Subgroups — #20288956

Order 6784.

1014 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811374	C₂	1	2	yes	maximal, complemented
2	#16811372	C₂	1	2	yes	maximal, complemented
3	#16811315	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811314	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811313	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811307	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811302	C₂	1	2	yes	maximal, complemented
8	#16811300	C₂	1	2	yes	maximal, complemented
9	#16811288	C₂	1	1	yes	maximal, characteristic, complemented
10	Q₈×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811252	C₂	1	2	yes	maximal, complemented
13	#16811250	C₂	1	2	yes	maximal, complemented
14	#16811234	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811231	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811219	C₂	1	2	yes	maximal, complemented
17	#16811215	C₂	1	2	yes	maximal, complemented
18	#16811197	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811196	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811132	C₂	1	1	yes	maximal, characteristic
21	C₂×#13133193	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂²⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
23	D₄×Q₈×C₅₃	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	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
28	#13133216	C₂²	1	2	yes	complemented
29	#13133216	C₂²	1	2	yes	complemented
30	#13133215	C₂²	1	1	yes	characteristic, complemented
31	#13133215	C₂²	1	2	yes	complemented
32	#13133215	C₂²	1	2	yes	complemented
33	#13133215	C₂²	1	2	yes	complemented
34	#13133212	C₂²	1	2	yes	—
35	#13133212	C₂²	1	2	yes	—
36	#13133212	C₂²	1	2	yes	—
37	#13133212	C₂²	1	2	yes	complemented
38	#13133211	C₂²	1	2	yes	—
39	#13133211	C₂²	1	2	yes	complemented
40	#13133210	C₂²	1	2	yes	complemented
41	#13133210	C₂²	1	2	yes	complemented
42	#13133209	C₂²	1	1	yes	characteristic, complemented
43	#13133209	C₂²	1	2	yes	complemented
44	#13133208	C₂²	1	1	yes	characteristic, complemented
45	#13133208	C₂²	1	2	yes	complemented
46	#13133207	C₂²	1	2	yes	complemented
47	#13133207	C₂²	1	2	yes	complemented
48	#13133207	C₂²	1	2	yes	complemented
49	#13133207	C₂²	1	2	yes	complemented
50	#13133207	C₂²	1	2	yes	complemented
51	#13133207	C₂²	1	2	yes	complemented
52	#13133206	C₂²	1	2	yes	—
53	#13133203	C₂²	1	2	yes	—
54	#13133202	C₂²	1	2	yes	—
55	#13133202	C₂²	1	2	yes	—
56	#13133202	C₂²	1	1	yes	characteristic
57	#13133201	C₂²	1	2	yes	—
58	#13133200	C₂²	1	2	yes	—
59	#13133200	C₂²	1	2	yes	complemented
60	#13133199	C₂²	1	2	yes	—
61	#13133199	C₂²	1	1	yes	characteristic
62	#13133199	C₂²	1	2	yes	complemented
63	#13133198	C₂²	1	2	yes	—
64	#13133197	C₂²	1	1	yes	characteristic
65	#13133196	C₂²	1	2	yes	—
66	#13133196	C₂²	1	1	yes	characteristic
67	#13133196	C₂²	1	1	yes	characteristic, complemented
68	#13133196	C₂²	1	2	yes	complemented
69	#13133195	C₂²	1	2	yes	—
70	#13133195	C₂²	1	2	yes	—
71	#13133195	C₂²	1	1	yes	characteristic
72	#13133195	C₂²	1	2	yes	complemented
73	#13133194	C₂²	1	2	yes	—
74	#13133194	C₂²	1	1	yes	characteristic
75	#13133193	C₂²	1	2	yes	—
76	#13133193	C₂²	1	1	yes	characteristic
77	#13133193	C₂²	1	1	yes	characteristic
78	#13133193	C₂²	1	2	yes	—
79	#13133193	C₂²	1	1	yes	characteristic
80	#13133193	C₂²	1	1	yes	characteristic
81	#13133193	C₂²	1	1	yes	characteristic
82	#13133193	C₂²	1	2	yes	—
83	#13133193	C₂²	1	1	yes	characteristic, complemented
84	#13133193	C₂²	1	2	yes	complemented
85	#13133193	C₂²	1	1	yes	characteristic, complemented
86	#13133190	C₂²	1	1	yes	characteristic
87	#13133189	C₂²	1	1	yes	characteristic
88	#13133187	C₂²	1	1	yes	characteristic
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	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
93	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
94	#13133179	C₂²	1	2	yes	—
95	C₄×C₂²⋊D₅₃	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	#13133176	C₂²	1	1	yes	characteristic, complemented
100	#13133175	C₂²	1	2	yes	—
101	#13133175	C₂²	1	1	yes	characteristic, complemented
102	#13133174	C₂²	1	2	yes	—
103	#13133174	C₂²	1	1	yes	characteristic
104	#13133173	C₂²	1	2	yes	—
105	#13133173	C₂²	1	2	yes	—
106	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
107	C₂×#11719919	C₂²	1	2	yes	—
108	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
109	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
110	C₂×#11719921	C₂²	1	2	yes	—
111	C₂²⋊C₄×D₅₃	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	2	yes	—
122	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
137	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
140	C₂²×Dic₅₃	C₂³	1	2	yes	—
141	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
142	Q₈×D₅₃	—	4	2	no	—
143	Q₈×D₅₃	—	4	2	no	—
144	#11719945	—	4	2	no	—
145	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
146	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
147	C₂×C₄×D₅₃	—	4	2	no	—
148	C₂×C₄×D₅₃	C₂³	1	2	yes	—
149	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
150	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
151	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₅₃	C₂³	1	2	yes	—
153	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
154	C₂×C₄×D₅₃	C₂³	1	2	yes	—
155	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
156	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
157	C₂×Dic₁₀₆	C₂³	1	2	yes	—
158	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
159	C₂×Dic₁₀₆	C₂³	1	2	yes	—
160	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
161	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
162	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
163	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
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	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
168	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
169	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
170	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₄²×C₅₃	C₂³	1	1	yes	characteristic
172	C₄²×C₅₃	C₂³	1	2	yes	—
173	C₂²⋊Dic₅₃	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	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	2	yes	—
183	#11719921	C₂³	1	2	yes	—
184	#11719921	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
190	#11719921	C₂³	1	2	yes	—
191	#11719921	C₂³	1	2	yes	—
192	#11719921	C₂³	1	1	yes	characteristic
193	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
194	C₄⋊Dic₅₃	C₂³	1	2	yes	—
195	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
196	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
197	C₄⋊Dic₅₃	C₂³	1	2	yes	—
198	C₄⋊Dic₅₃	C₂³	1	2	yes	—
199	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
200	C₄⋊Dic₅₃	C₂³	1	2	yes	—
201	C₄⋊Dic₅₃	C₂³	1	2	yes	—
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	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	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	#11719919	C₂³	1	1	yes	characteristic
221	#11719919	C₂³	1	2	yes	—
222	C₄×Dic₅₃	C₂³	1	2	yes	—
223	C₄×Dic₅₃	C₂³	1	2	yes	—
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₄×Dic₅₃	C₂³	1	2	yes	—
226	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
232	C₂²×D₅₃	—	4	1	no	—
233	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
234	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
235	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
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	2	yes	—
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	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	—	2	2	no	—
257	C₂²⋊D₅₃	—	4	1	no	—
258	C₂²⋊D₅₃	—	4	1	no	—
259	C₂×Dic₅₃	—	2	4	no	—
260	C₂×Dic₅₃	—	4	1	no	—
261	C₂×Dic₅₃	C₂⁴	1	2	yes	—
262	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
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	2	yes	—
267	C₂×Dic₅₃	C₂⁴	1	2	yes	—
268	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
269	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
270	D₂₁₂	—	2	2	no	—
271	D₂₁₂	—	4	1	no	—
272	C₄×D₅₃	—	2	4	no	—
273	C₄×D₅₃	—	2	2	no	—
274	C₄×D₅₃	—	2	2	no	—
275	C₄×D₅₃	—	2	2	no	—
276	C₄×D₅₃	—	4	1	no	—
277	C₄×D₅₃	—	4	1	no	—
278	C₄×D₅₃	—	4	2	no	—
279	C₄×D₅₃	—	4	2	no	—
280	Dic₁₀₆	—	2	2	no	—
281	Dic₁₀₆	—	4	1	no	—
282	Dic₁₀₆	—	4	2	no	—
283	Dic₁₀₆	—	4	2	no	—
284	C₂²×C₅₃	—	2	1	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⁺⁴	1	2	yes	—
290	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
291	C₂×D₅₃	—	2	1	no	—
292	C₂×D₅₃	—	2	1	no	—
293	C₂×D₅₃	—	2	1	no	—
294	C₂×D₅₃	—	2	1	no	—
295	C₂×D₅₃	—	2	1	no	—
296	C₂×D₅₃	—	2	1	no	—
297	C₂×D₅₃	—	2	1	no	—
298	C₂×D₅₃	—	2	1	no	—
299	C₂×D₅₃	—	4	4	no	—
300	C₂×D₅₃	2₋¹⁺⁴	1	2	yes	—
301	C₄×C₅₃	—	2	2	no	—
302	C₄×C₅₃	—	2	2	no	—
303	C₄×C₅₃	—	2	1	no	—
304	C₄×C₅₃	—	2	1	no	—
305	C₄×C₅₃	—	2	2	no	—
306	C₄×C₅₃	—	2	2	no	—
307	C₄×C₅₃	—	2	1	no	—
308	C₄×C₅₃	—	2	1	no	—
309	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
310	Dic₅₃	—	2	2	no	—
311	Dic₅₃	—	2	1	no	—
312	Dic₅₃	—	2	1	no	—
313	Dic₅₃	—	2	1	no	—
314	Dic₅₃	—	2	2	no	—
315	Dic₅₃	—	2	2	no	—
316	Dic₅₃	—	2	2	no	—
317	Dic₅₃	—	2	1	no	—
318	Dic₅₃	—	2	1	no	—
319	#3538	—	53	1	no	maximal, Sylow
320	C₂×C₅₃	—	2	2	no	—
321	C₂×C₅₃	—	4	1	no	—
322	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
323	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
324	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
325	D₅₃	—	2	2	no	—
326	D₅₃	—	4	1	no	—
327	D₅₃	—	4	1	no	—
328	#563	—	53	2	no	—
329	#563	—	53	2	no	—
330	#552	—	53	1	no	—
331	#552	—	53	1	no	—
332	#552	—	53	1	no	—
333	#552	—	53	1	no	—
334	#551	—	53	2	no	—
335	#551	—	53	2	no	—
336	D₄×Q₈	—	53	1	no	—
337	D₄×Q₈	—	53	1	no	—
338	D₄×Q₈	—	53	1	no	—
339	D₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
340	#544	—	53	2	no	—
341	#544	—	53	2	no	—
342	#541	—	53	1	no	—
343	#541	—	53	1	no	—
344	#539	—	53	2	no	—
345	#539	—	53	2	no	—
346	#536	—	53	1	no	—
347	#536	—	53	1	no	—
348	#525	—	53	1	no	—
349	C₂×C₂²⋊Q₈	—	53	1	no	—
350	C₂×C₂²⋊Q₈	—	53	1	no	—
351	C₅₃	#3538	1	1	yes	characteristic, complemented, Sylow
352	C₂×#41	—	53	1	no	—
353	C₂²×Q₈	—	53	1	no	—
354	C₂²×Q₈	—	53	1	no	—
355	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
356	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
357	C₂³×C₄	—	53	1	no	—
358	C₄⋊Q₈	—	53	2	no	—
359	C₄⋊Q₈	—	53	1	no	—
360	C₄⋊Q₈	—	53	2	no	—
361	C₄⋊Q₈	—	53	2	no	—
362	C₄⋊Q₈	—	53	2	no	—
363	C₄⋊Q₈	—	53	2	no	—
364	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
365	C₄⋊Q₈	C₂×D₅₃	1	2	yes	complemented
366	#126	—	53	2	no	—
367	#126	—	53	2	no	—
368	#126	—	53	2	no	—
369	#126	—	53	2	no	—
370	#126	—	53	2	no	—
371	#126	—	53	2	no	—
372	#126	—	53	2	no	—
373	#126	—	53	2	no	—
374	#125	—	53	2	no	—
375	#125	—	53	2	no	—
376	#125	—	53	2	no	—
377	#125	—	53	1	no	—
378	#125	—	53	2	no	—
379	#125	—	53	2	no	—
380	#125	—	53	2	no	—
381	#125	—	53	2	no	—
382	#125	—	53	2	no	—
383	#125	—	53	1	no	—
384	#124	—	53	2	no	—
385	#124	—	53	2	no	—
386	#123	—	53	2	no	—
387	#123	—	53	2	no	—
388	#123	—	53	2	no	—
389	#123	—	53	1	no	—
390	#123	—	53	2	no	—
391	#123	—	53	2	no	—
392	#123	—	53	1	no	—
393	#123	—	53	2	no	—
394	#123	—	53	2	no	—
395	#123	—	53	2	no	—
396	C₂²⋊Q₈	—	53	1	no	—
397	C₂²⋊Q₈	—	53	1	no	—
398	C₂²⋊Q₈	—	53	1	no	—
399	C₂²⋊Q₈	—	53	2	no	—
400	C₂²⋊Q₈	—	53	1	no	—
401	C₂²⋊Q₈	—	53	1	no	—
402	C₂²⋊Q₈	—	53	2	no	—
403	C₂²⋊Q₈	—	53	1	no	—
404	C₂²⋊Q₈	—	53	1	no	—
405	C₂²⋊Q₈	—	53	2	no	—
406	C₂²⋊Q₈	—	53	2	no	—
407	C₂²⋊Q₈	—	53	1	no	—
408	C₂²⋊Q₈	—	53	1	no	—
409	C₂²⋊Q₈	—	53	2	no	—
410	C₂²⋊Q₈	—	53	2	no	—
411	C₂²⋊Q₈	—	53	1	no	—
412	C₂²⋊Q₈	—	53	2	no	—
413	C₂²⋊Q₈	—	53	1	no	—
414	C₂²⋊Q₈	—	53	2	no	—
415	C₂²⋊Q₈	—	53	2	no	—
416	C₂²⋊Q₈	—	53	2	no	—
417	C₂²⋊Q₈	—	53	1	no	—
418	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
419	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
420	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
421	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
422	#121	—	53	1	no	—
423	#121	—	53	1	no	—
424	#120	—	53	1	no	—
425	#120	—	53	1	no	—
426	C₄×Q₈	—	53	1	no	—
427	C₄×Q₈	—	53	1	no	—
428	C₄×Q₈	—	53	1	no	—
429	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
430	C₄×D₄	—	53	1	no	—
431	C₄×D₄	—	53	1	no	—
432	C₄×D₄	—	53	2	no	—
433	C₄×D₄	—	53	2	no	—
434	C₄×D₄	—	53	2	no	—
435	C₄×D₄	—	53	1	no	—
436	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
437	C₄×D₄	C₂×D₅₃	1	2	yes	—
438	#117	—	53	1	no	—
439	#117	—	53	1	no	—
440	#117	—	53	2	no	—
441	#117	—	53	2	no	—
442	#117	—	53	2	no	—
443	#117	—	53	1	no	—
444	#117	—	53	2	no	—
445	C₂×C₄⋊C₄	—	53	2	no	—
446	C₂×C₄⋊C₄	—	53	1	no	—
447	C₂×C₄⋊C₄	—	53	2	no	—
448	C₂×C₄⋊C₄	—	53	1	no	—
449	C₂×C₂²⋊C₄	—	53	2	no	—
450	C₂×C₂²⋊C₄	—	53	2	no	—
451	C₂⁴	—	53	1	no	—
452	#41	—	212	2	no	—
453	C₂×Q₈	—	4	2	no	—
454	C₂×Q₈	—	4	2	no	—
455	C₂×Q₈	—	53	1	no	—
456	C₂×Q₈	—	53	2	no	—
457	C₂×Q₈	—	53	2	no	—
458	C₂×Q₈	—	53	1	no	—
459	C₂×Q₈	—	212	2	no	—
460	C₂×Q₈	—	212	2	no	—
461	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
462	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
463	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
464	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
465	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
466	C₂×D₄	—	53	1	no	—
467	C₂×D₄	—	53	1	no	—
468	C₂×D₄	—	53	1	no	—
469	C₂×D₄	—	53	1	no	—
470	C₂×D₄	—	53	1	no	—
471	C₂×D₄	—	53	1	no	—
472	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
473	C₂²×C₄	—	53	2	no	—
474	C₂²×C₄	—	53	1	no	—
475	C₂²×C₄	—	53	2	no	—
476	C₂²×C₄	—	53	1	no	—
477	C₂²×C₄	—	53	1	no	—
478	C₂²×C₄	—	53	1	no	—
479	C₂²×C₄	—	53	2	no	—
480	C₂²×C₄	—	53	1	no	—
481	C₂²×C₄	—	53	2	no	—
482	C₂²×C₄	—	53	1	no	—
483	C₂²×C₄	—	212	2	no	—
484	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
485	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
486	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
487	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
488	C₄⋊C₄	—	53	1	no	—
489	C₄⋊C₄	—	53	1	no	—
490	C₄⋊C₄	—	53	1	no	—
491	C₄⋊C₄	—	53	2	no	—
492	C₄⋊C₄	—	53	2	no	—
493	C₄⋊C₄	—	53	2	no	—
494	C₄⋊C₄	—	53	2	no	—
495	C₄⋊C₄	—	53	2	no	—
496	C₄⋊C₄	—	53	2	no	—
497	C₄⋊C₄	—	53	2	no	—
498	C₄⋊C₄	—	53	2	no	—
499	C₄⋊C₄	—	53	2	no	—
500	C₄⋊C₄	—	53	1	no	—
501	C₄⋊C₄	—	53	1	no	—
502	C₄⋊C₄	—	53	2	no	—
503	C₄⋊C₄	—	53	2	no	—
504	C₄⋊C₄	—	53	2	no	—
505	C₄⋊C₄	—	53	2	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	2	no	—
516	C₄⋊C₄	—	53	2	no	—
517	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
518	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
519	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
520	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
521	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
522	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
523	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
524	C₂²⋊C₄	—	53	2	no	—
525	C₂²⋊C₄	—	53	1	no	—
526	C₂²⋊C₄	—	53	2	no	—
527	C₂²⋊C₄	—	53	1	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	1	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	1	no	—
541	C₂²⋊C₄	—	53	1	no	—
542	C₂²⋊C₄	—	53	2	no	—
543	C₂²⋊C₄	—	53	2	no	—
544	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
545	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
546	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
547	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
548	C₄²	—	53	2	no	—
549	C₄²	—	53	2	no	—
550	C₄²	—	53	1	no	—
551	C₄²	—	53	2	no	—
552	C₄²	—	53	2	no	—
553	C₄²	—	53	2	no	—
554	C₄²	C₂²×D₅₃	1	1	yes	characteristic
555	C₄²	C₂²×D₅₃	1	2	yes	—
556	C₂³	—	53	1	no	—
557	C₂³	—	53	1	no	—
558	C₂³	—	53	1	no	—
559	C₂³	—	106	2	no	—
560	C₂³	—	106	2	no	—
561	C₂³	—	212	1	no	—
562	C₂³	C₂³×D₅₃	1	1	yes	characteristic
563	C₂³	C₂³×D₅₃	1	1	yes	characteristic
564	Q₈	—	2	2	no	—
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	4	2	no	—
568	Q₈	—	4	2	no	—
569	Q₈	—	106	2	no	—
570	Q₈	—	212	2	no	—
571	Q₈	—	212	2	no	—
572	Q₈	—	212	1	no	—
573	D₄	—	2	2	no	—
574	D₄	—	106	2	no	—
575	D₄	—	106	2	no	—
576	D₄	—	106	2	no	—
577	D₄	—	212	1	no	—
578	D₄	—	212	1	no	—
579	D₄	—	212	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	53	1	no	—
585	C₂×C₄	—	53	1	no	—
586	C₂×C₄	—	53	2	no	—
587	C₂×C₄	—	53	2	no	—
588	C₂×C₄	—	53	2	no	—
589	C₂×C₄	—	53	1	no	—
590	C₂×C₄	—	53	2	no	—
591	C₂×C₄	—	53	1	no	—
592	C₂×C₄	—	53	1	no	—
593	C₂×C₄	—	106	4	no	—
594	C₂×C₄	—	106	4	no	—
595	C₂×C₄	—	106	2	no	—
596	C₂×C₄	—	106	2	no	—
597	C₂×C₄	—	106	2	no	—
598	C₂×C₄	—	212	1	no	—
599	C₂×C₄	—	212	1	no	—
600	C₂×C₄	—	212	2	no	—
601	C₂×C₄	—	212	2	no	—
602	C₂×C₄	—	212	1	no	—
603	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
604	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
605	C₂×C₄	C₂³×D₅₃	1	2	yes	—
606	C₂×C₄	C₂³×D₅₃	1	2	yes	—
607	C₂×C₄	C₂³×D₅₃	1	2	yes	—
608	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
609	C₂×C₄	C₂³×D₅₃	1	2	yes	—
610	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
611	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
612	C₂²	—	2	1	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	53	2	no	—
618	C₂²	—	106	1	no	—
619	C₂²	—	106	1	no	—
620	C₂²	—	106	1	no	—
621	C₂²	—	106	1	no	—
622	C₂²	—	106	1	no	—
623	C₂²	—	106	1	no	—
624	C₂²	—	106	1	no	—
625	C₂²	—	106	1	no	—
626	C₂²	—	212	4	no	—
627	C₂²	#13133265	1	2	yes	—
628	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	1	no	—
636	C₄	—	2	1	no	—
637	C₄	—	106	1	no	—
638	C₄	—	106	1	no	—
639	C₄	—	106	1	no	—
640	C₄	—	106	2	no	—
641	C₄	—	106	2	no	—
642	C₄	—	106	1	no	—
643	C₄	—	106	2	no	—
644	C₄	—	106	2	no	—
645	C₄	—	106	1	no	—
646	C₄	#13133262	1	2	yes	—
647	C₂	—	2	2	no	—
648	C₂	—	4	1	no	—
649	C₂	—	106	2	no	—
650	C₂	—	212	1	no	—
651	C₂	—	212	1	no	—
652	C₂	C₂×#13133262	1	1	yes	characteristic
653	C₂	2₋¹⁺⁴×D₅₃	1	1	yes	characteristic
654	C₂	C₂×#13133265	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	#3538
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. #16811374	C₂	semidirect
2. #16811372	C₂	semidirect
3. #16811315	C₂	semidirect
4. #16811314	C₂	semidirect
5. #16811313	C₂	semidirect
6. #16811307	C₂	semidirect
7. #16811302	C₂	semidirect
8. #16811300	C₂	semidirect
9. #16811288	C₂	semidirect
10. Q₈×C₂²⋊D₅₃	C₂	semidirect
11. Q₈×C₂²⋊D₅₃	C₂	semidirect
12. #16811252	C₂	semidirect
13. #16811250	C₂	semidirect
14. #16811234	C₂	semidirect
15. #16811231	C₂	semidirect
16. #16811219	C₂	semidirect
17. #16811215	C₂	semidirect
18. #16811197	C₂	semidirect
19. #16811196	C₂	semidirect
21. C₂×#13133193	C₂	semidirect
22. C₂²⋊Q₈×D₅₃	C₂	semidirect
23. D₄×Q₈×C₅₃	C₂	semidirect
26. C₂×Q₈×D₅₃	C₂²	semidirect
28. #13133216	C₂²	semidirect
29. #13133216	C₂²	semidirect
30. #13133215	C₂²	semidirect
31. #13133215	C₂²	semidirect
32. #13133215	C₂²	semidirect
33. #13133215	C₂²	semidirect
37. #13133212	C₂²	semidirect
39. #13133211	C₂²	semidirect
40. #13133210	C₂²	semidirect
41. #13133210	C₂²	semidirect
42. #13133209	C₂²	semidirect
43. #13133209	C₂²	semidirect
44. #13133208	C₂²	semidirect
45. #13133208	C₂²	semidirect
46. #13133207	C₂²	semidirect
47. #13133207	C₂²	semidirect
48. #13133207	C₂²	semidirect
49. #13133207	C₂²	semidirect
50. #13133207	C₂²	semidirect
51. #13133207	C₂²	semidirect
59. #13133200	C₂²	semidirect
62. #13133199	C₂²	semidirect
67. #13133196	C₂²	semidirect
68. #13133196	C₂²	semidirect
72. #13133195	C₂²	semidirect
83. #13133193	C₂²	semidirect
84. #13133193	C₂²	semidirect
85. #13133193	C₂²	semidirect
90. Q₈×Dic₅₃	C₂²	semidirect
91. Q₈×Dic₅₃	C₂²	semidirect
92. C₄×Dic₁₀₆	C₂²	semidirect
99. #13133176	C₂²	semidirect
101. #13133175	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
339. D₄×Q₈	D₅₃	semidirect
351. C₅₃	#3538	semidirect
356. C₂²×Q₈	C₂×D₅₃	semidirect
364. C₄⋊Q₈	C₂×D₅₃	semidirect
365. C₄⋊Q₈	C₂×D₅₃	semidirect
420. C₂²⋊Q₈	C₂×D₅₃	semidirect
421. C₂²⋊Q₈	C₂×D₅₃	semidirect
429. C₄×Q₈	C₂×D₅₃	semidirect