Subgroups — #20288563

Order 6784.

1140 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811358	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811356	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811355	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811345	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811342	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811335	C₂	1	2	yes	maximal, complemented
7	#16811322	C₂	1	2	yes	maximal, complemented
8	#16811318	C₂	1	2	yes	maximal, complemented
9	#16811294	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811293	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811271	C₂	1	2	yes	maximal, complemented
12	#16811209	C₂	1	2	yes	maximal, complemented
13	#16811207	C₂	1	2	yes	maximal, complemented
14	#16811191	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811186	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811177	C₂	1	1	yes	maximal, characteristic, complemented
17	#16811172	C₂	1	1	yes	maximal, characteristic, complemented
18	#16811132	C₂	1	2	yes	maximal, complemented
19	#16811131	C₂	1	2	yes	maximal, complemented
20	#16811105	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811102	C₂	1	1	yes	maximal, characteristic, complemented
22	C₄×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	1	yes	characteristic
25	C₂×#11719945	C₂²	1	2	yes	complemented
26	C₂×#11719947	C₂²	1	1	yes	characteristic
27	C₂×#11719947	C₂²	1	2	yes	complemented
28	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
30	#13133216	C₂²	1	1	yes	characteristic, complemented
31	#13133215	C₂²	1	1	yes	characteristic, complemented
32	#13133215	C₂²	1	1	yes	characteristic, complemented
33	#13133214	C₂²	1	1	yes	characteristic
34	#13133213	C₂²	1	1	yes	characteristic
35	#13133213	C₂²	1	1	yes	characteristic
36	#13133213	C₂²	1	1	yes	characteristic, complemented
37	#13133212	C₂²	1	2	yes	complemented
38	#13133211	C₂²	1	2	yes	complemented
39	#13133210	C₂²	1	2	yes	complemented
40	#13133210	C₂²	1	2	yes	complemented
41	#13133207	C₂²	1	1	yes	characteristic, complemented
42	#13133207	C₂²	1	1	yes	characteristic, complemented
43	#13133206	C₂²	1	1	yes	characteristic
44	#13133206	C₂²	1	2	yes	complemented
45	#13133205	C₂²	1	1	yes	characteristic
46	#13133205	C₂²	1	1	yes	characteristic, complemented
47	#13133204	C₂²	1	1	yes	characteristic
48	#13133203	C₂²	1	2	yes	complemented
49	#13133203	C₂²	1	2	yes	complemented
50	#13133202	C₂²	1	2	yes	complemented
51	#13133202	C₂²	1	2	yes	complemented
52	#13133201	C₂²	1	2	yes	complemented
53	#13133200	C₂²	1	2	yes	complemented
54	#13133199	C₂²	1	2	yes	complemented
55	#13133199	C₂²	1	2	yes	complemented
56	#13133197	C₂²	1	1	yes	characteristic
57	#13133195	C₂²	1	1	yes	characteristic
58	#13133195	C₂²	1	1	yes	characteristic
59	#13133195	C₂²	1	2	yes	complemented
60	#13133194	C₂²	1	2	yes	complemented
61	#13133193	C₂²	1	1	yes	characteristic
62	#13133191	C₂²	1	2	yes	complemented
63	#13133190	C₂²	1	2	yes	—
64	#13133190	C₂²	1	2	yes	—
65	#13133190	C₂²	1	2	yes	—
66	#13133190	C₂²	1	2	yes	—
67	#13133190	C₂²	1	2	yes	complemented
68	#13133190	C₂²	1	2	yes	complemented
69	#13133189	C₂²	1	2	yes	—
70	#13133189	C₂²	1	2	yes	—
71	#13133189	C₂²	1	2	yes	complemented
72	#13133189	C₂²	1	2	yes	complemented
73	#13133188	C₂²	1	2	yes	complemented
74	#13133187	C₂²	1	2	yes	—
75	#13133187	C₂²	1	2	yes	—
76	#13133185	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	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
81	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
82	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
83	#13133182	C₂²	1	1	yes	characteristic, complemented
84	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
85	C₄×D₂₁₂	C₂²	1	2	yes	—
86	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
87	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
88	D₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
89	#13133179	C₂²	1	1	yes	characteristic
90	#13133179	C₂²	1	2	yes	—
91	#13133179	C₂²	1	1	yes	characteristic, complemented
92	#13133179	C₂²	1	2	yes	complemented
93	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
94	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
95	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
96	#13133177	C₂²	1	2	yes	—
97	#13133177	C₂²	1	2	yes	—
98	#13133177	C₂²	1	2	yes	complemented
99	#13133177	C₂²	1	2	yes	complemented
100	#13133177	C₂²	1	2	yes	complemented
101	#13133174	C₂²	1	1	yes	characteristic
102	#13133174	C₂²	1	1	yes	characteristic
103	#13133174	C₂²	1	2	yes	complemented
104	#13133174	C₂²	1	2	yes	complemented
105	#13133173	C₂²	1	2	yes	complemented
106	#13133173	C₂²	1	2	yes	complemented
107	C₂×#11719919	C₂²	1	2	yes	complemented
108	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
109	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
110	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
111	C₄²×D₅₃	C₂²	1	1	yes	characteristic
112	#127×C₅₃	C₂²	1	1	yes	characteristic
113	#124×C₅₃	C₂²	1	1	yes	characteristic
114	#124×C₅₃	C₂²	1	2	yes	complemented
115	#124×C₅₃	C₂²	1	1	yes	characteristic, complemented
116	#123×C₅₃	C₂²	1	2	yes	complemented
117	#123×C₅₃	C₂²	1	2	yes	complemented
118	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
120	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
121	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
122	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
123	C₂³×D₅₃	C₂³	1	2	yes	—
124	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×D₄×C₅₃	C₂³	1	2	yes	—
129	C₂×D₄×C₅₃	C₂³	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
131	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
132	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
133	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
134	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
137	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
138	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
139	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
140	C₂²×Dic₅₃	C₂³	1	2	yes	—
141	C₂²×Dic₅₃	C₂³	1	2	yes	—
142	C₂²×Dic₅₃	C₂³	1	2	yes	—
143	#11719949	—	2	4	no	—
144	#11719947	—	2	4	no	—
145	#11719947	—	4	4	no	—
146	D₄×D₅₃	—	2	4	no	—
147	#11719945	—	4	4	no	—
148	C₂×D₂₁₂	C₂³	1	2	yes	—
149	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
150	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
151	C₂×D₂₁₂	C₂³	1	2	yes	—
152	C₂×D₂₁₂	C₂³	1	2	yes	—
153	C₂×C₄×D₅₃	—	4	4	no	—
154	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
155	C₂×C₄×D₅₃	C₂³	1	2	yes	—
156	C₂×C₄×D₅₃	C₂³	1	2	yes	—
157	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
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	2	yes	—
162	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
163	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
164	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
165	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
166	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
167	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
168	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
169	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
170	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
171	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
172	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
173	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
174	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
175	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
176	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
177	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
178	C₄²×C₅₃	C₂³	1	1	yes	characteristic
179	C₄²×C₅₃	C₂³	1	1	yes	characteristic
180	C₄²×C₅₃	C₂³	1	2	yes	—
181	C₄²×C₅₃	C₂³	1	1	yes	characteristic
182	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
183	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
184	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
185	C₂²⋊Dic₅₃	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	#11719921	C₂³	1	1	yes	characteristic
192	#11719921	C₂³	1	2	yes	—
193	#11719921	C₂³	1	2	yes	—
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	2	yes	—
199	#11719921	C₂³	1	2	yes	—
200	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
201	C₄⋊Dic₅₃	C₂³	1	2	yes	—
202	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
203	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
204	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
205	#11719919	C₂³	1	2	yes	—
206	#11719919	C₂³	1	1	yes	characteristic
207	#11719919	C₂³	1	1	yes	characteristic
208	#11719919	C₂³	1	2	yes	—
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	1	yes	characteristic
216	#11719919	C₂³	1	2	yes	—
217	#11719919	C₂³	1	1	yes	characteristic
218	#11719919	C₂³	1	1	yes	characteristic
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	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	1	yes	characteristic
228	C₂³×C₅₃	C₂⁴	1	2	yes	—
229	C₂³×C₅₃	C₂⁴	1	2	yes	—
230	C₂²×D₅₃	—	2	4	no	—
231	C₂²×D₅₃	—	4	2	no	—
232	C₂²×D₅₃	—	4	2	no	—
233	C₂²×D₅₃	C₂⁴	1	2	yes	—
234	C₂²×D₅₃	C₂⁴	1	2	yes	—
235	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
236	Q₈×C₅₃	—	2	2	no	—
237	Q₈×C₅₃	—	4	1	no	—
238	D₄×C₅₃	—	2	2	no	—
239	D₄×C₅₃	—	2	2	no	—
240	D₄×C₅₃	—	4	2	no	—
241	D₄×C₅₃	—	4	2	no	—
242	C₂×C₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	—	4	2	no	—
244	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
249	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
250	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
253	C₂²⋊D₅₃	—	2	4	no	—
254	C₂²⋊D₅₃	—	2	4	no	—
255	C₂²⋊D₅₃	—	2	4	no	—
256	C₂²⋊D₅₃	—	4	2	no	—
257	C₂²⋊D₅₃	—	4	2	no	—
258	C₂²⋊D₅₃	—	4	2	no	—
259	C₂²⋊D₅₃	—	4	2	no	—
260	C₂²⋊D₅₃	—	4	2	no	—
261	C₂×Dic₅₃	—	2	4	no	—
262	C₂×Dic₅₃	—	2	4	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	2	yes	—
267	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
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	1	yes	characteristic
272	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
273	D₂₁₂	—	2	4	no	—
274	D₂₁₂	—	2	2	no	—
275	D₂₁₂	—	2	2	no	—
276	D₂₁₂	—	4	2	no	—
277	D₂₁₂	—	4	2	no	—
278	C₄×D₅₃	—	2	4	no	—
279	C₄×D₅₃	—	2	2	no	—
280	C₄×D₅₃	—	2	4	no	—
281	C₄×D₅₃	—	4	2	no	—
282	C₄×D₅₃	—	4	2	no	—
283	C₄×D₅₃	—	4	1	no	—
284	C₄×D₅₃	—	4	2	no	—
285	C₄×D₅₃	—	4	2	no	—
286	C₄×D₅₃	#41	1	4	yes	complemented
287	Dic₁₀₆	—	2	2	no	—
288	Dic₁₀₆	—	2	2	no	—
289	Dic₁₀₆	—	2	2	no	—
290	Dic₁₀₆	—	4	1	no	—
291	C₂²×C₅₃	—	2	2	no	—
292	C₂²×C₅₃	—	2	2	no	—
293	C₂²×C₅₃	—	2	2	no	—
294	C₂²×C₅₃	—	2	2	no	—
295	C₂²×C₅₃	—	2	2	no	—
296	C₂²×C₅₃	—	2	2	no	—
297	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
298	C₂×D₅₃	—	2	1	no	—
299	C₂×D₅₃	—	2	1	no	—
300	C₂×D₅₃	—	2	2	no	—
301	C₂×D₅₃	—	2	2	no	—
302	C₂×D₅₃	—	2	2	no	—
303	C₂×D₅₃	—	2	2	no	—
304	C₂×D₅₃	—	2	2	no	—
305	C₂×D₅₃	—	2	2	no	—
306	C₂×D₅₃	—	4	8	no	—
307	C₂×D₅₃	C₂×#41	1	2	yes	—
308	C₄×C₅₃	—	2	2	no	—
309	C₄×C₅₃	—	2	1	no	—
310	C₄×C₅₃	—	2	1	no	—
311	C₄×C₅₃	—	2	1	no	—
312	C₄×C₅₃	—	2	2	no	—
313	C₄×C₅₃	—	2	1	no	—
314	C₄×C₅₃	—	2	1	no	—
315	C₄×C₅₃	—	2	1	no	—
316	C₄×C₅₃	C₂×#41	1	2	yes	—
317	Dic₅₃	—	2	2	no	—
318	Dic₅₃	—	2	1	no	—
319	Dic₅₃	—	2	1	no	—
320	Dic₅₃	—	2	2	no	—
321	Dic₅₃	—	2	1	no	—
322	Dic₅₃	—	2	1	no	—
323	Dic₅₃	—	2	1	no	—
324	Dic₅₃	C₂×#41	1	2	yes	—
325	Dic₅₃	H(5,𝔽₂)	1	2	yes	—
326	#3506	—	53	1	no	maximal, Sylow
327	C₂×C₅₃	—	4	2	no	—
328	C₂×C₅₃	—	4	2	no	—
329	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
330	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
331	C₂×C₅₃	#585	1	1	yes	characteristic
332	D₅₃	—	2	2	no	—
333	D₅₃	—	4	2	no	—
334	D₅₃	—	4	2	no	—
335	#559	—	53	1	no	—
336	#559	—	53	1	no	—
337	#559	—	53	1	no	—
338	#559	D₅₃	1	1	yes	characteristic, complemented
339	#556	—	53	1	no	—
340	#556	—	53	1	no	—
341	#555	—	53	2	no	—
342	#553	—	53	2	no	—
343	#553	—	53	2	no	—
344	#550	—	53	1	no	—
345	#550	—	53	1	no	—
346	#547	—	53	2	no	—
347	#538	—	53	2	no	—
348	#538	—	53	2	no	—
349	#535	—	53	1	no	—
350	#535	—	53	1	no	—
351	#533	—	53	1	no	—
352	#532	—	53	1	no	—
353	#525	—	53	2	no	—
354	#525	—	53	2	no	—
355	#520	—	53	1	no	—
356	#520	—	53	1	no	—
357	C₂×C₄×D₄	—	53	1	no	—
358	C₅₃	#3506	1	1	yes	characteristic, complemented, Sylow
359	C₂×#41	—	53	1	no	—
360	C₂×#41	—	53	1	no	—
361	C₂×#41	—	53	2	no	—
362	C₂×#41	—	53	2	no	—
363	C₂²×D₄	—	53	1	no	—
364	C₂³×C₄	—	53	2	no	—
365	C₄⋊Q₈	—	53	1	no	—
366	C₄⋊Q₈	—	53	1	no	—
367	C₄⋊Q₈	—	53	1	no	—
368	#127	—	53	1	no	—
369	#127	—	53	1	no	—
370	#127	—	53	1	no	—
371	#127	—	53	1	no	—
372	#127	C₂×D₅₃	1	1	yes	characteristic
373	#126	—	53	2	no	—
374	#126	—	53	2	no	—
375	#126	—	53	2	no	—
376	#126	—	53	2	no	—
377	#125	—	53	1	no	—
378	#125	—	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	2	no	—
384	#124	—	53	1	no	—
385	#124	—	53	1	no	—
386	#124	C₂×D₅₃	1	1	yes	characteristic
387	#124	C₂×D₅₃	1	1	yes	characteristic, complemented
388	#124	C₂×D₅₃	1	2	yes	complemented
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	C₂×D₅₃	1	2	yes	complemented
396	#123	C₂×D₅₃	1	2	yes	complemented
397	C₂²⋊Q₈	—	53	1	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	2	no	—
403	#121	—	53	2	no	—
404	#121	—	53	2	no	—
405	#121	—	53	2	no	—
406	#121	—	53	2	no	—
407	#121	—	53	2	no	—
408	#121	—	53	2	no	—
409	#121	—	53	2	no	—
410	#121	—	53	2	no	—
411	#121	—	53	2	no	—
412	#121	—	53	2	no	—
413	#121	—	53	2	no	—
414	#121	—	53	2	no	—
415	#120	—	53	2	no	—
416	#120	—	53	2	no	—
417	#120	—	53	2	no	—
418	#120	—	53	2	no	—
419	C₄×Q₈	—	53	1	no	—
420	C₄×Q₈	—	53	1	no	—
421	C₄×Q₈	—	53	1	no	—
422	C₄×Q₈	—	53	1	no	—
423	C₄×Q₈	—	53	1	no	—
424	C₄×Q₈	—	53	1	no	—
425	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
426	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
427	C₄×D₄	—	53	1	no	—
428	C₄×D₄	—	53	1	no	—
429	C₄×D₄	—	53	2	no	—
430	C₄×D₄	—	53	2	no	—
431	C₄×D₄	—	53	2	no	—
432	C₄×D₄	—	53	1	no	—
433	C₄×D₄	—	53	2	no	—
434	C₄×D₄	—	53	1	no	—
435	C₄×D₄	—	53	2	no	—
436	C₄×D₄	—	53	2	no	—
437	C₄×D₄	—	53	2	no	—
438	C₄×D₄	—	53	2	no	—
439	C₄×D₄	—	53	2	no	—
440	C₄×D₄	—	53	2	no	—
441	C₄×D₄	—	53	1	no	—
442	C₄×D₄	—	53	2	no	—
443	C₄×D₄	—	53	1	no	—
444	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
445	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
446	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
447	#117	—	53	2	no	—
448	#117	—	53	2	no	—
449	#117	—	53	1	no	—
450	#117	—	53	1	no	—
451	#117	—	53	2	no	—
452	#117	—	53	2	no	—
453	C₂×C₄⋊C₄	—	53	2	no	—
454	C₂×C₄⋊C₄	—	53	1	no	—
455	C₂×C₄⋊C₄	—	53	2	no	—
456	C₂×C₂²⋊C₄	—	53	2	no	—
457	C₂×C₄²	—	53	1	no	—
458	C₂⁴	—	53	2	no	—
459	#41	—	106	4	no	—
460	#41	—	106	4	no	—
461	#41	—	212	4	no	—
462	#41	—	212	4	no	—
463	C₂×Q₈	—	53	1	no	—
464	C₂×Q₈	—	53	1	no	—
465	C₂×Q₈	—	53	1	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×D₄	—	53	2	no	—
470	C₂×D₄	—	53	2	no	—
471	C₂×D₄	—	53	2	no	—
472	C₂×D₄	—	53	2	no	—
473	C₂×D₄	—	53	2	no	—
474	C₂×D₄	—	53	2	no	—
475	C₂×D₄	—	53	2	no	—
476	C₂×D₄	—	53	2	no	—
477	C₂×D₄	—	53	2	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	—	106	4	no	—
483	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
484	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
485	C₂×D₄	C₂²×D₅₃	1	2	yes	—
486	C₂×D₄	C₂²×D₅₃	1	2	yes	—
487	C₂²×C₄	—	53	1	no	—
488	C₂²×C₄	—	53	2	no	—
489	C₂²×C₄	—	53	2	no	—
490	C₂²×C₄	—	53	2	no	—
491	C₂²×C₄	—	53	1	no	—
492	C₂²×C₄	—	53	2	no	—
493	C₂²×C₄	—	53	2	no	—
494	C₂²×C₄	—	53	1	no	—
495	C₂²×C₄	—	53	2	no	—
496	C₂²×C₄	—	53	2	no	—
497	C₂²×C₄	—	53	2	no	—
498	C₂²×C₄	—	53	2	no	—
499	C₂²×C₄	—	212	4	no	—
500	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
501	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
502	C₄⋊C₄	—	53	2	no	—
503	C₄⋊C₄	—	53	1	no	—
504	C₄⋊C₄	—	53	1	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	1	no	—
510	C₄⋊C₄	—	53	1	no	—
511	C₄⋊C₄	—	53	2	no	—
512	C₄⋊C₄	—	53	1	no	—
513	C₄⋊C₄	—	53	1	no	—
514	C₄⋊C₄	—	53	1	no	—
515	C₄⋊C₄	—	53	1	no	—
516	C₄⋊C₄	—	53	2	no	—
517	C₄⋊C₄	—	53	1	no	—
518	C₄⋊C₄	—	53	1	no	—
519	C₄⋊C₄	—	53	1	no	—
520	C₄⋊C₄	—	53	1	no	—
521	C₄⋊C₄	—	53	1	no	—
522	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
523	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
524	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
525	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
526	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
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	1	no	—
534	C₂²⋊C₄	—	53	1	no	—
535	C₂²⋊C₄	—	53	1	no	—
536	C₂²⋊C₄	—	53	1	no	—
537	C₂²⋊C₄	—	53	2	no	—
538	C₂²⋊C₄	—	53	2	no	—
539	C₂²⋊C₄	—	53	2	no	—
540	C₂²⋊C₄	—	53	2	no	—
541	C₂²⋊C₄	—	53	2	no	—
542	C₂²⋊C₄	—	53	2	no	—
543	C₂²⋊C₄	—	53	2	no	—
544	C₂²⋊C₄	—	53	2	no	—
545	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
546	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
547	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
548	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
549	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
550	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
551	C₄²	—	53	1	no	—
552	C₄²	—	53	1	no	—
553	C₄²	—	53	1	no	—
554	C₄²	—	53	2	no	—
555	C₄²	—	53	2	no	—
556	C₄²	—	53	2	no	—
557	C₄²	—	53	1	no	—
558	C₄²	—	53	1	no	—
559	C₄²	C₂²×D₅₃	1	1	yes	characteristic
560	C₄²	C₂²×D₅₃	1	2	yes	—
561	C₄²	C₂²×D₅₃	1	1	yes	characteristic
562	C₄²	C₂²×D₅₃	1	1	yes	characteristic
563	C₂³	—	53	1	no	—
564	C₂³	—	53	2	no	—
565	C₂³	—	53	2	no	—
566	C₂³	—	106	4	no	—
567	C₂³	—	212	2	no	—
568	C₂³	—	212	2	no	—
569	C₂³	C₂³×D₅₃	1	2	yes	—
570	C₂³	C₂³×D₅₃	1	2	yes	—
571	Q₈	—	2	2	no	—
572	Q₈	—	4	1	no	—
573	Q₈	—	106	2	no	—
574	Q₈	—	106	2	no	—
575	Q₈	—	106	2	no	—
576	Q₈	—	212	1	no	—
577	D₄	—	2	2	no	—
578	D₄	—	2	2	no	—
579	D₄	—	4	2	no	—
580	D₄	—	4	2	no	—
581	D₄	—	106	4	no	—
582	D₄	—	106	4	no	—
583	D₄	—	106	4	no	—
584	D₄	—	106	4	no	—
585	D₄	—	106	2	no	—
586	D₄	—	106	2	no	—
587	D₄	—	212	2	no	—
588	D₄	—	212	2	no	—
589	D₄	—	212	2	no	—
590	D₄	—	212	2	no	—
591	D₄	—	212	2	no	—
592	D₄	—	212	2	no	—
593	D₄	—	212	2	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	53	1	no	—
597	C₂×C₄	—	53	1	no	—
598	C₂×C₄	—	53	4	no	—
599	C₂×C₄	—	53	1	no	—
600	C₂×C₄	—	53	1	no	—
601	C₂×C₄	—	53	2	no	—
602	C₂×C₄	—	53	1	no	—
603	C₂×C₄	—	53	2	no	—
604	C₂×C₄	—	53	1	no	—
605	C₂×C₄	—	53	1	no	—
606	C₂×C₄	—	106	2	no	—
607	C₂×C₄	—	106	4	no	—
608	C₂×C₄	—	106	4	no	—
609	C₂×C₄	—	106	4	no	—
610	C₂×C₄	—	106	4	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	2	no	—
616	C₂×C₄	—	212	2	no	—
617	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
618	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
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	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
626	C₂²	—	2	2	no	—
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	53	2	no	—
633	C₂²	—	106	1	no	—
634	C₂²	—	106	1	no	—
635	C₂²	—	106	2	no	—
636	C₂²	—	106	2	no	—
637	C₂²	—	106	2	no	—
638	C₂²	—	106	2	no	—
639	C₂²	—	106	2	no	—
640	C₂²	—	106	2	no	—
641	C₂²	—	212	8	no	—
642	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
643	C₄	—	2	2	no	—
644	C₄	—	2	1	no	—
645	C₄	—	2	2	no	—
646	C₄	—	2	1	no	—
647	C₄	—	2	1	no	—
648	C₄	—	2	1	no	—
649	C₄	—	2	1	no	—
650	C₄	—	2	1	no	—
651	C₄	—	53	2	no	—
652	C₄	—	53	2	no	—
653	C₄	—	106	2	no	—
654	C₄	—	106	2	no	—
655	C₄	—	106	1	no	—
656	C₄	—	106	1	no	—
657	C₄	—	106	1	no	—
658	C₄	—	106	1	no	—
659	C₄	—	106	1	no	—
660	C₄	#41×D₅₃	1	2	yes	—
661	C₂	—	4	2	no	—
662	C₂	—	4	2	no	—
663	C₂	—	106	2	no	—
664	C₂	—	212	2	no	—
665	C₂	—	212	2	no	—
666	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
667	C₂	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
668	C₂	#16811459	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#559	C₂
hypercenter	#559	D₅₃
hypocenter	C₅₃	#3506
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #16811358	C₂	semidirect
2. #16811356	C₂	semidirect
3. #16811355	C₂	semidirect
4. #16811345	C₂	semidirect
5. #16811342	C₂	semidirect
6. #16811335	C₂	semidirect
7. #16811322	C₂	semidirect
8. #16811318	C₂	semidirect
9. #16811294	C₂	semidirect
10. #16811293	C₂	semidirect
11. #16811271	C₂	semidirect
12. #16811209	C₂	semidirect
13. #16811207	C₂	semidirect
14. #16811191	C₂	semidirect
15. #16811186	C₂	semidirect
16. #16811177	C₂	semidirect
17. #16811172	C₂	semidirect
18. #16811132	C₂	semidirect
19. #16811131	C₂	semidirect
20. #16811105	C₂	semidirect
21. #16811102	C₂	semidirect
22. C₄×D₄×D₅₃	C₂	semidirect
23. C₅₃×#559	C₂	semidirect
25. C₂×#11719945	C₂²	semidirect
27. C₂×#11719947	C₂²	semidirect
30. #13133216	C₂²	semidirect
31. #13133215	C₂²	semidirect
32. #13133215	C₂²	semidirect
36. #13133213	C₂²	semidirect
37. #13133212	C₂²	semidirect
38. #13133211	C₂²	semidirect
39. #13133210	C₂²	semidirect
40. #13133210	C₂²	semidirect
41. #13133207	C₂²	semidirect
42. #13133207	C₂²	semidirect
44. #13133206	C₂²	semidirect
46. #13133205	C₂²	semidirect
48. #13133203	C₂²	semidirect
49. #13133203	C₂²	semidirect
50. #13133202	C₂²	semidirect
51. #13133202	C₂²	semidirect
52. #13133201	C₂²	semidirect
53. #13133200	C₂²	semidirect
54. #13133199	C₂²	semidirect
55. #13133199	C₂²	semidirect
59. #13133195	C₂²	semidirect
60. #13133194	C₂²	semidirect
62. #13133191	C₂²	semidirect
67. #13133190	C₂²	semidirect
68. #13133190	C₂²	semidirect
71. #13133189	C₂²	semidirect
72. #13133189	C₂²	semidirect
73. #13133188	C₂²	semidirect
78. Q₈×Dic₅₃	C₂²	semidirect
79. C₄×Dic₁₀₆	C₂²	semidirect
80. C₄×Dic₁₀₆	C₂²	semidirect
81. C₄×Dic₁₀₆	C₂²	semidirect
82. C₄×Dic₁₀₆	C₂²	semidirect
83. #13133182	C₂²	semidirect
86. C₄×D₂₁₂	C₂²	semidirect
88. D₄×Dic₅₃	C₂²	semidirect
91. #13133179	C₂²	semidirect
92. #13133179	C₂²	semidirect
95. C₄×C₂²⋊D₅₃	C₂²	semidirect
98. #13133177	C₂²	semidirect
99. #13133177	C₂²	semidirect
100. #13133177	C₂²	semidirect
103. #13133174	C₂²	semidirect
104. #13133174	C₂²	semidirect
105. #13133173	C₂²	semidirect
106. #13133173	C₂²	semidirect
107. C₂×#11719919	C₂²	semidirect
109. C₄⋊C₄×D₅₃	C₂²	semidirect
114. #124×C₅₃	C₂²	semidirect
115. #124×C₅₃	C₂²	semidirect
116. #123×C₅₃	C₂²	semidirect
117. #123×C₅₃	C₂²	semidirect
118. C₄×Q₈×C₅₃	C₂²	semidirect
119. C₄×Q₈×C₅₃	C₂²	semidirect
121. C₄×D₄×C₅₃	C₂²	semidirect
122. C₄×D₄×C₅₃	C₂²	semidirect
286. C₄×D₅₃	#41	semidirect
338. #559	D₅₃	semidirect
358. C₅₃	#3506	semidirect
387. #124	C₂×D₅₃	semidirect
388. #124	C₂×D₅₃	semidirect
395. #123	C₂×D₅₃	semidirect
396. #123	C₂×D₅₃	semidirect
425. C₄×Q₈	C₂×D₅₃	semidirect
426. C₄×Q₈	C₂×D₅₃	semidirect
445. C₄×D₄	C₂×D₅₃	semidirect
446. C₄×D₄	C₂×D₅₃	semidirect