Subgroups — C₄×#16810899 #27390985

Order 13568.

882 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810899	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133250	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
5	#20275628	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275501	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275371	C₂	1	2	yes	maximal, complemented
8	#20275229	C₂	1	2	yes	maximal
9	#20274776	C₂	1	1	yes	maximal, characteristic
10	C₄×#13133135	C₂	1	2	yes	maximal, complemented
11	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133250	C₂²	1	2	yes	—
13	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
14	C₂×#13133223	C₂²	1	1	yes	characteristic
15	C₂×#13133223	C₂²	1	1	yes	characteristic
16	C₄×#11719945	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
18	#16810899	C₄	1	8	yes	direct product factor
19	#16810731	—	2	2	no	—
20	#16810725	—	2	2	no	—
21	C₄×Dic₂₁₂	—	2	2	no	—
22	C₄×Dic₂₁₂	—	2	1	no	—
23	C₄×Dic₂₁₂	—	2	1	no	—
24	C₄×Dic₂₁₂	—	2	1	no	—
25	C₄×Dic₂₁₂	—	2	1	no	—
26	C₄×#11719913	—	2	2	no	—
27	#16810665	C₂²	1	2	yes	—
28	C₂×#13133151	C₂²	1	1	yes	characteristic
29	#16810633	C₂²	1	2	yes	—
30	#16810625	C₂²	1	2	yes	—
31	C₂×#13133142	C₂²	1	2	yes	—
32	C₂×#13133135	C₂²	1	2	yes	—
33	C₂×#13133135	C₂²	1	2	yes	—
34	#16810578	C₂²	1	1	yes	characteristic
35	#16810578	C₂²	1	1	yes	characteristic
36	#16810577	—	2	1	no	—
37	#16810577	—	2	1	no	—
38	C₄×#11719917	C₂²	1	1	yes	characteristic
39	C₄×#11719912	—	2	1	no	—
40	C₄×C₈×D₅₃	—	2	1	no	—
41	#16810391	C₂²	1	2	yes	—
42	#16810391	C₂²	1	2	yes	—
43	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#341	C₂²	1	2	yes	—
48	C₂×#11719945	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
50	#13133250	—	2	4	no	—
51	#13133250	C₂×C₄	1	8	yes	—
52	C₂×Dic₂₁₂	—	2	1	no	—
53	C₂×Dic₂₁₂	—	2	1	no	—
54	C₂×Dic₂₁₂	—	2	2	no	—
55	C₂×Dic₂₁₂	—	2	2	no	—
56	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	complemented
57	C₂×#11719913	—	2	2	no	—
58	#13133223	—	2	2	no	—
59	#13133223	—	2	2	no	—
60	#13133223	—	2	2	no	—
61	#13133223	C₂×C₄	1	4	yes	—
62	C₂×#11719917	C₂³	1	1	yes	characteristic
63	C₂×#11719917	C₂³	1	1	yes	characteristic
64	C₂×#11719912	—	2	1	no	—
65	C₂×#11719912	—	2	1	no	—
66	C₂×C₈×D₅₃	—	2	1	no	—
67	C₂×C₈×D₅₃	—	2	1	no	—
68	C₄×Dic₁₀₆	—	2	2	no	—
69	C₄×Dic₁₀₆	—	2	2	no	—
70	C₄×Dic₁₀₆	—	2	2	no	—
71	C₄×Dic₁₀₆	—	2	2	no	—
72	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
73	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
74	C₄×D₂₁₂	D₄	1	1	yes	characteristic
75	C₄×D₂₁₂	D₄	1	1	yes	characteristic
76	C₄×C₂²⋊D₅₃	—	2	1	no	—
77	C₄×C₂²⋊D₅₃	—	2	1	no	—
78	#13133176	C₂³	1	1	yes	characteristic
79	#13133174	—	2	1	no	—
80	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
81	C₂×#11719919	—	2	2	no	—
82	C₂×C₄×Dic₅₃	—	2	2	no	—
83	C₄²×D₅₃	—	2	1	no	—
84	#13133151	—	2	1	no	—
85	#13133151	—	2	1	no	—
86	#13133151	—	2	2	no	—
87	#13133149	—	2	2	no	—
88	#13133142	—	2	2	no	—
89	#13133142	—	2	2	no	—
90	#13133142	—	2	2	no	—
91	#13133142	—	2	2	no	—
92	#13133142	—	2	2	no	—
93	#13133139	—	2	2	no	—
94	#13133135	—	2	4	no	—
95	#13133135	C₂×C₄	1	8	yes	complemented
96	#13133128	—	2	1	no	—
97	#13133127	—	2	1	no	—
98	#13133127	—	2	1	no	—
99	C₄×#88322	—	2	1	no	—
100	C₈×Dic₅₃	—	2	1	no	—
101	C₈×Dic₅₃	—	2	1	no	—
102	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
103	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
104	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
107	#108×C₅₃	—	2	2	no	—
108	#108×C₅₃	C₂×C₄	1	4	yes	complemented
109	#97×C₅₃	—	2	2	no	—
110	C₄×C₈×C₅₃	—	2	2	no	—
111	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
112	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
113	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
114	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂×C₂²⋊D₅₃	—	2	1	no	—
119	C₂²×Dic₅₃	—	2	2	no	—
120	C₂²×Dic₅₃	—	2	2	no	—
121	#11719945	—	2	1	no	—
122	#11719945	—	2	1	no	—
123	#11719945	C₂²×C₄	1	2	yes	—
124	#11719945	#41	1	2	yes	—
125	C₂×D₂₁₂	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×D₅₃	—	2	1	no	—
127	C₂×C₄×D₅₃	—	2	1	no	—
128	C₂×C₄×D₅₃	—	2	1	no	—
129	C₂×Dic₁₀₆	—	2	2	no	—
130	C₂×Dic₁₀₆	—	2	2	no	—
131	C₂×Dic₁₀₆	—	2	2	no	—
132	C₂×Dic₁₀₆	—	2	2	no	—
133	C₂×Dic₁₀₆	—	2	2	no	—
134	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
135	C₂×Dic₁₀₆	C₂×D₄	1	1	yes	characteristic
136	M₁₆×C₅₃	—	2	2	no	—
137	M₁₆×C₅₃	—	2	2	no	—
138	M₁₆×C₅₃	—	2	2	no	—
139	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
140	C₂×C₈×C₅₃	—	2	2	no	—
141	C₂×C₈×C₅₃	—	2	1	no	—
142	C₂×C₈×C₅₃	—	2	1	no	—
143	C₂×C₈×C₅₃	—	2	2	no	—
144	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
145	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₅₃	#41	1	2	yes	—
150	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₂²⋊Dic₅₃	—	2	1	no	—
155	#11719921	—	2	1	no	—
156	#11719921	—	2	1	no	—
157	C₄⋊Dic₅₃	—	2	2	no	—
158	C₄⋊Dic₅₃	—	2	2	no	—
159	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
160	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
161	#11719919	—	2	1	no	—
162	#11719919	—	2	1	no	—
163	#11719919	—	4	2	no	—
164	#11719919	—	4	2	no	—
165	C₄×Dic₅₃	—	2	1	no	—
166	C₄×Dic₅₃	—	2	1	no	—
167	C₄×Dic₅₃	—	2	1	no	—
168	C₄×Dic₅₃	—	4	2	no	—
169	C₄×Dic₅₃	—	4	2	no	—
170	#11719917	—	2	1	no	—
171	#11719917	—	2	1	no	—
172	#11719917	C₂²×C₄	1	2	yes	—
173	#11719917	#41	1	2	yes	—
174	C₂×#88322	—	2	1	no	—
175	C₂×#88322	—	2	1	no	—
176	Dic₂₁₂	—	2	8	no	—
177	Dic₂₁₂	—	2	4	no	—
178	Dic₂₁₂	—	2	4	no	—
179	#11719913	—	2	8	no	—
180	#11719912	—	2	4	no	—
181	#11719912	—	2	4	no	—
182	C₈×D₅₃	—	2	4	no	—
183	C₈×D₅₃	—	2	4	no	—
184	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
185	C₂²×D₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	4	1	no	—
188	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
189	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
190	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
191	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
198	C₂²⋊D₅₃	—	2	2	no	—
199	C₂²⋊D₅₃	—	2	2	no	—
200	C₂×Dic₅₃	—	2	1	no	—
201	C₂×Dic₅₃	—	2	4	no	—
202	C₂×Dic₅₃	—	2	1	no	—
203	C₂×Dic₅₃	—	2	1	no	—
204	C₂×Dic₅₃	—	4	2	no	—
205	C₂×Dic₅₃	—	4	2	no	—
206	C₂×Dic₅₃	—	4	2	no	—
207	C₂×Dic₅₃	—	8	2	no	—
208	D₂₁₂	—	2	1	no	—
209	D₂₁₂	C₄×D₄	1	2	yes	—
210	C₄×D₅₃	—	2	2	no	—
211	C₄×D₅₃	—	2	2	no	—
212	C₄×D₅₃	—	4	2	no	—
213	C₄×D₅₃	—	4	2	no	—
214	Dic₁₀₆	—	2	1	no	—
215	Dic₁₀₆	—	2	4	no	—
216	Dic₁₀₆	—	2	4	no	—
217	Dic₁₀₆	—	4	2	no	—
218	Dic₁₀₆	—	4	2	no	—
219	Dic₁₀₆	C₄×D₄	1	2	yes	—
220	C₈×C₅₃	—	2	4	no	—
221	C₈×C₅₃	—	2	1	no	—
222	C₈×C₅₃	—	2	1	no	—
223	C₈×C₅₃	—	4	2	no	—
224	C₈×C₅₃	C₄×D₄	1	2	yes	—
225	C₈×C₅₃	C₄×D₄	1	2	yes	—
226	#88322	—	2	2	no	—
227	#88322	—	2	2	no	—
228	C₄×#473	—	53	1	no	maximal, Sylow
229	C₂²×C₅₃	—	2	1	no	—
230	C₂²×C₅₃	—	2	1	no	—
231	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
232	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
233	C₂²×C₅₃	#529	1	1	yes	characteristic
234	C₂×D₅₃	—	2	2	no	—
235	C₂×D₅₃	—	4	1	no	—
236	C₂×D₅₃	—	4	1	no	—
237	C₄×C₅₃	—	2	1	no	—
238	C₄×C₅₃	—	2	1	no	—
239	C₄×C₅₃	—	2	1	no	—
240	C₄×C₅₃	#473	1	2	yes	complemented
241	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
242	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
243	C₄×C₅₃	#525	1	1	yes	characteristic
244	C₄×C₅₃	#532	1	1	yes	characteristic
245	Dic₅₃	—	2	2	no	—
246	Dic₅₃	—	4	1	no	—
247	Dic₅₃	—	4	1	no	—
248	Dic₅₃	—	4	4	no	—
249	Dic₅₃	—	8	2	no	—
250	C₂×#473	—	53	1	no	—
251	C₄×#137	—	53	2	no	—
252	C₂×C₄×Q₁₆	—	53	1	no	—
253	C₄×#131	—	53	1	no	—
254	#1979	—	53	1	no	—
255	#1952	—	53	1	no	—
256	#1930	—	53	2	no	—
257	#1908	—	53	2	no	—
258	#1816	—	53	1	no	—
259	C₄×#108	D₅₃	1	1	yes	characteristic, complemented
260	C₄×#101	—	53	2	no	—
261	C₂×C₅₃	—	2	1	no	—
262	C₂×C₅₃	—	2	1	no	—
263	C₂×C₅₃	C₄×#121	1	1	yes	characteristic
264	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
265	C₂×C₅₃	#3067	1	1	yes	characteristic
266	D₅₃	—	4	2	no	—
267	C₂×#137	—	53	2	no	—
268	C₂²×Q₁₆	—	53	1	no	—
269	C₂×#131	—	53	1	no	—
270	C₂×#131	—	53	1	no	—
271	C₄×#41	—	53	1	no	—
272	C₂×C₄×Q₈	—	53	2	no	—
273	#473	—	53	8	no	—
274	#441	—	106	2	no	—
275	#440	—	106	2	no	—
276	C₄×Q₁₆	—	106	2	no	—
277	C₄×Q₁₆	—	106	1	no	—
278	C₄×Q₁₆	—	106	1	no	—
279	C₄×Q₁₆	—	106	1	no	—
280	C₄×Q₁₆	—	106	1	no	—
281	C₄×SD₁₆	—	106	2	no	—
282	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
283	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
284	#428	—	53	2	no	—
285	C₂×#107	—	53	1	no	—
286	#419	—	53	2	no	—
287	#417	—	53	2	no	—
288	C₂×#103	—	53	2	no	—
289	C₂×#101	—	53	2	no	—
290	C₂×#101	—	53	2	no	—
291	#405	—	53	1	no	—
292	#405	—	53	1	no	—
293	#405	—	106	1	no	—
294	#405	—	106	1	no	—
295	C₄×M₁₆	—	53	1	no	—
296	C₄×M₁₆	—	106	1	no	—
297	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
298	C₂×C₄×C₈	—	106	1	no	—
299	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
300	#343	—	53	2	no	—
301	#343	—	53	2	no	—
302	#341	C₂×D₅₃	1	2	yes	—
303	C₅₃	C₄×#473	1	1	yes	characteristic, complemented, Sylow
304	C₂×#41	—	53	1	no	—
305	C₂²×Q₈	—	53	2	no	—
306	#137	—	53	8	no	—
307	#137	—	106	4	no	—
308	C₂×Q₁₆	—	53	4	no	—
309	C₂×Q₁₆	—	106	2	no	—
310	C₂×Q₁₆	—	106	1	no	—
311	C₂×Q₁₆	—	106	1	no	—
312	C₂×Q₁₆	—	106	2	no	—
313	C₂×SD₁₆	—	106	2	no	—
314	#131	—	53	4	no	—
315	#131	—	106	2	no	—
316	#131	—	106	2	no	—
317	#131	—	106	2	no	—
318	C₂×M₁₆	—	53	1	no	—
319	C₂×M₁₆	—	53	1	no	—
320	C₂×M₁₆	—	106	1	no	—
321	C₂×M₁₆	—	106	1	no	—
322	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
323	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
324	C₂²×C₈	—	106	1	no	—
325	C₂²×C₈	—	106	1	no	—
326	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
327	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
328	C₄×Q₈	—	53	1	no	—
329	C₄×Q₈	—	53	1	no	—
330	C₄×Q₈	—	106	2	no	—
331	C₄×Q₈	—	106	2	no	—
332	C₄×Q₈	—	106	2	no	—
333	C₄×Q₈	—	106	2	no	—
334	C₄×D₄	—	53	1	no	—
335	C₄×D₄	—	53	1	no	—
336	C₄×D₄	—	106	1	no	—
337	C₄×D₄	—	106	1	no	—
338	#117	—	53	1	no	—
339	#117	—	106	1	no	—
340	C₂×C₄⋊C₄	—	53	2	no	—
341	C₂×C₄⋊C₄	—	106	2	no	—
342	C₂×C₄²	—	106	1	no	—
343	C₂×C₄²	—	106	2	no	—
344	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
345	#108	—	2	2	no	—
346	#108	C₄×D₅₃	1	4	yes	complemented
347	#107	—	106	2	no	—
348	#107	—	106	1	no	—
349	#107	—	106	1	no	—
350	#106	—	106	2	no	—
351	#103	—	106	2	no	—
352	#103	—	106	2	no	—
353	#103	—	106	2	no	—
354	#103	—	106	2	no	—
355	#103	—	106	2	no	—
356	#102	—	106	2	no	—
357	#101	—	53	8	no	—
358	#101	—	106	4	no	—
359	#97	—	2	2	no	—
360	#97	—	106	1	no	—
361	#97	—	106	1	no	—
362	#97	—	106	1	no	—
363	C₄×C₈	—	2	2	no	—
364	C₄×C₈	—	106	1	no	—
365	C₄×C₈	—	106	1	no	—
366	C₄×C₈	—	106	1	no	—
367	C₄×C₈	D₂₁₂	1	1	yes	characteristic
368	C₄×C₈	D₂₁₂	1	1	yes	characteristic
369	C₄×C₈	D₂₁₂	1	1	yes	characteristic
370	C₄×C₈	D₂₁₂	1	1	yes	characteristic
371	#41	—	53	2	no	—
372	#41	—	53	2	no	—
373	#41	—	106	1	no	—
374	#41	—	106	1	no	—
375	C₂×Q₈	—	53	4	no	—
376	C₂×Q₈	—	53	1	no	—
377	C₂×Q₈	—	106	2	no	—
378	C₂×Q₈	—	106	2	no	—
379	C₂×Q₈	—	106	2	no	—
380	C₂×Q₈	—	106	2	no	—
381	C₂×Q₈	—	106	2	no	—
382	C₂×D₄	—	53	1	no	—
383	C₂×D₄	—	106	1	no	—
384	C₂²×C₄	—	106	1	no	—
385	C₂²×C₄	—	106	2	no	—
386	C₂²×C₄	—	106	1	no	—
387	C₂²×C₄	—	106	1	no	—
388	C₂²×C₄	—	106	2	no	—
389	C₂²×C₄	#11719949	1	1	yes	characteristic
390	C₂²×C₄	#11719949	1	1	yes	characteristic
391	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
392	Q₁₆	—	106	8	no	—
393	Q₁₆	—	106	4	no	—
394	Q₁₆	—	106	4	no	—
395	SD₁₆	—	106	8	no	—
396	M₁₆	—	2	2	no	—
397	M₁₆	—	2	2	no	—
398	M₁₆	—	2	2	no	—
399	M₁₆	—	53	2	no	—
400	M₁₆	—	53	2	no	—
401	M₁₆	—	106	4	no	—
402	M₁₆	—	106	1	no	—
403	M₁₆	—	106	1	no	—
404	M₁₆	—	106	4	no	—
405	M₁₆	C₂×C₄×D₅₃	1	4	yes	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	106	4	no	—
411	C₂×C₈	—	106	1	no	—
412	C₂×C₈	—	106	1	no	—
413	C₂×C₈	—	106	4	no	—
414	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
415	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
416	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
417	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
418	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
419	C₂×C₈	#11719945	1	2	yes	—
420	C₄⋊C₄	—	53	1	no	—
421	C₄⋊C₄	—	53	1	no	—
422	C₄⋊C₄	—	106	1	no	—
423	C₄⋊C₄	—	106	1	no	—
424	C₄⋊C₄	—	106	2	no	—
425	C₄⋊C₄	—	106	2	no	—
426	C₄⋊C₄	—	212	2	no	—
427	C₄⋊C₄	—	212	2	no	—
428	C₂²⋊C₄	—	106	1	no	—
429	C₂²⋊C₄	—	106	1	no	—
430	C₂²⋊C₄	—	106	1	no	—
431	C₄²	—	106	1	no	—
432	C₄²	—	106	1	no	—
433	C₄²	—	106	1	no	—
434	C₄²	—	212	2	no	—
435	C₄²	—	212	2	no	—
436	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
437	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
438	C₄²	D₄×D₅₃	1	1	yes	characteristic
439	C₄²	D₄×D₅₃	1	1	yes	characteristic
440	C₂³	—	106	1	no	—
441	C₂³	C₂×#11719949	1	1	yes	characteristic
442	Q₈	—	53	2	no	—
443	Q₈	—	106	1	no	—
444	Q₈	—	106	4	no	—
445	Q₈	—	106	4	no	—
446	Q₈	—	212	2	no	—
447	Q₈	—	212	2	no	—
448	D₄	—	53	2	no	—
449	D₄	—	106	2	no	—
450	D₄	—	106	2	no	—
451	D₄	—	106	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	106	1	no	—
455	C₂×C₄	—	106	2	no	—
456	C₂×C₄	—	106	2	no	—
457	C₂×C₄	—	106	4	no	—
458	C₂×C₄	—	106	1	no	—
459	C₂×C₄	—	106	1	no	—
460	C₂×C₄	—	424	2	no	—
461	C₂×C₄	—	212	2	no	—
462	C₂×C₄	—	212	2	no	—
463	C₂×C₄	—	212	2	no	—
464	C₂×C₄	—	212	2	no	—
465	C₂×C₄	—	212	2	no	—
466	C₂×C₄	#13133192	1	1	yes	characteristic
467	C₂×C₄	#13133192	1	1	yes	characteristic
468	C₂×C₄	#13133192	1	1	yes	characteristic
469	C₂×C₄	#13133192	1	1	yes	characteristic
470	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
471	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
473	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
474	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
475	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
476	C₈	—	2	4	no	—
477	C₈	—	2	1	no	—
478	C₈	—	2	1	no	—
479	C₈	—	4	2	no	—
480	C₈	—	106	2	no	—
481	C₈	—	106	2	no	—
482	C₈	C₄×D₂₁₂	1	2	yes	—
483	C₈	C₄×D₂₁₂	1	2	yes	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	106	2	no	—
487	C₂²	—	212	1	no	—
488	C₂²	—	212	1	no	—
489	C₂²	C₄×#11719949	1	1	yes	characteristic
490	C₂²	C₂×#13133192	1	1	yes	characteristic
491	C₂²	#16811160	1	1	yes	characteristic
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	106	2	no	—
496	C₄	—	424	2	no	—
497	C₄	—	212	1	no	—
498	C₄	—	212	1	no	—
499	C₄	—	212	4	no	—
500	C₄	#16810899	1	2	yes	direct product factor
501	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
502	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
503	C₄	#16811135	1	1	yes	characteristic
504	C₄	#16811170	1	1	yes	characteristic
505	C₂	—	2	1	no	—
506	C₂	—	2	1	no	—
507	C₂	—	212	2	no	—
508	C₂	C₄×#13133192	1	1	yes	characteristic
509	C₂	C₂×#16810899	1	1	yes	characteristic
510	C₂	#20284436	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#11719945	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	C₄×#473
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₂₁₂	C₂	semidirect
5. #20275628	C₂	semidirect
6. #20275501	C₂	semidirect
7. #20275371	C₂	semidirect
10. C₄×#13133135	C₂	semidirect
11. C₄×#108×C₅₃	C₂	semidirect
18. #16810899	C₄	direct (N × H)
56. C₂×Dic₂₁₂	C₂×C₄	semidirect
95. #13133135	C₂×C₄	semidirect
108. #108×C₅₃	C₂×C₄	semidirect
240. C₄×C₅₃	#473	semidirect
259. C₄×#108	D₅₃	semidirect
303. C₅₃	C₄×#473	semidirect
346. #108	C₄×D₅₃	semidirect
500. C₄	#16810899	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810899	2
C₄×#13133250	2
C₂×C₄×Dic₂₁₂	2
C₄×#13133223	2
#20275628	2
#20275501	2
#20275371	2
#20275229	2
#20274776	2
C₄×#13133135	2
C₄×#108×C₅₃	2
C₄×#473	53

Subgroups — C4×#16810899 #27390985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810899 #27390985