Subgroups — #27353140

Order 13568.

666 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	#20283832	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133224	C₂	1	1	yes	maximal, characteristic
3	#20273483	C₂	1	2	yes	maximal, complemented
4	#20273480	C₂	1	2	yes	maximal, complemented
5	#20273476	C₂	1	2	yes	maximal, complemented
6	#20273473	C₂	1	2	yes	maximal, complemented
7	#20273468	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273465	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273461	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16810375	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2989	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133224	C₄	1	1	yes	characteristic
13	C₂×#13133224	C₄	1	1	yes	characteristic
14	#16810754	—	2	1	no	—
15	#16810701	—	2	1	no	—
16	#16810695	—	2	2	no	—
17	#16810680	C₂²	1	2	yes	—
18	#16810676	C₂²	1	1	yes	characteristic
19	#16810647	C₂²	1	2	yes	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	#16810579	C₄	1	2	yes	—
22	#16810579	C₄	1	2	yes	—
23	#16810579	C₄	1	1	yes	characteristic
24	#16810579	C₄	1	1	yes	characteristic
25	C₄×#11719917	C₂²	1	1	yes	characteristic
26	C₄×#11719917	C₂²	1	2	yes	—
27	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
28	C₂×C₄×#88322	C₂²	1	2	yes	—
29	#16810375	C₂²	1	2	yes	—
30	#16810375	C₂²	1	2	yes	—
31	#16810375	C₂²	1	1	yes	characteristic
32	#16810375	C₂²	1	1	yes	characteristic
33	#16810375	C₂²	1	2	yes	—
34	#16810375	C₂²	1	2	yes	—
35	#16810373	C₂²	1	1	yes	characteristic
36	#16810373	C₂²	1	1	yes	characteristic
37	#16810373	C₂²	1	2	yes	—
38	#16810373	C₂²	1	2	yes	—
39	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#446	—	2	1	no	—
41	C₅₃×#436	—	2	1	no	—
42	C₅₃×#435	—	2	2	no	—
43	C₅₃×#432	C₂²	1	2	yes	—
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₅₃×#423	C₂²	1	2	yes	—
46	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
47	#13133224	—	2	4	no	—
48	#13133224	—	2	4	no	—
49	C₂×#11719917	—	2	2	no	—
50	C₂×#11719917	C₂×C₄	1	2	yes	—
51	C₂×#11719917	C₂×C₄	1	2	yes	—
52	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
53	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
54	C₂²×#88322	—	2	1	no	—
55	C₂²×#88322	C₂×C₄	1	2	yes	—
56	C₂²×#88322	C₂×C₄	1	2	yes	—
57	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
58	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#88322	—	2	1	no	—
60	C₄⋊#88322	—	2	2	no	—
61	C₄⋊#88322	C₂³	1	1	yes	characteristic
62	C₄⋊#88322	C₂³	1	2	yes	—
63	C₄⋊#88322	C₂³	1	1	yes	characteristic
64	C₂²⋊#88322	—	2	1	no	—
65	C₂²⋊#88322	—	2	2	no	—
66	#13133128	—	2	2	no	—
67	#13133128	C₂×C₄	1	2	yes	—
68	#13133128	C₂×C₄	1	2	yes	—
69	#13133128	C₂×C₄	1	2	yes	—
70	C₄×#88322	—	2	1	no	—
71	C₄×#88322	C₂×C₄	1	2	yes	—
72	C₄×#88322	C₂×C₄	1	1	yes	characteristic
73	C₄×#88322	C₂×C₄	1	1	yes	characteristic
74	C₄×#88322	C₂×C₄	1	2	yes	—
75	C₄×#88322	C₂³	1	1	yes	characteristic
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	2	yes	—
78	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×M₁₆×C₅₃	—	2	2	no	—
80	C₂²×C₈×C₅₃	—	2	1	no	—
81	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
83	C₄×D₄×C₅₃	D₄	1	2	yes	—
84	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
85	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
86	C₄×D₄×C₅₃	Q₈	1	2	yes	—
87	#117×C₅₃	C₂×C₄	1	2	yes	—
88	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
91	C₄⋊C₈×C₅₃	—	2	1	no	—
92	C₄⋊C₈×C₅₃	—	2	2	no	—
93	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
95	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₅₃	—	2	1	no	—
97	C₂²⋊C₈×C₅₃	—	2	2	no	—
98	#97×C₅₃	—	2	2	no	—
99	C₄×C₈×C₅₃	—	2	1	no	—
100	#41×C₅₃	—	2	2	no	—
101	#41×C₅₃	—	2	2	no	—
102	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
104	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
111	M₁₆×C₅₃	—	8	2	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	8	1	no	—
116	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
117	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
118	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
119	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
120	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
121	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
122	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
123	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
125	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅₃	C₂×D₄	1	2	yes	—
127	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
128	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
129	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
130	#11719917	—	2	2	no	—
131	#11719917	—	2	4	no	—
132	#11719917	—	2	4	no	—
133	#11719917	—	8	2	no	—
134	#11719917	M₁₆	1	4	yes	complemented
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	—	2	4	no	—
137	C₂×#88322	—	2	4	no	—
138	C₂×#88322	—	2	1	no	—
139	C₂×#88322	—	2	1	no	—
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	—	8	1	no	—
142	C₂×#88322	M₁₆	1	4	yes	complemented
143	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
144	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
145	C₂×#88322	C₂²×C₄	1	2	yes	—
146	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
147	C₂×#88322	C₂²×C₄	1	2	yes	—
148	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
149	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
150	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	Q₈×C₅₃	—	2	2	no	—
152	D₄×C₅₃	—	2	4	no	—
153	D₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	1	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	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
169	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
170	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	—	4	1	no	—
175	C₈×C₅₃	—	4	1	no	—
176	#88322	—	2	1	no	—
177	#88322	—	2	1	no	—
178	#88322	—	2	2	no	—
179	#88322	—	4	2	no	—
180	#88322	—	4	1	no	—
181	#88322	—	4	1	no	—
182	#88322	C₂×M₁₆	1	2	yes	—
183	#88322	C₂×M₁₆	1	4	yes	—
184	#88322	C₂×M₁₆	1	2	yes	—
185	#17799	—	53	1	no	maximal, Sylow
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	—	2	2	no	—
189	C₂²×C₅₃	—	2	2	no	—
190	C₂²×C₅₃	—	2	2	no	—
191	C₂²×C₅₃	#423	1	2	yes	—
192	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	2	2	no	—
199	C₄×C₅₃	#423	1	2	yes	—
200	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
201	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
202	C₄×C₅₃	#576	1	1	yes	characteristic
203	C₄×C₅₃	#578	1	1	yes	characteristic
204	#2989	—	53	1	no	—
205	#2989	D₅₃	1	1	yes	characteristic
206	C₄×#131	—	53	1	no	—
207	#1569	—	53	2	no	—
208	#1569	—	53	2	no	—
209	#1568	—	53	2	no	—
210	#1567	—	53	2	no	—
211	#1565	—	53	1	no	—
212	#1565	—	53	1	no	—
213	#1564	—	53	1	no	—
214	C₂×#335	—	53	1	no	—
215	C₂×C₅₃	—	2	2	no	—
216	C₂×C₅₃	—	4	2	no	—
217	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
218	C₂×C₅₃	#2913	1	1	yes	characteristic
219	C₂×C₅₃	#2916	1	1	yes	characteristic
220	C₂×#131	—	53	1	no	—
221	C₂×#131	—	53	1	no	—
222	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
223	#446	—	2	1	no	—
224	#446	—	106	1	no	—
225	#436	—	2	1	no	—
226	#436	—	106	1	no	—
227	#435	—	2	2	no	—
228	#435	—	106	2	no	—
229	#432	—	53	2	no	—
230	#432	C₂×D₅₃	1	2	yes	—
231	#431	—	53	1	no	—
232	#431	C₂×D₅₃	1	1	yes	characteristic
233	#423	—	53	2	no	—
234	#423	C₂×D₅₃	1	2	yes	—
235	C₂×C₄⋊C₈	—	53	1	no	—
236	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
237	#405	—	53	1	no	—
238	#405	—	53	2	no	—
239	#405	—	53	2	no	—
240	#405	—	53	1	no	—
241	C₄×M₁₆	—	53	1	no	—
242	C₄×M₁₆	—	53	2	no	—
243	C₂×C₄×C₈	—	53	1	no	—
244	C₂×C₄×C₈	—	53	2	no	—
245	#335	—	53	2	no	—
246	#335	—	53	2	no	—
247	#335	—	53	1	no	—
248	#335	—	53	1	no	—
249	#335	—	53	2	no	—
250	#335	—	53	2	no	—
251	#334	—	53	1	no	—
252	#334	—	53	1	no	—
253	#334	—	53	2	no	—
254	#334	—	53	2	no	—
255	C₅₃	#17799	1	1	yes	characteristic, complemented, Sylow
256	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
257	#131	—	106	4	no	—
258	#131	—	106	4	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	53	1	no	—
261	C₂×M₁₆	—	53	2	no	—
262	C₂×M₁₆	—	53	2	no	—
263	C₂×M₁₆	—	53	1	no	—
264	C₂×M₁₆	—	106	2	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	53	1	no	—
267	C₂²×C₈	—	53	2	no	—
268	C₂²×C₈	—	53	2	no	—
269	C₂²×C₈	—	53	1	no	—
270	C₂²×C₈	—	106	1	no	—
271	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
272	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
273	C₄×D₄	Dic₁₀₆	1	2	yes	—
274	C₄×D₄	Dic₁₀₆	1	1	yes	characteristic
275	C₄×D₄	C₂²⋊D₅₃	1	2	yes	—
276	C₄×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
277	#117	C₄×D₅₃	1	2	yes	—
278	#117	C₄×D₅₃	1	1	yes	characteristic
279	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
280	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	53	1	no	—
284	C₄⋊C₈	—	53	2	no	—
285	C₄⋊C₈	—	53	1	no	—
286	C₄⋊C₈	—	106	1	no	—
287	C₄⋊C₈	—	106	2	no	—
288	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
289	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
290	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
291	C₂²⋊C₈	—	2	1	no	—
292	C₂²⋊C₈	—	2	2	no	—
293	C₂²⋊C₈	—	106	1	no	—
294	C₂²⋊C₈	—	106	2	no	—
295	#97	—	2	2	no	—
296	#97	—	53	2	no	—
297	#97	—	53	2	no	—
298	#97	—	53	2	no	—
299	#97	—	106	2	no	—
300	C₄×C₈	—	2	1	no	—
301	C₄×C₈	—	53	2	no	—
302	C₄×C₈	—	53	1	no	—
303	C₄×C₈	—	53	1	no	—
304	C₄×C₈	—	53	2	no	—
305	C₄×C₈	—	53	1	no	—
306	C₄×C₈	—	53	2	no	—
307	C₄×C₈	—	53	1	no	—
308	C₄×C₈	—	106	1	no	—
309	#41	—	2	2	no	—
310	#41	—	2	2	no	—
311	C₂×Q₈	#11719919	1	1	yes	characteristic
312	C₂×D₄	#11719919	1	2	yes	—
313	C₂×D₄	#11719919	1	1	yes	characteristic
314	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
315	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
316	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
317	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
318	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
319	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
320	M₁₆	—	8	2	no	—
321	M₁₆	—	53	4	no	—
322	M₁₆	—	106	4	no	—
323	M₁₆	—	106	4	no	—
324	M₁₆	—	106	2	no	—
325	M₁₆	—	424	2	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	8	1	no	—
330	C₂×C₈	—	53	4	no	—
331	C₂×C₈	—	53	1	no	—
332	C₂×C₈	—	53	1	no	—
333	C₂×C₈	—	53	2	no	—
334	C₂×C₈	—	53	1	no	—
335	C₂×C₈	—	53	2	no	—
336	C₂×C₈	—	53	1	no	—
337	C₂×C₈	—	106	4	no	—
338	C₂×C₈	—	106	4	no	—
339	C₂×C₈	—	106	2	no	—
340	C₂×C₈	—	106	1	no	—
341	C₂×C₈	—	106	1	no	—
342	C₂×C₈	—	106	2	no	—
343	C₂×C₈	—	424	1	no	—
344	C₄⋊C₄	#11719919	1	2	yes	—
345	C₄⋊C₄	#11719919	1	1	yes	characteristic
346	C₄⋊C₄	#11719919	1	2	yes	—
347	C₄⋊C₄	#11719919	1	1	yes	characteristic
348	C₂²⋊C₄	#11719919	1	2	yes	—
349	C₂²⋊C₄	#11719919	1	2	yes	—
350	C₂²⋊C₄	#11719919	1	2	yes	—
351	C₄²	C₂×Dic₁₀₆	1	2	yes	—
352	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
353	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
354	C₄²	C₂×C₄×D₅₃	1	2	yes	—
355	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
356	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
357	C₄²	C₂³×D₅₃	1	1	yes	characteristic
358	C₂³	C₂×#11719919	1	2	yes	—
359	C₂³	C₂×#11719919	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
372	C₂×C₄	C₂×#11719919	1	2	yes	—
373	C₂×C₄	C₂×#11719919	1	2	yes	—
374	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
375	C₂×C₄	C₂×#11719919	1	2	yes	—
376	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
377	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
378	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
379	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
382	C₈	—	4	2	no	—
383	C₈	—	4	1	no	—
384	C₈	—	4	1	no	—
385	C₈	—	53	2	no	—
386	C₈	—	53	2	no	—
387	C₈	—	53	4	no	—
388	C₈	—	106	1	no	—
389	C₈	—	106	2	no	—
390	C₈	—	106	1	no	—
391	C₈	—	212	2	no	—
392	C₈	—	212	1	no	—
393	C₈	—	212	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	#16810646	1	2	yes	—
400	C₂²	C₂²×#11719919	1	1	yes	characteristic
401	C₄	—	2	2	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	2	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	2	2	no	—
407	C₄	#16810646	1	2	yes	—
408	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
409	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
410	C₄	#16811426	1	1	yes	characteristic
411	C₄	#16811438	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	4	2	no	—
414	C₂	C₂×#16810646	1	1	yes	characteristic
415	C₂	#20283313	1	1	yes	characteristic
416	C₂	#20283324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2989	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2989	D₅₃
hypocenter	C₅₃	#17799
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2989

Complements

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

N	H ≅ G/N	type
3. #20273483	C₂	semidirect
4. #20273480	C₂	semidirect
5. #20273476	C₂	semidirect
6. #20273473	C₂	semidirect
7. #20273468	C₂	semidirect
8. #20273465	C₂	semidirect
9. #20273461	C₂	semidirect
10. C₂×#16810375	C₂	semidirect
134. #11719917	M₁₆	semidirect
142. C₂×#88322	M₁₆	semidirect
255. C₅₃	#17799	semidirect

Maximal subgroups

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

subgroup	index
#20283832	2
C₄×#13133224	2
#20273483	2
#20273480	2
#20273476	2
#20273473	2
#20273468	2
#20273465	2
#20273461	2
C₂×#16810375	2
C₅₃×#2989	2
#17799	53