Subgroups — #20289013

Order 6784.

1032 conjugacy classes of subgroups.

The subgroup lattice has 672 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	#16811344	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811331	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811327	C₂	1	2	yes	maximal, complemented
6	#16811322	C₂	1	2	yes	maximal, complemented
7	#16811319	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811314	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811284	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811284	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811276	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811247	C₂	1	2	yes	maximal, complemented
13	#16811245	C₂	1	1	yes	maximal, characteristic, complemented
14	#16811234	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811219	C₂	1	2	yes	maximal, complemented
16	#16811213	C₂	1	2	yes	maximal, complemented
17	#16811206	C₂	1	2	yes	maximal, complemented
18	#16811203	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811197	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811136	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811132	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×#13133193	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×#11719949	C₂²	1	1	yes	characteristic, complemented
26	C₂×#11719945	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
28	#13133216	C₂²	1	1	yes	characteristic, complemented
29	#13133215	C₂²	1	2	yes	complemented
30	#13133212	C₂²	1	2	yes	—
31	#13133212	C₂²	1	2	yes	—
32	#13133212	C₂²	1	2	yes	—
33	#13133212	C₂²	1	2	yes	complemented
34	#13133211	C₂²	1	2	yes	—
35	#13133211	C₂²	1	2	yes	complemented
36	#13133210	C₂²	1	2	yes	complemented
37	#13133209	C₂²	1	1	yes	characteristic, complemented
38	#13133207	C₂²	1	2	yes	complemented
39	#13133207	C₂²	1	2	yes	complemented
40	#13133207	C₂²	1	2	yes	complemented
41	#13133207	C₂²	1	2	yes	complemented
42	#13133206	C₂²	1	2	yes	—
43	#13133206	C₂²	1	2	yes	—
44	#13133206	C₂²	1	1	yes	characteristic, complemented
45	#13133206	C₂²	1	2	yes	complemented
46	#13133205	C₂²	1	2	yes	—
47	#13133203	C₂²	1	2	yes	—
48	#13133202	C₂²	1	2	yes	—
49	#13133202	C₂²	1	2	yes	—
50	#13133202	C₂²	1	2	yes	—
51	#13133202	C₂²	1	1	yes	characteristic
52	#13133202	C₂²	1	2	yes	—
53	#13133202	C₂²	1	2	yes	complemented
54	#13133200	C₂²	1	2	yes	—
55	#13133200	C₂²	1	2	yes	complemented
56	#13133199	C₂²	1	1	yes	characteristic
57	#13133199	C₂²	1	2	yes	complemented
58	#13133197	C₂²	1	2	yes	complemented
59	#13133195	C₂²	1	1	yes	characteristic
60	#13133195	C₂²	1	2	yes	complemented
61	#13133195	C₂²	1	2	yes	complemented
62	#13133194	C₂²	1	1	yes	characteristic
63	#13133194	C₂²	1	2	yes	—
64	#13133193	C₂²	1	1	yes	characteristic
65	#13133193	C₂²	1	1	yes	characteristic
66	#13133193	C₂²	1	2	yes	—
67	#13133193	C₂²	1	1	yes	characteristic
68	#13133193	C₂²	1	1	yes	characteristic
69	#13133193	C₂²	1	2	yes	—
70	#13133193	C₂²	1	1	yes	characteristic
71	#13133193	C₂²	1	1	yes	characteristic, complemented
72	#13133193	C₂²	1	1	yes	characteristic, complemented
73	#13133193	C₂²	1	2	yes	complemented
74	#13133193	C₂²	1	2	yes	complemented
75	#13133192	C₂²	1	2	yes	—
76	#13133192	C₂²	1	1	yes	characteristic
77	#13133192	C₂²	1	2	yes	—
78	#13133191	C₂²	1	1	yes	characteristic
79	#13133191	C₂²	1	2	yes	—
80	#13133191	C₂²	1	2	yes	—
81	#13133191	C₂²	1	1	yes	characteristic, complemented
82	#13133190	C₂²	1	1	yes	characteristic
83	#13133190	C₂²	1	2	yes	—
84	#13133190	C₂²	1	2	yes	complemented
85	#13133189	C₂²	1	2	yes	—
86	#13133189	C₂²	1	1	yes	characteristic
87	#13133187	C₂²	1	1	yes	characteristic
88	#13133187	C₂²	1	2	yes	—
89	#13133185	C₂²	1	1	yes	characteristic
90	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
91	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
92	#13133182	C₂²	1	2	yes	complemented
93	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
94	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
95	#13133179	C₂²	1	2	yes	—
96	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
97	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
98	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
99	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
100	#13133177	C₂²	1	2	yes	—
101	#13133177	C₂²	1	2	yes	—
102	#13133177	C₂²	1	2	yes	—
103	#13133176	C₂²	1	1	yes	characteristic, complemented
104	#13133175	C₂²	1	1	yes	characteristic
105	#13133174	C₂²	1	1	yes	characteristic
106	#13133174	C₂²	1	2	yes	complemented
107	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
108	C₂×#11719919	C₂²	1	2	yes	—
109	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
110	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
111	C₂×#11719921	C₂²	1	2	yes	—
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	2	yes	—
117	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
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	2	yes	—
137	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
140	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
141	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
142	C₂²×Dic₅₃	C₂³	1	2	yes	—
143	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
144	#11719949	—	4	2	no	—
145	#11719949	—	4	2	no	—
146	#11719945	—	4	2	no	—
147	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
148	C₂×D₂₁₂	C₂³	1	2	yes	—
149	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
150	C₂×D₂₁₂	C₂³	1	2	yes	—
151	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
152	C₂×D₂₁₂	C₂³	1	2	yes	—
153	C₂×C₄×D₅₃	—	4	2	no	—
154	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
155	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
156	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
157	C₂×C₄×D₅₃	C₂³	1	2	yes	—
158	C₂×C₄×D₅₃	C₂³	1	2	yes	—
159	C₂×C₄×D₅₃	C₂³	1	2	yes	—
160	C₂×C₄×D₅₃	C₂³	1	2	yes	—
161	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
162	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
163	C₂×Dic₁₀₆	C₂³	1	2	yes	—
164	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
165	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
166	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
167	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
168	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
169	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
170	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
172	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
173	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
174	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
175	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
176	C₄²×C₅₃	C₂³	1	1	yes	characteristic
177	C₄²×C₅₃	C₂³	1	2	yes	—
178	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
179	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
180	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
186	#11719921	C₂³	1	1	yes	characteristic
187	#11719921	C₂³	1	1	yes	characteristic
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	2	yes	—
193	#11719921	C₂³	1	1	yes	characteristic
194	#11719921	C₂³	1	1	yes	characteristic
195	#11719921	C₂³	1	2	yes	—
196	#11719921	C₂³	1	2	yes	—
197	#11719921	C₂³	1	2	yes	—
198	#11719921	C₂³	1	2	yes	—
199	#11719921	C₂³	1	1	yes	characteristic
200	#11719921	C₂³	1	2	yes	—
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	1	yes	characteristic
206	C₄⋊Dic₅₃	C₂³	1	2	yes	—
207	C₄⋊Dic₅₃	C₂³	1	2	yes	—
208	#11719919	C₂³	1	2	yes	—
209	#11719919	C₂³	1	2	yes	—
210	#11719919	C₂³	1	2	yes	—
211	#11719919	C₂³	1	2	yes	—
212	#11719919	C₂³	1	2	yes	—
213	#11719919	C₂³	1	2	yes	—
214	#11719919	C₂³	1	1	yes	characteristic
215	#11719919	C₂³	1	1	yes	characteristic
216	#11719919	C₂³	1	2	yes	—
217	#11719919	C₂³	1	2	yes	—
218	#11719919	C₂³	1	2	yes	—
219	#11719919	C₂³	1	2	yes	—
220	#11719919	C₂³	1	1	yes	characteristic
221	#11719919	C₂³	1	1	yes	characteristic
222	#11719919	C₂³	1	2	yes	—
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	2	yes	—
228	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
229	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
230	C₂²×D₅₃	—	2	2	no	—
231	C₂²×D₅₃	—	4	1	no	—
232	C₂²×D₅₃	—	4	1	no	—
233	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
234	C₂²×D₅₃	C₂⁴	1	2	yes	—
235	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
236	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
237	Q₈×C₅₃	—	2	2	no	—
238	Q₈×C₅₃	—	2	2	no	—
239	Q₈×C₅₃	—	2	2	no	—
240	Q₈×C₅₃	—	4	2	no	—
241	Q₈×C₅₃	—	4	2	no	—
242	D₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	2	2	no	—
244	C₂×C₄×C₅₃	—	2	2	no	—
245	C₂×C₄×C₅₃	—	4	2	no	—
246	C₂×C₄×C₅₃	—	4	2	no	—
247	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
248	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
252	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
253	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
254	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
255	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
256	C₂²⋊D₅₃	—	2	4	no	—
257	C₂²⋊D₅₃	—	2	2	no	—
258	C₂²⋊D₅₃	—	4	1	no	—
259	C₂²⋊D₅₃	—	4	1	no	—
260	C₂²⋊D₅₃	—	4	1	no	—
261	C₂²⋊D₅₃	—	4	2	no	—
262	C₂×Dic₅₃	—	4	1	no	—
263	C₂×Dic₅₃	—	4	2	no	—
264	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
265	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
266	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
267	C₂×Dic₅₃	C₂⁴	1	2	yes	—
268	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
269	C₂×Dic₅₃	C₂⁴	1	2	yes	—
270	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
271	C₂×Dic₅₃	C₂⁴	1	2	yes	—
272	D₂₁₂	—	2	2	no	—
273	D₂₁₂	—	4	2	no	—
274	D₂₁₂	—	4	1	no	—
275	D₂₁₂	—	4	1	no	—
276	D₂₁₂	—	4	2	no	—
277	D₂₁₂	—	4	2	no	—
278	C₄×D₅₃	—	2	2	no	—
279	C₄×D₅₃	—	2	2	no	—
280	C₄×D₅₃	—	2	2	no	—
281	C₄×D₅₃	—	4	1	no	—
282	C₄×D₅₃	—	4	1	no	—
283	C₄×D₅₃	—	4	2	no	—
284	C₄×D₅₃	—	4	2	no	—
285	C₄×D₅₃	—	4	2	no	—
286	C₄×D₅₃	—	4	2	no	—
287	Dic₁₀₆	—	4	2	no	—
288	Dic₁₀₆	—	4	1	no	—
289	C₂²×C₅₃	—	2	1	no	—
290	C₂²×C₅₃	—	2	1	no	—
291	C₂²×C₅₃	—	2	1	no	—
292	C₂²×C₅₃	—	2	1	no	—
293	C₂²×C₅₃	—	2	1	no	—
294	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
295	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
296	C₂×D₅₃	—	2	1	no	—
297	C₂×D₅₃	—	2	1	no	—
298	C₂×D₅₃	—	2	1	no	—
299	C₂×D₅₃	—	2	2	no	—
300	C₂×D₅₃	—	2	1	no	—
301	C₂×D₅₃	—	2	2	no	—
302	C₂×D₅₃	—	2	1	no	—
303	C₂×D₅₃	—	2	1	no	—
304	C₂×D₅₃	—	2	1	no	—
305	C₂×D₅₃	—	2	2	no	—
306	C₂×D₅₃	—	2	1	no	—
307	C₂×D₅₃	—	2	1	no	—
308	C₂×D₅₃	—	4	4	no	—
309	C₄×C₅₃	—	2	1	no	—
310	C₄×C₅₃	—	2	1	no	—
311	C₄×C₅₃	—	2	2	no	—
312	C₄×C₅₃	—	2	2	no	—
313	C₄×C₅₃	—	2	2	no	—
314	C₄×C₅₃	—	2	1	no	—
315	C₄×C₅₃	—	2	2	no	—
316	C₄×C₅₃	—	2	1	no	—
317	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
318	Dic₅₃	—	2	1	no	—
319	Dic₅₃	—	2	1	no	—
320	Dic₅₃	—	2	1	no	—
321	Dic₅₃	—	2	1	no	—
322	Dic₅₃	—	2	2	no	—
323	Dic₅₃	—	2	1	no	—
324	Dic₅₃	—	2	2	no	—
325	Dic₅₃	—	2	2	no	—
326	#3541	—	53	1	no	maximal, Sylow
327	C₂×C₅₃	—	2	2	no	—
328	C₂×C₅₃	—	4	1	no	—
329	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
330	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
331	C₂×C₅₃	#585	1	1	yes	characteristic
332	D₅₃	—	4	1	no	—
333	D₅₃	—	4	2	no	—
334	D₅₃	—	4	1	no	—
335	D₅₃	—	4	1	no	—
336	#563	—	53	2	no	—
337	#562	—	53	2	no	—
338	#556	—	53	1	no	—
339	#554	—	53	2	no	—
340	#554	—	53	1	no	—
341	#553	—	53	1	no	—
342	#553	—	53	2	no	—
343	#552	—	53	1	no	—
344	D₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
345	#548	—	53	1	no	—
346	#548	—	53	1	no	—
347	#547	—	53	1	no	—
348	#542	—	53	2	no	—
349	#542	—	53	1	no	—
350	#541	—	53	1	no	—
351	#539	—	53	2	no	—
352	#538	—	53	2	no	—
353	#538	—	53	2	no	—
354	#537	—	53	1	no	—
355	#536	—	53	1	no	—
356	#525	—	53	1	no	—
357	#525	—	53	1	no	—
358	C₂×C₂²⋊Q₈	—	53	1	no	—
359	C₅₃	#3541	1	1	yes	characteristic, complemented, Sylow
360	C₂×#41	—	53	1	no	—
361	C₂×#41	—	53	1	no	—
362	C₂×#41	—	53	1	no	—
363	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
364	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
365	C₂³×C₄	—	53	1	no	—
366	C₄⋊Q₈	—	53	2	no	—
367	C₄⋊Q₈	—	53	1	no	—
368	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
369	C₄⋊Q₈	C₂×D₅₃	1	2	yes	complemented
370	#126	—	53	2	no	—
371	#126	—	53	2	no	—
372	#126	—	53	2	no	—
373	#126	—	53	2	no	—
374	#126	—	53	2	no	—
375	#126	—	53	2	no	—
376	#126	—	53	2	no	—
377	#125	—	53	2	no	—
378	#125	—	53	2	no	—
379	#125	—	53	2	no	—
380	#125	—	53	2	no	—
381	#125	—	53	1	no	—
382	#124	—	53	2	no	—
383	#124	—	53	2	no	—
384	#124	—	53	1	no	—
385	#124	—	53	2	no	—
386	#124	—	53	2	no	—
387	#124	—	53	2	no	—
388	#123	—	53	2	no	—
389	#123	—	53	2	no	—
390	#123	—	53	1	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	1	no	—
399	C₂²⋊Q₈	—	53	1	no	—
400	C₂²⋊Q₈	—	53	2	no	—
401	C₂²⋊Q₈	—	53	1	no	—
402	C₂²⋊Q₈	—	53	1	no	—
403	C₂²⋊Q₈	—	53	2	no	—
404	C₂²⋊Q₈	—	53	1	no	—
405	C₂²⋊Q₈	—	53	1	no	—
406	C₂²⋊Q₈	—	53	1	no	—
407	C₂²⋊Q₈	—	53	2	no	—
408	C₂²⋊Q₈	—	53	2	no	—
409	C₂²⋊Q₈	—	53	1	no	—
410	C₂²⋊Q₈	—	53	2	no	—
411	C₂²⋊Q₈	—	53	2	no	—
412	C₂²⋊Q₈	—	53	2	no	—
413	C₂²⋊Q₈	—	53	1	no	—
414	C₂²⋊Q₈	—	53	2	no	—
415	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
416	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
417	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
418	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
419	#121	—	53	1	no	—
420	#121	—	53	2	no	—
421	#121	—	53	1	no	—
422	#121	—	53	1	no	—
423	#121	—	53	2	no	—
424	#121	—	53	2	no	—
425	#121	—	53	1	no	—
426	#121	—	53	2	no	—
427	#121	—	53	2	no	—
428	#121	—	53	1	no	—
429	#121	—	53	2	no	—
430	#121	—	53	2	no	—
431	#120	—	53	1	no	—
432	#120	—	53	1	no	—
433	#120	—	53	2	no	—
434	C₄×Q₈	—	53	1	no	—
435	C₄×Q₈	—	53	1	no	—
436	C₄×Q₈	—	53	2	no	—
437	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
438	C₄×D₄	—	53	1	no	—
439	C₄×D₄	—	53	1	no	—
440	C₄×D₄	—	53	2	no	—
441	C₄×D₄	—	53	2	no	—
442	C₄×D₄	—	53	2	no	—
443	C₄×D₄	—	53	2	no	—
444	C₄×D₄	—	53	2	no	—
445	C₄×D₄	—	53	2	no	—
446	C₄×D₄	—	53	1	no	—
447	C₄×D₄	—	53	1	no	—
448	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
449	C₄×D₄	C₂×D₅₃	1	2	yes	—
450	#117	—	53	1	no	—
451	#117	—	53	1	no	—
452	#117	—	53	1	no	—
453	#117	—	53	2	no	—
454	C₂×C₄⋊C₄	—	53	2	no	—
455	C₂×C₄⋊C₄	—	53	1	no	—
456	C₂×C₄⋊C₄	—	53	1	no	—
457	C₂×C₄⋊C₄	—	53	2	no	—
458	C₂×C₂²⋊C₄	—	53	2	no	—
459	C₂⁴	—	53	1	no	—
460	#41	—	212	2	no	—
461	#41	—	212	2	no	—
462	#41	—	212	2	no	—
463	C₂×Q₈	—	4	2	no	—
464	C₂×Q₈	—	4	2	no	—
465	C₂×Q₈	—	53	2	no	—
466	C₂×Q₈	—	53	1	no	—
467	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
468	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
469	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
470	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
471	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
472	C₂×D₄	—	53	1	no	—
473	C₂×D₄	—	53	2	no	—
474	C₂×D₄	—	53	1	no	—
475	C₂×D₄	—	53	2	no	—
476	C₂×D₄	—	53	1	no	—
477	C₂×D₄	—	53	1	no	—
478	C₂×D₄	—	53	1	no	—
479	C₂×D₄	—	53	1	no	—
480	C₂×D₄	—	53	2	no	—
481	C₂×D₄	—	53	2	no	—
482	C₂×D₄	—	53	1	no	—
483	C₂×D₄	—	53	2	no	—
484	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
485	C₂²×C₄	—	53	2	no	—
486	C₂²×C₄	—	53	1	no	—
487	C₂²×C₄	—	53	1	no	—
488	C₂²×C₄	—	53	1	no	—
489	C₂²×C₄	—	53	1	no	—
490	C₂²×C₄	—	53	2	no	—
491	C₂²×C₄	—	53	2	no	—
492	C₂²×C₄	—	53	2	no	—
493	C₂²×C₄	—	53	2	no	—
494	C₂²×C₄	—	53	1	no	—
495	C₂²×C₄	—	53	1	no	—
496	C₂²×C₄	—	212	2	no	—
497	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
498	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
499	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
500	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
501	C₄⋊C₄	—	53	1	no	—
502	C₄⋊C₄	—	53	1	no	—
503	C₄⋊C₄	—	53	2	no	—
504	C₄⋊C₄	—	53	2	no	—
505	C₄⋊C₄	—	53	1	no	—
506	C₄⋊C₄	—	53	2	no	—
507	C₄⋊C₄	—	53	2	no	—
508	C₄⋊C₄	—	53	2	no	—
509	C₄⋊C₄	—	53	2	no	—
510	C₄⋊C₄	—	53	2	no	—
511	C₄⋊C₄	—	53	1	no	—
512	C₄⋊C₄	—	53	1	no	—
513	C₄⋊C₄	—	53	1	no	—
514	C₄⋊C₄	—	53	2	no	—
515	C₄⋊C₄	—	53	1	no	—
516	C₄⋊C₄	—	53	1	no	—
517	C₄⋊C₄	—	53	2	no	—
518	C₄⋊C₄	—	53	2	no	—
519	C₄⋊C₄	—	53	2	no	—
520	C₄⋊C₄	—	53	2	no	—
521	C₄⋊C₄	—	53	2	no	—
522	C₄⋊C₄	—	53	2	no	—
523	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
524	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
525	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
526	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
527	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
528	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
529	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
530	C₂²⋊C₄	—	53	2	no	—
531	C₂²⋊C₄	—	53	1	no	—
532	C₂²⋊C₄	—	53	1	no	—
533	C₂²⋊C₄	—	53	2	no	—
534	C₂²⋊C₄	—	53	2	no	—
535	C₂²⋊C₄	—	53	2	no	—
536	C₂²⋊C₄	—	53	2	no	—
537	C₂²⋊C₄	—	53	1	no	—
538	C₂²⋊C₄	—	53	1	no	—
539	C₂²⋊C₄	—	53	2	no	—
540	C₂²⋊C₄	—	53	1	no	—
541	C₂²⋊C₄	—	53	2	no	—
542	C₂²⋊C₄	—	53	1	no	—
543	C₂²⋊C₄	—	53	2	no	—
544	C₂²⋊C₄	—	53	1	no	—
545	C₂²⋊C₄	—	53	2	no	—
546	C₂²⋊C₄	—	53	2	no	—
547	C₂²⋊C₄	—	53	2	no	—
548	C₂²⋊C₄	—	53	1	no	—
549	C₂²⋊C₄	—	53	2	no	—
550	C₂²⋊C₄	—	53	2	no	—
551	C₂²⋊C₄	—	53	2	no	—
552	C₂²⋊C₄	—	53	2	no	—
553	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
554	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
555	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
556	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
557	C₄²	—	53	2	no	—
558	C₄²	—	53	1	no	—
559	C₄²	—	53	2	no	—
560	C₄²	—	53	2	no	—
561	C₄²	—	53	2	no	—
562	C₄²	C₂²×D₅₃	1	1	yes	characteristic
563	C₄²	C₂²×D₅₃	1	2	yes	—
564	C₂³	—	53	1	no	—
565	C₂³	—	53	1	no	—
566	C₂³	—	53	1	no	—
567	C₂³	—	53	2	no	—
568	C₂³	—	106	2	no	—
569	C₂³	—	212	1	no	—
570	C₂³	—	212	1	no	—
571	C₂³	C₂³×D₅₃	1	1	yes	characteristic
572	C₂³	C₂³×D₅₃	1	1	yes	characteristic
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	2	2	no	—
576	Q₈	—	4	2	no	—
577	Q₈	—	4	2	no	—
578	Q₈	—	212	2	no	—
579	Q₈	—	212	1	no	—
580	D₄	—	2	2	no	—
581	D₄	—	106	2	no	—
582	D₄	—	106	4	no	—
583	D₄	—	106	2	no	—
584	D₄	—	212	1	no	—
585	D₄	—	212	2	no	—
586	D₄	—	212	1	no	—
587	D₄	—	212	2	no	—
588	D₄	—	212	2	no	—
589	D₄	—	212	2	no	—
590	D₄	—	212	1	no	—
591	D₄	—	212	1	no	—
592	D₄	—	212	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	53	2	no	—
598	C₂×C₄	—	53	1	no	—
599	C₂×C₄	—	53	1	no	—
600	C₂×C₄	—	53	1	no	—
601	C₂×C₄	—	53	2	no	—
602	C₂×C₄	—	53	2	no	—
603	C₂×C₄	—	53	1	no	—
604	C₂×C₄	—	53	1	no	—
605	C₂×C₄	—	106	2	no	—
606	C₂×C₄	—	106	2	no	—
607	C₂×C₄	—	106	2	no	—
608	C₂×C₄	—	212	2	no	—
609	C₂×C₄	—	212	2	no	—
610	C₂×C₄	—	212	1	no	—
611	C₂×C₄	—	212	1	no	—
612	C₂×C₄	—	212	2	no	—
613	C₂×C₄	—	212	2	no	—
614	C₂×C₄	—	212	2	no	—
615	C₂×C₄	—	212	1	no	—
616	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
617	C₂×C₄	C₂³×D₅₃	1	2	yes	—
618	C₂×C₄	C₂³×D₅₃	1	2	yes	—
619	C₂×C₄	C₂³×D₅₃	1	2	yes	—
620	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
621	C₂×C₄	C₂³×D₅₃	1	2	yes	—
622	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
623	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
624	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
625	C₂²	—	2	1	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	2	1	no	—
628	C₂²	—	2	1	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	106	1	no	—
631	C₂²	—	106	1	no	—
632	C₂²	—	106	1	no	—
633	C₂²	—	106	2	no	—
634	C₂²	—	106	1	no	—
635	C₂²	—	106	2	no	—
636	C₂²	—	106	1	no	—
637	C₂²	—	106	1	no	—
638	C₂²	—	106	1	no	—
639	C₂²	—	106	2	no	—
640	C₂²	—	106	1	no	—
641	C₂²	—	106	1	no	—
642	C₂²	—	212	4	no	—
643	C₂²	#13133265	1	2	yes	—
644	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
645	C₄	—	2	1	no	—
646	C₄	—	2	1	no	—
647	C₄	—	2	2	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	1	no	—
651	C₄	—	2	1	no	—
652	C₄	—	2	2	no	—
653	C₄	—	106	1	no	—
654	C₄	—	106	1	no	—
655	C₄	—	106	1	no	—
656	C₄	—	106	2	no	—
657	C₄	—	106	1	no	—
658	C₄	—	106	2	no	—
659	C₄	—	106	2	no	—
660	C₄	—	106	1	no	—
661	C₄	#13133262	1	2	yes	—
662	C₂	—	2	2	no	—
663	C₂	—	4	1	no	—
664	C₂	—	212	1	no	—
665	C₂	—	212	1	no	—
666	C₂	—	212	1	no	—
667	C₂	—	212	2	no	—
668	C₂	C₂×#13133262	1	1	yes	characteristic
669	C₂	C₂×#13133265	1	1	yes	characteristic
670	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	#3541
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. #16811344	C₂	semidirect
4. #16811331	C₂	semidirect
5. #16811327	C₂	semidirect
6. #16811322	C₂	semidirect
7. #16811319	C₂	semidirect
8. #16811314	C₂	semidirect
9. #16811284	C₂	semidirect
10. #16811284	C₂	semidirect
11. #16811276	C₂	semidirect
12. #16811247	C₂	semidirect
13. #16811245	C₂	semidirect
14. #16811234	C₂	semidirect
15. #16811219	C₂	semidirect
16. #16811213	C₂	semidirect
17. #16811206	C₂	semidirect
18. #16811203	C₂	semidirect
19. #16811197	C₂	semidirect
20. #16811136	C₂	semidirect
21. #16811132	C₂	semidirect
22. C₂×#13133193	C₂	semidirect
23. D₄×Q₈×C₅₃	C₂	semidirect
25. C₂×#11719949	C₂²	semidirect
28. #13133216	C₂²	semidirect
29. #13133215	C₂²	semidirect
33. #13133212	C₂²	semidirect
35. #13133211	C₂²	semidirect
36. #13133210	C₂²	semidirect
37. #13133209	C₂²	semidirect
38. #13133207	C₂²	semidirect
39. #13133207	C₂²	semidirect
40. #13133207	C₂²	semidirect
41. #13133207	C₂²	semidirect
44. #13133206	C₂²	semidirect
45. #13133206	C₂²	semidirect
53. #13133202	C₂²	semidirect
55. #13133200	C₂²	semidirect
57. #13133199	C₂²	semidirect
58. #13133197	C₂²	semidirect
60. #13133195	C₂²	semidirect
61. #13133195	C₂²	semidirect
71. #13133193	C₂²	semidirect
72. #13133193	C₂²	semidirect
73. #13133193	C₂²	semidirect
74. #13133193	C₂²	semidirect
81. #13133191	C₂²	semidirect
84. #13133190	C₂²	semidirect
90. Q₈×Dic₅₃	C₂²	semidirect
91. Q₈×Dic₅₃	C₂²	semidirect
92. #13133182	C₂²	semidirect
94. C₄×D₂₁₂	C₂²	semidirect
99. C₄×C₂²⋊D₅₃	C₂²	semidirect
103. #13133176	C₂²	semidirect
106. #13133174	C₂²	semidirect
109. C₄⋊C₄×D₅₃	C₂²	semidirect
110. C₄⋊C₄×D₅₃	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
344. D₄×Q₈	D₅₃	semidirect
359. C₅₃	#3541	semidirect
364. C₂²×Q₈	C₂×D₅₃	semidirect
368. C₄⋊Q₈	C₂×D₅₃	semidirect
369. C₄⋊Q₈	C₂×D₅₃	semidirect
417. C₂²⋊Q₈	C₂×D₅₃	semidirect
418. C₂²⋊Q₈	C₂×D₅₃	semidirect
437. C₄×Q₈	C₂×D₅₃	semidirect