Subgroups — #20288456

Order 6784.

1022 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811346	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811341	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811331	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811330	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811315	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811311	C₂	1	2	yes	maximal, complemented
7	#16811311	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811309	C₂	1	2	yes	maximal, complemented
9	#16811308	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811297	C₂	1	2	yes	maximal, complemented
11	#16811296	C₂	1	2	yes	maximal, complemented
12	#16811253	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811239	C₂	1	2	yes	maximal, complemented
14	#16811232	C₂	1	2	yes	maximal, complemented
15	#16811196	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811176	C₂	1	2	yes	maximal, complemented
17	#16811150	C₂	1	2	yes	maximal, complemented
18	#16811132	C₂	1	1	yes	maximal, characteristic
19	C₂²⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811103	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811099	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×#13133175	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#552	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719945	C₂²	1	1	yes	characteristic
25	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
27	#13133217	C₂²	1	1	yes	characteristic, complemented
28	#13133216	C₂²	1	1	yes	characteristic, complemented
29	#13133216	C₂²	1	2	yes	complemented
30	#13133216	C₂²	1	2	yes	complemented
31	#13133212	C₂²	1	2	yes	—
32	#13133212	C₂²	1	2	yes	—
33	#13133212	C₂²	1	2	yes	—
34	#13133212	C₂²	1	2	yes	complemented
35	#13133212	C₂²	1	2	yes	complemented
36	#13133211	C₂²	1	2	yes	—
37	#13133210	C₂²	1	2	yes	complemented
38	#13133208	C₂²	1	1	yes	characteristic, complemented
39	#13133208	C₂²	1	2	yes	complemented
40	#13133208	C₂²	1	2	yes	complemented
41	#13133208	C₂²	1	2	yes	complemented
42	#13133207	C₂²	1	2	yes	complemented
43	#13133207	C₂²	1	2	yes	complemented
44	#13133207	C₂²	1	2	yes	complemented
45	#13133205	C₂²	1	2	yes	—
46	#13133203	C₂²	1	2	yes	—
47	#13133203	C₂²	1	2	yes	complemented
48	#13133202	C₂²	1	2	yes	—
49	#13133202	C₂²	1	2	yes	—
50	#13133202	C₂²	1	1	yes	characteristic
51	#13133201	C₂²	1	2	yes	—
52	#13133200	C₂²	1	2	yes	—
53	#13133199	C₂²	1	2	yes	—
54	#13133199	C₂²	1	1	yes	characteristic
55	#13133197	C₂²	1	2	yes	—
56	#13133197	C₂²	1	1	yes	characteristic
57	#13133197	C₂²	1	1	yes	characteristic, complemented
58	#13133197	C₂²	1	2	yes	complemented
59	#13133196	C₂²	1	1	yes	characteristic
60	#13133196	C₂²	1	1	yes	characteristic, complemented
61	#13133195	C₂²	1	2	yes	—
62	#13133195	C₂²	1	2	yes	—
63	#13133195	C₂²	1	1	yes	characteristic
64	#13133194	C₂²	1	2	yes	—
65	#13133194	C₂²	1	2	yes	—
66	#13133194	C₂²	1	2	yes	—
67	#13133194	C₂²	1	1	yes	characteristic
68	#13133194	C₂²	1	2	yes	complemented
69	#13133193	C₂²	1	1	yes	characteristic
70	#13133193	C₂²	1	1	yes	characteristic, complemented
71	#13133190	C₂²	1	1	yes	characteristic
72	#13133189	C₂²	1	1	yes	characteristic
73	#13133187	C₂²	1	1	yes	characteristic
74	#13133185	C₂²	1	1	yes	characteristic
75	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
76	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
77	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
78	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
79	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
80	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
81	#13133182	C₂²	1	1	yes	characteristic, complemented
82	#13133182	C₂²	1	2	yes	complemented
83	C₄×D₂₁₂	C₂²	1	2	yes	—
84	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
85	D₄×Dic₅₃	C₂²	1	2	yes	—
86	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
87	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
88	#13133177	C₂²	1	2	yes	—
89	#13133177	C₂²	1	2	yes	—
90	#13133177	C₂²	1	2	yes	complemented
91	#13133176	C₂²	1	2	yes	—
92	#13133175	C₂²	1	2	yes	—
93	#13133175	C₂²	1	2	yes	—
94	#13133175	C₂²	1	1	yes	characteristic
95	#13133175	C₂²	1	1	yes	characteristic
96	#13133175	C₂²	1	1	yes	characteristic
97	#13133175	C₂²	1	1	yes	characteristic
98	#13133175	C₂²	1	2	yes	—
99	#13133175	C₂²	1	1	yes	characteristic, complemented
100	#13133175	C₂²	1	2	yes	complemented
101	#13133174	C₂²	1	2	yes	—
102	#13133174	C₂²	1	1	yes	characteristic
103	#13133173	C₂²	1	2	yes	complemented
104	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
105	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
106	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
107	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
108	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
109	C₂×#11719921	C₂²	1	2	yes	—
110	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
111	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
112	#126×C₅₃	C₂²	1	2	yes	complemented
113	#125×C₅₃	C₂²	1	2	yes	complemented
114	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
115	#123×C₅₃	C₂²	1	2	yes	—
116	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
117	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
118	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
120	#117×C₅₃	C₂²	1	2	yes	—
121	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
122	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
123	C₂³×D₅₃	C₂³	1	1	yes	characteristic
124	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
130	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
131	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
132	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
133	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
134	C₂²×Dic₅₃	C₂³	1	2	yes	—
135	C₂²×Dic₅₃	C₂³	1	2	yes	—
136	Q₈×D₅₃	—	2	4	no	—
137	#11719945	—	2	4	no	—
138	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
139	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
140	C₂×C₄×D₅₃	—	2	4	no	—
141	C₂×C₄×D₅₃	C₂³	1	2	yes	—
142	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
143	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
144	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
145	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
146	C₂×C₄×D₅₃	C₂³	1	2	yes	—
147	C₂×C₄×D₅₃	C₂³	1	2	yes	—
148	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
149	C₂×Dic₁₀₆	C₂³	1	2	yes	—
150	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
151	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
152	C₂×Dic₁₀₆	C₂³	1	2	yes	—
153	C₂×Dic₁₀₆	C₂³	1	2	yes	—
154	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
155	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
156	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
157	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
158	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
159	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
160	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
161	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
162	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
163	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
164	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
165	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
166	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
167	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
168	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
169	C₄²×C₅₃	C₂³	1	2	yes	—
170	C₄²×C₅₃	C₂³	1	2	yes	—
171	C₄²×C₅₃	C₂³	1	1	yes	characteristic
172	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
173	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
174	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
175	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
176	#11719921	C₂³	1	2	yes	—
177	#11719921	C₂³	1	2	yes	—
178	#11719921	C₂³	1	2	yes	—
179	#11719921	C₂³	1	2	yes	—
180	#11719921	C₂³	1	2	yes	—
181	#11719921	C₂³	1	2	yes	—
182	#11719921	C₂³	1	1	yes	characteristic
183	#11719921	C₂³	1	1	yes	characteristic
184	#11719921	C₂³	1	1	yes	characteristic
185	#11719921	C₂³	1	2	yes	—
186	#11719921	C₂³	1	2	yes	—
187	#11719921	C₂³	1	2	yes	—
188	#11719921	C₂³	1	1	yes	characteristic
189	#11719921	C₂³	1	1	yes	characteristic
190	#11719921	C₂³	1	1	yes	characteristic
191	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
192	C₄⋊Dic₅₃	C₂³	1	2	yes	—
193	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
194	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
195	C₄⋊Dic₅₃	C₂³	1	2	yes	—
196	C₄⋊Dic₅₃	C₂³	1	2	yes	—
197	C₄⋊Dic₅₃	C₂³	1	2	yes	—
198	C₄⋊Dic₅₃	C₂³	1	2	yes	—
199	C₄⋊Dic₅₃	C₂³	1	2	yes	—
200	C₄⋊Dic₅₃	C₂³	1	2	yes	—
201	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
202	#11719919	C₂³	1	2	yes	—
203	#11719919	C₂³	1	2	yes	—
204	#11719919	C₂³	1	2	yes	—
205	#11719919	C₂³	1	2	yes	—
206	#11719919	C₂³	1	2	yes	—
207	#11719919	C₂³	1	1	yes	characteristic
208	#11719919	C₂³	1	2	yes	—
209	#11719919	C₂³	1	1	yes	characteristic
210	#11719919	C₂³	1	2	yes	—
211	#11719919	C₂³	1	2	yes	—
212	#11719919	C₂³	1	1	yes	characteristic
213	#11719919	C₂³	1	1	yes	characteristic
214	#11719919	C₂³	1	2	yes	—
215	#11719919	C₂³	1	2	yes	—
216	C₄×Dic₅₃	C₂³	1	2	yes	—
217	C₄×Dic₅₃	C₂³	1	2	yes	—
218	C₄×Dic₅₃	C₂³	1	2	yes	—
219	C₄×Dic₅₃	C₂³	1	2	yes	—
220	C₄×Dic₅₃	C₂³	1	2	yes	—
221	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
222	C₄×Dic₅₃	C₂³	1	2	yes	—
223	C₄×Dic₅₃	C₂³	1	2	yes	—
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
226	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
227	C₂²×D₅₃	—	2	2	no	—
228	C₂²×D₅₃	—	2	2	no	—
229	C₂²×D₅₃	—	4	1	no	—
230	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
231	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
232	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
233	Q₈×C₅₃	—	2	2	no	—
234	D₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	2	4	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
240	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
241	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
242	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
244	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
245	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₂²⋊D₅₃	—	2	2	no	—
248	C₂²⋊D₅₃	—	2	2	no	—
249	C₂²⋊D₅₃	—	2	2	no	—
250	C₂²⋊D₅₃	—	4	1	no	—
251	C₂×Dic₅₃	—	2	2	no	—
252	C₂×Dic₅₃	—	4	2	no	—
253	C₂×Dic₅₃	—	4	2	no	—
254	C₂×Dic₅₃	C₂⁴	1	2	yes	—
255	C₂×Dic₅₃	C₂⁴	1	2	yes	—
256	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
257	C₂×Dic₅₃	C₂⁴	1	2	yes	—
258	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
259	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
260	C₂×Dic₅₃	C₂⁴	1	2	yes	—
261	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
262	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
263	D₂₁₂	—	2	2	no	—
264	D₂₁₂	—	4	1	no	—
265	C₄×D₅₃	—	2	2	no	—
266	C₄×D₅₃	—	2	4	no	—
267	C₄×D₅₃	—	2	2	no	—
268	C₄×D₅₃	—	2	2	no	—
269	C₄×D₅₃	—	2	2	no	—
270	C₄×D₅₃	—	4	2	no	—
271	C₄×D₅₃	—	4	2	no	—
272	C₄×D₅₃	#41	1	4	yes	—
273	Dic₁₀₆	—	2	2	no	—
274	Dic₁₀₆	—	2	2	no	—
275	Dic₁₀₆	—	2	4	no	—
276	Dic₁₀₆	—	4	2	no	—
277	Dic₁₀₆	—	4	2	no	—
278	C₂²×C₅₃	—	2	1	no	—
279	C₂²×C₅₃	—	2	1	no	—
280	C₂²×C₅₃	—	2	1	no	—
281	C₂²×C₅₃	—	2	1	no	—
282	C₂²×C₅₃	—	2	1	no	—
283	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
284	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
285	C₂×D₅₃	—	2	1	no	—
286	C₂×D₅₃	—	2	1	no	—
287	C₂×D₅₃	—	2	1	no	—
288	C₂×D₅₃	—	2	1	no	—
289	C₂×D₅₃	—	2	1	no	—
290	C₂×D₅₃	—	2	1	no	—
291	C₂×D₅₃	—	2	1	no	—
292	C₂×D₅₃	—	2	1	no	—
293	C₂×D₅₃	—	4	4	no	—
294	C₂×D₅₃	C₂×#41	1	2	yes	—
295	C₄×C₅₃	—	2	2	no	—
296	C₄×C₅₃	—	2	1	no	—
297	C₄×C₅₃	—	2	1	no	—
298	C₄×C₅₃	—	2	2	no	—
299	C₄×C₅₃	—	2	2	no	—
300	C₄×C₅₃	—	2	1	no	—
301	C₄×C₅₃	—	2	1	no	—
302	C₄×C₅₃	—	2	2	no	—
303	C₄×C₅₃	C₂×#41	1	2	yes	—
304	Dic₅₃	—	2	2	no	—
305	Dic₅₃	—	2	1	no	—
306	Dic₅₃	—	2	1	no	—
307	Dic₅₃	—	2	1	no	—
308	Dic₅₃	—	2	2	no	—
309	Dic₅₃	—	2	1	no	—
310	Dic₅₃	—	2	2	no	—
311	Dic₅₃	—	2	2	no	—
312	Dic₅₃	C₂×#41	1	2	yes	—
313	#3495	—	53	1	no	maximal, Sylow
314	C₂×C₅₃	—	2	2	no	—
315	C₂×C₅₃	—	4	1	no	—
316	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
317	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
318	C₂×C₅₃	#585	1	1	yes	characteristic
319	D₅₃	—	2	2	no	—
320	D₅₃	—	4	1	no	—
321	D₅₃	—	4	1	no	—
322	#556	—	53	1	no	—
323	#556	—	53	1	no	—
324	#554	—	53	1	no	—
325	#554	—	53	1	no	—
326	#552	—	53	2	no	—
327	#552	—	53	1	no	—
328	#552	—	53	2	no	—
329	#552	—	53	1	no	—
330	#552	—	53	1	no	—
331	#552	D₅₃	1	1	yes	characteristic, complemented
332	#551	—	53	2	no	—
333	#551	—	53	2	no	—
334	#544	—	53	1	no	—
335	#542	—	53	2	no	—
336	#541	—	53	2	no	—
337	#536	—	53	1	no	—
338	#533	—	53	2	no	—
339	#529	—	53	2	no	—
340	#525	—	53	1	no	—
341	C₂×C₂²⋊Q₈	—	53	1	no	—
342	#520	—	53	1	no	—
343	#519	—	53	1	no	—
344	C₂×#117	—	53	1	no	—
345	C₅₃	#3495	1	1	yes	characteristic, complemented, Sylow
346	C₂×#41	—	53	1	no	—
347	C₂²×Q₈	—	53	1	no	—
348	C₂³×C₄	—	53	1	no	—
349	C₄⋊Q₈	—	53	2	no	—
350	C₄⋊Q₈	—	53	2	no	—
351	C₄⋊Q₈	—	53	1	no	—
352	C₄⋊Q₈	—	53	1	no	—
353	#126	—	53	2	no	—
354	#126	—	53	2	no	—
355	#126	—	53	2	no	—
356	#126	—	53	2	no	—
357	#126	—	53	2	no	—
358	#126	—	53	2	no	—
359	#126	—	53	2	no	—
360	#126	C₂×D₅₃	1	2	yes	complemented
361	#125	—	53	2	no	—
362	#125	—	53	2	no	—
363	#125	—	53	2	no	—
364	#125	—	53	2	no	—
365	#125	—	53	2	no	—
366	#125	—	53	2	no	—
367	#125	—	53	1	no	—
368	#125	C₂×D₅₃	1	2	yes	complemented
369	#125	C₂×D₅₃	1	1	yes	characteristic, complemented
370	#124	—	53	2	no	—
371	#124	—	53	2	no	—
372	#124	—	53	2	no	—
373	#123	—	53	2	no	—
374	#123	—	53	1	no	—
375	#123	—	53	2	no	—
376	#123	—	53	2	no	—
377	#123	—	53	1	no	—
378	#123	—	53	2	no	—
379	#123	—	53	2	no	—
380	#123	C₂×D₅₃	1	2	yes	—
381	C₂²⋊Q₈	—	53	1	no	—
382	C₂²⋊Q₈	—	53	1	no	—
383	C₂²⋊Q₈	—	53	1	no	—
384	C₂²⋊Q₈	—	53	1	no	—
385	C₂²⋊Q₈	—	53	2	no	—
386	C₂²⋊Q₈	—	53	1	no	—
387	C₂²⋊Q₈	—	53	1	no	—
388	C₂²⋊Q₈	—	53	1	no	—
389	C₂²⋊Q₈	—	53	2	no	—
390	C₂²⋊Q₈	—	53	2	no	—
391	C₂²⋊Q₈	—	53	2	no	—
392	C₂²⋊Q₈	—	53	2	no	—
393	C₂²⋊Q₈	—	53	1	no	—
394	C₂²⋊Q₈	—	53	2	no	—
395	C₂²⋊Q₈	—	53	2	no	—
396	C₂²⋊Q₈	—	53	2	no	—
397	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
398	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
399	#121	—	53	1	no	—
400	#121	—	53	1	no	—
401	#120	—	53	1	no	—
402	#120	—	53	1	no	—
403	C₄×Q₈	—	53	1	no	—
404	C₄×Q₈	—	53	2	no	—
405	C₄×Q₈	—	53	1	no	—
406	C₄×Q₈	—	53	2	no	—
407	C₄×Q₈	—	53	1	no	—
408	C₄×Q₈	—	53	2	no	—
409	C₄×Q₈	—	53	2	no	—
410	C₄×Q₈	—	53	2	no	—
411	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
412	C₄×D₄	—	53	2	no	—
413	C₄×D₄	—	53	1	no	—
414	C₄×D₄	—	53	1	no	—
415	C₄×D₄	—	53	2	no	—
416	C₄×D₄	—	53	2	no	—
417	C₄×D₄	—	53	2	no	—
418	C₄×D₄	—	53	2	no	—
419	C₄×D₄	—	53	1	no	—
420	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
421	#117	—	53	2	no	—
422	#117	—	53	1	no	—
423	#117	—	53	1	no	—
424	#117	—	53	2	no	—
425	#117	—	53	1	no	—
426	#117	—	53	2	no	—
427	#117	—	53	1	no	—
428	#117	—	53	2	no	—
429	#117	—	53	1	no	—
430	#117	—	53	2	no	—
431	#117	—	53	2	no	—
432	#117	—	53	1	no	—
433	#117	—	53	2	no	—
434	#117	C₂×D₅₃	1	2	yes	—
435	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
436	C₂×C₄⋊C₄	—	53	1	no	—
437	C₂×C₄⋊C₄	—	53	1	no	—
438	C₂×C₄⋊C₄	—	53	1	no	—
439	C₂×C₄⋊C₄	—	53	1	no	—
440	C₂×C₄⋊C₄	—	53	2	no	—
441	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
442	C₂×C₂²⋊C₄	—	53	2	no	—
443	C₂×C₂²⋊C₄	—	53	2	no	—
444	C₂×C₄²	—	53	2	no	—
445	C₂⁴	—	53	1	no	—
446	#41	—	106	4	no	—
447	C₂×Q₈	—	53	1	no	—
448	C₂×Q₈	—	53	2	no	—
449	C₂×Q₈	—	53	1	no	—
450	C₂×Q₈	—	53	2	no	—
451	C₂×Q₈	—	53	2	no	—
452	C₂×Q₈	—	106	4	no	—
453	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
454	C₂×D₄	—	53	1	no	—
455	C₂×D₄	—	53	1	no	—
456	C₂×D₄	—	53	1	no	—
457	C₂×D₄	—	53	1	no	—
458	C₂×D₄	—	53	1	no	—
459	C₂×D₄	—	53	1	no	—
460	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
461	C₂²×C₄	—	53	1	no	—
462	C₂²×C₄	—	53	1	no	—
463	C₂²×C₄	—	53	2	no	—
464	C₂²×C₄	—	53	2	no	—
465	C₂²×C₄	—	53	2	no	—
466	C₂²×C₄	—	53	1	no	—
467	C₂²×C₄	—	53	1	no	—
468	C₂²×C₄	—	53	1	no	—
469	C₂²×C₄	—	53	1	no	—
470	C₂²×C₄	—	53	2	no	—
471	C₂²×C₄	—	53	2	no	—
472	C₂²×C₄	—	106	4	no	—
473	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
474	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
475	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
476	C₄⋊C₄	—	53	1	no	—
477	C₄⋊C₄	—	53	1	no	—
478	C₄⋊C₄	—	53	2	no	—
479	C₄⋊C₄	—	53	1	no	—
480	C₄⋊C₄	—	53	2	no	—
481	C₄⋊C₄	—	53	2	no	—
482	C₄⋊C₄	—	53	2	no	—
483	C₄⋊C₄	—	53	2	no	—
484	C₄⋊C₄	—	53	2	no	—
485	C₄⋊C₄	—	53	2	no	—
486	C₄⋊C₄	—	53	1	no	—
487	C₄⋊C₄	—	53	1	no	—
488	C₄⋊C₄	—	53	2	no	—
489	C₄⋊C₄	—	53	2	no	—
490	C₄⋊C₄	—	53	2	no	—
491	C₄⋊C₄	—	53	2	no	—
492	C₄⋊C₄	—	53	2	no	—
493	C₄⋊C₄	—	53	2	no	—
494	C₄⋊C₄	—	53	1	no	—
495	C₄⋊C₄	—	53	1	no	—
496	C₄⋊C₄	—	53	1	no	—
497	C₄⋊C₄	—	53	2	no	—
498	C₄⋊C₄	—	53	2	no	—
499	C₄⋊C₄	—	53	2	no	—
500	C₄⋊C₄	—	53	2	no	—
501	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
502	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
503	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
504	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
505	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
506	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
507	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
508	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
509	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
510	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
511	C₂²⋊C₄	—	53	2	no	—
512	C₂²⋊C₄	—	53	1	no	—
513	C₂²⋊C₄	—	53	1	no	—
514	C₂²⋊C₄	—	53	2	no	—
515	C₂²⋊C₄	—	53	2	no	—
516	C₂²⋊C₄	—	53	2	no	—
517	C₂²⋊C₄	—	53	2	no	—
518	C₂²⋊C₄	—	53	2	no	—
519	C₂²⋊C₄	—	53	2	no	—
520	C₂²⋊C₄	—	53	2	no	—
521	C₂²⋊C₄	—	53	1	no	—
522	C₂²⋊C₄	—	53	1	no	—
523	C₂²⋊C₄	—	53	1	no	—
524	C₂²⋊C₄	—	53	2	no	—
525	C₂²⋊C₄	—	53	2	no	—
526	C₂²⋊C₄	—	53	1	no	—
527	C₂²⋊C₄	—	53	1	no	—
528	C₂²⋊C₄	—	53	1	no	—
529	C₂²⋊C₄	—	53	2	no	—
530	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
531	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
532	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
533	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
534	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
535	C₄²	—	53	2	no	—
536	C₄²	—	53	2	no	—
537	C₄²	—	53	2	no	—
538	C₄²	—	53	2	no	—
539	C₄²	—	53	2	no	—
540	C₄²	—	53	2	no	—
541	C₄²	—	53	2	no	—
542	C₄²	—	53	2	no	—
543	C₄²	—	53	1	no	—
544	C₄²	C₂²×D₅₃	1	2	yes	—
545	C₄²	C₂²×D₅₃	1	1	yes	characteristic
546	C₄²	C₂²×D₅₃	1	2	yes	—
547	C₂³	—	53	1	no	—
548	C₂³	—	53	1	no	—
549	C₂³	—	53	1	no	—
550	C₂³	—	106	2	no	—
551	C₂³	—	106	2	no	—
552	C₂³	—	212	1	no	—
553	C₂³	C₂³×D₅₃	1	1	yes	characteristic
554	C₂³	C₂³×D₅₃	1	1	yes	characteristic
555	Q₈	—	2	2	no	—
556	Q₈	—	106	4	no	—
557	Q₈	—	106	2	no	—
558	Q₈	—	106	2	no	—
559	Q₈	—	212	2	no	—
560	Q₈	—	212	2	no	—
561	D₄	—	2	2	no	—
562	D₄	—	106	2	no	—
563	D₄	—	106	2	no	—
564	D₄	—	106	2	no	—
565	D₄	—	106	2	no	—
566	D₄	—	212	1	no	—
567	D₄	—	212	1	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	53	1	no	—
572	C₂×C₄	—	53	4	no	—
573	C₂×C₄	—	53	1	no	—
574	C₂×C₄	—	53	2	no	—
575	C₂×C₄	—	53	1	no	—
576	C₂×C₄	—	53	1	no	—
577	C₂×C₄	—	53	2	no	—
578	C₂×C₄	—	53	2	no	—
579	C₂×C₄	—	53	1	no	—
580	C₂×C₄	—	53	2	no	—
581	C₂×C₄	—	106	2	no	—
582	C₂×C₄	—	106	2	no	—
583	C₂×C₄	—	106	4	no	—
584	C₂×C₄	—	106	2	no	—
585	C₂×C₄	—	106	2	no	—
586	C₂×C₄	—	106	2	no	—
587	C₂×C₄	—	212	2	no	—
588	C₂×C₄	—	212	2	no	—
589	C₂×C₄	—	212	2	no	—
590	C₂×C₄	—	212	2	no	—
591	C₂×C₄	C₂³×D₅₃	1	2	yes	—
592	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
593	C₂×C₄	C₂³×D₅₃	1	2	yes	—
594	C₂×C₄	C₂³×D₅₃	1	2	yes	—
595	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
596	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
597	C₂×C₄	C₂³×D₅₃	1	2	yes	—
598	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
599	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	53	2	no	—
606	C₂²	—	106	1	no	—
607	C₂²	—	106	1	no	—
608	C₂²	—	106	1	no	—
609	C₂²	—	106	1	no	—
610	C₂²	—	106	1	no	—
611	C₂²	—	106	1	no	—
612	C₂²	—	106	1	no	—
613	C₂²	—	106	1	no	—
614	C₂²	—	212	4	no	—
615	C₂²	#13133264	1	2	yes	—
616	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	1	no	—
623	C₄	—	2	2	no	—
624	C₄	—	2	1	no	—
625	C₄	—	53	2	no	—
626	C₄	—	106	1	no	—
627	C₄	—	106	1	no	—
628	C₄	—	106	2	no	—
629	C₄	—	106	1	no	—
630	C₄	—	106	2	no	—
631	C₄	—	106	2	no	—
632	C₄	—	106	2	no	—
633	C₄	—	106	1	no	—
634	C₄	#41×D₅₃	1	2	yes	—
635	C₂	—	2	2	no	—
636	C₂	—	4	1	no	—
637	C₂	—	106	2	no	—
638	C₂	—	212	1	no	—
639	C₂	—	212	1	no	—
640	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
641	C₂	C₂×#13133264	1	1	yes	characteristic
642	C₂	#16811460	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#552	C₂
hypercenter	#552	D₅₃
hypocenter	C₅₃	#3495
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #552

Complements

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

N	H ≅ G/N	type
1. #16811346	C₂	semidirect
2. #16811341	C₂	semidirect
3. #16811331	C₂	semidirect
4. #16811330	C₂	semidirect
5. #16811315	C₂	semidirect
6. #16811311	C₂	semidirect
7. #16811311	C₂	semidirect
8. #16811309	C₂	semidirect
9. #16811308	C₂	semidirect
10. #16811297	C₂	semidirect
11. #16811296	C₂	semidirect
12. #16811253	C₂	semidirect
13. #16811239	C₂	semidirect
14. #16811232	C₂	semidirect
15. #16811196	C₂	semidirect
16. #16811176	C₂	semidirect
17. #16811150	C₂	semidirect
19. C₂²⋊Q₈×D₅₃	C₂	semidirect
20. #16811103	C₂	semidirect
21. #16811099	C₂	semidirect
22. C₂×#13133175	C₂	semidirect
23. C₅₃×#552	C₂	semidirect
27. #13133217	C₂²	semidirect
28. #13133216	C₂²	semidirect
29. #13133216	C₂²	semidirect
30. #13133216	C₂²	semidirect
34. #13133212	C₂²	semidirect
35. #13133212	C₂²	semidirect
37. #13133210	C₂²	semidirect
38. #13133208	C₂²	semidirect
39. #13133208	C₂²	semidirect
40. #13133208	C₂²	semidirect
41. #13133208	C₂²	semidirect
42. #13133207	C₂²	semidirect
43. #13133207	C₂²	semidirect
44. #13133207	C₂²	semidirect
47. #13133203	C₂²	semidirect
57. #13133197	C₂²	semidirect
58. #13133197	C₂²	semidirect
60. #13133196	C₂²	semidirect
68. #13133194	C₂²	semidirect
70. #13133193	C₂²	semidirect
75. Q₈×Dic₅₃	C₂²	semidirect
76. C₄×Dic₁₀₆	C₂²	semidirect
77. C₄×Dic₁₀₆	C₂²	semidirect
78. C₄×Dic₁₀₆	C₂²	semidirect
79. C₄×Dic₁₀₆	C₂²	semidirect
80. C₄×Dic₁₀₆	C₂²	semidirect
81. #13133182	C₂²	semidirect
82. #13133182	C₂²	semidirect
90. #13133177	C₂²	semidirect
99. #13133175	C₂²	semidirect
100. #13133175	C₂²	semidirect
103. #13133173	C₂²	semidirect
105. C₂×C₄⋊Dic₅₃	C₂²	semidirect
108. C₄⋊C₄×D₅₃	C₂²	semidirect
112. #126×C₅₃	C₂²	semidirect
113. #125×C₅₃	C₂²	semidirect
114. #125×C₅₃	C₂²	semidirect
117. C₂²⋊Q₈×C₅₃	C₂²	semidirect
118. C₄×Q₈×C₅₃	C₂²	semidirect
121. #117×C₅₃	C₂²	semidirect
331. #552	D₅₃	semidirect
345. C₅₃	#3495	semidirect
360. #126	C₂×D₅₃	semidirect
368. #125	C₂×D₅₃	semidirect
369. #125	C₂×D₅₃	semidirect
398. C₂²⋊Q₈	C₂×D₅₃	semidirect
411. C₄×Q₈	C₂×D₅₃	semidirect
435. #117	C₂×D₅₃	semidirect