Subgroups — #28316112

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291391	C₂	1	2	yes	maximal
2	#20290277	C₂	1	2	yes	maximal, complemented
3	#20290275	C₂	1	2	yes	maximal, complemented
4	#20290272	C₂	1	2	yes	maximal, complemented
5	#20290269	C₂	1	2	yes	maximal, complemented
6	#20290264	C₂	1	2	yes	maximal, complemented
7	#20290262	C₂	1	2	yes	maximal, complemented
8	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811644	C₂²	1	4	yes	—
10	#16811643	—	2	2	no	—
11	#16811640	—	2	2	no	—
12	#16811639	C₂²	1	4	yes	—
13	#16811626	C₂²	1	4	yes	—
14	#16811593	—	2	2	no	—
15	C₄×#11719940	—	2	2	no	—
16	#16811487	C₂²	1	4	yes	complemented
17	#16811486	C₂²	1	4	yes	complemented
18	#16811485	C₂²	1	4	yes	complemented
19	#16811484	C₂²	1	4	yes	complemented
20	C₂³×#88322	C₄	1	1	yes	characteristic
21	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
23	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
24	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
25	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
26	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
27	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
35	C₂×#11719940	—	2	4	no	—
36	#13133284	—	2	4	no	—
37	C₄⋊#88324	C₂³	1	8	yes	—
38	C₂²⋊#88324	—	2	4	no	—
39	C₂²⋊#88324	—	2	4	no	—
40	#13133272	—	2	4	no	—
41	C₄×#88324	—	2	4	no	—
42	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
43	C₂³×Dic₅₃	Q₈	1	1	yes	characteristic
44	C₂²×#88322	—	2	4	no	—
45	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
46	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
47	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
48	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
49	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
50	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
58	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
59	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
60	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
63	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
76	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
77	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
78	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₈×C₅₃	—	2	4	no	—
81	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	—	2	2	no	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×Dic₅₃	—	2	4	no	—
97	C₂²×Dic₅₃	—	2	4	no	—
98	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
108	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
109	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
110	#11719940	—	4	8	no	—
111	C₂×#88324	—	2	8	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
118	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
125	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
128	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
135	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
136	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
137	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	—	2	2	no	—
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	—	2	2	no	—
142	C₂×#88322	—	2	2	no	—
143	C₂×#88322	M₁₆	1	4	yes	complemented
144	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
145	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
146	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
147	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	2	2	no	—
150	C₂³×C₅₃	—	2	2	no	—
151	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
165	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
166	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
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	—	2	2	no	—
176	C₂×Dic₅₃	—	2	2	no	—
177	C₂×Dic₅₃	#108	1	4	yes	—
178	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×Dic₅₃	#135	1	4	yes	—
185	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
187	#88324	—	4	8	no	—
188	C₈×C₅₃	—	2	1	no	—
189	C₈×C₅₃	—	2	1	no	—
190	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
191	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
192	#88322	—	2	1	no	—
193	#88322	—	2	1	no	—
194	#88322	C₂×M₁₆	1	2	yes	—
195	#88322	C₂×M₁₆	1	2	yes	—
196	#17793	—	53	1	no	maximal, Sylow
197	C₂²×C₅₃	—	2	8	no	—
198	C₂²×C₅₃	—	2	1	no	—
199	C₂²×C₅₃	—	2	1	no	—
200	C₂²×C₅₃	—	2	1	no	—
201	C₂²×C₅₃	—	2	1	no	—
202	C₂²×C₅₃	—	2	1	no	—
203	C₂²×C₅₃	—	2	1	no	—
204	C₂²×C₅₃	#423	1	2	yes	—
205	C₂²×C₅₃	#427	1	2	yes	—
206	C₂²×C₅₃	C₂×#108	1	2	yes	—
207	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
208	C₄×C₅₃	—	2	1	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	#423	1	2	yes	—
211	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
212	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
213	Dic₅₃	—	2	1	no	—
214	Dic₅₃	—	2	1	no	—
215	Dic₅₃	—	2	1	no	—
216	Dic₅₃	—	2	1	no	—
217	Dic₅₃	#427	1	2	yes	—
218	Dic₅₃	C₂×#108	1	2	yes	—
219	Dic₅₃	C₂×#108	1	2	yes	—
220	Dic₅₃	C₂×#135	1	2	yes	—
221	#2920	—	53	2	no	—
222	C₂²×C₄×C₈	—	53	1	no	—
223	#1568	—	53	2	no	—
224	#1568	—	53	2	no	—
225	#1567	—	53	2	no	—
226	#1566	—	53	2	no	—
227	#1564	—	53	2	no	—
228	#1564	—	53	2	no	—
229	C₂×C₅₃	—	2	2	no	—
230	C₂×C₅₃	—	2	2	no	—
231	C₂×C₅₃	—	2	2	no	—
232	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
233	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
234	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
235	C₂³×C₈	—	53	1	no	—
236	C₂³×C₈	#6157	1	1	yes	characteristic
237	C₂²×C₄²	—	53	1	no	—
238	#432	—	53	4	no	—
239	#432	—	106	2	no	—
240	#431	—	53	4	no	—
241	#431	—	106	2	no	—
242	#423	—	53	4	no	—
243	#407	—	106	2	no	—
244	C₄×M₁₆	—	106	2	no	—
245	C₂×C₄×C₈	—	53	1	no	—
246	C₂×C₄×C₈	—	53	1	no	—
247	C₂×C₄×C₈	—	53	1	no	—
248	C₂×C₄×C₈	—	53	1	no	—
249	C₂×C₄×C₈	—	53	1	no	—
250	C₂×C₄×C₈	—	53	1	no	—
251	C₂×C₄×C₈	—	53	1	no	—
252	C₂×C₄×C₈	—	53	1	no	—
253	C₂×C₄×C₈	—	53	1	no	—
254	C₂×C₄×C₈	—	53	1	no	—
255	C₂×C₄×C₈	—	53	1	no	—
256	C₂×C₄×C₈	—	53	1	no	—
257	#335	—	53	4	no	—
258	#335	—	53	4	no	—
259	#334	—	53	4	no	—
260	#334	—	53	4	no	—
261	C₅₃	#17793	1	1	yes	characteristic, complemented, Sylow
262	C₂³×C₄	—	53	1	no	—
263	C₂³×C₄	—	53	1	no	—
264	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
265	C₂×M₁₆	—	106	4	no	—
266	C₂²×C₈	—	2	4	no	—
267	C₂²×C₈	—	53	1	no	—
268	C₂²×C₈	—	53	1	no	—
269	C₂²×C₈	—	53	1	no	—
270	C₂²×C₈	—	53	1	no	—
271	C₂²×C₈	—	53	1	no	—
272	C₂²×C₈	—	53	1	no	—
273	C₂²×C₈	—	106	4	no	—
274	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
275	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
276	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
277	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
278	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
279	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
280	C₂×C₄²	—	53	1	no	—
281	C₂×C₄²	—	53	1	no	—
282	C₂×C₄²	—	53	1	no	—
283	C₂×C₄²	—	53	1	no	—
284	C₂×C₄²	—	53	1	no	—
285	C₂×C₄²	—	53	1	no	—
286	C₂×C₄²	—	53	1	no	—
287	C₂×C₄²	—	53	1	no	—
288	C₂×C₄²	—	53	1	no	—
289	C₂×C₄²	—	53	1	no	—
290	C₂×C₄²	—	53	1	no	—
291	C₂×C₄²	—	53	1	no	—
292	C₄⋊C₈	—	53	8	no	—
293	C₄⋊C₈	—	106	4	no	—
294	C₂²⋊C₈	—	106	4	no	—
295	C₂²⋊C₈	—	106	4	no	—
296	#97	—	106	4	no	—
297	C₄×C₈	—	53	1	no	—
298	C₄×C₈	—	53	1	no	—
299	C₄×C₈	—	53	1	no	—
300	C₄×C₈	—	53	1	no	—
301	C₄×C₈	—	53	1	no	—
302	C₄×C₈	—	53	1	no	—
303	C₄×C₈	—	53	1	no	—
304	C₄×C₈	—	53	1	no	—
305	C₄×C₈	—	53	1	no	—
306	C₄×C₈	—	53	1	no	—
307	C₄×C₈	—	53	1	no	—
308	C₄×C₈	—	53	1	no	—
309	C₄×C₈	—	53	1	no	—
310	C₄×C₈	—	53	1	no	—
311	C₄×C₈	—	53	1	no	—
312	C₄×C₈	—	53	1	no	—
313	C₄×C₈	—	106	4	no	—
314	C₂⁴	C₄⋊#6157	1	1	yes	characteristic
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	53	1	no	—
318	C₂²×C₄	—	53	1	no	—
319	C₂²×C₄	—	53	1	no	—
320	C₂²×C₄	—	53	1	no	—
321	C₂²×C₄	—	53	1	no	—
322	C₂²×C₄	—	53	1	no	—
323	C₂²×C₄	—	53	1	no	—
324	C₂²×C₄	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	53	1	no	—
328	C₂²×C₄	—	53	1	no	—
329	C₂²×C₄	—	106	4	no	—
330	C₂²×C₄	—	106	4	no	—
331	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
332	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
333	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
334	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
335	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
336	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
337	M₁₆	—	212	8	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	53	4	no	—
344	C₂×C₈	—	53	1	no	—
345	C₂×C₈	—	53	1	no	—
346	C₂×C₈	—	53	1	no	—
347	C₂×C₈	—	53	1	no	—
348	C₂×C₈	—	106	2	no	—
349	C₂×C₈	—	106	2	no	—
350	C₂×C₈	—	106	2	no	—
351	C₂×C₈	—	106	2	no	—
352	C₂×C₈	—	106	2	no	—
353	C₂×C₈	—	106	8	no	—
354	C₂×C₈	#11719940	1	4	yes	complemented
355	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
356	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
357	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
358	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
359	C₄²	—	53	1	no	—
360	C₄²	—	53	1	no	—
361	C₄²	—	53	1	no	—
362	C₄²	—	53	1	no	—
363	C₄²	—	53	1	no	—
364	C₄²	—	53	1	no	—
365	C₄²	—	53	1	no	—
366	C₄²	—	53	1	no	—
367	C₄²	—	53	1	no	—
368	C₄²	—	53	1	no	—
369	C₄²	—	53	1	no	—
370	C₄²	—	53	1	no	—
371	C₄²	—	53	1	no	—
372	C₄²	—	53	1	no	—
373	C₄²	—	53	1	no	—
374	C₄²	—	53	1	no	—
375	C₂³	—	2	2	no	—
376	C₂³	—	2	2	no	—
377	C₂³	—	2	2	no	—
378	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
379	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
380	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	53	4	no	—
389	C₂×C₄	—	53	4	no	—
390	C₂×C₄	—	53	1	no	—
391	C₂×C₄	—	53	1	no	—
392	C₂×C₄	—	53	1	no	—
393	C₂×C₄	—	53	1	no	—
394	C₂×C₄	—	53	1	no	—
395	C₂×C₄	—	53	1	no	—
396	C₂×C₄	—	53	1	no	—
397	C₂×C₄	—	53	1	no	—
398	C₂×C₄	—	106	2	no	—
399	C₂×C₄	—	106	2	no	—
400	C₂×C₄	—	106	2	no	—
401	C₂×C₄	—	106	2	no	—
402	C₂×C₄	—	106	2	no	—
403	C₂×C₄	—	106	2	no	—
404	C₂×C₄	—	106	2	no	—
405	C₂×C₄	—	106	2	no	—
406	C₂×C₄	—	106	2	no	—
407	C₂×C₄	—	106	2	no	—
408	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
409	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
410	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
411	C₂×C₄	C₂×#11719940	1	2	yes	—
412	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
413	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
414	C₈	—	2	1	no	—
415	C₈	—	2	1	no	—
416	C₈	—	53	2	no	—
417	C₈	—	53	2	no	—
418	C₈	—	106	1	no	—
419	C₈	—	106	1	no	—
420	C₈	—	212	8	no	—
421	C₈	C₂×#11719940	1	2	yes	—
422	C₈	C₂×#11719940	1	2	yes	—
423	C₂²	—	2	8	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	#16811626	1	2	yes	—
431	C₂²	#16811633	1	2	yes	—
432	C₂²	D₅₃⋊#108	1	2	yes	—
433	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
434	C₄	—	2	1	no	—
435	C₄	—	2	1	no	—
436	C₄	—	53	2	no	—
437	C₄	—	53	2	no	—
438	C₄	—	53	2	no	—
439	C₄	—	53	2	no	—
440	C₄	—	106	1	no	—
441	C₄	—	106	1	no	—
442	C₄	—	106	1	no	—
443	C₄	—	106	1	no	—
444	C₄	#16811626	1	2	yes	—
445	C₄	C₂²×#11719940	1	1	yes	characteristic
446	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
447	C₂	—	2	2	no	—
448	C₂	—	2	2	no	—
449	C₂	—	2	2	no	—
450	C₂	C₂×#16811626	1	1	yes	characteristic
451	C₂	C₂×#16811633	1	1	yes	characteristic
452	C₂	C₂×D₅₃⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	#17793
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #20290277	C₂	semidirect
3. #20290275	C₂	semidirect
4. #20290272	C₂	semidirect
5. #20290269	C₂	semidirect
6. #20290264	C₂	semidirect
7. #20290262	C₂	semidirect
16. #16811487	C₂²	semidirect
17. #16811486	C₂²	semidirect
18. #16811485	C₂²	semidirect
19. #16811484	C₂²	semidirect
117. C₂×C₈×C₅₃	M₁₆	semidirect
143. C₂×#88322	M₁₆	semidirect
261. C₅₃	#17793	semidirect
354. C₂×C₈	#11719940	semidirect

Maximal subgroups

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

subgroup	index
#20291391	2
#20290277	2
#20290275	2
#20290272	2
#20290269	2
#20290264	2
#20290262	2
C₂²×C₈×Dic₅₃	2
#17793	53