Subgroups — #27353722

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 470 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, complemented
2	#20283739	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273580	C₂	1	2	yes	maximal, complemented
4	#20273573	C₂	1	2	yes	maximal, complemented
5	#20273552	C₂	1	2	yes	maximal, complemented
6	#20273545	C₂	1	2	yes	maximal, complemented
7	#20273467	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273097	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273002	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	1	yes	maximal, characteristic
12	#16810754	C₂²	1	2	yes	complemented
13	#16810701	C₂²	1	2	yes	complemented
14	#16810695	—	2	2	no	—
15	D₄×#88322	—	2	1	no	—
16	D₄×#88322	—	2	1	no	—
17	D₄×#88322	C₂²	1	2	yes	—
18	D₄×#88322	C₂²	1	2	yes	—
19	#16810680	—	2	1	no	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810676	C₂²	1	1	yes	characteristic
22	#16810647	—	2	1	no	—
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊#88322	—	2	2	no	—
25	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
26	#16810376	C₂²	1	2	yes	complemented
27	#16810374	C₂²	1	2	yes	complemented
28	Q₈⋊#88322	C₂²	1	1	yes	characteristic, complemented
29	Q₈⋊#88322	C₂²	1	1	yes	characteristic, complemented
30	D₄⋊#88322	C₂²	1	2	yes	—
31	D₄⋊#88322	C₂²	1	2	yes	—
32	D₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
33	D₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
34	#16810345	C₂²	1	2	yes	complemented
35	#16810345	C₂²	1	2	yes	complemented
36	C₂×AGL(1,ℤ/8)×C₅₃	C₄	1	1	yes	characteristic
37	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#442	C₄	1	2	yes	—
40	C₅₃×#440	C₄	1	2	yes	—
41	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
42	C₄×D₈×C₅₃	C₂²	1	2	yes	—
43	C₅₃×#428	C₄	1	1	yes	characteristic
44	C₅₃×#418	C₄	1	1	yes	characteristic
45	C₅₃×#417	C₄	1	1	yes	characteristic
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#11719917	—	2	2	no	—
48	C₂²×#88322	—	2	2	no	—
49	C₂²×#88322	—	2	2	no	—
50	C₂²×#88322	—	2	1	no	—
51	C₂²×#88322	—	2	1	no	—
52	C₄⋊#88322	—	2	1	no	—
53	C₄⋊#88322	—	2	2	no	—
54	C₄⋊#88322	D₄	1	1	yes	characteristic
55	C₄⋊#88322	D₄	1	1	yes	characteristic
56	C₄⋊#88322	C₂³	1	2	yes	—
57	C₄⋊#88322	C₂³	1	1	yes	characteristic
58	C₄⋊#88322	C₂³	1	1	yes	characteristic
59	C₂²⋊#88322	—	2	2	no	—
60	C₂²⋊#88322	—	2	2	no	—
61	C₂²⋊#88322	—	2	2	no	—
62	C₂²⋊#88322	—	2	1	no	—
63	C₂²⋊#88322	D₄	1	2	yes	—
64	#13133128	—	2	2	no	—
65	C₄×#88322	—	2	1	no	—
66	C₄×#88322	—	2	1	no	—
67	C₄×#88322	C₂³	1	1	yes	characteristic
68	C₄×#88322	C₂³	1	1	yes	characteristic
69	C₄×#88322	C₂³	1	2	yes	—
70	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂³×C₄×C₅₃	—	2	1	no	—
73	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
74	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
75	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₄×D₄×C₅₃	—	2	1	no	—
81	C₄×D₄×C₅₃	—	2	1	no	—
82	C₄×D₄×C₅₃	—	2	1	no	—
83	C₄×D₄×C₅₃	—	2	1	no	—
84	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×D₄×C₅₃	C₂³	1	2	yes	—
87	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
88	#117×C₅₃	—	2	1	no	—
89	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
92	C₂×C₄²×C₅₃	—	2	1	no	—
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	#107×C₅₃	C₂×C₄	1	2	yes	—
95	#106×C₅₃	C₂×C₄	1	2	yes	—
96	#103×C₅₃	C₂×C₄	1	2	yes	—
97	#102×C₅₃	C₂×C₄	1	2	yes	—
98	#102×C₅₃	C₂×C₄	1	2	yes	—
99	#102×C₅₃	C₂×C₄	1	2	yes	—
100	#97×C₅₃	C₂×C₄	1	2	yes	—
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂⁴×C₅₃	—	2	1	no	—
103	#41×C₅₃	—	2	2	no	—
104	#41×C₅₃	—	2	2	no	—
105	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	—	2	2	no	—
109	C₂×D₄×C₅₃	M₁₆	1	4	yes	complemented
110	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₅₃	—	2	1	no	—
113	C₂²×C₄×C₅₃	—	2	1	no	—
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	2	2	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	2	no	—
118	C₂²×C₄×C₅₃	—	8	1	no	—
119	C₂²×C₄×C₅₃	—	8	1	no	—
120	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
123	SD₁₆×C₅₃	—	2	4	no	—
124	D₈×C₅₃	—	2	4	no	—
125	M₁₆×C₅₃	—	2	2	no	—
126	M₁₆×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	—	2	1	no	—
132	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
135	C₂²⋊C₄×C₅₃	—	2	1	no	—
136	C₂²⋊C₄×C₅₃	—	2	2	no	—
137	C₂²⋊C₄×C₅₃	—	2	1	no	—
138	C₂²⋊C₄×C₅₃	—	2	2	no	—
139	C₂²⋊C₄×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₅₃	#41	1	1	yes	characteristic
146	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
147	#11719917	—	8	2	no	—
148	C₂×#88322	—	2	1	no	—
149	C₂×#88322	—	2	1	no	—
150	C₂×#88322	—	2	1	no	—
151	C₂×#88322	—	2	2	no	—
152	C₂×#88322	—	2	1	no	—
153	C₂×#88322	—	4	2	no	—
154	C₂×#88322	—	8	2	no	—
155	C₂×#88322	—	8	2	no	—
156	C₂×#88322	—	8	1	no	—
157	C₂×#88322	C₂×D₄	1	2	yes	—
158	C₂×#88322	C₂×D₄	1	2	yes	—
159	C₂³×C₅₃	—	2	2	no	—
160	C₂³×C₅₃	—	2	4	no	—
161	C₂³×C₅₃	—	2	1	no	—
162	C₂³×C₅₃	—	8	1	no	—
163	C₂³×C₅₃	—	8	1	no	—
164	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
165	Q₈×C₅₃	—	2	1	no	—
166	Q₈×C₅₃	—	2	1	no	—
167	D₄×C₅₃	—	2	1	no	—
168	D₄×C₅₃	—	2	2	no	—
169	D₄×C₅₃	—	2	1	no	—
170	D₄×C₅₃	—	2	4	no	—
171	D₄×C₅₃	—	4	2	no	—
172	D₄×C₅₃	C₂×M₁₆	1	4	yes	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	—	4	2	no	—
181	C₂×C₄×C₅₃	—	4	1	no	—
182	C₂×C₄×C₅₃	—	4	1	no	—
183	C₂×C₄×C₅₃	—	4	2	no	—
184	C₂×C₄×C₅₃	—	4	2	no	—
185	C₂×C₄×C₅₃	—	8	1	no	—
186	C₂×C₄×C₅₃	—	8	2	no	—
187	C₂×C₄×C₅₃	—	8	1	no	—
188	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₅₃	#131	1	2	yes	—
193	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₈×C₅₃	—	2	2	no	—
197	C₈×C₅₃	—	2	2	no	—
198	#88322	—	2	2	no	—
199	#88322	—	2	2	no	—
200	#88322	—	2	2	no	—
201	#88322	—	4	1	no	—
202	#88322	—	4	1	no	—
203	#88322	—	4	2	no	—
204	#88322	—	4	1	no	—
205	#17873	—	53	1	no	maximal, Sylow
206	C₂²×C₅₃	—	2	1	no	—
207	C₂²×C₅₃	—	2	1	no	—
208	C₂²×C₅₃	—	2	4	no	—
209	C₂²×C₅₃	—	4	2	no	—
210	C₂²×C₅₃	—	4	2	no	—
211	C₂²×C₅₃	—	4	1	no	—
212	C₂²×C₅₃	—	4	1	no	—
213	C₂²×C₅₃	—	4	1	no	—
214	C₂²×C₅₃	—	8	4	no	—
215	C₂²×C₅₃	#435	1	2	yes	—
216	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	—	4	1	no	—
221	C₄×C₅₃	—	4	1	no	—
222	C₄×C₅₃	—	4	2	no	—
223	C₄×C₅₃	—	4	1	no	—
224	C₄×C₅₃	#435	1	2	yes	—
225	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
226	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
227	C₄×C₅₃	#576	1	1	yes	characteristic
228	C₄×C₅₃	#577	1	1	yes	characteristic
229	#2989	—	53	1	no	—
230	#2972	—	53	1	no	—
231	C₄×AGL(1,ℤ/8)	D₅₃	1	1	yes	characteristic
232	#1586	—	53	2	no	—
233	#1585	—	53	2	no	—
234	#1582	—	53	2	no	—
235	#1581	—	53	2	no	—
236	#1565	—	53	1	no	—
237	#1476	—	53	1	no	—
238	C₂×D₄⋊C₈	—	53	1	no	—
239	#1451	—	53	1	no	—
240	C₂×C₅₃	—	2	2	no	—
241	C₂×C₅₃	—	4	4	no	—
242	C₂×C₅₃	—	8	1	no	—
243	C₂×C₅₃	C₂×#435	1	1	yes	characteristic
244	C₂×C₅₃	#2946	1	1	yes	characteristic
245	C₂×C₅₃	#2956	1	1	yes	characteristic
246	C₂×AGL(1,ℤ/8)	Dic₅₃	1	1	yes	characteristic
247	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
248	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
249	#446	—	53	2	no	—
250	#442	Dic₅₃	1	2	yes	—
251	#440	Dic₅₃	1	2	yes	—
252	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
253	C₄×D₈	C₂×D₅₃	1	2	yes	—
254	#436	—	53	2	no	—
255	#435	—	106	2	no	—
256	C₈×D₄	—	53	2	no	—
257	C₈×D₄	—	53	2	no	—
258	C₈×D₄	—	106	1	no	—
259	C₈×D₄	—	106	1	no	—
260	#432	—	106	1	no	—
261	#431	—	53	1	no	—
262	#431	—	53	1	no	—
263	#428	Dic₅₃	1	1	yes	characteristic
264	#423	—	106	1	no	—
265	C₂×C₄⋊C₈	—	53	1	no	—
266	#418	Dic₅₃	1	1	yes	characteristic
267	#417	Dic₅₃	1	1	yes	characteristic
268	C₂×C₂²⋊C₈	—	106	2	no	—
269	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
270	C₂×C₄×C₈	—	53	1	no	—
271	#335	—	53	2	no	—
272	#334	—	53	2	no	—
273	Q₈⋊C₈	—	53	1	no	—
274	Q₈⋊C₈	—	53	1	no	—
275	D₄⋊C₈	—	53	1	no	—
276	D₄⋊C₈	—	53	2	no	—
277	D₄⋊C₈	—	53	2	no	—
278	D₄⋊C₈	—	53	1	no	—
279	#322	—	53	2	no	—
280	#322	—	53	2	no	—
281	C₅₃	#17873	1	1	yes	characteristic, complemented, Sylow
282	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
283	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
284	C₂³×C₄	—	2	1	no	—
285	AGL(1,ℤ/8)	—	2	4	no	—
286	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
287	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
288	C₂×M₁₆	—	106	2	no	—
289	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
290	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
291	C₂²×C₈	—	106	2	no	—
292	C₂²×C₈	—	106	2	no	—
293	C₂²×C₈	—	106	1	no	—
294	C₂²×C₈	—	106	1	no	—
295	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
296	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
297	C₄×D₄	—	2	1	no	—
298	C₄×D₄	—	2	1	no	—
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
302	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
303	C₄×D₄	C₂²×D₅₃	1	2	yes	—
304	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
305	#117	—	2	1	no	—
306	#117	C₂×Dic₅₃	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
311	#107	C₂×Dic₅₃	1	2	yes	—
312	#106	C₂×Dic₅₃	1	2	yes	—
313	C₄⋊C₈	—	53	1	no	—
314	C₄⋊C₈	—	53	1	no	—
315	C₄⋊C₈	—	53	2	no	—
316	C₄⋊C₈	—	53	1	no	—
317	C₄⋊C₈	—	53	1	no	—
318	C₄⋊C₈	—	106	2	no	—
319	C₄⋊C₈	—	106	1	no	—
320	#103	C₂×Dic₅₃	1	2	yes	—
321	#102	C₂×Dic₅₃	1	2	yes	—
322	#102	C₂×Dic₅₃	1	2	yes	—
323	#102	C₂×Dic₅₃	1	2	yes	—
324	C₂²⋊C₈	—	53	2	no	—
325	C₂²⋊C₈	—	106	2	no	—
326	C₂²⋊C₈	—	106	2	no	—
327	C₂²⋊C₈	—	106	2	no	—
328	C₂²⋊C₈	—	106	1	no	—
329	#97	—	106	2	no	—
330	#97	C₂×Dic₅₃	1	2	yes	—
331	C₄×C₈	—	53	2	no	—
332	C₄×C₈	—	53	1	no	—
333	C₄×C₈	—	53	1	no	—
334	C₄×C₈	—	106	1	no	—
335	C₄×C₈	—	106	1	no	—
336	C₄×C₈	C₂²×D₅₃	1	2	yes	—
337	C₂⁴	—	2	1	no	—
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	#11719917	1	4	yes	complemented
345	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
346	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	8	1	no	—
354	C₂²×C₄	—	8	1	no	—
355	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
356	C₂²×C₄	#11719947	1	1	yes	characteristic
357	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
358	SD₁₆	—	2	4	no	—
359	D₈	—	2	4	no	—
360	M₁₆	—	2	2	no	—
361	M₁₆	—	2	2	no	—
362	M₁₆	—	424	2	no	—
363	C₂×C₈	—	53	2	no	—
364	C₂×C₈	—	53	2	no	—
365	C₂×C₈	—	106	1	no	—
366	C₂×C₈	—	106	1	no	—
367	C₂×C₈	—	106	1	no	—
368	C₂×C₈	—	106	2	no	—
369	C₂×C₈	—	106	1	no	—
370	C₂×C₈	—	424	2	no	—
371	C₂×C₈	—	424	2	no	—
372	C₂×C₈	—	424	1	no	—
373	C₂×C₈	—	212	2	no	—
374	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
375	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
380	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
381	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
382	C₂²⋊C₄	—	2	1	no	—
383	C₂²⋊C₄	—	2	2	no	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	2	no	—
386	C₂²⋊C₄	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	D₄×D₅₃	1	1	yes	characteristic
391	C₄²	#11719947	1	1	yes	characteristic
392	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
393	C₄²	C₂³×D₅₃	1	1	yes	characteristic
394	C₂³	—	2	2	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	8	1	no	—
398	C₂³	—	8	1	no	—
399	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
400	Q₈	—	2	1	no	—
401	Q₈	—	2	1	no	—
402	D₄	—	2	1	no	—
403	D₄	—	2	2	no	—
404	D₄	—	2	1	no	—
405	D₄	—	2	4	no	—
406	D₄	—	4	2	no	—
407	D₄	C₂×#11719917	1	4	yes	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	8	1	no	—
421	C₂×C₄	—	8	2	no	—
422	C₂×C₄	—	8	1	no	—
423	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
424	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
425	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
426	C₂×C₄	C₂×#11719917	1	2	yes	—
427	C₂×C₄	#13133224	1	2	yes	—
428	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
430	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	106	2	no	—
434	C₈	—	106	2	no	—
435	C₈	—	106	2	no	—
436	C₈	—	212	1	no	—
437	C₈	—	212	1	no	—
438	C₈	—	212	2	no	—
439	C₈	—	212	1	no	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	—	4	1	no	—
448	C₂²	—	8	4	no	—
449	C₂²	#16810695	1	2	yes	—
450	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	1	no	—
457	C₄	—	4	1	no	—
458	C₄	#16810695	1	2	yes	—
459	C₄	C₂²×#11719917	1	1	yes	characteristic
460	C₄	C₂×#13133224	1	1	yes	characteristic
461	C₄	#16811427	1	1	yes	characteristic
462	C₄	#16811433	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	4	4	no	—
465	C₂	—	8	1	no	—
466	C₂	C₂×#16810695	1	1	yes	characteristic
467	C₂	#20283548	1	1	yes	characteristic
468	C₂	#20283625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₄×AGL(1,ℤ/8)×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×AGL(1,ℤ/8)×C₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₄×AGL(1,ℤ/8)	D₅₃
hypocenter	C₅₃	#17873
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #20283832	C₂	semidirect
2. #20283739	C₂	semidirect
3. #20273580	C₂	semidirect
4. #20273573	C₂	semidirect
5. #20273552	C₂	semidirect
6. #20273545	C₂	semidirect
7. #20273467	C₂	semidirect
8. #20273097	C₂	semidirect
9. C₂×D₄⋊#88322	C₂	semidirect
10. #20273002	C₂	semidirect
12. #16810754	C₂²	semidirect
13. #16810701	C₂²	semidirect
26. #16810376	C₂²	semidirect
27. #16810374	C₂²	semidirect
28. Q₈⋊#88322	C₂²	semidirect
29. Q₈⋊#88322	C₂²	semidirect
32. D₄⋊#88322	C₂²	semidirect
33. D₄⋊#88322	C₂²	semidirect
34. #16810345	C₂²	semidirect
35. #16810345	C₂²	semidirect
109. C₂×D₄×C₅₃	M₁₆	semidirect
281. C₅₃	#17873	semidirect
344. C₂×D₄	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283832	2
#20283739	2
#20273580	2
#20273573	2
#20273552	2
#20273545	2
#20273467	2
#20273097	2
C₂×D₄⋊#88322	2
#20273002	2
C₄×AGL(1,ℤ/8)×C₅₃	2
#17873	53