Subgroups — #27353954

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283828	C₂	1	2	yes	maximal, complemented
2	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
3	#20273630	C₂	1	2	yes	maximal, complemented
4	#20273625	C₂	1	2	yes	maximal, complemented
5	#20273607	C₂	1	4	yes	maximal, complemented
6	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273103	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₄	1	1	yes	characteristic
10	C₄×#11719945	C₂²	1	2	yes	—
11	#16810753	C₂²	1	4	yes	—
12	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
13	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
14	C₄×#11719913	C₄	1	2	yes	—
15	C₄×#11719913	C₂²	1	2	yes	—
16	C₄×D₄₂₄	C₄	1	1	yes	characteristic
17	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
18	#16810702	C₂²	1	4	yes	—
19	#16810691	—	2	4	no	—
20	#16810679	—	2	2	no	—
21	#16810661	C₄	1	1	yes	characteristic
22	#16810645	—	2	2	no	—
23	C₂×C₄⋊#88322	C₂²	1	2	yes	—
24	#16810620	C₄	1	2	yes	—
25	#16810374	C₂²	1	2	yes	—
26	#16810374	C₂²	1	2	yes	—
27	#16810355	C₂²	1	4	yes	—
28	#16810352	C₂²	1	4	yes	—
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₅₃×#431	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₈²×C₅₃	C₂²	1	2	yes	—
34	C₂×#11719945	C₂×C₄	1	2	yes	—
35	#13133239	—	2	4	no	—
36	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719913	C₂×C₄	1	2	yes	—
38	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719912	—	2	4	no	—
40	C₂²×#88322	—	2	2	no	—
41	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₀₆	C₂³	1	2	yes	—
43	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
44	C₄×D₂₁₂	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₅₃	—	2	2	no	—
46	C₄×C₂²⋊D₅₃	—	2	2	no	—
47	#13133176	C₂×C₄	1	2	yes	—
48	#13133174	—	2	2	no	—
49	C₄²×D₅₃	—	2	2	no	—
50	#13133151	C₂×C₄	1	1	yes	characteristic
51	#13133151	C₂×C₄	1	1	yes	characteristic
52	#13133149	C₂×C₄	1	2	yes	—
53	C₄⋊#88322	—	2	2	no	—
54	C₄⋊#88322	C₂³	1	4	yes	—
55	#13133146	—	2	4	no	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133142	C₂×C₄	1	2	yes	—
58	#13133139	C₂×C₄	1	2	yes	—
59	#13133139	C₂×C₄	1	2	yes	—
60	#13133129	—	2	4	no	—
61	#13133127	—	2	4	no	—
62	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₅₃	—	2	1	no	—
66	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
67	C₂²⋊C₈×C₅₃	—	2	1	no	—
68	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
69	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂³	1	2	yes	—
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₃	—	2	2	no	—
79	#11719945	—	2	4	no	—
80	#11719945	—	2	4	no	—
81	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₅₃	—	2	2	no	—
83	C₂×C₄×D₅₃	—	2	2	no	—
84	C₂×C₄×D₅₃	—	2	2	no	—
85	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
94	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	#41	1	1	yes	characteristic
97	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₅₃	—	2	2	no	—
99	#11719921	—	2	2	no	—
100	#11719921	—	2	2	no	—
101	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
103	#11719919	—	2	2	no	—
104	#11719919	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₂×#88322	—	2	4	no	—
109	C₂×#88322	—	8	2	no	—
110	Dic₂₁₂	—	2	2	no	—
111	D₄₂₄	—	2	2	no	—
112	#11719913	—	2	4	no	—
113	#11719912	—	8	4	no	—
114	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂²×D₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	4	1	no	—
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₅₃	#131	1	2	yes	—
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
126	C₂²⋊D₅₃	—	4	4	no	—
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	2	2	no	—
130	D₂₁₂	—	2	2	no	—
131	D₂₁₂	—	2	2	no	—
132	C₄×D₅₃	—	4	2	no	—
133	C₄×D₅₃	—	4	2	no	—
134	C₄×D₅₃	—	4	4	no	—
135	C₄×D₅₃	—	8	2	no	—
136	Dic₁₀₆	—	2	2	no	—
137	Dic₁₀₆	—	2	2	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	4	no	—
141	C₈×C₅₃	—	4	2	no	—
142	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
143	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
144	#88322	—	4	4	no	—
145	#17899	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	#436	1	2	yes	—
149	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×D₅₃	—	4	2	no	—
151	C₂×D₅₃	—	4	2	no	—
152	C₂×D₅₃	—	4	2	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	#436	1	2	yes	—
157	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
159	C₄×C₅₃	#577	1	2	yes	—
160	Dic₅₃	—	4	2	no	—
161	Dic₅₃	—	4	2	no	—
162	Dic₅₃	—	4	2	no	—
163	#2989	—	53	2	no	—
164	C₄×#135	—	53	1	no	—
165	#1594	—	53	2	no	—
166	#1593	—	53	2	no	—
167	#1590	—	53	4	no	—
168	C₂×#334	—	53	1	no	—
169	#1477	—	53	2	no	—
170	#1450	D₅₃	1	1	yes	characteristic, complemented
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
173	C₂×C₅₃	#2952	1	1	yes	characteristic
174	C₂×C₅₃	#2957	1	1	yes	characteristic
175	D₅₃	—	8	2	no	—
176	C₂×#135	—	53	1	no	—
177	C₄×#41	—	53	2	no	—
178	#446	—	53	4	no	—
179	C₄×Q₁₆	—	53	1	no	—
180	C₄×Q₁₆	—	53	1	no	—
181	C₄×SD₁₆	—	53	2	no	—
182	C₄×SD₁₆	—	53	2	no	—
183	C₄×D₈	—	53	1	no	—
184	C₄×D₈	—	53	1	no	—
185	#436	—	53	4	no	—
186	#435	—	106	4	no	—
187	#432	—	106	2	no	—
188	#431	C₂×D₅₃	1	1	yes	characteristic
189	#431	C₂×D₅₃	1	1	yes	characteristic
190	#427	—	53	1	no	—
191	#423	—	106	2	no	—
192	C₂×C₄⋊C₈	—	53	2	no	—
193	#416	—	53	2	no	—
194	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
195	#334	—	53	2	no	—
196	#334	—	53	2	no	—
197	Q₈⋊C₈	—	53	4	no	—
198	D₄⋊C₈	—	53	4	no	—
199	#322	C₂×D₅₃	1	2	yes	—
200	C₈²	C₂×D₅₃	1	2	yes	—
201	C₅₃	#17899	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	—	53	2	no	—
203	#135	—	106	4	no	—
204	C₂×Q₁₆	—	53	1	no	—
205	C₂×SD₁₆	—	53	2	no	—
206	C₂×D₈	—	53	1	no	—
207	C₂×M₁₆	—	106	4	no	—
208	C₂²×C₈	—	106	2	no	—
209	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
211	C₄×Q₈	—	53	2	no	—
212	C₄×Q₈	—	53	2	no	—
213	C₄×D₄	—	53	2	no	—
214	C₄×D₄	—	53	2	no	—
215	C₄×D₄	—	106	2	no	—
216	C₄×D₄	—	106	2	no	—
217	#117	—	53	2	no	—
218	#117	—	106	2	no	—
219	C₂×C₄²	—	106	2	no	—
220	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
221	#107	—	53	1	no	—
222	#107	—	53	1	no	—
223	#106	—	53	2	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	53	4	no	—
226	C₄⋊C₈	—	106	4	no	—
227	C₄⋊C₈	—	106	2	no	—
228	C₄⋊C₈	D₂₁₂	1	2	yes	—
229	#103	—	53	2	no	—
230	#103	—	53	2	no	—
231	#102	—	53	2	no	—
232	#102	—	53	2	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	106	4	no	—
235	C₂²⋊C₈	D₂₁₂	1	2	yes	—
236	#97	—	106	4	no	—
237	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
238	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₅₃	1	2	yes	—
240	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₅₃	1	2	yes	—
243	#41	—	106	4	no	—
244	#41	—	106	4	no	—
245	C₂×Q₈	—	53	2	no	—
246	C₂×D₄	—	53	2	no	—
247	C₂×D₄	—	106	2	no	—
248	C₂²×C₄	—	106	2	no	—
249	C₂²×C₄	—	106	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
253	C₂²×C₄	#11719945	1	1	yes	characteristic
254	Q₁₆	—	106	2	no	—
255	SD₁₆	—	106	4	no	—
256	D₈	—	106	2	no	—
257	M₁₆	—	424	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	106	4	no	—
261	C₂×C₈	—	424	2	no	—
262	C₂×C₈	#11719912	1	4	yes	complemented
263	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
268	C₄⋊C₄	—	53	2	no	—
269	C₄⋊C₄	—	53	2	no	—
270	C₄⋊C₄	—	106	2	no	—
271	C₄⋊C₄	—	106	2	no	—
272	C₂²⋊C₄	—	106	2	no	—
273	C₂²⋊C₄	—	106	2	no	—
274	C₂²⋊C₄	—	106	2	no	—
275	C₄²	—	106	2	no	—
276	C₄²	—	106	2	no	—
277	C₄²	—	106	2	no	—
278	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
280	C₄²	#11719945	1	1	yes	characteristic
281	C₄²	C₂³×D₅₃	1	1	yes	characteristic
282	C₂³	—	106	2	no	—
283	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
284	Q₈	—	106	2	no	—
285	Q₈	—	106	2	no	—
286	D₄	—	106	2	no	—
287	D₄	—	106	2	no	—
288	D₄	—	212	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	106	2	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	106	2	no	—
294	C₂×C₄	—	424	2	no	—
295	C₂×C₄	—	212	2	no	—
296	C₂×C₄	—	212	2	no	—
297	C₂×C₄	—	212	4	no	—
298	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
299	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
300	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
301	C₂×C₄	C₂×#11719912	1	2	yes	—
302	C₂×C₄	#13133223	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
305	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	4	no	—
309	C₈	—	4	2	no	—
310	C₈	—	212	4	no	—
311	C₈	C₂×#11719912	1	2	yes	—
312	C₈	C₂×#11719912	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	212	2	no	—
316	C₂²	—	212	2	no	—
317	C₂²	—	212	2	no	—
318	C₂²	#16810702	1	2	yes	—
319	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	212	2	no	—
324	C₄	—	212	2	no	—
325	C₄	—	212	2	no	—
326	C₄	#16810702	1	2	yes	—
327	C₄	C₂²×#11719912	1	1	yes	characteristic
328	C₄	C₂×#13133223	1	1	yes	characteristic
329	C₄	#16811435	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	424	2	no	—
332	C₂	C₂×#16810702	1	1	yes	characteristic
333	C₂	#20283600	1	1	yes	characteristic
334	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1450	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#17899
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #20283828	C₂	semidirect
3. #20273630	C₂	semidirect
4. #20273625	C₂	semidirect
5. #20273607	C₂	semidirect
6. C₂×#16810374	C₂	semidirect
7. #20273103	C₂	semidirect
8. C₅₃×#1450	C₂	semidirect
88. C₂×C₈×C₅₃	M₁₆	semidirect
170. #1450	D₅₃	semidirect
201. C₅₃	#17899	semidirect
262. C₂×C₈	#11719912	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20283828	2
C₄×#13133239	2
#20273630	2
#20273625	2
#20273607	2
C₂×#16810374	2
#20273103	2
C₅₃×#1450	2
#17899	53