	semidirect
91. C₂²×D₂₁₂	C₂³	semidirect
96. C₂×D₄×D₅₃	C₂³	semidirect
105. C₂²×C₄×D₅₃	C₂³	semidirect
106. #13133217	C₂³	semidirect
107. #13133214	C₂³	semidirect
108. #13133205	C₂³	semidirect
109. #13133205	C₂³	semidirect
112. #13133201	C₂³	semidirect
113. #13133201	C₂³	semidirect
116. #13133197	C₂³	semidirect
117. #13133197	C₂³	semidirect
118. #13133197	C₂³	semidirect
121. #13133192	C₂³	semidirect
122. #13133192	C₂³	semidirect
123. #13133192	C₂³	semidirect
124. #13133191	C₂³	semidirect
128. #13133187	C₂³	semidirect
131. C₄×D₂₁₂	C₂³	semidirect
132. C₄×D₂₁₂	C₂³	semidirect
133. C₂×C₄⋊Dic₅₃	C₂³	semidirect
135. C₂×#11719921	C₂³	semidirect
136. C₂×#41×C₅₃	D₄	semidirect
142. C₄×Q₈×C₅₃	C₂³	semidirect
143. C₄×D₄×C₅₃	C₂³	semidirect
146. #117×C₅₃	C₂³	semidirect
147. #117×C₅₃	C₂³	semidirect
148. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
150. C₂×C₄²×C₅₃	C₂³	semidirect
158. #41×C₅₃	C₂×D₄	semidirect
159. #41×C₅₃	C₂×D₄	semidirect
160. C₂×Q₈×C₅₃	C₂×D₄	semidirect
162. C₂×D₄×C₅₃	C₂×D₄	semidirect
164. C₂²×C₄×C₅₃	C₂×D₄	semidirect
305. C₂×D₅₃	#585	semidirect
324. #3428	D₅₃	semidirect
375. #520	C₂×D₅₃	semidirect
376. #519	C₂×D₅₃	semidirect
377. #518	C₂×D₅₃	semidirect
378. C₄×#41	C₂×D₅₃	semidirect
380. C₂×#117	C₂×D₅₃	semidirect
381. C₅₃	#62811	semidirect
404. C₂×#41	D₂₁₂	semidirect
455. C₄×Q₈	C₂²×D₅₃	semidirect
460. C₄×D₄	C₂²×D₅₃	semidirect
463. #117	C₂²×D₅₃	semidirect
464. #117	C₂²×D₅₃	semidirect
466. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
470. C₂×C₄²	C₂²×D₅₃	semidirect
488. #41	C₂×D₂₁₂	semidirect
489. #41	C₂×D₂₁₂	semidirect
497. C₂×Q₈	C₂×D₂₁₂	semidirect
514. C₂×D₄	C₂×D₂₁₂	semidirect
536. C₂²×C₄	C₂×D₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811459	2
C₂²×#41×D₅₃	2
#20288396	2
#20288385	2
#20288374	2
#20288367	2
#20288360	2
#20288351	2
#41×D₂₁₂	2
#20288048	2
C₂×#16811135	2
C₅₃×#3428	2
#62811	53