Subgroups — #27496239

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276178	C₂	1	2	yes	maximal, complemented
2	#20276174	C₂	1	2	yes	maximal, complemented
3	#20276173	C₂	1	2	yes	maximal, complemented
4	#20276112	C₂	1	2	yes	maximal, complemented
5	#20275625	C₂	1	1	yes	maximal, characteristic
6	#20275623	C₂	1	1	yes	maximal, characteristic
7	#20275623	C₂	1	1	yes	maximal, characteristic
8	#20275506	C₂	1	1	yes	maximal, characteristic, complemented
9	#20275503	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274470	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
13	#16810672	C₂²	1	2	yes	—
14	#16810671	C₂²	1	2	yes	—
15	C₂×#13133153	C₂²	1	1	yes	characteristic
16	#16810629	C₂²	1	2	yes	—
17	#16810629	C₂²	1	2	yes	—
18	#16810627	C₂²	1	2	yes	—
19	C₂×#13133133	C₂²	1	2	yes	—
20	C₂×#13133133	C₂²	1	2	yes	—
21	C₂×#13133133	C₂²	1	2	yes	—
22	#16810578	C₂²	1	1	yes	characteristic
23	#16810578	C₂²	1	1	yes	characteristic
24	#16810395	C₂²	1	2	yes	—
25	#16810394	C₂²	1	2	yes	—
26	#16810381	C₂²	1	1	yes	characteristic
27	#16810380	C₂²	1	1	yes	characteristic
28	#16810379	C₂²	1	1	yes	characteristic
29	#16810379	C₂²	1	1	yes	characteristic
30	C₅₃×#430	C₂²	1	2	yes	—
31	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#405	C₂²	1	1	yes	characteristic
34	C₅₃×#344	C₂²	1	2	yes	—
35	C₂²×D₂₁₂	C₂³	1	2	yes	—
36	C₂×D₄₂₄	—	2	1	no	—
37	C₂×D₄₂₄	—	2	1	no	—
38	C₂×D₄₂₄	—	2	1	no	—
39	C₂×D₄₂₄	—	2	1	no	—
40	C₂×D₄₂₄	—	2	1	no	—
41	C₂×D₄₂₄	—	2	1	no	—
42	C₂×#11719917	C₂³	1	1	yes	characteristic
43	C₂×#11719917	C₂³	1	1	yes	characteristic
44	C₂×#11719917	C₂³	1	2	yes	—
45	#13133176	C₂³	1	1	yes	characteristic
46	#13133176	C₂³	1	1	yes	characteristic
47	#13133153	—	2	1	no	—
48	#13133153	—	2	1	no	—
49	#13133153	—	4	2	no	—
50	#13133152	—	4	2	no	—
51	#13133139	—	4	2	no	—
52	#13133139	—	4	2	no	—
53	#13133137	—	4	2	no	—
54	#13133133	—	4	2	no	—
55	#13133133	—	4	2	no	—
56	#13133133	—	4	2	no	—
57	#13133127	—	2	1	no	—
58	#13133127	—	2	1	no	—
59	C₈×Dic₅₃	—	2	1	no	—
60	C₈×Dic₅₃	—	2	1	no	—
61	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
63	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	#117×C₅₃	C₂³	1	1	yes	characteristic
66	#108×C₅₃	—	2	1	no	—
67	#108×C₅₃	—	2	1	no	—
68	#108×C₅₃	—	2	1	no	—
69	#108×C₅₃	—	2	1	no	—
70	#108×C₅₃	—	4	2	no	—
71	#97×C₅₃	—	2	1	no	—
72	C₄×C₈×C₅₃	—	2	1	no	—
73	C₂³×D₅₃	—	2	2	no	—
74	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
75	C₂×D₂₁₂	—	2	2	no	—
76	C₂×D₂₁₂	—	2	2	no	—
77	C₂×D₂₁₂	—	2	2	no	—
78	C₂×D₂₁₂	—	2	2	no	—
79	C₂×D₂₁₂	—	2	2	no	—
80	C₂×D₂₁₂	—	2	2	no	—
81	M₁₆×C₅₃	—	2	2	no	—
82	M₁₆×C₅₃	—	2	2	no	—
83	M₁₆×C₅₃	—	2	1	no	—
84	M₁₆×C₅₃	—	2	2	no	—
85	M₁₆×C₅₃	—	2	1	no	—
86	M₁₆×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₂²⋊C₄×C₅₃	—	2	1	no	—
98	C₄²×C₅₃	—	2	1	no	—
99	C₂²⋊Dic₅₃	—	2	1	no	—
100	C₂²⋊Dic₅₃	—	2	1	no	—
101	C₄⋊Dic₅₃	—	2	1	no	—
102	C₄⋊Dic₅₃	—	2	1	no	—
103	C₄×Dic₅₃	—	2	1	no	—
104	C₄×Dic₅₃	—	2	1	no	—
105	#11719917	—	2	2	no	—
106	#11719917	—	2	1	no	—
107	#11719917	—	2	1	no	—
108	#11719917	—	2	2	no	—
109	#11719917	—	2	1	no	—
110	#11719917	—	2	1	no	—
111	C₂×#88322	—	2	1	no	—
112	C₂×#88322	—	2	1	no	—
113	C₂×#88322	—	2	2	no	—
114	D₄₂₄	—	4	1	no	—
115	D₄₂₄	—	4	1	no	—
116	D₄₂₄	—	4	1	no	—
117	D₄₂₄	—	4	1	no	—
118	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂²×D₅₃	—	4	2	no	—
120	C₂²×D₅₃	—	4	2	no	—
121	C₂²×D₅₃	—	4	2	no	—
122	C₂²×D₅₃	—	16	2	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	1	no	—
132	D₂₁₂	—	4	2	no	—
133	D₂₁₂	—	4	2	no	—
134	D₂₁₂	—	4	2	no	—
135	D₂₁₂	—	4	2	no	—
136	C₈×C₅₃	—	4	2	no	—
137	C₈×C₅₃	—	4	1	no	—
138	C₈×C₅₃	—	4	2	no	—
139	C₈×C₅₃	#126	1	1	yes	characteristic
140	C₈×C₅₃	#126	1	1	yes	characteristic
141	C₈×C₅₃	#126	1	1	yes	characteristic
142	C₈×C₅₃	#126	1	1	yes	characteristic
143	#88322	—	4	1	no	—
144	#88322	—	4	1	no	—
145	#88322	—	4	2	no	—
146	#30237	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	4	1	no	—
148	C₂²×C₅₃	#542	1	1	yes	characteristic
149	C₂²×C₅₃	#542	1	1	yes	characteristic
150	C₂²×C₅₃	#542	1	1	yes	characteristic
151	C₂×D₅₃	—	8	2	no	—
152	C₂×D₅₃	—	16	2	no	—
153	C₂×D₅₃	—	16	2	no	—
154	C₂×D₅₃	—	16	2	no	—
155	C₄×C₅₃	—	8	1	no	—
156	C₄×C₅₃	C₂×#126	1	1	yes	characteristic
157	C₄×C₅₃	#540	1	1	yes	characteristic
158	C₄×C₅₃	#540	1	1	yes	characteristic
159	C₄×C₅₃	#540	1	1	yes	characteristic
160	Dic₅₃	—	8	1	no	—
161	Dic₅₃	—	8	1	no	—
162	#2081	—	53	2	no	—
163	#2081	—	53	2	no	—
164	#2081	—	53	2	no	—
165	#2069	—	53	2	no	—
166	#1978	—	53	1	no	—
167	#1978	—	53	1	no	—
168	#1978	—	53	1	no	—
169	#1953	—	53	1	no	—
170	#1953	—	53	1	no	—
171	#1953	D₅₃	1	1	yes	characteristic, complemented
172	#1747	—	53	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	2	1	no	—
175	C₂×C₅₃	—	2	1	no	—
176	C₂×C₅₃	#2644	1	1	yes	characteristic
177	D₅₃	—	16	2	no	—
178	C₂²×D₈	—	53	1	no	—
179	#430	—	53	2	no	—
180	#430	—	53	2	no	—
181	#430	C₂×D₅₃	1	2	yes	—
182	C₂×#108	—	53	1	no	—
183	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
184	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
185	#418	—	53	2	no	—
186	#418	—	53	2	no	—
187	#418	—	53	2	no	—
188	C₂×#100	—	53	2	no	—
189	C₂×#100	—	53	2	no	—
190	C₂×#100	—	53	2	no	—
191	#405	—	53	1	no	—
192	#405	—	53	1	no	—
193	#405	C₂×D₅₃	1	1	yes	characteristic
194	#344	—	53	2	no	—
195	#344	—	53	2	no	—
196	#344	C₂×D₅₃	1	2	yes	—
197	#338	—	53	1	no	—
198	H(3,ℤ/4)	—	53	1	no	—
199	H(3,ℤ/4)	—	53	1	no	—
200	H(3,ℤ/4)	—	53	1	no	—
201	C₅₃	#30237	1	1	yes	characteristic, complemented, Sylow
202	C₂²×D₄	—	53	2	no	—
203	C₂×D₈	—	106	1	no	—
204	C₂×D₈	—	106	1	no	—
205	C₂×D₈	—	106	1	no	—
206	C₂×D₈	—	106	1	no	—
207	C₂×D₈	—	106	1	no	—
208	C₂×D₈	—	106	1	no	—
209	C₂×M₁₆	—	53	1	no	—
210	C₂×M₁₆	—	53	1	no	—
211	C₂×M₁₆	—	53	2	no	—
212	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
213	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
214	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
215	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	#117	—	53	1	no	—
217	#117	—	53	1	no	—
218	#117	C₂²×D₅₃	1	1	yes	characteristic
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	—	2	1	no	—
222	#108	—	2	1	no	—
223	#108	—	4	2	no	—
224	#108	—	106	1	no	—
225	#108	—	106	1	no	—
226	#108	—	212	2	no	—
227	#108	—	212	2	no	—
228	#102	—	212	2	no	—
229	#102	—	212	2	no	—
230	#102	—	212	2	no	—
231	#100	—	212	2	no	—
232	#100	—	212	2	no	—
233	#100	—	212	2	no	—
234	#97	—	2	1	no	—
235	#97	—	106	1	no	—
236	#97	—	106	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	106	1	no	—
239	C₄×C₈	—	106	1	no	—
240	C₂⁴	—	106	2	no	—
241	C₂×D₄	—	106	2	no	—
242	C₂×D₄	—	106	2	no	—
243	C₂×D₄	—	106	2	no	—
244	C₂×D₄	—	106	2	no	—
245	C₂×D₄	—	106	2	no	—
246	C₂×D₄	—	106	2	no	—
247	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
248	D₈	—	212	1	no	—
249	D₈	—	212	1	no	—
250	D₈	—	212	1	no	—
251	D₈	—	212	1	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	106	2	no	—
259	M₁₆	—	106	1	no	—
260	M₁₆	—	106	1	no	—
261	M₁₆	—	106	2	no	—
262	M₁₆	—	106	1	no	—
263	M₁₆	—	106	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	106	1	no	—
268	C₂×C₈	—	106	1	no	—
269	C₂×C₈	—	106	2	no	—
270	C₂×C₈	#11719945	1	1	yes	characteristic
271	C₂×C₈	#11719945	1	1	yes	characteristic
272	C₂×C₈	#11719947	1	1	yes	characteristic
273	C₂×C₈	#11719947	1	1	yes	characteristic
274	C₂×C₈	#11719949	1	1	yes	characteristic
275	C₂×C₈	#11719949	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	106	1	no	—
278	C₄⋊C₄	—	106	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	106	1	no	—
281	C₂²⋊C₄	—	106	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	106	1	no	—
284	C₄²	—	106	1	no	—
285	C₂³	—	848	2	no	—
286	C₂³	—	212	2	no	—
287	C₂³	—	212	2	no	—
288	C₂³	—	212	2	no	—
289	C₂³	#13133265	1	1	yes	characteristic
290	D₄	—	212	2	no	—
291	D₄	—	212	2	no	—
292	D₄	—	212	2	no	—
293	D₄	—	212	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	212	1	no	—
296	C₂×C₄	—	212	1	no	—
297	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
298	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
299	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
300	C₂×C₄	#13133262	1	1	yes	characteristic
301	C₂×C₄	#13133263	1	1	yes	characteristic
302	C₂×C₄	#13133263	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	—	4	1	no	—
305	C₈	—	4	2	no	—
306	C₈	—	212	1	no	—
307	C₈	—	212	1	no	—
308	C₈	—	212	2	no	—
309	C₈	#13133212	1	1	yes	characteristic
310	C₈	#13133212	1	1	yes	characteristic
311	C₈	#13133212	1	1	yes	characteristic
312	C₈	#13133212	1	1	yes	characteristic
313	C₂²	—	4	1	no	—
314	C₂²	—	848	2	no	—
315	C₂²	—	848	2	no	—
316	C₂²	—	848	2	no	—
317	C₂²	—	424	2	no	—
318	C₂²	#16811245	1	1	yes	characteristic
319	C₂²	#16811246	1	1	yes	characteristic
320	C₂²	#16811247	1	1	yes	characteristic
321	C₄	—	8	1	no	—
322	C₄	—	424	1	no	—
323	C₄	—	424	1	no	—
324	C₄	C₂×#13133212	1	1	yes	characteristic
325	C₄	#16811224	1	1	yes	characteristic
326	C₄	#16811226	1	1	yes	characteristic
327	C₄	#16811229	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	848	2	no	—
332	C₂	#20280678	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20280678
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1953	C₂
socle	C₂×C₅₃	#2644
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	#1953	D₅₃
hypocenter	C₅₃	#30237
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1953

Complements

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

N	H ≅ G/N	type
1. #20276178	C₂	semidirect
2. #20276174	C₂	semidirect
3. #20276173	C₂	semidirect
4. #20276112	C₂	semidirect
8. #20275506	C₂	semidirect
9. #20275503	C₂	semidirect
10. #20274470	C₂	semidirect
11. C₅₃×#1953	C₂	semidirect
171. #1953	D₅₃	semidirect
201. C₅₃	#30237	semidirect

Maximal subgroups

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

subgroup	index
#20276178	2
#20276174	2
#20276173	2
#20276112	2
#20275625	2
#20275623	2
#20275506	2
#20275503	2
#20274470	2
C₅₃×#1953	2
#30237	53