Subgroups — #27300049

Order 13568.

440 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283137	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810443	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810443	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276344	C₂	1	1	yes	maximal, characteristic
5	#20276342	C₂	1	1	yes	maximal, characteristic
6	#20272801	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133225	C₄	1	1	yes	characteristic
9	C₂×#13133223	C₂²	1	1	yes	characteristic
10	#16811091	C₄	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	#16810580	—	2	1	no	—
14	#16810579	C₄	1	1	yes	characteristic
15	#16810577	—	2	1	no	—
16	#16810575	—	2	1	no	—
17	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₅₃	—	2	1	no	—
20	#16810443	—	2	1	no	—
21	#16810443	—	2	1	no	—
22	#16810443	—	2	1	no	—
23	#16810443	—	2	1	no	—
24	#16810408	—	2	2	no	—
25	#16810406	—	2	2	no	—
26	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#405	C₂²	1	1	yes	characteristic
29	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
30	#13133225	—	2	2	no	—
31	#13133225	C₈	1	4	yes	—
32	#13133223	—	2	1	no	—
33	#13133223	—	2	1	no	—
34	#13133223	D₄	1	2	yes	—
35	#13133223	Q₈	1	2	yes	—
36	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719912	—	2	1	no	—
38	C₂×#11719912	—	2	1	no	—
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	—	2	1	no	—
41	C₂×C₈×D₅₃	—	2	1	no	—
42	#13133182	—	2	1	no	—
43	#13133179	—	2	1	no	—
44	#13133177	—	2	2	no	—
45	#13133174	—	2	1	no	—
46	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₅₃	—	2	1	no	—
48	#13133157	—	2	1	no	—
49	#13133157	—	2	1	no	—
50	#13133157	—	2	1	no	—
51	#13133157	—	2	1	no	—
52	C₂×#11719908	—	2	1	no	—
53	C₂×#11719908	—	2	1	no	—
54	#13133128	—	2	1	no	—
55	#13133127	—	2	1	no	—
56	C₄×#88322	—	2	1	no	—
57	C₈×Dic₅₃	—	2	1	no	—
58	C₈×Dic₅₃	D₄	1	1	yes	characteristic
59	C₈×Dic₅₃	D₄	1	1	yes	characteristic
60	C₈×Dic₅₃	D₄	1	1	yes	characteristic
61	C₈×Dic₅₃	D₄	1	1	yes	characteristic
62	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₅₃	—	2	1	no	—
66	M₃₂×C₅₃	—	2	1	no	—
67	M₃₂×C₅₃	—	2	1	no	—
68	M₃₂×C₅₃	—	2	1	no	—
69	C₂×C₁₆×C₅₃	—	2	1	no	—
70	C₂×C₁₆×C₅₃	—	2	1	no	—
71	#97×C₅₃	—	2	1	no	—
72	C₄×C₈×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₅₃	—	2	1	no	—
75	C₂²×Dic₅₃	—	2	1	no	—
76	#11719945	—	2	1	no	—
77	#11719945	—	2	1	no	—
78	#11719945	C₄⋊C₄	1	2	yes	—
79	#11719945	C₂×C₈	1	2	yes	—
80	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
81	C₂×C₄×D₅₃	—	2	1	no	—
82	C₂×C₄×D₅₃	—	2	2	no	—
83	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
84	M₁₆×C₅₃	—	2	1	no	—
85	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₅₃	—	2	1	no	—
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	1	no	—
94	C₂²⋊C₄×C₅₃	M₁₆	1	2	yes	—
95	C₄²×C₅₃	—	2	1	no	—
96	#11719921	—	2	2	no	—
97	#11719919	—	2	2	no	—
98	C₄×Dic₅₃	—	2	2	no	—
99	C₄×Dic₅₃	—	2	1	no	—
100	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	#11719917	—	2	1	no	—
103	#11719917	C₄⋊C₄	1	2	yes	—
104	C₂×#88322	—	2	1	no	—
105	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#88322	C₂×C₈	1	2	yes	—
108	C₂×#88322	M₁₆	1	2	yes	—
109	#11719912	—	2	2	no	—
110	#11719912	—	2	2	no	—
111	#11719912	—	2	4	no	—
112	C₈×D₅₃	—	2	2	no	—
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	—	2	4	no	—
115	C₁₆×C₅₃	—	4	1	no	—
116	C₁₆×C₅₃	—	4	1	no	—
117	#11719908	—	4	1	no	—
118	#11719908	—	4	1	no	—
119	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂²×D₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂²⋊D₅₃	—	2	2	no	—
129	C₂²⋊D₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	2	1	no	—
131	C₂×Dic₅₃	—	2	1	no	—
132	C₂×Dic₅₃	—	2	1	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	D₂₁₂	—	2	1	no	—
135	D₂₁₂	C₄⋊C₈	1	2	yes	—
136	C₄×D₅₃	—	2	2	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	4	4	no	—
139	Dic₁₀₆	—	2	1	no	—
140	Dic₁₀₆	C₄⋊C₈	1	2	yes	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	#121	1	1	yes	characteristic
143	C₈×C₅₃	#121	1	1	yes	characteristic
144	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
146	#88322	—	2	2	no	—
147	#88322	C₂²⋊C₈	1	2	yes	—
148	#88322	C₂²⋊C₈	1	2	yes	—
149	#12792	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	4	1	no	—
151	C₂²×C₅₃	#407	1	1	yes	characteristic
152	C₂²×C₅₃	#423	1	1	yes	characteristic
153	C₂²×C₅₃	#431	1	1	yes	characteristic
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	—	4	2	no	—
156	C₄×C₅₃	—	4	2	no	—
157	C₄×C₅₃	#380	1	1	yes	characteristic
158	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₄×C₅₃	#432	1	1	yes	characteristic
161	Dic₅₃	—	2	2	no	—
162	Dic₅₃	—	2	2	no	—
163	Dic₅₃	—	4	1	no	—
164	C₄×#131	—	53	1	no	—
165	C₂×#364	—	53	1	no	—
166	C₂×#364	—	53	1	no	—
167	#2117	—	53	1	no	—
168	#2117	—	53	1	no	—
169	#1378	—	53	1	no	—
170	#1378	D₅₃	1	1	yes	characteristic, complemented
171	C₂×C₅₃	—	2	1	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	#1797	1	1	yes	characteristic
175	D₅₃	—	4	2	no	—
176	C₂×#131	—	53	1	no	—
177	C₂×#131	—	53	1	no	—
178	C₄×#41	—	53	1	no	—
179	C₂×M₃₂	—	53	1	no	—
180	C₂×M₃₂	—	53	1	no	—
181	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
182	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
183	#405	—	53	1	no	—
184	#405	—	106	1	no	—
185	#405	—	106	1	no	—
186	#405	C₂×D₅₃	1	1	yes	characteristic
187	C₄×M₁₆	—	53	1	no	—
188	C₄×M₁₆	—	106	1	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	106	1	no	—
191	#364	—	106	1	no	—
192	#364	—	106	1	no	—
193	#364	—	106	1	no	—
194	#364	—	106	1	no	—
195	#350	—	106	2	no	—
196	#350	—	106	2	no	—
197	C₅₃	#12792	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	53	1	no	—
199	#131	—	53	4	no	—
200	#131	—	53	2	no	—
201	#131	—	53	2	no	—
202	#131	—	106	2	no	—
203	#131	—	106	1	no	—
204	#131	—	106	1	no	—
205	C₂×M₁₆	—	53	1	no	—
206	C₂×M₁₆	—	106	1	no	—
207	C₂×M₁₆	—	106	1	no	—
208	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	106	1	no	—
211	C₂²×C₈	—	106	1	no	—
212	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	C₄×Q₈	—	106	1	no	—
214	C₄×D₄	—	106	1	no	—
215	C₄×D₄	—	106	2	no	—
216	#117	—	106	1	no	—
217	#117	C₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄²	—	53	1	no	—
219	C₂×C₄²	—	106	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	106	1	no	—
225	M₃₂	—	106	1	no	—
226	M₃₂	—	106	1	no	—
227	M₃₂	—	106	1	no	—
228	C₂×C₁₆	—	2	1	no	—
229	C₂×C₁₆	—	2	1	no	—
230	C₂×C₁₆	—	106	1	no	—
231	C₂×C₁₆	—	106	1	no	—
232	#97	—	2	1	no	—
233	#97	—	106	1	no	—
234	#97	—	106	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	—	106	1	no	—
241	C₄×C₈	—	106	1	no	—
242	#41	—	53	2	no	—
243	#41	—	53	2	no	—
244	#41	—	106	1	no	—
245	#41	—	106	1	no	—
246	C₂×Q₈	—	53	1	no	—
247	C₂×D₄	—	53	1	no	—
248	C₂×D₄	—	106	1	no	—
249	C₂²×C₄	—	106	2	no	—
250	C₂²×C₄	—	106	1	no	—
251	C₂²×C₄	—	106	1	no	—
252	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
253	M₁₆	—	2	1	no	—
254	M₁₆	—	53	2	no	—
255	M₁₆	—	106	4	no	—
256	M₁₆	—	106	1	no	—
257	M₁₆	—	106	2	no	—
258	M₁₆	—	106	2	no	—
259	M₁₆	C₈×D₅₃	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	53	1	no	—
262	C₂×C₈	—	53	1	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	106	4	no	—
266	C₂×C₈	—	106	1	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	#11719945	1	1	yes	characteristic
270	C₂×C₈	#11719945	1	1	yes	characteristic
271	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
272	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
273	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₂²⋊C₄	—	106	2	no	—
278	C₂²⋊C₄	#11719912	1	2	yes	—
279	C₄²	—	2	1	no	—
280	C₄²	—	53	1	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	106	2	no	—
283	C₄²	—	106	1	no	—
284	C₁₆	—	4	1	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	212	1	no	—
287	C₁₆	—	212	1	no	—
288	C₂³	—	106	1	no	—
289	C₂³	C₂×#11719912	1	1	yes	characteristic
290	Q₈	—	53	2	no	—
291	Q₈	—	106	1	no	—
292	D₄	—	53	2	no	—
293	D₄	—	106	2	no	—
294	D₄	—	106	2	no	—
295	D₄	—	106	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	106	1	no	—
298	C₂×C₄	—	106	1	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	106	2	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	—	212	4	no	—
303	C₂×C₄	—	212	2	no	—
304	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
306	C₂×C₄	#13133174	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	53	2	no	—
312	C₈	—	53	2	no	—
313	C₈	—	106	2	no	—
314	C₈	#13133190	1	1	yes	characteristic
315	C₈	#13133190	1	1	yes	characteristic
316	C₈	#13133195	1	1	yes	characteristic
317	C₈	#13133195	1	1	yes	characteristic
318	C₂²	—	4	1	no	—
319	C₂²	—	106	2	no	—
320	C₂²	—	212	2	no	—
321	C₂²	#16810585	1	1	yes	characteristic
322	C₂²	#16810645	1	1	yes	characteristic
323	C₂²	#16810674	1	1	yes	characteristic
324	C₄	—	4	2	no	—
325	C₄	—	106	2	no	—
326	C₄	—	106	2	no	—
327	C₄	—	212	1	no	—
328	C₄	#16810482	1	1	yes	characteristic
329	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
330	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
331	C₄	#16810678	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	212	2	no	—
336	C₂	#20274694	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133190
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1378	C₂
socle	C₂×C₅₃	#1797
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1378	D₅₃
hypocenter	C₅₃	#12792
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
2. C₂×#16810443	C₂	semidirect
3. C₂×#16810443	C₂	semidirect
6. #20272801	C₂	semidirect
7. C₅₃×#1378	C₂	semidirect
170. #1378	D₅₃	semidirect
197. C₅₃	#12792	semidirect

Maximal subgroups

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

subgroup	index
#20283137	2
C₂×#16810443	2
#20276344	2
#20276342	2
#20272801	2
C₅₃×#1378	2
#12792	53