Subgroups — #27402890

Order 13568.

760 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20277192	C₂	1	1	yes	maximal, characteristic
4	#20275437	C₂	1	2	yes	maximal
5	#20275433	C₂	1	1	yes	maximal, characteristic
6	#20275433	C₂	1	1	yes	maximal, characteristic
7	#20275432	C₂	1	2	yes	maximal
8	#20274868	C₂	1	2	yes	maximal
9	#20274867	C₂	1	1	yes	maximal, characteristic
10	#20274574	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	#16810753	—	2	4	no	—
14	C₈×Dic₁₀₆	—	2	4	no	—
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#88322	C₂²	1	2	yes	—
19	C₂×#13133146	C₂²	1	2	yes	—
20	C₂×#13133146	C₂²	1	2	yes	—
21	C₂×#13133146	C₂²	1	2	yes	—
22	C₂×#13133146	C₂²	1	2	yes	—
23	C₂×#13133127	C₂²	1	2	yes	—
24	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
25	#16810529	C₄	1	1	yes	characteristic
26	#16810529	C₄	1	1	yes	characteristic
27	#16810521	C₄	1	1	yes	characteristic
28	#16810521	C₄	1	1	yes	characteristic
29	#16810507	C₄	1	2	yes	—
30	#16810507	C₄	1	2	yes	—
31	C₄×#11719919	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
33	#16810469	C₂²	1	2	yes	—
34	#16810377	C₂²	1	2	yes	—
35	#16810377	C₂²	1	2	yes	—
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄³×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#336	C₂²	1	1	yes	characteristic
42	C₅₃×#336	C₂²	1	1	yes	characteristic
43	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
44	C₂²×#88322	C₂³	1	2	yes	—
45	C₂²×#88322	C₂³	1	2	yes	—
46	C₄×Dic₁₀₆	D₄	1	4	yes	—
47	C₄×Dic₁₀₆	Q₈	1	4	yes	—
48	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×#11719919	C₂×C₄	1	2	yes	—
52	C₂×#11719919	C₂×C₄	1	2	yes	—
53	C₂×#11719919	C₂×C₄	1	2	yes	—
54	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
55	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
56	C₄⋊#88322	—	2	2	no	—
57	C₄⋊#88322	—	2	2	no	—
58	C₄⋊#88322	—	2	4	no	—
59	C₄⋊#88322	—	4	2	no	—
60	#13133146	—	2	4	no	—
61	#13133146	—	2	4	no	—
62	#13133146	—	4	2	no	—
63	#13133146	—	4	2	no	—
64	#13133127	—	2	4	no	—
65	C₈×Dic₅₃	—	2	4	no	—
66	#13133124	C₂×C₄	1	2	yes	—
67	#13133124	C₂×C₄	1	2	yes	—
68	#13133124	C₂×C₄	1	2	yes	—
69	#13133124	C₂×C₄	1	2	yes	—
70	#13133124	C₂×C₄	1	2	yes	—
71	#13133124	C₂×C₄	1	2	yes	—
72	#13133124	C₂×C₄	1	2	yes	—
73	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	—	2	2	no	—
85	C₄⋊C₈×C₅₃	Q₈	1	4	yes	complemented
86	C₄×C₈×C₅₃	—	2	2	no	—
87	C₄×C₈×C₅₃	—	2	2	no	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
96	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
98	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₀₆	—	2	2	no	—
100	C₂×Dic₁₀₆	—	2	2	no	—
101	C₂×Dic₁₀₆	C₄⋊C₄	1	4	yes	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	1	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	1	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	—	2	1	no	—
118	C₄²×C₅₃	—	2	1	no	—
119	C₄²×C₅₃	—	4	1	no	—
120	C₄²×C₅₃	C₂×D₄	1	2	yes	—
121	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
122	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
123	C₄²×C₅₃	#41	1	2	yes	—
124	C₄⋊Dic₅₃	—	2	2	no	—
125	C₄⋊Dic₅₃	—	2	2	no	—
126	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
127	#11719919	—	2	4	no	—
128	#11719919	—	4	2	no	—
129	#11719919	C₄⋊C₄	1	8	yes	—
130	C₄×Dic₅₃	—	2	2	no	—
131	C₄×Dic₅₃	—	2	2	no	—
132	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
133	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
134	C₂×#88322	—	2	2	no	—
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	—	2	2	no	—
137	C₂×#88322	—	2	2	no	—
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	—	2	2	no	—
140	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	1	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
155	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₅₃	#131	1	2	yes	—
157	C₂×C₄×C₅₃	#131	1	2	yes	—
158	C₂×C₄×C₅₃	#131	1	2	yes	—
159	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
165	C₂×Dic₅₃	—	2	2	no	—
166	C₂×Dic₅₃	—	2	2	no	—
167	C₂×Dic₅₃	—	2	2	no	—
168	C₂×Dic₅₃	—	2	2	no	—
169	C₂×Dic₅₃	—	2	2	no	—
170	C₂×Dic₅₃	—	2	2	no	—
171	C₂×Dic₅₃	—	2	2	no	—
172	C₂×Dic₅₃	—	2	2	no	—
173	C₂×Dic₅₃	—	2	2	no	—
174	C₂×Dic₅₃	—	2	2	no	—
175	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
176	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
177	Dic₁₀₆	—	2	8	no	—
178	C₈×C₅₃	—	2	2	no	—
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	—	4	1	no	—
181	C₈×C₅₃	—	4	1	no	—
182	#88322	—	4	2	no	—
183	#88322	—	4	2	no	—
184	#22721	—	53	1	no	maximal, Sylow
185	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₂²×C₅₃	#520	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
191	C₂²×C₅₃	#568	1	1	yes	characteristic
192	C₄×C₅₃	—	2	2	no	—
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	2	2	no	—
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	—	4	1	no	—
197	C₄×C₅₃	#424	1	4	yes	—
198	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
199	C₄×C₅₃	#550	1	1	yes	characteristic
200	C₄×C₅₃	#557	1	1	yes	characteristic
201	C₄×C₅₃	Q₈²	1	1	yes	characteristic
202	Dic₅₃	—	2	4	no	—
203	Dic₅₃	—	2	4	no	—
204	Dic₅₃	—	4	2	no	—
205	Dic₅₃	—	4	2	no	—
206	C₂×#446	—	53	1	no	—
207	C₂×C₈×Q₈	—	53	1	no	—
208	#2295	—	53	1	no	—
209	#1940	—	53	2	no	—
210	#1939	—	53	1	no	—
211	#1939	—	53	1	no	—
212	#1939	—	53	2	no	—
213	#1843	—	53	2	no	—
214	#1842	—	53	1	no	—
215	#1768	—	53	2	no	—
216	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
217	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
218	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
219	C₂×C₅₃	#2906	1	1	yes	characteristic
220	C₂×C₅₃	#2960	1	1	yes	characteristic
221	C₂×C₅₃	#2962	1	1	yes	characteristic
222	C₂×C₅₃	#2993	1	1	yes	characteristic
223	C₂×C₅₃	#2994	1	1	yes	characteristic
224	C₂×C₄×Q₈	—	53	1	no	—
225	#446	—	106	4	no	—
226	C₈×Q₈	—	106	4	no	—
227	C₂×C₄⋊C₈	—	53	2	no	—
228	C₂×C₄⋊C₈	—	53	2	no	—
229	C₂×C₄⋊C₈	—	53	2	no	—
230	C₂×C₄⋊C₈	—	53	1	no	—
231	C₂×C₄⋊C₈	—	53	1	no	—
232	C₂×C₄⋊C₈	—	53	2	no	—
233	C₂×C₄⋊C₈	—	53	2	no	—
234	C₂×C₄⋊C₈	—	53	2	no	—
235	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
237	C₂×#97	—	53	2	no	—
238	C₂×C₄×C₈	—	53	2	no	—
239	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
240	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
241	#391	—	53	1	no	—
242	#391	—	53	1	no	—
243	#389	—	53	1	no	—
244	#389	—	53	1	no	—
245	#387	—	53	2	no	—
246	#387	—	53	2	no	—
247	C₄×C₄⋊C₄	—	53	1	no	—
248	C₄×C₄⋊C₄	—	53	2	no	—
249	#376	—	53	2	no	—
250	C₄³	C₂×D₅₃	1	1	yes	characteristic
251	#336	—	53	2	no	—
252	#336	—	53	2	no	—
253	#336	C₂×D₅₃	1	1	yes	characteristic
254	#336	C₂×D₅₃	1	1	yes	characteristic
255	C₅₃	#22721	1	1	yes	characteristic, complemented, Sylow
256	C₂²×Q₈	—	53	1	no	—
257	C₂²×C₈	—	53	2	no	—
258	C₂²×C₈	—	53	2	no	—
259	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
260	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
261	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
263	C₄×Q₈	—	53	4	no	—
264	C₄×Q₈	—	53	4	no	—
265	C₂×C₄⋊C₄	—	53	2	no	—
266	C₂×C₄⋊C₄	—	53	1	no	—
267	C₂×C₄⋊C₄	—	53	2	no	—
268	C₂×C₄⋊C₄	—	53	1	no	—
269	C₂×C₄⋊C₄	—	53	1	no	—
270	C₂×C₄⋊C₄	—	53	2	no	—
271	C₂×C₄²	—	53	2	no	—
272	C₂×C₄²	—	53	2	no	—
273	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
274	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
275	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
276	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
277	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
278	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
279	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
280	C₄⋊C₈	—	2	2	no	—
281	C₄⋊C₈	—	106	4	no	—
282	C₄⋊C₈	—	106	4	no	—
283	C₄⋊C₈	—	106	4	no	—
284	C₄⋊C₈	—	106	2	no	—
285	C₄⋊C₈	—	106	2	no	—
286	C₄⋊C₈	—	212	2	no	—
287	C₄⋊C₈	—	212	2	no	—
288	C₄⋊C₈	—	212	2	no	—
289	C₄⋊C₈	Dic₁₀₆	1	4	yes	complemented
290	#97	—	106	4	no	—
291	C₄×C₈	—	2	2	no	—
292	C₄×C₈	—	2	2	no	—
293	C₄×C₈	—	106	4	no	—
294	#95	—	53	2	no	—
295	#95	—	53	2	no	—
296	#95	—	53	2	no	—
297	#95	—	53	2	no	—
298	#95	—	53	2	no	—
299	#95	—	53	2	no	—
300	#95	—	53	2	no	—
301	C₂×Q₈	—	53	4	no	—
302	C₂×Q₈	—	106	2	no	—
303	C₂×Q₈	—	106	2	no	—
304	C₂²×C₄	—	53	2	no	—
305	C₂²×C₄	—	53	2	no	—
306	C₂²×C₄	—	53	2	no	—
307	C₂²×C₄	—	53	2	no	—
308	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
310	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	106	2	no	—
326	C₂×C₈	—	106	2	no	—
327	C₂×C₈	—	106	2	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	—	106	2	no	—
330	C₂×C₈	—	106	2	no	—
331	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
332	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
333	C₄⋊C₄	—	53	8	no	—
334	C₄⋊C₄	—	53	4	no	—
335	C₄⋊C₄	—	106	4	no	—
336	C₄⋊C₄	—	106	2	no	—
337	C₄⋊C₄	—	106	2	no	—
338	C₄⋊C₄	—	212	2	no	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	4	1	no	—
345	C₄²	—	53	4	no	—
346	C₄²	—	53	4	no	—
347	C₄²	—	106	2	no	—
348	C₄²	—	106	2	no	—
349	C₄²	C₂×Dic₁₀₆	1	2	yes	—
350	C₄²	#11719945	1	2	yes	—
351	C₄²	D₄×D₅₃	1	2	yes	—
352	C₄²	Q₈×D₅₃	1	2	yes	—
353	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
354	Q₈	—	106	8	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	53	4	no	—
367	C₂×C₄	—	53	4	no	—
368	C₂×C₄	—	106	2	no	—
369	C₂×C₄	—	106	2	no	—
370	C₂×C₄	—	106	2	no	—
371	C₂×C₄	—	106	2	no	—
372	C₂×C₄	—	106	2	no	—
373	C₂×C₄	—	106	2	no	—
374	C₂×C₄	—	106	2	no	—
375	C₂×C₄	—	106	2	no	—
376	C₂×C₄	—	106	2	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
379	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
380	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
381	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
382	C₂×C₄	#13133223	1	2	yes	—
383	C₂×C₄	#13133223	1	2	yes	—
384	C₂×C₄	#13133225	1	2	yes	—
385	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
386	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
387	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
388	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
389	C₂×C₄	#13133263	1	1	yes	characteristic
390	C₂×C₄	#13133264	1	1	yes	characteristic
391	C₈	—	2	2	no	—
392	C₈	—	2	2	no	—
393	C₈	—	4	1	no	—
394	C₈	—	4	1	no	—
395	C₈	—	212	2	no	—
396	C₈	—	212	2	no	—
397	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
398	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
399	C₂²	#16811103	1	1	yes	characteristic
400	C₂²	C₂×#13133223	1	1	yes	characteristic
401	C₂²	C₂×#13133223	1	1	yes	characteristic
402	C₂²	C₂×#13133225	1	1	yes	characteristic
403	C₂²	#16811389	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	2	2	no	—
406	C₄	—	2	2	no	—
407	C₄	—	2	2	no	—
408	C₄	—	4	1	no	—
409	C₄	—	106	4	no	—
410	C₄	—	106	4	no	—
411	C₄	—	212	2	no	—
412	C₄	—	212	2	no	—
413	C₄	#16810648	1	4	yes	—
414	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
415	C₄	#16811294	1	1	yes	characteristic
416	C₄	#16811351	1	1	yes	characteristic
417	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
418	C₂	C₄⋊C₄×Dic₁₀₆	1	1	yes	characteristic
419	C₂	C₂×#16810648	1	1	yes	characteristic
420	C₂	#20283280	1	1	yes	characteristic
421	C₂	#20283649	1	1	yes	characteristic
422	C₂	#20283662	1	1	yes	characteristic
423	C₂	#20283866	1	1	yes	characteristic
424	C₂	#20283877	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#22721
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
85. C₄⋊C₈×C₅₃	Q₈	semidirect
255. C₅₃	#22721	semidirect
289. C₄⋊C₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810753	2
C₂×C₈×Dic₁₀₆	2
#20277192	2
#20275437	2
#20275433	2
#20275432	2
#20274868	2
#20274867	2
#20274574	2
C₄×C₄⋊C₈×C₅₃	2
#22721	53