Subgroups — C₄×#16811064 #27537421

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811064	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
3	#20276584	C₂	1	2	yes	maximal, complemented
4	#20276577	C₂	1	2	yes	maximal, complemented
5	C₄×#13133163	C₂	1	2	yes	maximal, complemented
6	C₄×#13133160	C₂	1	2	yes	maximal, complemented
7	#20276501	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276435	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276425	C₂	1	1	yes	maximal, characteristic
10	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₂²	1	1	yes	characteristic
13	C₄×#11719945	—	2	1	no	—
14	#16811064	C₄	1	8	yes	direct product factor
15	C₂×#13133163	C₂²	1	2	yes	—
16	C₂×#13133160	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
20	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
21	C₄×#11719913	—	2	1	no	—
22	C₄×#11719913	—	2	1	no	—
23	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
24	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
25	#16810661	C₂²	1	1	yes	characteristic
26	#16810620	—	2	1	no	—
27	#16810450	C₂²	1	2	yes	—
28	#16810448	C₂²	1	2	yes	—
29	#16810427	C₂²	1	2	yes	—
30	#16810427	C₂²	1	2	yes	—
31	#16810426	C₂²	1	2	yes	—
32	#16810423	C₂²	1	2	yes	—
33	#16810399	C₂²	1	2	yes	—
34	C₄×#11719908	C₂²	1	2	yes	—
35	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
37	C₄×Q₁₆×C₅₃	—	2	1	no	—
38	C₄×Q₁₆×C₅₃	—	2	1	no	—
39	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
40	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#103×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	—
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	#13133163	C₂×C₄	1	8	yes	complemented
60	#13133160	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×Q₈×C₅₃	—	2	1	no	—
73	C₂×Q₁₆×C₅₃	—	2	1	no	—
74	C₂×Q₁₆×C₅₃	—	2	2	no	—
75	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
76	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×Q₈×C₅₃	—	2	2	no	—
80	C₄×Q₈×C₅₃	—	2	2	no	—
81	C₄×Q₈×C₅₃	—	4	1	no	—
82	C₄×Q₈×C₅₃	—	4	1	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
85	C₂×C₄²×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	#103×C₅₃	—	2	2	no	—
90	#103×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	C₂×Q₈×C₅₃	—	2	4	no	—
96	C₂×Q₈×C₅₃	—	2	2	no	—
97	C₂×Q₈×C₅₃	—	4	1	no	—
98	C₂×Q₈×C₅₃	—	4	2	no	—
99	C₂²×C₄×C₅₃	—	4	1	no	—
100	C₂²×C₄×C₅₃	—	4	1	no	—
101	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₂²⋊D₅₃	—	4	1	no	—
105	#11719945	—	2	4	no	—
106	#11719945	—	4	2	no	—
107	C₂×D₂₁₂	—	2	1	no	—
108	C₂×C₄×D₅₃	—	4	1	no	—
109	C₂×C₄×D₅₃	—	4	1	no	—
110	C₂×C₄×D₅₃	—	4	1	no	—
111	C₂×Dic₁₀₆	—	2	1	no	—
112	Q₁₆×C₅₃	—	2	4	no	—
113	Q₁₆×C₅₃	—	2	2	no	—
114	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	#41	1	2	yes	—
121	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	—	4	1	no	—
124	C₄⋊C₄×C₅₃	—	4	2	no	—
125	C₄⋊C₄×C₅₃	—	4	2	no	—
126	C₄²×C₅₃	—	4	2	no	—
127	C₄²×C₅₃	—	4	2	no	—
128	C₄²×C₅₃	D₈	1	1	yes	characteristic
129	C₄²×C₅₃	D₈	1	1	yes	characteristic
130	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²⋊Dic₅₃	—	4	1	no	—
133	#11719921	—	4	1	no	—
134	#11719921	—	4	1	no	—
135	C₄⋊Dic₅₃	—	2	1	no	—
136	C₄⋊Dic₅₃	—	2	1	no	—
137	#11719919	—	4	1	no	—
138	#11719919	—	4	1	no	—
139	C₄×Dic₅₃	—	4	1	no	—
140	C₄×Dic₅₃	—	4	1	no	—
141	C₄×Dic₅₃	—	4	1	no	—
142	Dic₂₁₂	—	2	1	no	—
143	Dic₂₁₂	C₂²×C₄	1	2	yes	—
144	D₄₂₄	—	2	1	no	—
145	D₄₂₄	C₂²×C₄	1	2	yes	—
146	#11719913	—	2	4	no	—
147	#11719908	—	2	2	no	—
148	#11719908	C₂²×C₄	1	4	yes	—
149	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
150	C₂²×D₅₃	—	4	1	no	—
151	Q₈×C₅₃	—	2	4	no	—
152	Q₈×C₅₃	—	4	2	no	—
153	Q₈×C₅₃	—	4	4	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	4	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	8	2	no	—
160	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
161	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅₃	#135	1	2	yes	—
167	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
168	C₂²⋊D₅₃	—	4	4	no	—
169	C₂×Dic₅₃	—	4	1	no	—
170	C₂×Dic₅₃	—	4	1	no	—
171	C₂×Dic₅₃	—	4	1	no	—
172	D₂₁₂	—	2	2	no	—
173	D₂₁₂	—	4	1	no	—
174	C₄×D₅₃	—	4	4	no	—
175	C₄×D₅₃	—	8	2	no	—
176	C₄×D₅₃	—	8	2	no	—
177	Dic₁₀₆	—	2	2	no	—
178	Dic₁₀₆	—	4	1	no	—
179	C₈×C₅₃	—	2	1	no	—
180	C₈×C₅₃	—	2	1	no	—
181	C₈×C₅₃	C₄×D₄	1	2	yes	—
182	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
183	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
184	C₄×#510	—	53	1	no	maximal, Sylow
185	C₂²×C₅₃	—	2	1	no	—
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	C₄×D₈	1	2	yes	—
188	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
189	C₂×D₅₃	—	4	2	no	—
190	C₂×D₅₃	—	8	1	no	—
191	C₂×D₅₃	—	8	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	4	4	no	—
196	C₄×C₅₃	—	8	2	no	—
197	C₄×C₅₃	C₄×D₈	1	2	yes	—
198	C₄×C₅₃	#510	1	2	yes	complemented
199	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
201	Dic₅₃	—	4	2	no	—
202	Dic₅₃	—	8	1	no	—
203	Dic₅₃	—	8	1	no	—
204	C₂×#510	—	53	1	no	—
205	C₄×#135	—	53	1	no	—
206	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
207	#2176	—	53	2	no	—
208	#2175	—	53	2	no	—
209	C₄×Q₃₂	—	53	2	no	—
210	C₄×SD₃₂	—	53	2	no	—
211	#2159	—	53	1	no	—
212	#2141	—	53	1	no	—
213	#2139	—	53	1	no	—
214	C₄×M₃₂	—	53	1	no	—
215	C₂×C₅₃	—	2	2	no	—
216	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
217	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
218	C₂×C₅₃	#3414	1	1	yes	characteristic
219	D₅₃	—	8	2	no	—
220	C₂×#135	—	53	1	no	—
221	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
222	C₄×#41	—	106	1	no	—
223	C₂×C₄×Q₈	—	2	1	no	—
224	#510	—	53	8	no	—
225	C₂×Q₃₂	—	53	2	no	—
226	C₂×SD₃₂	—	53	2	no	—
227	C₂×M₃₂	—	53	1	no	—
228	C₂×M₃₂	—	53	1	no	—
229	C₄×Q₁₆	—	2	1	no	—
230	C₄×Q₁₆	—	2	1	no	—
231	C₄×Q₁₆	—	53	1	no	—
232	C₄×Q₁₆	—	53	1	no	—
233	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
234	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
235	C₄×SD₁₆	—	106	1	no	—
236	C₄×SD₁₆	—	106	1	no	—
237	C₄×D₈	—	53	1	no	—
238	C₄×D₈	—	53	1	no	—
239	#427	—	53	1	no	—
240	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
241	#416	—	106	1	no	—
242	C₂×#103	—	2	1	no	—
243	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
244	#367	—	53	2	no	—
245	#366	—	53	2	no	—
246	#358	—	53	2	no	—
247	#358	—	53	2	no	—
248	#358	—	53	2	no	—
249	#357	—	53	2	no	—
250	#346	—	53	2	no	—
251	C₄×C₁₆	—	53	2	no	—
252	C₅₃	C₄×#510	1	1	yes	characteristic, complemented, Sylow
253	C₂×#41	—	106	1	no	—
254	C₂²×Q₈	—	2	1	no	—
255	#135	—	53	4	no	—
256	#135	—	106	2	no	—
257	C₂×Q₁₆	—	2	2	no	—
258	C₂×Q₁₆	—	2	1	no	—
259	C₂×Q₁₆	—	53	1	no	—
260	C₂×Q₁₆	C₄×D₅₃	1	4	yes	complemented
261	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
262	C₂×SD₁₆	—	106	1	no	—
263	C₂×D₈	—	53	1	no	—
264	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
265	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
266	C₄×Q₈	—	2	2	no	—
267	C₄×Q₈	—	2	2	no	—
268	C₄×Q₈	—	4	1	no	—
269	C₄×Q₈	—	4	1	no	—
270	C₄×Q₈	—	106	1	no	—
271	C₄×Q₈	—	106	1	no	—
272	C₄×D₄	—	106	1	no	—
273	C₄×D₄	—	106	1	no	—
274	C₄×D₄	—	212	1	no	—
275	C₄×D₄	—	212	1	no	—
276	#117	—	106	1	no	—
277	#117	—	212	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	4	1	no	—
280	C₂×C₄²	—	4	1	no	—
281	C₂×C₄²	—	212	1	no	—
282	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
283	Q₃₂	—	53	8	no	—
284	SD₃₂	—	53	8	no	—
285	M₃₂	—	53	4	no	—
286	M₃₂	—	106	2	no	—
287	C₂×C₁₆	—	53	2	no	—
288	C₂×C₁₆	—	53	2	no	—
289	#107	—	2	1	no	—
290	#107	—	53	1	no	—
291	#107	—	53	1	no	—
292	#107	C₂²×D₅₃	1	2	yes	—
293	#106	—	106	1	no	—
294	#103	—	2	2	no	—
295	#103	—	2	2	no	—
296	#103	—	106	1	no	—
297	#103	—	106	1	no	—
298	#102	—	106	1	no	—
299	#102	—	106	1	no	—
300	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
301	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
302	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
303	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
304	#41	—	106	4	no	—
305	#41	—	212	2	no	—
306	C₂×Q₈	—	2	4	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	4	1	no	—
309	C₂×Q₈	—	4	2	no	—
310	C₂×Q₈	—	106	1	no	—
311	C₂×D₄	—	106	1	no	—
312	C₂×D₄	—	212	1	no	—
313	C₂²×C₄	—	4	1	no	—
314	C₂²×C₄	—	4	1	no	—
315	C₂²×C₄	—	212	1	no	—
316	C₂²×C₄	—	212	1	no	—
317	C₂²×C₄	—	212	1	no	—
318	C₂²×C₄	#11719922	1	1	yes	characteristic
319	C₂²×C₄	#11719922	1	1	yes	characteristic
320	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
321	Q₁₆	—	2	4	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	53	2	no	—
324	Q₁₆	—	106	1	no	—
325	Q₁₆	C₂×C₄×D₅₃	1	4	yes	—
326	SD₁₆	—	106	4	no	—
327	D₈	—	53	2	no	—
328	D₈	—	106	1	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
331	C₂×C₈	#11719945	1	2	yes	—
332	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
333	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
334	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
335	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	1	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	106	1	no	—
341	C₄⋊C₄	—	106	1	no	—
342	C₄⋊C₄	—	212	1	no	—
343	C₄⋊C₄	—	212	1	no	—
344	C₂²⋊C₄	—	212	1	no	—
345	C₂²⋊C₄	—	212	1	no	—
346	C₂²⋊C₄	—	212	1	no	—
347	C₄²	—	4	2	no	—
348	C₄²	—	4	2	no	—
349	C₄²	—	212	1	no	—
350	C₄²	—	212	1	no	—
351	C₄²	—	212	1	no	—
352	C₄²	#11719922	1	1	yes	characteristic
353	C₄²	#11719922	1	1	yes	characteristic
354	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
355	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
356	C₁₆	—	53	4	no	—
357	C₁₆	—	106	2	no	—
358	C₂³	—	212	1	no	—
359	C₂³	C₂×#11719922	1	1	yes	characteristic
360	Q₈	—	2	4	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	4	4	no	—
363	Q₈	—	106	2	no	—
364	Q₈	—	212	1	no	—
365	D₄	—	106	2	no	—
366	D₄	—	212	4	no	—
367	D₄	—	212	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	8	2	no	—
374	C₂×C₄	—	424	2	no	—
375	C₂×C₄	—	424	2	no	—
376	C₂×C₄	—	212	1	no	—
377	C₂×C₄	—	212	1	no	—
378	C₂×C₄	—	212	1	no	—
379	C₂×C₄	—	212	4	no	—
380	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
381	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
382	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
383	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
384	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
385	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
386	C₂×C₄	#13133237	1	2	yes	—
387	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
391	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
392	C₈	C₂×#11719945	1	1	yes	characteristic
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	424	1	no	—
396	C₂²	—	424	1	no	—
397	C₂²	—	212	2	no	—
398	C₂²	C₄×#11719922	1	2	yes	—
399	C₂²	C₂²×#11719922	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	4	4	no	—
404	C₄	—	8	2	no	—
405	C₄	—	424	1	no	—
406	C₄	—	424	1	no	—
407	C₄	—	212	2	no	—
408	C₄	C₄×#11719922	1	2	yes	—
409	C₄	#16811064	1	2	yes	direct product factor
410	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
411	C₄	C₂×#13133237	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	424	2	no	—
414	C₂	C₂×C₄×#11719922	1	1	yes	characteristic
415	C₂	C₂×#16811064	1	1	yes	characteristic
416	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×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133239	C₂²
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	C₄×#510
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20276584	C₂	semidirect
4. #20276577	C₂	semidirect
5. C₄×#13133163	C₂	semidirect
6. C₄×#13133160	C₂	semidirect
7. #20276501	C₂	semidirect
8. #20276435	C₂	semidirect
10. C₄×#13133157	C₂	semidirect
11. C₂×C₄×Q₁₆×C₅₃	C₂	semidirect
14. #16811064	C₄	direct (N × H)
59. #13133163	C₂×C₄	semidirect
60. #13133160	C₂×C₄	semidirect
62. #13133157	C₂×C₄	semidirect
75. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
198. C₄×C₅₃	#510	semidirect
206. C₂×C₄×Q₁₆	D₅₃	semidirect
252. C₅₃	C₄×#510	semidirect
260. C₂×Q₁₆	C₄×D₅₃	semidirect
409. C₄	#16811064	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×#16811064	2
C₄×#13133239	2
#20276584	2
#20276577	2
C₄×#13133163	2
C₄×#13133160	2
#20276501	2
#20276435	2
#20276425	2
C₄×#13133157	2
C₂×C₄×Q₁₆×C₅₃	2
C₄×#510	53

Subgroups — C4×#16811064 #27537421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16811064 #27537421