Subgroups — C₄×#16811058 #27537408

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811058	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276589	C₂	1	2	yes	maximal, complemented
4	#20276577	C₂	1	2	yes	maximal, complemented
5	C₄×#13133160	C₂	1	2	yes	maximal, complemented
6	C₄×#13133158	C₂	1	2	yes	maximal, complemented
7	#20276504	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276430	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₂₁₂	—	2	1	no	—
14	#16811058	C₄	1	8	yes	direct product factor
15	C₂×#13133160	C₂²	1	2	yes	—
16	C₂×#13133158	C₂²	1	2	yes	—
17	C₂×#13133157	C₂²	1	1	yes	characteristic
18	C₂×#13133157	C₂²	1	1	yes	characteristic
19	C₄×D₄₂₄	—	2	1	no	—
20	C₄×D₄₂₄	—	2	1	no	—
21	C₄×D₄₂₄	C₂²	1	2	yes	—
22	C₄×D₄₂₄	C₂²	1	2	yes	—
23	C₂×#13133151	C₂²	1	1	yes	characteristic
24	C₂×#13133139	—	2	1	no	—
25	#16810450	C₂²	1	2	yes	complemented
26	#16810448	C₂²	1	2	yes	complemented
27	#16810427	C₂²	1	2	yes	complemented
28	#16810424	C₂²	1	2	yes	complemented
29	#16810423	C₂²	1	2	yes	—
30	#16810423	C₂²	1	2	yes	—
31	#16810399	C₂²	1	2	yes	complemented
32	C₄×#11719908	C₂²	1	2	yes	complemented
33	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×#41×C₅₃	—	2	1	no	—
35	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
36	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₄×SD₁₆×C₅₃	—	2	1	no	—
38	C₄×SD₁₆×C₅₃	—	2	1	no	—
39	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#427	C₂²	1	1	yes	characteristic
42	C₅₃×#416	—	2	1	no	—
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₂²×D₂₁₂	—	2	1	no	—
45	C₂²×C₄×D₅₃	—	4	1	no	—
46	C₂×D₄₂₄	—	2	1	no	—
47	C₂×D₄₂₄	—	2	2	no	—
48	C₂×D₄₂₄	C₂×C₄	1	4	yes	complemented
49	C₂×D₄₂₄	C₂³	1	2	yes	—
50	C₄×D₂₁₂	—	2	2	no	—
51	C₄×D₂₁₂	—	2	2	no	—
52	C₄×D₂₁₂	—	4	1	no	—
53	C₄×D₂₁₂	—	4	1	no	—
54	C₂×C₄⋊Dic₅₃	—	2	1	no	—
55	C₂×#11719921	—	4	1	no	—
56	#13133160	C₂×C₄	1	8	yes	complemented
57	#13133158	C₂×C₄	1	8	yes	complemented
58	#13133157	—	2	2	no	—
59	#13133157	C₂×C₄	1	4	yes	complemented
60	C₂×#11719908	C₂³	1	2	yes	—
61	C₂×#11719908	C₂³	1	2	yes	—
62	#13133151	—	2	1	no	—
63	#13133151	C₂³	1	2	yes	—
64	#13133139	—	2	2	no	—
65	#13133139	—	2	2	no	—
66	C₂×#41×C₅₃	—	2	1	no	—
67	#135×C₅₃	—	2	2	no	—
68	#135×C₅₃	C₂×C₄	1	4	yes	complemented
69	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×SD₁₆×C₅₃	—	2	1	no	—
71	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×Q₈×C₅₃	—	2	1	no	—
75	C₄×Q₈×C₅₃	—	2	1	no	—
76	C₄×D₄×C₅₃	—	2	1	no	—
77	C₄×D₄×C₅₃	—	2	1	no	—
78	C₄×D₄×C₅₃	—	4	1	no	—
79	C₄×D₄×C₅₃	—	4	1	no	—
80	#117×C₅₃	—	2	1	no	—
81	#117×C₅₃	—	4	1	no	—
82	C₂×C₄²×C₅₃	—	4	1	no	—
83	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
84	#107×C₅₃	C₂³	1	1	yes	characteristic
85	#107×C₅₃	C₂³	1	1	yes	characteristic
86	#106×C₅₃	—	2	1	no	—
87	#103×C₅₃	—	2	1	no	—
88	#103×C₅₃	—	2	1	no	—
89	#102×C₅₃	—	2	1	no	—
90	#102×C₅₃	—	2	1	no	—
91	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂³×D₅₃	—	4	1	no	—
96	#41×C₅₃	—	2	4	no	—
97	#41×C₅₃	—	4	2	no	—
98	C₂×Q₈×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	2	1	no	—
100	C₂×D₄×C₅₃	—	4	1	no	—
101	C₂²×C₄×C₅₃	—	4	1	no	—
102	C₂²×C₄×C₅₃	—	4	1	no	—
103	C₂²×C₄×C₅₃	—	4	1	no	—
104	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×Dic₅₃	—	4	1	no	—
108	C₂×D₂₁₂	—	2	4	no	—
109	C₂×D₂₁₂	—	2	2	no	—
110	C₂×D₂₁₂	—	4	1	no	—
111	C₂×D₂₁₂	—	4	2	no	—
112	C₂×C₄×D₅₃	—	4	2	no	—
113	C₂×C₄×D₅₃	—	4	2	no	—
114	C₂×C₄×D₅₃	—	8	4	no	—
115	Q₁₆×C₅₃	—	2	1	no	—
116	Q₁₆×C₅₃	C₂²×C₄	1	2	yes	complemented
117	SD₁₆×C₅₃	—	2	4	no	—
118	D₈×C₅₃	—	2	1	no	—
119	D₈×C₅₃	C₂²×C₄	1	2	yes	complemented
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₅₃	#41	1	2	yes	—
126	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₅₃	—	2	1	no	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	4	1	no	—
130	C₄⋊C₄×C₅₃	—	4	1	no	—
131	C₂²⋊C₄×C₅₃	—	4	1	no	—
132	C₂²⋊C₄×C₅₃	—	4	1	no	—
133	C₂²⋊C₄×C₅₃	—	4	1	no	—
134	C₄²×C₅₃	—	4	1	no	—
135	C₄²×C₅₃	—	4	1	no	—
136	C₄²×C₅₃	—	4	1	no	—
137	C₄²×C₅₃	D₈	1	1	yes	characteristic
138	C₄²×C₅₃	D₈	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	#11719921	—	4	2	no	—
142	#11719921	—	4	2	no	—
143	C₄⋊Dic₅₃	—	2	2	no	—
144	C₄⋊Dic₅₃	—	4	1	no	—
145	D₄₂₄	—	2	4	no	—
146	D₄₂₄	—	2	2	no	—
147	D₄₂₄	C₂²×C₄	1	4	yes	—
148	#11719908	—	2	2	no	—
149	#11719908	C₂²×C₄	1	4	yes	complemented
150	C₂³×C₅₃	—	4	1	no	—
151	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
152	C₂²×D₅₃	—	4	2	no	—
153	C₂²×D₅₃	—	4	4	no	—
154	C₂²×D₅₃	—	8	2	no	—
155	C₂²×D₅₃	—	8	2	no	—
156	Q₈×C₅₃	—	2	2	no	—
157	Q₈×C₅₃	—	4	1	no	—
158	D₄×C₅₃	—	2	2	no	—
159	D₄×C₅₃	—	4	4	no	—
160	D₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	4	1	no	—
163	C₂×C₄×C₅₃	—	4	1	no	—
164	C₂×C₄×C₅₃	—	4	4	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	—	4	1	no	—
167	C₂×C₄×C₅₃	—	8	2	no	—
168	C₂×C₄×C₅₃	—	8	2	no	—
169	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₅₃	#135	1	2	yes	—
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×Dic₅₃	—	4	2	no	—
178	C₂×Dic₅₃	—	8	2	no	—
179	D₂₁₂	—	2	4	no	—
180	D₂₁₂	—	4	2	no	—
181	D₂₁₂	—	4	4	no	—
182	C₄×D₅₃	—	8	4	no	—
183	C₄×D₅₃	—	8	4	no	—
184	C₈×C₅₃	—	2	1	no	—
185	C₈×C₅₃	—	2	1	no	—
186	C₈×C₅₃	C₄×D₄	1	2	yes	—
187	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
188	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
189	C₄×#509	—	53	1	no	maximal, Sylow
190	C₂²×C₅₃	—	2	1	no	—
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	4	2	no	—
193	C₂²×C₅₃	—	8	1	no	—
194	C₂²×C₅₃	—	8	1	no	—
195	C₂²×C₅₃	C₄×D₈	1	2	yes	—
196	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
197	C₂×D₅₃	—	4	4	no	—
198	C₂×D₅₃	—	8	8	no	—
199	C₂×D₅₃	—	8	2	no	—
200	C₂×D₅₃	—	8	2	no	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	4	2	no	—
205	C₄×C₅₃	—	8	1	no	—
206	C₄×C₅₃	—	8	1	no	—
207	C₄×C₅₃	C₄×D₈	1	2	yes	—
208	C₄×C₅₃	#509	1	2	yes	complemented
209	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
210	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
211	Dic₅₃	—	8	2	no	—
212	C₂×#509	—	53	1	no	—
213	C₄×#135	D₅₃	1	1	yes	characteristic, complemented
214	C₂×C₄×D₈	—	53	1	no	—
215	#2177	—	53	2	no	—
216	#2175	—	53	2	no	—
217	C₄×SD₃₂	—	53	2	no	—
218	C₄×D₁₆	—	53	2	no	—
219	#2159	—	53	1	no	—
220	#2140	—	53	1	no	—
221	#2139	—	53	1	no	—
222	C₄×M₃₂	—	53	1	no	—
223	C₂×C₅₃	—	2	2	no	—
224	C₂×C₅₃	—	8	2	no	—
225	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
226	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
227	C₂×C₅₃	#3414	1	1	yes	characteristic
228	D₅₃	—	8	4	no	—
229	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
230	C₂²×D₈	—	53	1	no	—
231	C₄×#41	—	2	1	no	—
232	C₂×C₄×D₄	—	106	1	no	—
233	#509	—	53	8	no	—
234	C₂×SD₃₂	—	53	2	no	—
235	C₂×D₁₆	—	53	2	no	—
236	C₂×M₃₂	—	53	1	no	—
237	C₂×M₃₂	—	53	1	no	—
238	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
239	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
240	C₄×SD₁₆	—	2	1	no	—
241	C₄×SD₁₆	—	2	1	no	—
242	C₄×D₈	—	53	2	no	—
243	C₄×D₈	—	53	2	no	—
244	C₄×D₈	—	106	1	no	—
245	C₄×D₈	—	106	1	no	—
246	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic, complemented
247	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic, complemented
248	#427	C₂×D₅₃	1	1	yes	characteristic
249	C₂×#107	—	53	1	no	—
250	#416	—	2	1	no	—
251	C₂×#102	—	106	1	no	—
252	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
253	#367	—	53	2	no	—
254	#366	—	53	2	no	—
255	#358	—	53	2	no	—
256	#357	—	53	2	no	—
257	#357	—	53	2	no	—
258	#357	—	53	2	no	—
259	#346	—	53	2	no	—
260	C₄×C₁₆	—	53	2	no	—
261	C₅₃	C₄×#509	1	1	yes	characteristic, complemented, Sylow
262	C₂×#41	—	2	1	no	—
263	C₂²×D₄	—	106	1	no	—
264	C₂³×C₄	—	212	1	no	—
265	#135	—	2	2	no	—
266	#135	C₄×D₅₃	1	4	yes	complemented
267	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
268	C₂×SD₁₆	—	2	1	no	—
269	C₂×D₈	—	53	4	no	—
270	C₂×D₈	—	53	2	no	—
271	C₂×D₈	—	106	2	no	—
272	C₂×D₈	—	106	1	no	—
273	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
274	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
275	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
276	C₄×Q₈	—	2	1	no	—
277	C₄×Q₈	—	2	1	no	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	—	4	1	no	—
281	C₄×D₄	—	4	1	no	—
282	C₄×D₄	—	106	2	no	—
283	C₄×D₄	—	106	2	no	—
284	C₄×D₄	—	212	1	no	—
285	C₄×D₄	—	212	1	no	—
286	#117	—	2	1	no	—
287	#117	—	4	1	no	—
288	C₂×C₄⋊C₄	—	106	1	no	—
289	C₂×C₂²⋊C₄	—	212	1	no	—
290	C₂×C₄²	—	4	1	no	—
291	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
292	SD₃₂	—	53	8	no	—
293	D₁₆	—	53	8	no	—
294	M₃₂	—	53	4	no	—
295	M₃₂	—	106	2	no	—
296	C₂×C₁₆	—	53	2	no	—
297	C₂×C₁₆	—	53	2	no	—
298	#107	—	53	2	no	—
299	#107	—	106	1	no	—
300	#107	C₂²×D₅₃	1	1	yes	characteristic
301	#107	C₂²×D₅₃	1	1	yes	characteristic
302	#106	—	2	1	no	—
303	#103	—	2	1	no	—
304	#103	—	2	1	no	—
305	#102	—	2	1	no	—
306	#102	—	2	1	no	—
307	#102	—	106	2	no	—
308	#102	—	106	2	no	—
309	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
310	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
311	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
312	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
313	C₂⁴	—	212	1	no	—
314	#41	—	2	4	no	—
315	#41	—	4	2	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	106	4	no	—
320	C₂×D₄	—	106	2	no	—
321	C₂×D₄	—	212	1	no	—
322	C₂×D₄	—	212	2	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	—	4	1	no	—
325	C₂²×C₄	—	4	1	no	—
326	C₂²×C₄	—	424	4	no	—
327	C₂²×C₄	—	212	1	no	—
328	C₂²×C₄	—	212	2	no	—
329	C₂²×C₄	—	212	2	no	—
330	C₂²×C₄	#11719922	1	1	yes	characteristic
331	C₂²×C₄	#11719922	1	1	yes	characteristic
332	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
333	Q₁₆	—	2	1	no	—
334	Q₁₆	C₂×C₄×D₅₃	1	2	yes	complemented
335	SD₁₆	—	2	4	no	—
336	D₈	—	2	1	no	—
337	D₈	—	53	4	no	—
338	D₈	—	106	4	no	—
339	D₈	—	106	2	no	—
340	D₈	C₂×C₄×D₅₃	1	2	yes	complemented
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
343	C₂×C₈	#11719945	1	2	yes	—
344	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
345	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
346	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
347	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₄⋊C₄	—	4	1	no	—
352	C₄⋊C₄	—	106	2	no	—
353	C₄⋊C₄	—	212	1	no	—
354	C₂²⋊C₄	—	4	1	no	—
355	C₂²⋊C₄	—	4	1	no	—
356	C₂²⋊C₄	—	4	1	no	—
357	C₂²⋊C₄	—	212	2	no	—
358	C₂²⋊C₄	—	212	2	no	—
359	C₄²	—	4	1	no	—
360	C₄²	—	4	1	no	—
361	C₄²	—	4	1	no	—
362	C₄²	#11719922	1	1	yes	characteristic
363	C₄²	#11719922	1	1	yes	characteristic
364	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
365	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
366	C₁₆	—	53	4	no	—
367	C₁₆	—	106	2	no	—
368	C₂³	—	4	1	no	—
369	C₂³	—	424	2	no	—
370	C₂³	—	424	2	no	—
371	C₂³	—	212	2	no	—
372	C₂³	—	212	4	no	—
373	C₂³	C₂×#11719922	1	1	yes	characteristic
374	Q₈	—	2	2	no	—
375	Q₈	—	4	1	no	—
376	D₄	—	2	2	no	—
377	D₄	—	4	4	no	—
378	D₄	—	4	1	no	—
379	D₄	—	106	4	no	—
380	D₄	—	212	2	no	—
381	D₄	—	212	4	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	—	424	4	no	—
391	C₂×C₄	—	424	2	no	—
392	C₂×C₄	—	424	4	no	—
393	C₂×C₄	—	212	2	no	—
394	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
395	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
396	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
397	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
398	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
399	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
400	C₂×C₄	#13133237	1	2	yes	—
401	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
402	C₈	—	2	1	no	—
403	C₈	—	2	1	no	—
404	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
405	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
406	C₈	C₂×#11719945	1	1	yes	characteristic
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	8	1	no	—
411	C₂²	—	8	1	no	—
412	C₂²	—	424	8	no	—
413	C₂²	—	424	2	no	—
414	C₂²	—	424	2	no	—
415	C₂²	—	212	4	no	—
416	C₂²	C₄×#11719922	1	2	yes	—
417	C₂²	C₂²×#11719922	1	1	yes	characteristic
418	C₄	—	2	1	no	—
419	C₄	—	2	1	no	—
420	C₄	—	2	1	no	—
421	C₄	—	4	2	no	—
422	C₄	—	8	1	no	—
423	C₄	—	8	1	no	—
424	C₄	—	424	2	no	—
425	C₄	C₄×#11719922	1	2	yes	—
426	C₄	#16811058	1	2	yes	direct product factor
427	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
428	C₄	C₂×#13133237	1	1	yes	characteristic
429	C₂	—	2	2	no	—
430	C₂	—	8	2	no	—
431	C₂	—	424	4	no	—
432	C₂	C₂×C₄×#11719922	1	1	yes	characteristic
433	C₂	C₂×#16811058	1	1	yes	characteristic
434	C₂	#20287849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811058	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	C₄×#509
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₂₄	C₂	semidirect
3. #20276589	C₂	semidirect
4. #20276577	C₂	semidirect
5. C₄×#13133160	C₂	semidirect
6. C₄×#13133158	C₂	semidirect
7. #20276504	C₂	semidirect
8. #20276430	C₂	semidirect
9. #20276427	C₂	semidirect
10. C₄×#13133157	C₂	semidirect
11. C₄×#135×C₅₃	C₂	semidirect
14. #16811058	C₄	direct (N × H)
25. #16810450	C₂²	semidirect
26. #16810448	C₂²	semidirect
27. #16810427	C₂²	semidirect
28. #16810424	C₂²	semidirect
31. #16810399	C₂²	semidirect
32. C₄×#11719908	C₂²	semidirect
35. C₄×Q₁₆×C₅₃	C₂²	semidirect
36. C₄×Q₁₆×C₅₃	C₂²	semidirect
39. C₄×D₈×C₅₃	C₂²	semidirect
40. C₄×D₈×C₅₃	C₂²	semidirect
48. C₂×D₄₂₄	C₂×C₄	semidirect
56. #13133160	C₂×C₄	semidirect
57. #13133158	C₂×C₄	semidirect
59. #13133157	C₂×C₄	semidirect
68. #135×C₅₃	C₂×C₄	semidirect
116. Q₁₆×C₅₃	C₂²×C₄	semidirect
119. D₈×C₅₃	C₂²×C₄	semidirect
149. #11719908	C₂²×C₄	semidirect
208. C₄×C₅₃	#509	semidirect
213. C₄×#135	D₅₃	semidirect
238. C₄×Q₁₆	C₂×D₅₃	semidirect
239. C₄×Q₁₆	C₂×D₅₃	semidirect
246. C₄×D₈	C₂×D₅₃	semidirect
247. C₄×D₈	C₂×D₅₃	semidirect
261. C₅₃	C₄×#509	semidirect
266. #135	C₄×D₅₃	semidirect
334. Q₁₆	C₂×C₄×D₅₃	semidirect
340. D₈	C₂×C₄×D₅₃	semidirect
426. C₄	#16811058	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×#16811058	2
C₂×C₄×D₄₂₄	2
#20276589	2
#20276577	2
C₄×#13133160	2
C₄×#13133158	2
#20276504	2
#20276430	2
#20276427	2
C₄×#13133157	2
C₄×#135×C₅₃	2
C₄×#509	53

Subgroups — C4×#16811058 #27537408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16811058 #27537408