Subgroups — #20288760

Order 6784.

1124 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811345	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811328	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811305	C₂	1	2	yes	maximal, complemented
4	#16811302	C₂	1	2	yes	maximal, complemented
5	#16811300	C₂	1	2	yes	maximal, complemented
6	#16811288	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811283	C₂	1	2	yes	maximal, complemented
8	#16811275	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811274	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	#16811247	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811246	C₂	1	1	yes	maximal, characteristic, complemented
14	#16811221	C₂	1	2	yes	maximal, complemented
15	#16811219	C₂	1	2	yes	maximal, complemented
16	#16811175	C₂	1	1	yes	maximal, characteristic, complemented
17	#16811153	C₂	1	1	yes	maximal, characteristic, complemented
18	#16811132	C₂	1	2	yes	maximal, complemented
19	C₂²⋊Q₈×D₅₃	C₂	1	2	yes	maximal, complemented
20	#16811105	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811104	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×#13133179	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#554	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×#11719945	C₂²	1	2	yes	complemented
27	C₂×Q₈×D₅₃	C₂²	1	2	yes	complemented
28	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
30	#13133216	C₂²	1	1	yes	characteristic, complemented
31	#13133216	C₂²	1	1	yes	characteristic, complemented
32	#13133215	C₂²	1	1	yes	characteristic, 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	#13133210	C₂²	1	1	yes	characteristic, complemented
39	#13133207	C₂²	1	1	yes	characteristic, complemented
40	#13133207	C₂²	1	2	yes	complemented
41	#13133207	C₂²	1	2	yes	complemented
42	#13133206	C₂²	1	1	yes	characteristic
43	#13133206	C₂²	1	2	yes	complemented
44	#13133205	C₂²	1	1	yes	characteristic
45	#13133203	C₂²	1	1	yes	characteristic, complemented
46	#13133203	C₂²	1	2	yes	complemented
47	#13133202	C₂²	1	2	yes	—
48	#13133202	C₂²	1	2	yes	—
49	#13133202	C₂²	1	2	yes	complemented
50	#13133202	C₂²	1	2	yes	complemented
51	#13133202	C₂²	1	2	yes	complemented
52	#13133201	C₂²	1	1	yes	characteristic
53	#13133200	C₂²	1	2	yes	complemented
54	#13133199	C₂²	1	2	yes	complemented
55	#13133199	C₂²	1	2	yes	complemented
56	#13133198	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	2	yes	—
61	#13133195	C₂²	1	2	yes	complemented
62	#13133195	C₂²	1	2	yes	complemented
63	#13133195	C₂²	1	2	yes	complemented
64	#13133195	C₂²	1	2	yes	complemented
65	#13133195	C₂²	1	2	yes	complemented
66	#13133194	C₂²	1	1	yes	characteristic
67	#13133194	C₂²	1	2	yes	—
68	#13133193	C₂²	1	2	yes	complemented
69	#13133193	C₂²	1	2	yes	complemented
70	#13133192	C₂²	1	1	yes	characteristic
71	#13133192	C₂²	1	2	yes	—
72	#13133191	C₂²	1	1	yes	characteristic
73	#13133190	C₂²	1	2	yes	—
74	#13133190	C₂²	1	2	yes	complemented
75	#13133189	C₂²	1	1	yes	characteristic
76	#13133187	C₂²	1	2	yes	—
77	#13133185	C₂²	1	2	yes	—
78	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
79	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
80	#13133182	C₂²	1	2	yes	complemented
81	#13133182	C₂²	1	1	yes	characteristic, complemented
82	#13133182	C₂²	1	1	yes	characteristic, complemented
83	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
84	C₄×D₂₁₂	C₂²	1	2	yes	complemented
85	#13133179	C₂²	1	1	yes	characteristic
86	#13133179	C₂²	1	1	yes	characteristic
87	#13133179	C₂²	1	1	yes	characteristic
88	#13133179	C₂²	1	2	yes	—
89	#13133179	C₂²	1	1	yes	characteristic
90	#13133179	C₂²	1	2	yes	—
91	#13133179	C₂²	1	1	yes	characteristic
92	#13133179	C₂²	1	2	yes	complemented
93	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
94	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
95	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic, complemented
96	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
97	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
98	#13133176	C₂²	1	1	yes	characteristic, complemented
99	#13133175	C₂²	1	2	yes	complemented
100	#13133174	C₂²	1	2	yes	complemented
101	#13133174	C₂²	1	2	yes	complemented
102	#13133174	C₂²	1	1	yes	characteristic, complemented
103	#13133173	C₂²	1	2	yes	—
104	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
105	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
106	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
107	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
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	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
113	#125×C₅₃	C₂²	1	2	yes	complemented
114	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
115	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
116	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
117	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
118	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
120	C₄×D₄×C₅₃	C₂²	1	2	yes	—
121	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₅₃	C₂³	1	2	yes	—
124	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
126	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
131	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
132	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
133	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
134	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
135	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
136	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
137	#11719949	—	2	4	no	—
138	Q₈×D₅₃	—	4	4	no	—
139	#11719945	—	2	4	no	—
140	#11719945	—	4	4	no	—
141	C₂×D₂₁₂	—	2	4	no	—
142	C₂×D₂₁₂	C₂³	1	2	yes	—
143	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
144	C₂×D₂₁₂	C₂³	1	2	yes	—
145	C₂×D₂₁₂	C₂³	1	2	yes	—
146	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
147	C₂×C₄×D₅₃	—	4	4	no	—
148	C₂×C₄×D₅₃	C₂³	1	2	yes	—
149	C₂×C₄×D₅₃	C₂³	1	2	yes	—
150	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
151	C₂×C₄×D₅₃	C₂³	1	2	yes	—
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×C₄×D₅₃	C₂³	1	2	yes	—
158	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
159	C₂×Dic₁₀₆	C₂³	1	2	yes	—
160	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
161	C₂×Dic₁₀₆	C₂³	1	2	yes	—
162	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
163	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
164	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
165	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
166	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
173	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
174	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
175	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
176	C₄²×C₅₃	C₂³	1	2	yes	—
177	C₄²×C₅₃	C₂³	1	1	yes	characteristic
178	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
179	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
180	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
181	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
182	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
192	#11719921	C₂³	1	2	yes	—
193	#11719921	C₂³	1	1	yes	characteristic
194	#11719921	C₂³	1	2	yes	—
195	#11719921	C₂³	1	2	yes	—
196	#11719921	C₂³	1	2	yes	—
197	#11719921	C₂³	1	2	yes	—
198	#11719921	C₂³	1	1	yes	characteristic
199	#11719921	C₂³	1	1	yes	characteristic
200	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
201	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
202	C₄⋊Dic₅₃	C₂³	1	2	yes	—
203	C₄⋊Dic₅₃	C₂³	1	2	yes	—
204	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
205	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
206	#11719919	C₂³	1	1	yes	characteristic
207	#11719919	C₂³	1	2	yes	—
208	#11719919	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
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	—	2	4	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	—	4	2	no	—
238	D₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	4	2	no	—
242	C₂×C₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
244	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
246	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
247	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
249	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
252	C₂²⋊D₅₃	—	2	2	no	—
253	C₂²⋊D₅₃	—	2	4	no	—
254	C₂²⋊D₅₃	—	2	2	no	—
255	C₂²⋊D₅₃	—	4	2	no	—
256	C₂×Dic₅₃	—	2	2	no	—
257	C₂×Dic₅₃	—	2	2	no	—
258	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
259	C₂×Dic₅₃	C₂⁴	1	2	yes	—
260	C₂×Dic₅₃	C₂⁴	1	2	yes	—
261	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
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	1	yes	characteristic
266	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
267	D₂₁₂	—	2	4	no	—
268	D₂₁₂	—	2	4	no	—
269	D₂₁₂	—	2	2	no	—
270	D₂₁₂	—	4	2	no	—
271	D₂₁₂	#41	1	4	yes	complemented
272	C₄×D₅₃	—	2	4	no	—
273	C₄×D₅₃	—	2	4	no	—
274	C₄×D₅₃	—	2	2	no	—
275	C₄×D₅₃	—	2	4	no	—
276	C₄×D₅₃	—	2	4	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	C₄×D₅₃	—	4	2	no	—
282	Dic₁₀₆	—	2	2	no	—
283	Dic₁₀₆	—	2	4	no	—
284	Dic₁₀₆	—	2	2	no	—
285	Dic₁₀₆	—	4	2	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	no	—
291	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
292	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
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	1	no	—
299	C₂×D₅₃	—	2	1	no	—
300	C₂×D₅₃	—	2	2	no	—
301	C₂×D₅₃	—	4	8	no	—
302	C₂×D₅₃	C₂×#41	1	4	yes	—
303	C₄×C₅₃	—	2	1	no	—
304	C₄×C₅₃	—	2	2	no	—
305	C₄×C₅₃	—	2	1	no	—
306	C₄×C₅₃	—	2	1	no	—
307	C₄×C₅₃	—	2	2	no	—
308	C₄×C₅₃	—	2	2	no	—
309	C₄×C₅₃	—	2	2	no	—
310	C₄×C₅₃	—	2	1	no	—
311	C₄×C₅₃	C₂×#41	1	2	yes	—
312	Dic₅₃	—	2	2	no	—
313	Dic₅₃	—	2	1	no	—
314	Dic₅₃	—	2	1	no	—
315	Dic₅₃	—	2	2	no	—
316	Dic₅₃	—	2	1	no	—
317	Dic₅₃	—	2	1	no	—
318	Dic₅₃	—	2	1	no	—
319	Dic₅₃	—	2	1	no	—
320	Dic₅₃	H(5,𝔽₂)	1	2	yes	—
321	#3520	—	53	1	no	maximal, Sylow
322	C₂×C₅₃	—	2	2	no	—
323	C₂×C₅₃	—	4	1	no	—
324	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
325	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
326	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
327	D₅₃	—	2	4	no	—
328	D₅₃	—	4	1	no	—
329	D₅₃	—	4	2	no	—
330	#556	—	53	1	no	—
331	#554	—	53	1	no	—
332	#554	D₅₃	1	1	yes	characteristic, complemented
333	#552	—	53	2	no	—
334	#551	—	53	2	no	—
335	#551	—	53	2	no	—
336	D₄×Q₈	—	53	1	no	—
337	#548	—	53	2	no	—
338	#547	—	53	1	no	—
339	#547	—	53	1	no	—
340	#546	—	53	1	no	—
341	#546	—	53	1	no	—
342	#542	—	53	1	no	—
343	#542	—	53	1	no	—
344	#539	—	53	2	no	—
345	#539	—	53	2	no	—
346	#533	—	53	1	no	—
347	#529	—	53	1	no	—
348	#525	—	53	2	no	—
349	C₂×C₂²⋊Q₈	—	53	2	no	—
350	#520	—	53	1	no	—
351	#520	—	53	1	no	—
352	C₂×C₄×D₄	—	53	1	no	—
353	C₅₃	#3520	1	1	yes	characteristic, complemented, Sylow
354	C₂×#41	—	53	1	no	—
355	C₂×#41	—	53	1	no	—
356	C₂×#41	—	53	2	no	—
357	C₂²×Q₈	—	53	2	no	—
358	C₂²×D₄	—	53	1	no	—
359	C₂³×C₄	—	53	2	no	—
360	C₄⋊Q₈	—	53	1	no	—
361	C₄⋊Q₈	—	53	1	no	—
362	C₄⋊Q₈	—	53	1	no	—
363	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
364	#126	—	53	1	no	—
365	#126	—	53	2	no	—
366	#126	—	53	2	no	—
367	#126	—	53	2	no	—
368	#126	—	53	1	no	—
369	#126	—	53	2	no	—
370	#125	—	53	1	no	—
371	#125	—	53	2	no	—
372	#125	—	53	2	no	—
373	#125	C₂×D₅₃	1	2	yes	complemented
374	#124	—	53	1	no	—
375	#124	—	53	2	no	—
376	#124	—	53	1	no	—
377	#124	—	53	2	no	—
378	#124	—	53	1	no	—
379	#123	—	53	1	no	—
380	#123	—	53	2	no	—
381	#123	—	53	2	no	—
382	#123	—	53	2	no	—
383	#123	—	53	2	no	—
384	#123	—	53	2	no	—
385	#123	—	53	2	no	—
386	#123	—	53	2	no	—
387	#123	—	53	1	no	—
388	#123	—	53	2	no	—
389	C₂²⋊Q₈	—	53	2	no	—
390	C₂²⋊Q₈	—	53	2	no	—
391	C₂²⋊Q₈	—	53	2	no	—
392	C₂²⋊Q₈	—	53	1	no	—
393	C₂²⋊Q₈	—	53	2	no	—
394	C₂²⋊Q₈	—	53	2	no	—
395	C₂²⋊Q₈	—	53	2	no	—
396	C₂²⋊Q₈	—	53	2	no	—
397	C₂²⋊Q₈	—	53	2	no	—
398	C₂²⋊Q₈	—	53	2	no	—
399	C₂²⋊Q₈	—	53	1	no	—
400	C₂²⋊Q₈	—	53	1	no	—
401	C₂²⋊Q₈	—	53	2	no	—
402	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
403	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
404	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
405	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
406	#121	—	53	1	no	—
407	#121	—	53	1	no	—
408	#121	—	53	2	no	—
409	#121	—	53	1	no	—
410	#121	—	53	2	no	—
411	#121	—	53	2	no	—
412	#120	—	53	2	no	—
413	#120	—	53	2	no	—
414	C₄×Q₈	—	53	1	no	—
415	C₄×Q₈	—	53	1	no	—
416	C₄×Q₈	—	53	1	no	—
417	C₄×Q₈	—	53	2	no	—
418	C₄×Q₈	—	53	1	no	—
419	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
420	C₄×D₄	—	53	2	no	—
421	C₄×D₄	—	53	2	no	—
422	C₄×D₄	—	53	2	no	—
423	C₄×D₄	—	53	1	no	—
424	C₄×D₄	—	53	1	no	—
425	C₄×D₄	—	53	1	no	—
426	C₄×D₄	—	53	2	no	—
427	C₄×D₄	—	53	1	no	—
428	C₄×D₄	—	53	1	no	—
429	C₄×D₄	—	53	1	no	—
430	C₄×D₄	—	53	2	no	—
431	C₄×D₄	—	53	1	no	—
432	C₄×D₄	—	53	2	no	—
433	C₄×D₄	—	53	2	no	—
434	C₄×D₄	—	53	1	no	—
435	C₄×D₄	C₂×D₅₃	1	2	yes	—
436	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
437	#117	—	53	1	no	—
438	#117	—	53	2	no	—
439	#117	—	53	2	no	—
440	#117	—	53	1	no	—
441	#117	—	53	2	no	—
442	#117	—	53	2	no	—
443	C₂×C₄⋊C₄	—	53	2	no	—
444	C₂×C₄⋊C₄	—	53	2	no	—
445	C₂×C₄⋊C₄	—	53	1	no	—
446	C₂×C₄⋊C₄	—	53	2	no	—
447	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
448	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
449	C₂×C₂²⋊C₄	—	53	2	no	—
450	C₂×C₂²⋊C₄	—	53	2	no	—
451	C₂×C₂²⋊C₄	—	53	2	no	—
452	C₂×C₄²	—	53	1	no	—
453	C₂⁴	—	53	2	no	—
454	#41	—	106	4	no	—
455	#41	—	106	4	no	—
456	#41	—	212	4	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	4	no	—
462	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
463	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
464	C₂×D₄	—	53	1	no	—
465	C₂×D₄	—	53	2	no	—
466	C₂×D₄	—	53	1	no	—
467	C₂×D₄	—	53	2	no	—
468	C₂×D₄	—	53	1	no	—
469	C₂×D₄	—	53	2	no	—
470	C₂×D₄	—	53	2	no	—
471	C₂×D₄	—	53	1	no	—
472	C₂×D₄	—	53	2	no	—
473	C₂×D₄	—	106	4	no	—
474	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
475	C₂²×C₄	—	53	1	no	—
476	C₂²×C₄	—	53	1	no	—
477	C₂²×C₄	—	53	2	no	—
478	C₂²×C₄	—	53	2	no	—
479	C₂²×C₄	—	53	1	no	—
480	C₂²×C₄	—	53	2	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	—	53	2	no	—
487	C₂²×C₄	—	212	4	no	—
488	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
489	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
490	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
491	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
492	C₄⋊C₄	—	53	1	no	—
493	C₄⋊C₄	—	53	1	no	—
494	C₄⋊C₄	—	53	2	no	—
495	C₄⋊C₄	—	53	1	no	—
496	C₄⋊C₄	—	53	2	no	—
497	C₄⋊C₄	—	53	1	no	—
498	C₄⋊C₄	—	53	2	no	—
499	C₄⋊C₄	—	53	2	no	—
500	C₄⋊C₄	—	53	2	no	—
501	C₄⋊C₄	—	53	2	no	—
502	C₄⋊C₄	—	53	1	no	—
503	C₄⋊C₄	—	53	1	no	—
504	C₄⋊C₄	—	53	1	no	—
505	C₄⋊C₄	—	53	2	no	—
506	C₄⋊C₄	—	53	2	no	—
507	C₄⋊C₄	—	53	2	no	—
508	C₄⋊C₄	—	53	1	no	—
509	C₄⋊C₄	—	53	2	no	—
510	C₄⋊C₄	—	53	1	no	—
511	C₄⋊C₄	—	53	1	no	—
512	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
513	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
514	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
515	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
516	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
517	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
518	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
519	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
520	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
521	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
522	C₂²⋊C₄	—	53	1	no	—
523	C₂²⋊C₄	—	53	2	no	—
524	C₂²⋊C₄	—	53	1	no	—
525	C₂²⋊C₄	—	53	1	no	—
526	C₂²⋊C₄	—	53	1	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	2	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	—	53	1	no	—
544	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
545	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
546	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
547	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
548	C₄²	—	53	1	no	—
549	C₄²	—	53	1	no	—
550	C₄²	—	53	1	no	—
551	C₄²	—	53	1	no	—
552	C₄²	—	53	2	no	—
553	C₄²	—	53	2	no	—
554	C₄²	—	53	2	no	—
555	C₄²	—	53	1	no	—
556	C₄²	C₂²×D₅₃	1	2	yes	—
557	C₄²	C₂²×D₅₃	1	1	yes	characteristic
558	C₂³	—	53	2	no	—
559	C₂³	—	53	2	no	—
560	C₂³	—	53	1	no	—
561	C₂³	—	106	4	no	—
562	C₂³	—	106	4	no	—
563	C₂³	—	212	2	no	—
564	C₂³	C₂³×D₅₃	1	1	yes	characteristic
565	C₂³	C₂³×D₅₃	1	1	yes	characteristic
566	Q₈	—	2	2	no	—
567	Q₈	—	4	2	no	—
568	Q₈	—	106	2	no	—
569	Q₈	—	106	4	no	—
570	Q₈	—	106	2	no	—
571	Q₈	—	212	2	no	—
572	D₄	—	2	2	no	—
573	D₄	—	53	4	no	—
574	D₄	—	106	4	no	—
575	D₄	—	106	4	no	—
576	D₄	—	106	2	no	—
577	D₄	—	106	4	no	—
578	D₄	—	106	2	no	—
579	D₄	—	106	2	no	—
580	D₄	—	212	2	no	—
581	D₄	—	212	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	53	1	no	—
587	C₂×C₄	—	53	1	no	—
588	C₂×C₄	—	53	1	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	—	53	2	no	—
594	C₂×C₄	—	53	1	no	—
595	C₂×C₄	—	106	4	no	—
596	C₂×C₄	—	106	4	no	—
597	C₂×C₄	—	106	4	no	—
598	C₂×C₄	—	106	4	no	—
599	C₂×C₄	—	106	2	no	—
600	C₂×C₄	—	106	2	no	—
601	C₂×C₄	—	106	2	no	—
602	C₂×C₄	—	212	2	no	—
603	C₂×C₄	—	212	2	no	—
604	C₂×C₄	—	212	2	no	—
605	C₂×C₄	—	212	2	no	—
606	C₂×C₄	—	212	2	no	—
607	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
608	C₂×C₄	C₂³×D₅₃	1	2	yes	—
609	C₂×C₄	C₂³×D₅₃	1	2	yes	—
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	C₂×C₄	C₂³×D₅₃	1	2	yes	—
615	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
616	C₂²	—	2	1	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	1	no	—
619	C₂²	—	2	1	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	53	4	no	—
622	C₂²	—	106	2	no	—
623	C₂²	—	106	2	no	—
624	C₂²	—	106	1	no	—
625	C₂²	—	106	2	no	—
626	C₂²	—	106	2	no	—
627	C₂²	—	106	1	no	—
628	C₂²	—	106	1	no	—
629	C₂²	—	106	2	no	—
630	C₂²	—	212	8	no	—
631	C₂²	#13133265	1	2	yes	—
632	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	2	no	—
636	C₄	—	2	2	no	—
637	C₄	—	2	2	no	—
638	C₄	—	2	2	no	—
639	C₄	—	2	1	no	—
640	C₄	—	2	1	no	—
641	C₄	—	53	2	no	—
642	C₄	—	106	1	no	—
643	C₄	—	106	2	no	—
644	C₄	—	106	1	no	—
645	C₄	—	106	2	no	—
646	C₄	—	106	1	no	—
647	C₄	—	106	1	no	—
648	C₄	—	106	1	no	—
649	C₄	—	106	1	no	—
650	C₄	#41×D₅₃	1	2	yes	—
651	C₂	—	2	2	no	—
652	C₂	—	4	1	no	—
653	C₂	—	106	4	no	—
654	C₂	—	212	2	no	—
655	C₂	—	212	1	no	—
656	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
657	C₂	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
658	C₂	C₂×#13133265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#554	C₂
hypercenter	#554	D₅₃
hypocenter	C₅₃	#3520
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #554

Complements

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

N	H ≅ G/N	type
1. #16811345	C₂	semidirect
2. #16811328	C₂	semidirect
3. #16811305	C₂	semidirect
4. #16811302	C₂	semidirect
5. #16811300	C₂	semidirect
6. #16811288	C₂	semidirect
7. #16811283	C₂	semidirect
8. #16811275	C₂	semidirect
9. #16811274	C₂	semidirect
10. #16811269	C₂	semidirect
11. #16811264	C₂	semidirect
12. #16811247	C₂	semidirect
13. #16811246	C₂	semidirect
14. #16811221	C₂	semidirect
15. #16811219	C₂	semidirect
16. #16811175	C₂	semidirect
17. #16811153	C₂	semidirect
18. #16811132	C₂	semidirect
19. C₂²⋊Q₈×D₅₃	C₂	semidirect
20. #16811105	C₂	semidirect
21. #16811104	C₂	semidirect
22. C₂×#13133179	C₂	semidirect
23. C₅₃×#554	C₂	semidirect
26. C₂×#11719945	C₂²	semidirect
27. C₂×Q₈×D₅₃	C₂²	semidirect
30. #13133216	C₂²	semidirect
31. #13133216	C₂²	semidirect
32. #13133215	C₂²	semidirect
33. #13133212	C₂²	semidirect
34. #13133212	C₂²	semidirect
35. #13133212	C₂²	semidirect
36. #13133212	C₂²	semidirect
37. #13133211	C₂²	semidirect
38. #13133210	C₂²	semidirect
39. #13133207	C₂²	semidirect
40. #13133207	C₂²	semidirect
41. #13133207	C₂²	semidirect
43. #13133206	C₂²	semidirect
45. #13133203	C₂²	semidirect
46. #13133203	C₂²	semidirect
49. #13133202	C₂²	semidirect
50. #13133202	C₂²	semidirect
51. #13133202	C₂²	semidirect
53. #13133200	C₂²	semidirect
54. #13133199	C₂²	semidirect
55. #13133199	C₂²	semidirect
57. #13133197	C₂²	semidirect
58. #13133197	C₂²	semidirect
61. #13133195	C₂²	semidirect
62. #13133195	C₂²	semidirect
63. #13133195	C₂²	semidirect
64. #13133195	C₂²	semidirect
65. #13133195	C₂²	semidirect
68. #13133193	C₂²	semidirect
69. #13133193	C₂²	semidirect
74. #13133190	C₂²	semidirect
78. Q₈×Dic₅₃	C₂²	semidirect
79. C₄×Dic₁₀₆	C₂²	semidirect
80. #13133182	C₂²	semidirect
81. #13133182	C₂²	semidirect
82. #13133182	C₂²	semidirect
84. C₄×D₂₁₂	C₂²	semidirect
92. #13133179	C₂²	semidirect
95. C₄×C₂²⋊D₅₃	C₂²	semidirect
96. C₄×C₂²⋊D₅₃	C₂²	semidirect
97. C₄×C₂²⋊D₅₃	C₂²	semidirect
98. #13133176	C₂²	semidirect
99. #13133175	C₂²	semidirect
100. #13133174	C₂²	semidirect
101. #13133174	C₂²	semidirect
102. #13133174	C₂²	semidirect
104. C₄⋊C₄×D₅₃	C₂²	semidirect
105. C₄⋊C₄×D₅₃	C₂²	semidirect
106. C₄⋊C₄×D₅₃	C₂²	semidirect
107. C₄⋊C₄×D₅₃	C₂²	semidirect
112. C₄⋊Q₈×C₅₃	C₂²	semidirect
113. #125×C₅₃	C₂²	semidirect
116. C₂²⋊Q₈×C₅₃	C₂²	semidirect
117. C₂²⋊Q₈×C₅₃	C₂²	semidirect
118. C₄×Q₈×C₅₃	C₂²	semidirect
122. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
271. D₂₁₂	#41	semidirect
332. #554	D₅₃	semidirect
353. C₅₃	#3520	semidirect
363. C₄⋊Q₈	C₂×D₅₃	semidirect
373. #125	C₂×D₅₃	semidirect
404. C₂²⋊Q₈	C₂×D₅₃	semidirect
405. C₂²⋊Q₈	C₂×D₅₃	semidirect
419. C₄×Q₈	C₂×D₅₃	semidirect
448. C₂×C₄⋊C₄	C₂×D₅₃	semidirect