Subgroups — #27402892

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×#16810754	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
3	#20275439	C₂	1	2	yes	maximal
4	#20275434	C₂	1	1	yes	maximal, characteristic
5	#20275434	C₂	1	1	yes	maximal, characteristic
6	#20275434	C₂	1	2	yes	maximal
7	#20274869	C₂	1	2	yes	maximal
8	#20274867	C₂	1	1	yes	maximal, characteristic
9	#20274572	C₂	1	2	yes	maximal
10	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic
12	#16810754	—	2	4	no	—
13	Q₈×#88322	—	2	4	no	—
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
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×C₄⋊#88322	C₂²	1	2	yes	—
20	C₂×C₄⋊#88322	C₂²	1	2	yes	—
21	C₂×C₄⋊#88322	C₂²	1	2	yes	—
22	C₂×C₄⋊#88322	C₂²	1	2	yes	—
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×#13133128	C₂²	1	2	yes	—
25	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
26	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
27	C₂×C₄×#88322	C₂²	1	2	yes	—
28	#16810378	C₂²	1	1	yes	characteristic
29	#16810378	C₂²	1	1	yes	characteristic
30	#16810378	C₂²	1	2	yes	—
31	#16810378	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#391	C₄	1	1	yes	characteristic
34	C₅₃×#391	C₄	1	1	yes	characteristic
35	C₅₃×#389	C₄	1	1	yes	characteristic
36	C₅₃×#389	C₄	1	1	yes	characteristic
37	C₅₃×#387	C₄	1	2	yes	—
38	C₅₃×#387	C₄	1	2	yes	—
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
41	C₅₃×#376	C₂²	1	2	yes	—
42	C₄³×C₅₃	C₂²	1	1	yes	characteristic
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	C₂²×#88322	C₂³	1	1	yes	characteristic
45	C₂²×#88322	C₂³	1	2	yes	—
46	C₂²×#88322	C₂³	1	2	yes	—
47	C₂²×#88322	C₂³	1	1	yes	characteristic
48	C₂²×#88322	C₂³	1	1	yes	characteristic
49	C₄⋊#88322	—	2	4	no	—
50	C₄⋊#88322	—	2	2	no	—
51	C₄⋊#88322	—	2	2	no	—
52	C₄⋊#88322	—	2	4	no	—
53	C₄⋊#88322	—	2	2	no	—
54	C₄⋊#88322	—	2	4	no	—
55	C₄⋊#88322	—	4	2	no	—
56	C₄⋊#88322	—	4	2	no	—
57	C₄⋊#88322	—	4	2	no	—
58	C₄⋊#88322	Q₈	1	4	yes	complemented
59	#13133128	—	2	4	no	—
60	C₄×#88322	—	2	2	no	—
61	C₄×#88322	—	2	2	no	—
62	C₄×#88322	—	2	4	no	—
63	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈×C₅₃	D₄	1	4	yes	—
65	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
66	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#95×C₅₃	C₂×C₄	1	2	yes	—
82	#95×C₅₃	C₂×C₄	1	2	yes	—
83	#95×C₅₃	C₂×C₄	1	2	yes	—
84	#95×C₅₃	C₂×C₄	1	2	yes	—
85	#95×C₅₃	C₂×C₄	1	2	yes	—
86	#95×C₅₃	C₂×C₄	1	2	yes	—
87	#95×C₅₃	C₂×C₄	1	2	yes	—
88	C₂×Q₈×C₅₃	—	2	2	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
91	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₅₃	—	2	4	no	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	4	2	no	—
106	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
107	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
108	C₄²×C₅₃	—	2	2	no	—
109	C₄²×C₅₃	—	2	2	no	—
110	C₄²×C₅₃	—	2	2	no	—
111	C₄²×C₅₃	—	2	1	no	—
112	C₄²×C₅₃	—	2	2	no	—
113	C₄²×C₅₃	—	2	1	no	—
114	C₄²×C₅₃	—	2	2	no	—
115	C₄²×C₅₃	—	4	1	no	—
116	C₄²×C₅₃	C₂×D₄	1	2	yes	—
117	C₄²×C₅₃	C₂×D₄	1	4	yes	—
118	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
119	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
120	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
121	C₄²×C₅₃	#41	1	2	yes	—
122	C₂×#88322	—	2	1	no	—
123	C₂×#88322	—	2	1	no	—
124	C₂×#88322	—	2	1	no	—
125	C₂×#88322	—	2	1	no	—
126	C₂×#88322	—	2	1	no	—
127	C₂×#88322	—	2	1	no	—
128	C₂×#88322	—	2	2	no	—
129	C₂×#88322	—	2	2	no	—
130	C₂×#88322	—	2	1	no	—
131	C₂×#88322	—	2	1	no	—
132	C₂×#88322	—	2	2	no	—
133	C₂×#88322	—	2	2	no	—
134	C₂×#88322	—	2	2	no	—
135	C₂×#88322	—	2	1	no	—
136	C₂×#88322	—	2	1	no	—
137	C₂×#88322	—	2	2	no	—
138	C₂×#88322	C₂×Q₈	1	2	yes	—
139	C₂×#88322	C₂×Q₈	1	2	yes	—
140	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
141	Q₈×C₅₃	—	2	8	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	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	2	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	1	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
164	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
166	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
168	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
169	C₂×C₄×C₅₃	#131	1	2	yes	—
170	C₂×C₄×C₅₃	#131	1	2	yes	—
171	C₂×C₄×C₅₃	#131	1	2	yes	—
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
178	#88322	—	2	2	no	—
179	#88322	—	2	2	no	—
180	#88322	—	4	1	no	—
181	#88322	—	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	4	no	—
193	C₄×C₅₃	—	2	2	no	—
194	C₄×C₅₃	—	2	4	no	—
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	—	2	2	no	—
197	C₄×C₅₃	—	2	2	no	—
198	C₄×C₅₃	—	4	1	no	—
199	C₄×C₅₃	—	4	2	no	—
200	C₄×C₅₃	—	4	2	no	—
201	C₄×C₅₃	#424	1	4	yes	—
202	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
203	C₄×C₅₃	#550	1	1	yes	characteristic
204	C₄×C₅₃	#557	1	1	yes	characteristic
205	C₄×C₅₃	Q₈²	1	1	yes	characteristic
206	C₂×#446	—	53	1	no	—
207	C₂×C₈×Q₈	—	53	1	no	—
208	#2295	D₅₃	1	1	yes	characteristic
209	#1940	—	53	2	no	—
210	#1939	—	53	2	no	—
211	#1939	—	53	1	no	—
212	#1939	—	53	1	no	—
213	#1843	—	53	2	no	—
214	#1842	—	53	1	no	—
215	#1768	—	53	2	no	—
216	C₄×C₄⋊C₈	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
225	#446	—	106	4	no	—
226	C₈×Q₈	—	106	4	no	—
227	C₂×C₄⋊C₈	—	53	1	no	—
228	C₂×C₄⋊C₈	—	53	2	no	—
229	C₂×C₄⋊C₈	—	53	1	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	—	53	2	no	—
236	C₂×C₄⋊C₈	—	53	2	no	—
237	C₂×#97	—	53	2	no	—
238	C₂×C₄×C₈	—	53	1	no	—
239	C₂×C₄×C₈	—	53	2	no	—
240	C₂×C₄×C₈	—	53	1	no	—
241	#391	Dic₅₃	1	1	yes	characteristic
242	#391	Dic₅₃	1	1	yes	characteristic
243	#389	Dic₅₃	1	1	yes	characteristic
244	#389	Dic₅₃	1	1	yes	characteristic
245	#387	Dic₅₃	1	2	yes	—
246	#387	Dic₅₃	1	2	yes	—
247	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
248	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
249	#376	C₂×D₅₃	1	2	yes	—
250	C₄³	C₂×D₅₃	1	1	yes	characteristic
251	#336	—	53	1	no	—
252	#336	—	53	2	no	—
253	#336	—	53	2	no	—
254	#336	—	53	1	no	—
255	C₅₃	#22721	1	1	yes	characteristic, complemented, Sylow
256	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
257	C₂²×C₈	—	53	1	no	—
258	C₂²×C₈	—	53	1	no	—
259	C₂²×C₈	—	53	1	no	—
260	C₂²×C₈	—	53	2	no	—
261	C₂²×C₈	—	53	1	no	—
262	C₂²×C₈	—	53	2	no	—
263	C₄×Q₈	Dic₁₀₆	1	4	yes	—
264	C₄×Q₈	D₂₁₂	1	4	yes	—
265	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
268	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
269	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
271	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
272	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
273	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
274	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
275	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
276	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
277	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
278	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
279	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
280	C₄⋊C₈	—	53	4	no	—
281	C₄⋊C₈	—	106	2	no	—
282	C₄⋊C₈	—	106	4	no	—
283	C₄⋊C₈	—	106	2	no	—
284	C₄⋊C₈	—	106	4	no	—
285	C₄⋊C₈	—	106	4	no	—
286	C₄⋊C₈	—	106	2	no	—
287	C₄⋊C₈	—	212	2	no	—
288	C₄⋊C₈	—	212	2	no	—
289	C₄⋊C₈	—	212	2	no	—
290	#97	—	106	4	no	—
291	C₄×C₈	—	106	2	no	—
292	C₄×C₈	—	106	4	no	—
293	C₄×C₈	—	106	2	no	—
294	#95	C₂×Dic₅₃	1	2	yes	—
295	#95	C₂×Dic₅₃	1	2	yes	—
296	#95	C₂×Dic₅₃	1	2	yes	—
297	#95	C₂×Dic₅₃	1	2	yes	—
298	#95	C₂×Dic₅₃	1	2	yes	—
299	#95	C₂×Dic₅₃	1	2	yes	—
300	#95	C₂×Dic₅₃	1	2	yes	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
304	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
305	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
306	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
308	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
309	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
310	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
312	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
315	C₂×C₈	—	53	2	no	—
316	C₂×C₈	—	53	2	no	—
317	C₂×C₈	—	106	1	no	—
318	C₂×C₈	—	106	1	no	—
319	C₂×C₈	—	106	1	no	—
320	C₂×C₈	—	106	1	no	—
321	C₂×C₈	—	106	1	no	—
322	C₂×C₈	—	106	2	no	—
323	C₂×C₈	—	106	1	no	—
324	C₂×C₈	—	106	2	no	—
325	C₂×C₈	—	106	2	no	—
326	C₂×C₈	—	106	1	no	—
327	C₂×C₈	—	106	1	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	—	106	2	no	—
330	C₂×C₈	—	106	1	no	—
331	C₂×C₈	—	106	1	no	—
332	C₂×C₈	—	106	2	no	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
338	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	4	1	no	—
347	C₄²	C₂×Dic₁₀₆	1	4	yes	—
348	C₄²	C₂×Dic₁₀₆	1	2	yes	—
349	C₄²	C₂×D₂₁₂	1	2	yes	—
350	C₄²	C₂×D₂₁₂	1	4	yes	—
351	C₄²	Q₈×D₅₃	1	2	yes	—
352	C₄²	#11719949	1	2	yes	—
353	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
354	Q₈	—	2	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	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
377	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
378	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
379	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
380	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
381	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
382	C₂×C₄	#13133224	1	2	yes	—
383	C₂×C₄	#13133224	1	2	yes	—
384	C₂×C₄	#13133224	1	2	yes	—
385	C₂×C₄	C₂²×D₂₁₂	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×#11719949	1	1	yes	characteristic
389	C₂×C₄	#13133263	1	1	yes	characteristic
390	C₂×C₄	#13133264	1	1	yes	characteristic
391	C₈	—	106	2	no	—
392	C₈	—	106	2	no	—
393	C₈	—	212	1	no	—
394	C₈	—	212	2	no	—
395	C₈	—	212	1	no	—
396	C₈	—	212	2	no	—
397	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
398	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
399	C₂²	#16811106	1	1	yes	characteristic
400	C₂²	C₂×#13133224	1	1	yes	characteristic
401	C₂²	C₂×#13133224	1	1	yes	characteristic
402	C₂²	C₂×#13133224	1	1	yes	characteristic
403	C₂²	#16811390	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	2	2	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	2	no	—
408	C₄	—	2	4	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	#16810650	1	4	yes	—
414	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
415	C₄	#16811295	1	1	yes	characteristic
416	C₄	#16811352	1	1	yes	characteristic
417	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
418	C₂	Q₈×C₄⋊Dic₅₃	1	1	yes	characteristic
419	C₂	C₂×#16810650	1	1	yes	characteristic
420	C₂	#20283283	1	1	yes	characteristic
421	C₂	#20283650	1	1	yes	characteristic
422	C₂	#20283664	1	1	yes	characteristic
423	C₂	#20283864	1	1	yes	characteristic
424	C₂	#20283875	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#2295	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2295	D₅₃
hypocenter	C₅₃	#22721
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
58. C₄⋊#88322	Q₈	semidirect
255. C₅₃	#22721	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
C₂×Q₈×#88322	2
#20275439	2
#20275434	2
#20274869	2
#20274867	2
#20274572	2
C₄×C₄⋊#88322	2
C₅₃×#2295	2
#22721	53