Subgroups — C₄×#16811059 #27537410

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×#16811059	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276589	C₂	1	2	yes	maximal, complemented
4	#20276579	C₂	1	2	yes	maximal, complemented
5	C₄×#13133161	C₂	1	2	yes	maximal, complemented
6	C₄×#13133158	C₂	1	2	yes	maximal, complemented
7	#20276501	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276431	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276425	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₂²	1	1	yes	characteristic
13	C₄×#11719945	—	2	1	no	—
14	#16811059	C₄	1	8	yes	direct product factor
15	C₂×#13133161	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×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
20	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
21	C₄×#11719913	—	2	1	no	—
22	C₄×#11719913	—	2	1	no	—
23	C₄×D₄₂₄	C₂²	1	1	yes	characteristic, complemented
24	C₄×D₄₂₄	C₂²	1	1	yes	characteristic, complemented
25	#16810661	C₂²	1	1	yes	characteristic
26	#16810620	—	2	1	no	—
27	#16810450	C₂²	1	2	yes	complemented
28	#16810448	C₂²	1	2	yes	complemented
29	#16810426	C₂²	1	2	yes	complemented
30	#16810424	C₂²	1	2	yes	—
31	#16810424	C₂²	1	2	yes	—
32	#16810423	C₂²	1	2	yes	complemented
33	#16810399	C₂²	1	2	yes	complemented
34	C₄×#11719908	C₂²	1	2	yes	complemented
35	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₅₃	—	2	1	no	—
37	C₄×D₈×C₅₃	—	2	1	no	—
38	C₄×D₈×C₅₃	—	2	1	no	—
39	C₄×D₈×C₅₃	C₂²	1	2	yes	—
40	C₄×D₈×C₅₃	C₂²	1	2	yes	—
41	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#102×C₅₃	—	2	1	no	—
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#11719945	—	2	1	no	—
45	#13133239	—	2	2	no	—
46	#13133239	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
48	C₂×#11719913	—	2	1	no	—
49	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
50	C₄×Dic₁₀₆	—	2	1	no	—
51	C₄×Dic₁₀₆	—	2	1	no	—
52	C₄×D₂₁₂	—	2	1	no	—
53	C₄×D₂₁₂	—	2	1	no	—
54	C₄×C₂²⋊D₅₃	—	4	1	no	—
55	C₄×C₂²⋊D₅₃	—	4	1	no	—
56	#13133176	—	2	1	no	—
57	#13133174	—	4	1	no	—
58	C₄²×D₅₃	—	4	1	no	—
59	#13133161	C₂×C₄	1	8	yes	complemented
60	#13133158	C₂×C₄	1	8	yes	complemented
61	#13133157	—	2	2	no	—
62	#13133157	C₂×C₄	1	4	yes	complemented
63	C₂×#11719908	C₂³	1	2	yes	—
64	C₂×#11719908	C₂³	1	2	yes	—
65	#13133151	C₂³	1	1	yes	characteristic
66	#13133151	C₂³	1	1	yes	characteristic
67	#13133149	—	2	1	no	—
68	#13133142	—	2	1	no	—
69	#13133142	—	2	1	no	—
70	#13133139	—	2	1	no	—
71	#13133139	—	2	1	no	—
72	C₂²×D₄×C₅₃	—	2	1	no	—
73	C₂³×C₄×C₅₃	—	4	1	no	—
74	C₂×D₈×C₅₃	—	2	1	no	—
75	C₂×D₈×C₅₃	—	2	2	no	—
76	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
77	C₂×D₈×C₅₃	C₂³	1	2	yes	—
78	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₄×D₄×C₅₃	—	2	2	no	—
81	C₄×D₄×C₅₃	—	2	2	no	—
82	C₄×D₄×C₅₃	—	4	1	no	—
83	C₄×D₄×C₅₃	—	4	1	no	—
84	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
86	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
87	#107×C₅₃	—	2	1	no	—
88	#107×C₅₃	C₂³	1	2	yes	—
89	#102×C₅₃	—	2	2	no	—
90	#102×C₅₃	—	2	2	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	C₂⁴×C₅₃	—	4	1	no	—
96	C₂×D₄×C₅₃	—	2	4	no	—
97	C₂×D₄×C₅₃	—	2	2	no	—
98	C₂×D₄×C₅₃	—	4	1	no	—
99	C₂×D₄×C₅₃	—	4	2	no	—
100	C₂²×C₄×C₅₃	—	4	1	no	—
101	C₂²×C₄×C₅₃	—	4	2	no	—
102	C₂²×C₄×C₅₃	—	4	2	no	—
103	C₂²×C₄×C₅₃	—	8	4	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	C₂×C₂²⋊D₅₃	—	4	1	no	—
108	#11719945	—	2	4	no	—
109	#11719945	—	4	2	no	—
110	C₂×D₂₁₂	—	2	1	no	—
111	C₂×C₄×D₅₃	—	4	1	no	—
112	C₂×C₄×D₅₃	—	4	1	no	—
113	C₂×C₄×D₅₃	—	4	1	no	—
114	C₂×Dic₁₀₆	—	2	1	no	—
115	D₈×C₅₃	—	2	4	no	—
116	D₈×C₅₃	—	2	2	no	—
117	D₈×C₅₃	C₂²×C₄	1	4	yes	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	#41	1	2	yes	—
124	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	—	4	1	no	—
127	C₂²⋊C₄×C₅₃	—	4	2	no	—
128	C₂²⋊C₄×C₅₃	—	4	2	no	—
129	C₄²×C₅₃	D₈	1	1	yes	characteristic
130	C₄²×C₅₃	D₈	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂²⋊Dic₅₃	—	4	1	no	—
134	#11719921	—	4	1	no	—
135	#11719921	—	4	1	no	—
136	C₄⋊Dic₅₃	—	2	1	no	—
137	C₄⋊Dic₅₃	—	2	1	no	—
138	#11719919	—	4	1	no	—
139	#11719919	—	4	1	no	—
140	C₄×Dic₅₃	—	4	1	no	—
141	C₄×Dic₅₃	—	4	1	no	—
142	C₄×Dic₅₃	—	4	1	no	—
143	Dic₂₁₂	—	2	1	no	—
144	Dic₂₁₂	C₂²×C₄	1	2	yes	complemented
145	D₄₂₄	—	2	1	no	—
146	D₄₂₄	C₂²×C₄	1	2	yes	complemented
147	#11719913	—	2	4	no	—
148	#11719908	—	2	2	no	—
149	#11719908	C₂²×C₄	1	4	yes	complemented
150	C₂³×C₅₃	—	4	2	no	—
151	C₂³×C₅₃	—	4	4	no	—
152	C₂³×C₅₃	—	8	2	no	—
153	C₂³×C₅₃	—	8	2	no	—
154	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
155	C₂²×D₅₃	—	4	1	no	—
156	D₄×C₅₃	—	2	4	no	—
157	D₄×C₅₃	—	4	2	no	—
158	D₄×C₅₃	—	4	4	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	8	4	no	—
163	C₂×C₄×C₅₃	—	8	2	no	—
164	C₂×C₄×C₅₃	—	8	4	no	—
165	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₅₃	#135	1	2	yes	—
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂²⋊D₅₃	—	4	4	no	—
174	C₂×Dic₅₃	—	4	1	no	—
175	C₂×Dic₅₃	—	4	1	no	—
176	C₂×Dic₅₃	—	4	1	no	—
177	D₂₁₂	—	2	2	no	—
178	D₂₁₂	—	4	1	no	—
179	C₄×D₅₃	—	4	4	no	—
180	C₄×D₅₃	—	8	2	no	—
181	C₄×D₅₃	—	8	2	no	—
182	Dic₁₀₆	—	2	2	no	—
183	Dic₁₀₆	—	4	1	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	4	no	—
193	C₂²×C₅₃	—	8	8	no	—
194	C₂²×C₅₃	—	8	2	no	—
195	C₂²×C₅₃	—	8	2	no	—
196	C₂²×C₅₃	C₄×D₈	1	2	yes	—
197	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
198	C₂×D₅₃	—	4	2	no	—
199	C₂×D₅₃	—	8	1	no	—
200	C₂×D₅₃	—	8	1	no	—
201	C₄×C₅₃	—	2	1	no	—
202	C₄×C₅₃	—	2	1	no	—
203	C₄×C₅₃	—	2	1	no	—
204	C₄×C₅₃	—	8	2	no	—
205	C₄×C₅₃	C₄×D₈	1	2	yes	—
206	C₄×C₅₃	#509	1	2	yes	complemented
207	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
208	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
209	Dic₅₃	—	4	2	no	—
210	Dic₅₃	—	8	1	no	—
211	Dic₅₃	—	8	1	no	—
212	C₂×#509	—	53	1	no	—
213	C₄×#135	—	53	1	no	—
214	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic, complemented
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	4	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	2	no	—
229	C₂×#135	—	53	1	no	—
230	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
231	C₄×#41	—	106	1	no	—
232	C₂×C₄×D₄	—	2	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	—	53	1	no	—
239	C₄×Q₁₆	—	53	1	no	—
240	C₄×SD₁₆	—	106	1	no	—
241	C₄×SD₁₆	—	106	1	no	—
242	C₄×D₈	—	2	1	no	—
243	C₄×D₈	—	2	1	no	—
244	C₄×D₈	—	53	1	no	—
245	C₄×D₈	—	53	1	no	—
246	C₄×D₈	C₂×D₅₃	1	2	yes	—
247	C₄×D₈	C₂×D₅₃	1	2	yes	—
248	#427	—	53	1	no	—
249	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
250	#416	—	106	1	no	—
251	C₂×#102	—	2	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	—	106	1	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂³×C₄	—	4	1	no	—
265	#135	—	53	4	no	—
266	#135	—	106	2	no	—
267	C₂×Q₁₆	—	53	1	no	—
268	C₂×SD₁₆	—	106	1	no	—
269	C₂×D₈	—	2	2	no	—
270	C₂×D₈	—	2	1	no	—
271	C₂×D₈	—	53	1	no	—
272	C₂×D₈	C₄×D₅₃	1	4	yes	complemented
273	C₂×D₈	C₂²×D₅₃	1	2	yes	—
274	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
275	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
276	C₄×Q₈	—	106	1	no	—
277	C₄×Q₈	—	106	1	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	—	4	1	no	—
281	C₄×D₄	—	4	1	no	—
282	C₄×D₄	—	106	1	no	—
283	C₄×D₄	—	106	1	no	—
284	C₄×D₄	—	212	1	no	—
285	C₄×D₄	—	212	1	no	—
286	#117	—	106	1	no	—
287	#117	—	212	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	4	1	no	—
290	C₂×C₄²	—	212	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	—	2	1	no	—
299	#107	—	53	1	no	—
300	#107	—	53	1	no	—
301	#107	C₂²×D₅₃	1	2	yes	—
302	#106	—	106	1	no	—
303	#103	—	106	1	no	—
304	#103	—	106	1	no	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	#102	—	106	1	no	—
308	#102	—	106	1	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	—	4	1	no	—
314	#41	—	106	4	no	—
315	#41	—	212	2	no	—
316	C₂×Q₈	—	106	1	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	2	no	—
321	C₂×D₄	—	106	1	no	—
322	C₂×D₄	—	212	1	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	—	4	2	no	—
325	C₂²×C₄	—	4	2	no	—
326	C₂²×C₄	—	8	4	no	—
327	C₂²×C₄	—	212	1	no	—
328	C₂²×C₄	—	212	1	no	—
329	C₂²×C₄	—	212	1	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	—	53	2	no	—
334	Q₁₆	—	106	1	no	—
335	SD₁₆	—	106	4	no	—
336	D₈	—	2	4	no	—
337	D₈	—	2	2	no	—
338	D₈	—	53	2	no	—
339	D₈	—	106	1	no	—
340	D₈	C₂×C₄×D₅₃	1	4	yes	—
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	2	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	106	1	no	—
351	C₄⋊C₄	—	106	1	no	—
352	C₄⋊C₄	—	212	1	no	—
353	C₄⋊C₄	—	212	1	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₂²⋊C₄	—	212	1	no	—
357	C₂²⋊C₄	—	212	1	no	—
358	C₂²⋊C₄	—	212	1	no	—
359	C₄²	—	212	1	no	—
360	C₄²	—	212	1	no	—
361	C₄²	—	212	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	2	no	—
369	C₂³	—	4	4	no	—
370	C₂³	—	8	2	no	—
371	C₂³	—	8	2	no	—
372	C₂³	—	212	1	no	—
373	C₂³	C₂×#11719922	1	1	yes	characteristic
374	Q₈	—	106	2	no	—
375	Q₈	—	212	1	no	—
376	D₄	—	2	4	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	4	no	—
379	D₄	—	106	2	no	—
380	D₄	—	212	4	no	—
381	D₄	—	212	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	8	4	no	—
386	C₂×C₄	—	8	2	no	—
387	C₂×C₄	—	8	4	no	—
388	C₂×C₄	—	424	2	no	—
389	C₂×C₄	—	424	2	no	—
390	C₂×C₄	—	212	1	no	—
391	C₂×C₄	—	212	1	no	—
392	C₂×C₄	—	212	1	no	—
393	C₂×C₄	—	212	4	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	4	no	—
410	C₂²	—	8	8	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	8	2	no	—
413	C₂²	—	424	1	no	—
414	C₂²	—	424	1	no	—
415	C₂²	—	212	2	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	—	8	2	no	—
422	C₄	—	424	1	no	—
423	C₄	—	424	1	no	—
424	C₄	—	212	2	no	—
425	C₄	C₄×#11719922	1	2	yes	—
426	C₄	#16811059	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	4	no	—
431	C₂	—	424	2	no	—
432	C₂	C₂×C₄×#11719922	1	1	yes	characteristic
433	C₂	C₂×#16811059	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	C₂×C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811059	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	C₄×#509
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×#13133239	C₂	semidirect
3. #20276589	C₂	semidirect
4. #20276579	C₂	semidirect
5. C₄×#13133161	C₂	semidirect
6. C₄×#13133158	C₂	semidirect
7. #20276501	C₂	semidirect
8. #20276431	C₂	semidirect
9. #20276425	C₂	semidirect
10. C₄×#13133157	C₂	semidirect
11. C₂×C₄×D₈×C₅₃	C₂	semidirect
14. #16811059	C₄	direct (N × H)
19. C₄×Dic₂₁₂	C₂²	semidirect
20. C₄×Dic₂₁₂	C₂²	semidirect
23. C₄×D₄₂₄	C₂²	semidirect
24. C₄×D₄₂₄	C₂²	semidirect
27. #16810450	C₂²	semidirect
28. #16810448	C₂²	semidirect
29. #16810426	C₂²	semidirect
32. #16810423	C₂²	semidirect
33. #16810399	C₂²	semidirect
34. C₄×#11719908	C₂²	semidirect
46. #13133239	C₂×C₄	semidirect
59. #13133161	C₂×C₄	semidirect
60. #13133158	C₂×C₄	semidirect
62. #13133157	C₂×C₄	semidirect
76. C₂×D₈×C₅₃	C₂×C₄	semidirect
144. Dic₂₁₂	C₂²×C₄	semidirect
146. D₄₂₄	C₂²×C₄	semidirect
149. #11719908	C₂²×C₄	semidirect
206. C₄×C₅₃	#509	semidirect
214. C₂×C₄×D₈	D₅₃	semidirect
261. C₅₃	C₄×#509	semidirect
272. C₂×D₈	C₄×D₅₃	semidirect
426. C₄	#16811059	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×#16811059	2
C₄×#13133239	2
#20276589	2
#20276579	2
C₄×#13133161	2
C₄×#13133158	2
#20276501	2
#20276431	2
#20276425	2
C₄×#13133157	2
C₂×C₄×D₈×C₅₃	2
C₄×#509	53

Subgroups — C4×#16811059 #27537410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16811059 #27537410