Subgroups — #20289281

Order 6784.

1032 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811372	C₂	1	2	yes	maximal, complemented
2	#16811369	C₂	1	2	yes	maximal, complemented
3	#16811347	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811345	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811302	C₂	1	2	yes	maximal, complemented
6	#16811300	C₂	1	2	yes	maximal, complemented
7	Q₈×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811284	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811283	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811269	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811264	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811248	C₂	1	2	yes	maximal, complemented
13	#16811247	C₂	1	2	yes	maximal, complemented
14	#16811246	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811245	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811243	C₂	1	2	yes	maximal, complemented
17	#16811206	C₂	1	2	yes	maximal, complemented
18	#16811198	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811197	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811175	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811153	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×#13133206	C₂	1	1	yes	maximal, characteristic, complemented
23	D₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	1	yes	characteristic
25	C₂×#11719945	C₂²	1	1	yes	characteristic
26	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
28	#13133216	C₂²	1	1	yes	characteristic, complemented
29	#13133215	C₂²	1	1	yes	characteristic, complemented
30	#13133215	C₂²	1	2	yes	complemented
31	#13133215	C₂²	1	1	yes	characteristic, complemented
32	#13133212	C₂²	1	2	yes	complemented
33	#13133212	C₂²	1	2	yes	complemented
34	#13133212	C₂²	1	2	yes	complemented
35	#13133212	C₂²	1	2	yes	complemented
36	#13133212	C₂²	1	1	yes	characteristic, complemented
37	#13133211	C₂²	1	2	yes	complemented
38	#13133211	C₂²	1	2	yes	complemented
39	#13133210	C₂²	1	1	yes	characteristic, complemented
40	#13133207	C₂²	1	2	yes	complemented
41	#13133207	C₂²	1	1	yes	characteristic, complemented
42	#13133206	C₂²	1	1	yes	characteristic
43	#13133206	C₂²	1	1	yes	characteristic
44	#13133206	C₂²	1	1	yes	characteristic
45	#13133206	C₂²	1	2	yes	—
46	#13133206	C₂²	1	1	yes	characteristic
47	#13133206	C₂²	1	1	yes	characteristic
48	#13133206	C₂²	1	2	yes	—
49	#13133206	C₂²	1	2	yes	complemented
50	#13133205	C₂²	1	1	yes	characteristic
51	#13133205	C₂²	1	2	yes	complemented
52	#13133203	C₂²	1	2	yes	complemented
53	#13133203	C₂²	1	1	yes	characteristic, complemented
54	#13133203	C₂²	1	2	yes	complemented
55	#13133202	C₂²	1	2	yes	—
56	#13133202	C₂²	1	2	yes	—
57	#13133202	C₂²	1	2	yes	complemented
58	#13133202	C₂²	1	2	yes	complemented
59	#13133202	C₂²	1	2	yes	complemented
60	#13133201	C₂²	1	1	yes	characteristic
61	#13133200	C₂²	1	2	yes	complemented
62	#13133200	C₂²	1	2	yes	complemented
63	#13133199	C₂²	1	2	yes	complemented
64	#13133198	C₂²	1	1	yes	characteristic
65	#13133197	C₂²	1	2	yes	complemented
66	#13133197	C₂²	1	2	yes	complemented
67	#13133196	C₂²	1	1	yes	characteristic
68	#13133196	C₂²	1	2	yes	—
69	#13133195	C₂²	1	2	yes	complemented
70	#13133195	C₂²	1	2	yes	complemented
71	#13133195	C₂²	1	2	yes	complemented
72	#13133195	C₂²	1	2	yes	complemented
73	#13133195	C₂²	1	2	yes	complemented
74	#13133194	C₂²	1	1	yes	characteristic
75	#13133194	C₂²	1	2	yes	—
76	#13133193	C₂²	1	1	yes	characteristic, complemented
77	#13133193	C₂²	1	2	yes	complemented
78	#13133193	C₂²	1	2	yes	complemented
79	#13133193	C₂²	1	2	yes	complemented
80	#13133193	C₂²	1	2	yes	complemented
81	#13133192	C₂²	1	1	yes	characteristic
82	#13133192	C₂²	1	2	yes	—
83	#13133191	C₂²	1	1	yes	characteristic
84	#13133191	C₂²	1	2	yes	complemented
85	#13133190	C₂²	1	2	yes	—
86	#13133190	C₂²	1	2	yes	complemented
87	#13133189	C₂²	1	1	yes	characteristic
88	#13133187	C₂²	1	2	yes	—
89	#13133185	C₂²	1	2	yes	—
90	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
91	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
92	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
93	#13133182	C₂²	1	1	yes	characteristic, complemented
94	#13133182	C₂²	1	2	yes	complemented
95	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
96	#13133179	C₂²	1	1	yes	characteristic
97	#13133179	C₂²	1	2	yes	complemented
98	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
99	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
100	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic, complemented
101	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
102	#13133176	C₂²	1	1	yes	characteristic, complemented
103	#13133175	C₂²	1	2	yes	complemented
104	#13133174	C₂²	1	2	yes	complemented
105	#13133174	C₂²	1	1	yes	characteristic, complemented
106	#13133173	C₂²	1	2	yes	—
107	C₂×#11719921	C₂²	1	2	yes	—
108	C₂×#11719921	C₂²	1	2	yes	—
109	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
110	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
111	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
112	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
113	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
114	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
115	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
116	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
117	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
118	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
120	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
121	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
122	C₄×D₄×C₅₃	C₂²	1	2	yes	—
123	C₂³×D₅₃	C₂³	1	2	yes	—
124	C₂×Q₈×C₅₃	—	4	2	no	—
125	C₂×Q₈×C₅₃	—	4	2	no	—
126	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
130	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
131	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
137	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
138	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
140	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
141	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
142	#11719949	—	4	2	no	—
143	Q₈×D₅₃	—	4	2	no	—
144	#11719945	—	4	2	no	—
145	C₂×D₂₁₂	—	4	2	no	—
146	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
147	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
148	C₂×D₂₁₂	C₂³	1	2	yes	—
149	C₂×D₂₁₂	C₂³	1	2	yes	—
150	C₂×D₂₁₂	C₂³	1	2	yes	—
151	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₅₃	C₂³	1	2	yes	—
153	C₂×C₄×D₅₃	C₂³	1	2	yes	—
154	C₂×C₄×D₅₃	C₂³	1	2	yes	—
155	C₂×C₄×D₅₃	C₂³	1	2	yes	—
156	C₂×C₄×D₅₃	C₂³	1	2	yes	—
157	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
158	C₂×Dic₁₀₆	C₂³	1	2	yes	—
159	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
160	C₂×Dic₁₀₆	C₂³	1	2	yes	—
161	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
162	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
163	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
164	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
165	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
166	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
167	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
168	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
169	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
170	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
171	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
172	C₄²×C₅₃	C₂³	1	1	yes	characteristic
173	C₄²×C₅₃	C₂³	1	2	yes	—
174	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
175	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
176	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
177	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
178	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
179	#11719921	C₂³	1	2	yes	—
180	#11719921	C₂³	1	2	yes	—
181	#11719921	C₂³	1	2	yes	—
182	#11719921	C₂³	1	2	yes	—
183	#11719921	C₂³	1	2	yes	—
184	#11719921	C₂³	1	2	yes	—
185	#11719921	C₂³	1	2	yes	—
186	#11719921	C₂³	1	2	yes	—
187	#11719921	C₂³	1	2	yes	—
188	#11719921	C₂³	1	2	yes	—
189	#11719921	C₂³	1	2	yes	—
190	#11719921	C₂³	1	2	yes	—
191	#11719921	C₂³	1	2	yes	—
192	#11719921	C₂³	1	1	yes	characteristic
193	#11719921	C₂³	1	1	yes	characteristic
194	#11719921	C₂³	1	2	yes	—
195	#11719921	C₂³	1	2	yes	—
196	#11719921	C₂³	1	1	yes	characteristic
197	#11719921	C₂³	1	1	yes	characteristic
198	#11719921	C₂³	1	2	yes	—
199	#11719921	C₂³	1	2	yes	—
200	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
201	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
202	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
203	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
204	C₄⋊Dic₅₃	C₂³	1	2	yes	—
205	C₄⋊Dic₅₃	C₂³	1	2	yes	—
206	#11719919	C₂³	1	2	yes	—
207	#11719919	C₂³	1	2	yes	—
208	#11719919	C₂³	1	1	yes	characteristic
209	#11719919	C₂³	1	2	yes	—
210	#11719919	C₂³	1	1	yes	characteristic
211	#11719919	C₂³	1	1	yes	characteristic
212	#11719919	C₂³	1	2	yes	—
213	#11719919	C₂³	1	1	yes	characteristic
214	#11719919	C₂³	1	1	yes	characteristic
215	#11719919	C₂³	1	2	yes	—
216	#11719919	C₂³	1	2	yes	—
217	#11719919	C₂³	1	2	yes	—
218	#11719919	C₂³	1	2	yes	—
219	#11719919	C₂³	1	1	yes	characteristic
220	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
221	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
222	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
223	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₄×Dic₅₃	C₂³	1	2	yes	—
226	C₄×Dic₅₃	C₂³	1	2	yes	—
227	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
228	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
229	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
230	C₂²×D₅₃	—	2	4	no	—
231	C₂²×D₅₃	—	4	2	no	—
232	C₂²×D₅₃	—	4	2	no	—
233	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
234	C₂²×D₅₃	C₂⁴	1	2	yes	—
235	C₂²×D₅₃	C₂⁴	1	2	yes	—
236	Q₈×C₅₃	—	2	2	no	—
237	Q₈×C₅₃	—	2	2	no	—
238	Q₈×C₅₃	—	2	2	no	—
239	Q₈×C₅₃	—	4	2	no	—
240	Q₈×C₅₃	—	4	2	no	—
241	D₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	2	2	no	—
244	C₂×C₄×C₅₃	—	4	2	no	—
245	C₂×C₄×C₅₃	—	4	2	no	—
246	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
248	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
249	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
251	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
253	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
255	C₂²⋊D₅₃	—	2	2	no	—
256	C₂²⋊D₅₃	—	4	2	no	—
257	C₂²⋊D₅₃	—	4	2	no	—
258	C₂²⋊D₅₃	—	4	1	no	—
259	C₂×Dic₅₃	—	2	2	no	—
260	C₂×Dic₅₃	—	4	1	no	—
261	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
262	C₂×Dic₅₃	C₂⁴	1	2	yes	—
263	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
264	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
265	C₂×Dic₅₃	C₂⁴	1	2	yes	—
266	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
267	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
268	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
269	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
270	D₂₁₂	—	2	2	no	—
271	D₂₁₂	—	2	2	no	—
272	D₂₁₂	—	4	2	no	—
273	D₂₁₂	—	4	2	no	—
274	D₂₁₂	—	4	2	no	—
275	C₄×D₅₃	—	2	2	no	—
276	C₄×D₅₃	—	4	2	no	—
277	C₄×D₅₃	—	4	2	no	—
278	C₄×D₅₃	—	4	2	no	—
279	C₄×D₅₃	—	4	2	no	—
280	C₄×D₅₃	—	4	2	no	—
281	Dic₁₀₆	—	2	2	no	—
282	Dic₁₀₆	—	4	2	no	—
283	Dic₁₀₆	—	4	1	no	—
284	Dic₁₀₆	—	4	2	no	—
285	C₂²×C₅₃	—	2	1	no	—
286	C₂²×C₅₃	—	2	1	no	—
287	C₂²×C₅₃	—	2	1	no	—
288	C₂²×C₅₃	—	2	1	no	—
289	C₂²×C₅₃	—	2	1	no	—
290	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
291	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
292	C₂×D₅₃	—	2	1	no	—
293	C₂×D₅₃	—	2	2	no	—
294	C₂×D₅₃	—	2	2	no	—
295	C₂×D₅₃	—	2	1	no	—
296	C₂×D₅₃	—	2	2	no	—
297	C₂×D₅₃	—	2	2	no	—
298	C₂×D₅₃	—	2	2	no	—
299	C₂×D₅₃	—	2	2	no	—
300	C₂×D₅₃	—	2	1	no	—
301	C₂×D₅₃	—	8	4	no	—
302	C₄×C₅₃	—	2	2	no	—
303	C₄×C₅₃	—	2	1	no	—
304	C₄×C₅₃	—	2	2	no	—
305	C₄×C₅₃	—	2	1	no	—
306	C₄×C₅₃	—	2	2	no	—
307	C₄×C₅₃	—	2	1	no	—
308	C₄×C₅₃	—	2	1	no	—
309	C₄×C₅₃	—	2	2	no	—
310	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
311	Dic₅₃	—	2	2	no	—
312	Dic₅₃	—	2	1	no	—
313	Dic₅₃	—	2	1	no	—
314	Dic₅₃	—	2	1	no	—
315	Dic₅₃	—	2	1	no	—
316	Dic₅₃	—	2	1	no	—
317	Dic₅₃	—	2	1	no	—
318	Dic₅₃	—	2	2	no	—
319	Dic₅₃	—	2	1	no	—
320	#3561	—	53	1	no	maximal, Sylow
321	C₂×C₅₃	—	2	2	no	—
322	C₂×C₅₃	—	4	1	no	—
323	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
324	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
325	C₂×C₅₃	#585	1	1	yes	characteristic
326	D₅₃	—	4	1	no	—
327	D₅₃	—	4	2	no	—
328	D₅₃	—	4	2	no	—
329	#563	—	53	2	no	—
330	#562	—	53	2	no	—
331	#556	—	53	1	no	—
332	#556	—	53	1	no	—
333	#551	—	53	2	no	—
334	#551	—	53	2	no	—
335	D₄×Q₈	—	53	1	no	—
336	D₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
337	#548	—	53	1	no	—
338	#548	—	53	1	no	—
339	#546	—	53	1	no	—
340	#546	—	53	1	no	—
341	#543	—	53	2	no	—
342	#542	—	53	2	no	—
343	#542	—	53	2	no	—
344	#542	—	53	1	no	—
345	#542	—	53	1	no	—
346	#538	—	53	2	no	—
347	#536	—	53	1	no	—
348	#536	—	53	1	no	—
349	#533	—	53	1	no	—
350	#529	—	53	1	no	—
351	C₂×#124	—	53	1	no	—
352	C₅₃	#3561	1	1	yes	characteristic, complemented, Sylow
353	C₂×#41	—	53	1	no	—
354	C₂×#41	—	53	1	no	—
355	C₂²×Q₈	—	53	1	no	—
356	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
357	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
358	C₂²×D₄	—	53	1	no	—
359	C₄⋊Q₈	—	53	1	no	—
360	C₄⋊Q₈	—	53	1	no	—
361	C₄⋊Q₈	—	53	2	no	—
362	C₄⋊Q₈	—	53	1	no	—
363	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
364	C₄⋊Q₈	C₂×D₅₃	1	2	yes	complemented
365	#126	—	53	1	no	—
366	#126	—	53	2	no	—
367	#126	—	53	2	no	—
368	#126	—	53	2	no	—
369	#126	—	53	2	no	—
370	#126	—	53	1	no	—
371	#126	—	53	2	no	—
372	#126	—	53	2	no	—
373	#125	—	53	1	no	—
374	#125	—	53	2	no	—
375	#124	—	53	1	no	—
376	#124	—	53	1	no	—
377	#124	—	53	2	no	—
378	#124	—	53	1	no	—
379	#124	—	53	1	no	—
380	#124	—	53	2	no	—
381	#124	—	53	1	no	—
382	#124	—	53	2	no	—
383	#124	—	53	1	no	—
384	#124	—	53	2	no	—
385	#124	—	53	2	no	—
386	#124	—	53	1	no	—
387	#124	—	53	2	no	—
388	#123	—	53	1	no	—
389	#123	—	53	2	no	—
390	#123	—	53	2	no	—
391	#123	—	53	2	no	—
392	#123	—	53	2	no	—
393	#123	—	53	2	no	—
394	#123	—	53	2	no	—
395	#123	—	53	2	no	—
396	#123	—	53	1	no	—
397	#123	—	53	2	no	—
398	C₂²⋊Q₈	—	53	2	no	—
399	C₂²⋊Q₈	—	53	2	no	—
400	C₂²⋊Q₈	—	53	1	no	—
401	C₂²⋊Q₈	—	53	2	no	—
402	C₂²⋊Q₈	—	53	2	no	—
403	C₂²⋊Q₈	—	53	1	no	—
404	C₂²⋊Q₈	—	53	2	no	—
405	C₂²⋊Q₈	—	53	2	no	—
406	C₂²⋊Q₈	—	53	2	no	—
407	C₂²⋊Q₈	—	53	2	no	—
408	C₂²⋊Q₈	—	53	2	no	—
409	C₂²⋊Q₈	—	53	2	no	—
410	C₂²⋊Q₈	—	53	2	no	—
411	C₂²⋊Q₈	—	53	2	no	—
412	C₂²⋊Q₈	—	53	1	no	—
413	C₂²⋊Q₈	—	53	2	no	—
414	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
415	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
416	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
417	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
418	#121	—	53	1	no	—
419	#121	—	53	2	no	—
420	#121	—	53	1	no	—
421	#121	—	53	1	no	—
422	#121	—	53	2	no	—
423	#121	—	53	2	no	—
424	#121	—	53	2	no	—
425	#120	—	53	2	no	—
426	#120	—	53	2	no	—
427	C₄×Q₈	—	53	1	no	—
428	C₄×Q₈	—	53	1	no	—
429	C₄×Q₈	—	53	2	no	—
430	C₄×Q₈	—	53	1	no	—
431	C₄×Q₈	—	53	1	no	—
432	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
433	C₄×D₄	—	53	1	no	—
434	C₄×D₄	—	53	1	no	—
435	C₄×D₄	—	53	2	no	—
436	C₄×D₄	—	53	2	no	—
437	C₄×D₄	—	53	1	no	—
438	C₄×D₄	—	53	2	no	—
439	C₄×D₄	—	53	1	no	—
440	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
441	C₄×D₄	C₂×D₅₃	1	2	yes	—
442	#117	—	53	1	no	—
443	#117	—	53	2	no	—
444	#117	—	53	2	no	—
445	#117	—	53	1	no	—
446	#117	—	53	2	no	—
447	C₂×C₂²⋊C₄	—	53	2	no	—
448	C₂×C₂²⋊C₄	—	53	2	no	—
449	C₂×C₂²⋊C₄	—	53	2	no	—
450	C₂×C₂²⋊C₄	—	53	2	no	—
451	C₂×C₄²	—	53	1	no	—
452	C₂⁴	—	53	2	no	—
453	#41	—	212	2	no	—
454	#41	—	212	2	no	—
455	C₂×Q₈	—	4	2	no	—
456	C₂×Q₈	—	4	2	no	—
457	C₂×Q₈	—	53	1	no	—
458	C₂×Q₈	—	53	1	no	—
459	C₂×Q₈	—	53	2	no	—
460	C₂×Q₈	—	53	2	no	—
461	C₂×Q₈	—	212	2	no	—
462	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
463	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
464	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
465	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
466	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
467	C₂×D₄	—	53	1	no	—
468	C₂×D₄	—	53	2	no	—
469	C₂×D₄	—	53	1	no	—
470	C₂×D₄	—	53	2	no	—
471	C₂×D₄	—	53	1	no	—
472	C₂×D₄	—	53	1	no	—
473	C₂×D₄	—	53	2	no	—
474	C₂×D₄	—	53	2	no	—
475	C₂×D₄	—	53	2	no	—
476	C₂×D₄	—	212	2	no	—
477	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
478	C₂²×C₄	—	53	1	no	—
479	C₂²×C₄	—	53	1	no	—
480	C₂²×C₄	—	53	1	no	—
481	C₂²×C₄	—	53	2	no	—
482	C₂²×C₄	—	53	2	no	—
483	C₂²×C₄	—	53	2	no	—
484	C₂²×C₄	—	53	2	no	—
485	C₂²×C₄	—	53	2	no	—
486	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
487	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
488	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
489	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
490	C₄⋊C₄	—	53	1	no	—
491	C₄⋊C₄	—	53	1	no	—
492	C₄⋊C₄	—	53	1	no	—
493	C₄⋊C₄	—	53	1	no	—
494	C₄⋊C₄	—	53	2	no	—
495	C₄⋊C₄	—	53	2	no	—
496	C₄⋊C₄	—	53	1	no	—
497	C₄⋊C₄	—	53	1	no	—
498	C₄⋊C₄	—	53	2	no	—
499	C₄⋊C₄	—	53	1	no	—
500	C₄⋊C₄	—	53	1	no	—
501	C₄⋊C₄	—	53	2	no	—
502	C₄⋊C₄	—	53	2	no	—
503	C₄⋊C₄	—	53	1	no	—
504	C₄⋊C₄	—	53	2	no	—
505	C₄⋊C₄	—	53	2	no	—
506	C₄⋊C₄	—	53	2	no	—
507	C₄⋊C₄	—	53	1	no	—
508	C₄⋊C₄	—	53	2	no	—
509	C₄⋊C₄	—	53	2	no	—
510	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
511	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
512	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
513	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
514	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
516	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
517	C₂²⋊C₄	—	53	1	no	—
518	C₂²⋊C₄	—	53	2	no	—
519	C₂²⋊C₄	—	53	1	no	—
520	C₂²⋊C₄	—	53	1	no	—
521	C₂²⋊C₄	—	53	1	no	—
522	C₂²⋊C₄	—	53	2	no	—
523	C₂²⋊C₄	—	53	2	no	—
524	C₂²⋊C₄	—	53	2	no	—
525	C₂²⋊C₄	—	53	2	no	—
526	C₂²⋊C₄	—	53	2	no	—
527	C₂²⋊C₄	—	53	2	no	—
528	C₂²⋊C₄	—	53	2	no	—
529	C₂²⋊C₄	—	53	2	no	—
530	C₂²⋊C₄	—	53	2	no	—
531	C₂²⋊C₄	—	53	2	no	—
532	C₂²⋊C₄	—	53	2	no	—
533	C₂²⋊C₄	—	53	2	no	—
534	C₂²⋊C₄	—	53	2	no	—
535	C₂²⋊C₄	—	53	2	no	—
536	C₂²⋊C₄	—	53	1	no	—
537	C₂²⋊C₄	—	53	1	no	—
538	C₂²⋊C₄	—	53	1	no	—
539	C₂²⋊C₄	—	53	2	no	—
540	C₂²⋊C₄	—	53	2	no	—
541	C₂²⋊C₄	—	53	1	no	—
542	C₂²⋊C₄	—	53	2	no	—
543	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
544	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
545	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
546	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
547	C₄²	—	53	1	no	—
548	C₄²	—	53	1	no	—
549	C₄²	—	53	1	no	—
550	C₄²	—	53	1	no	—
551	C₄²	—	53	2	no	—
552	C₄²	—	53	2	no	—
553	C₄²	—	53	1	no	—
554	C₄²	—	53	2	no	—
555	C₄²	C₂²×D₅₃	1	1	yes	characteristic
556	C₄²	C₂²×D₅₃	1	2	yes	—
557	C₂³	—	53	2	no	—
558	C₂³	—	53	2	no	—
559	C₂³	—	53	1	no	—
560	C₂³	—	106	4	no	—
561	C₂³	—	212	2	no	—
562	C₂³	—	212	2	no	—
563	C₂³	C₂³×D₅₃	1	1	yes	characteristic
564	C₂³	C₂³×D₅₃	1	1	yes	characteristic
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	2	2	no	—
568	Q₈	—	4	2	no	—
569	Q₈	—	4	2	no	—
570	Q₈	—	106	2	no	—
571	Q₈	—	212	1	no	—
572	Q₈	—	212	2	no	—
573	Q₈	—	212	2	no	—
574	D₄	—	2	2	no	—
575	D₄	—	106	2	no	—
576	D₄	—	106	2	no	—
577	D₄	—	106	2	no	—
578	D₄	—	212	2	no	—
579	D₄	—	212	2	no	—
580	D₄	—	212	2	no	—
581	D₄	—	212	2	no	—
582	D₄	—	212	1	no	—
583	D₄	—	212	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	53	1	no	—
589	C₂×C₄	—	53	1	no	—
590	C₂×C₄	—	53	1	no	—
591	C₂×C₄	—	53	1	no	—
592	C₂×C₄	—	53	2	no	—
593	C₂×C₄	—	53	1	no	—
594	C₂×C₄	—	53	1	no	—
595	C₂×C₄	—	53	2	no	—
596	C₂×C₄	—	53	1	no	—
597	C₂×C₄	—	106	2	no	—
598	C₂×C₄	—	106	2	no	—
599	C₂×C₄	—	212	2	no	—
600	C₂×C₄	—	212	2	no	—
601	C₂×C₄	—	212	2	no	—
602	C₂×C₄	—	212	2	no	—
603	C₂×C₄	—	212	2	no	—
604	C₂×C₄	—	212	1	no	—
605	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
606	C₂×C₄	C₂³×D₅₃	1	2	yes	—
607	C₂×C₄	C₂³×D₅₃	1	2	yes	—
608	C₂×C₄	C₂³×D₅₃	1	2	yes	—
609	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
610	C₂×C₄	C₂³×D₅₃	1	2	yes	—
611	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
612	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
613	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	1	no	—
619	C₂²	—	106	1	no	—
620	C₂²	—	106	2	no	—
621	C₂²	—	106	2	no	—
622	C₂²	—	106	1	no	—
623	C₂²	—	106	2	no	—
624	C₂²	—	106	2	no	—
625	C₂²	—	106	2	no	—
626	C₂²	—	106	2	no	—
627	C₂²	—	106	1	no	—
628	C₂²	—	424	4	no	—
629	C₂²	#13133265	1	2	yes	—
630	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
631	C₄	—	2	1	no	—
632	C₄	—	2	1	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	2	no	—
636	C₄	—	2	1	no	—
637	C₄	—	2	1	no	—
638	C₄	—	2	2	no	—
639	C₄	—	106	1	no	—
640	C₄	—	106	2	no	—
641	C₄	—	106	1	no	—
642	C₄	—	106	1	no	—
643	C₄	—	106	1	no	—
644	C₄	—	106	1	no	—
645	C₄	—	106	2	no	—
646	C₄	—	106	1	no	—
647	C₄	—	106	1	no	—
648	C₄	#13133262	1	2	yes	—
649	C₂	—	2	2	no	—
650	C₂	—	4	1	no	—
651	C₂	—	212	2	no	—
652	C₂	—	212	2	no	—
653	C₂	—	212	1	no	—
654	C₂	C₂×#13133262	1	1	yes	characteristic
655	C₂	C₂×#13133265	1	1	yes	characteristic
656	C₂	#16811460	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	D₄×Q₈×C₅₃	C₂
hypercenter	D₄×Q₈	D₅₃
hypocenter	C₅₃	#3561
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #16811372	C₂	semidirect
2. #16811369	C₂	semidirect
3. #16811347	C₂	semidirect
4. #16811345	C₂	semidirect
5. #16811302	C₂	semidirect
6. #16811300	C₂	semidirect
7. Q₈×C₂²⋊D₅₃	C₂	semidirect
8. #16811284	C₂	semidirect
9. #16811283	C₂	semidirect
10. #16811269	C₂	semidirect
11. #16811264	C₂	semidirect
12. #16811248	C₂	semidirect
13. #16811247	C₂	semidirect
14. #16811246	C₂	semidirect
15. #16811245	C₂	semidirect
16. #16811243	C₂	semidirect
17. #16811206	C₂	semidirect
18. #16811198	C₂	semidirect
19. #16811197	C₂	semidirect
20. #16811175	C₂	semidirect
21. #16811153	C₂	semidirect
22. C₂×#13133206	C₂	semidirect
23. D₄×Q₈×C₅₃	C₂	semidirect
26. C₂×Q₈×D₅₃	C₂²	semidirect
28. #13133216	C₂²	semidirect
29. #13133215	C₂²	semidirect
30. #13133215	C₂²	semidirect
31. #13133215	C₂²	semidirect
32. #13133212	C₂²	semidirect
33. #13133212	C₂²	semidirect
34. #13133212	C₂²	semidirect
35. #13133212	C₂²	semidirect
36. #13133212	C₂²	semidirect
37. #13133211	C₂²	semidirect
38. #13133211	C₂²	semidirect
39. #13133210	C₂²	semidirect
40. #13133207	C₂²	semidirect
41. #13133207	C₂²	semidirect
49. #13133206	C₂²	semidirect
51. #13133205	C₂²	semidirect
52. #13133203	C₂²	semidirect
53. #13133203	C₂²	semidirect
54. #13133203	C₂²	semidirect
57. #13133202	C₂²	semidirect
58. #13133202	C₂²	semidirect
59. #13133202	C₂²	semidirect
61. #13133200	C₂²	semidirect
62. #13133200	C₂²	semidirect
63. #13133199	C₂²	semidirect
65. #13133197	C₂²	semidirect
66. #13133197	C₂²	semidirect
69. #13133195	C₂²	semidirect
70. #13133195	C₂²	semidirect
71. #13133195	C₂²	semidirect
72. #13133195	C₂²	semidirect
73. #13133195	C₂²	semidirect
76. #13133193	C₂²	semidirect
77. #13133193	C₂²	semidirect
78. #13133193	C₂²	semidirect
79. #13133193	C₂²	semidirect
80. #13133193	C₂²	semidirect
84. #13133191	C₂²	semidirect
86. #13133190	C₂²	semidirect
90. Q₈×Dic₅₃	C₂²	semidirect
91. Q₈×Dic₅₃	C₂²	semidirect
92. C₄×Dic₁₀₆	C₂²	semidirect
93. #13133182	C₂²	semidirect
94. #13133182	C₂²	semidirect
97. #13133179	C₂²	semidirect
100. C₄×C₂²⋊D₅₃	C₂²	semidirect
101. C₄×C₂²⋊D₅₃	C₂²	semidirect
102. #13133176	C₂²	semidirect
103. #13133175	C₂²	semidirect
104. #13133174	C₂²	semidirect
105. #13133174	C₂²	semidirect
113. C₂²×Q₈×C₅₃	C₂²	semidirect
114. C₄⋊Q₈×C₅₃	C₂²	semidirect
115. C₄⋊Q₈×C₅₃	C₂²	semidirect
118. C₂²⋊Q₈×C₅₃	C₂²	semidirect
119. C₂²⋊Q₈×C₅₃	C₂²	semidirect
120. C₄×Q₈×C₅₃	C₂²	semidirect
336. D₄×Q₈	D₅₃	semidirect
352. C₅₃	#3561	semidirect
357. C₂²×Q₈	C₂×D₅₃	semidirect
363. C₄⋊Q₈	C₂×D₅₃	semidirect
364. C₄⋊Q₈	C₂×D₅₃	semidirect
416. C₂²⋊Q₈	C₂×D₅₃	semidirect
417. C₂²⋊Q₈	C₂×D₅₃	semidirect
432. C₄×Q₈	C₂×D₅₃	semidirect