Subgroups — #27537558

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811066	C₂	1	1	yes	maximal, characteristic
2	#20283631	C₂	1	1	yes	maximal, characteristic
3	#20283582	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276605	C₂	1	8	yes	maximal, complemented
5	C₂×#16810438	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276450	C₂	1	1	yes	maximal, characteristic
7	#20276319	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
11	#16811389	—	2	1	no	—
12	#16811100	—	2	1	no	—
13	#16811066	C₄	1	4	yes	complemented
14	#16811066	C₂²	1	4	yes	—
15	C₂×Dic₄₂₄	C₄	1	2	yes	complemented
16	C₂×#13133162	C₄	1	2	yes	complemented
17	C₂×#13133157	C₂²	1	1	yes	characteristic
18	#16810744	C₂²	1	4	yes	—
19	#16810739	—	2	2	no	—
20	#16810729	C₂²	1	4	yes	complemented
21	#16810717	—	2	2	no	—
22	#16810639	—	2	1	no	—
23	#16810631	—	2	1	no	—
24	#16810578	C₂²	1	1	yes	characteristic
25	#16810438	C₂²	1	4	yes	complemented
26	#16810432	C₂²	1	4	yes	—
27	#16810400	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#427	C₂²	1	1	yes	characteristic
31	C₅₃×#365	C₂²	1	4	yes	complemented
32	C₂×#11719945	—	2	1	no	—
33	C₂²×Dic₁₀₆	—	2	1	no	—
34	#13133239	—	2	1	no	—
35	#13133239	—	2	1	no	—
36	#13133239	C₂×C₄	1	2	yes	—
37	#13133239	C₂³	1	2	yes	—
38	C₂×Dic₂₁₂	—	2	1	no	—
39	C₂×Dic₂₁₂	—	2	1	no	—
40	C₂×Dic₂₁₂	—	2	1	no	—
41	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
43	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
44	C₂×Dic₂₁₂	C₂³	1	2	yes	—
45	C₂×#11719913	—	2	1	no	—
46	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
47	#13133225	—	2	4	no	—
48	#13133223	—	4	2	no	—
49	C₂×#11719917	D₄	1	1	yes	characteristic
50	M₁₆×D₅₃	—	4	2	no	—
51	C₄×Dic₁₀₆	—	4	2	no	—
52	#13133182	—	2	4	no	—
53	#13133176	D₄	1	1	yes	characteristic
54	#13133173	—	4	2	no	—
55	Dic₄₂₄	C₂×C₄	1	8	yes	—
56	#13133162	C₂×C₄	1	8	yes	—
57	#13133157	—	2	1	no	—
58	#13133157	C₂³	1	2	yes	—
59	C₂×#11719908	—	2	1	no	—
60	#13133143	—	2	4	no	—
61	#13133140	—	2	4	no	—
62	#13133127	C₂³	1	2	yes	—
63	C₈×Dic₅₃	D₄	1	2	yes	—
64	C₂×M₁₆×C₅₃	—	2	1	no	—
65	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	#117×C₅₃	—	2	1	no	—
67	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
68	M₃₂×C₅₃	C₂³	1	2	yes	—
69	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
70	#108×C₅₃	—	2	1	no	—
71	#108×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	—	2	1	no	—
73	#106×C₅₃	C₂³	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₅₃	—	4	1	no	—
76	C₂²×Dic₅₃	—	4	1	no	—
77	#11719945	—	2	2	no	—
78	#11719945	—	2	2	no	—
79	#11719945	—	4	1	no	—
80	#11719945	—	4	1	no	—
81	C₂×D₂₁₂	—	2	1	no	—
82	C₂×C₄×D₅₃	—	4	1	no	—
83	C₂×Dic₁₀₆	—	2	1	no	—
84	C₂×Dic₁₀₆	—	2	2	no	—
85	C₂×Dic₁₀₆	—	2	2	no	—
86	C₂×Dic₁₀₆	—	2	2	no	—
87	C₂×Dic₁₀₆	—	4	1	no	—
88	C₂×Dic₁₀₆	—	4	1	no	—
89	C₂×Dic₁₀₆	—	4	1	no	—
90	M₁₆×C₅₃	—	2	2	no	—
91	M₁₆×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	—	4	1	no	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₂²⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	4	1	no	—
102	C₂²⋊Dic₅₃	—	2	1	no	—
103	C₄⋊Dic₅₃	—	2	1	no	—
104	#11719919	—	4	4	no	—
105	C₄×Dic₅₃	—	4	4	no	—
106	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
107	#11719917	—	2	1	no	—
108	#11719917	—	2	1	no	—
109	C₂×#88322	C₂×D₄	1	2	yes	—
110	Dic₂₁₂	—	2	2	no	—
111	Dic₂₁₂	—	2	2	no	—
112	Dic₂₁₂	—	2	1	no	—
113	Dic₂₁₂	—	2	2	no	—
114	Dic₂₁₂	C₂²×C₄	1	4	yes	—
115	Dic₂₁₂	C₂²×C₄	1	2	yes	—
116	D₄₂₄	—	2	1	no	—
117	D₄₂₄	C₂²×C₄	1	2	yes	—
118	#11719913	—	2	2	no	—
119	#11719913	—	2	2	no	—
120	#11719912	—	4	4	no	—
121	C₈×D₅₃	—	4	4	no	—
122	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
123	#11719908	—	2	2	no	—
124	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₅₃	—	4	2	no	—
134	C₂²⋊D₅₃	—	4	2	no	—
135	C₂×Dic₅₃	—	2	2	no	—
136	C₂×Dic₅₃	—	4	2	no	—
137	C₂×Dic₅₃	—	4	2	no	—
138	C₂×Dic₅₃	—	4	2	no	—
139	C₂×Dic₅₃	—	4	1	no	—
140	D₂₁₂	—	2	2	no	—
141	D₂₁₂	—	4	1	no	—
142	C₄×D₅₃	—	4	2	no	—
143	C₄×D₅₃	—	4	2	no	—
144	Dic₁₀₆	—	2	4	no	—
145	Dic₁₀₆	—	2	2	no	—
146	Dic₁₀₆	—	4	2	no	—
147	Dic₁₀₆	—	4	2	no	—
148	Dic₁₀₆	—	4	2	no	—
149	Dic₁₀₆	—	4	1	no	—
150	C₈×C₅₃	—	4	2	no	—
151	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
153	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
155	#88322	—	2	2	no	—
156	#33470	—	53	1	no	maximal, Sylow
157	C₂²×C₅₃	—	4	1	no	—
158	C₂²×C₅₃	#442	1	1	yes	characteristic
159	C₂²×C₅₃	#442	1	1	yes	characteristic
160	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×D₅₃	—	4	2	no	—
162	C₂×D₅₃	—	16	1	no	—
163	C₄×C₅₃	—	8	2	no	—
164	C₄×C₅₃	#442	1	1	yes	characteristic
165	C₄×C₅₃	#442	1	1	yes	characteristic
166	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	Dic₅₃	—	4	2	no	—
169	Dic₅₃	—	4	4	no	—
170	Dic₅₃	—	4	2	no	—
171	C₂×#510	—	53	1	no	—
172	#2957	—	53	1	no	—
173	#2950	—	53	1	no	—
174	#2180	—	53	8	no	—
175	#2155	D₅₃	1	1	yes	characteristic, complemented
176	C₂×#362	—	53	1	no	—
177	#2145	—	53	1	no	—
178	#2110	—	53	1	no	—
179	C₂×C₅₃	—	2	1	no	—
180	C₂×C₅₃	—	2	1	no	—
181	C₂×C₅₃	—	2	1	no	—
182	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
183	D₅₃	—	8	2	no	—
184	C₂×#135	—	53	1	no	—
185	C₂²×Q₁₆	—	53	1	no	—
186	#568	—	106	1	no	—
187	#519	—	106	1	no	—
188	#510	—	53	4	no	—
189	#510	—	53	4	no	—
190	C₂×Q₃₂	—	53	2	no	—
191	C₂×SD₃₂	—	53	2	no	—
192	C₂×M₃₂	—	53	1	no	—
193	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
194	#444	—	53	4	no	—
195	#443	—	106	2	no	—
196	#441	—	53	4	no	—
197	C₄×Q₁₆	—	106	2	no	—
198	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
199	#427	C₂×D₅₃	1	1	yes	characteristic
200	#421	—	106	1	no	—
201	#419	—	106	1	no	—
202	#405	—	53	1	no	—
203	#365	C₂×D₅₃	1	4	yes	complemented
204	#362	—	53	4	no	—
205	#360	—	53	4	no	—
206	#347	—	53	4	no	—
207	C₅₃	#33470	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	106	1	no	—
209	C₂²×Q₈	—	106	1	no	—
210	#135	—	53	2	no	—
211	#135	—	53	2	no	—
212	#135	—	106	1	no	—
213	#135	—	106	1	no	—
214	C₂×Q₁₆	—	53	2	no	—
215	C₂×Q₁₆	—	53	2	no	—
216	C₂×Q₁₆	—	53	1	no	—
217	C₂×Q₁₆	—	53	2	no	—
218	C₂×Q₁₆	—	106	1	no	—
219	C₂×Q₁₆	—	106	1	no	—
220	C₂×Q₁₆	—	106	1	no	—
221	C₂×SD₁₆	—	106	1	no	—
222	C₂×D₈	—	53	1	no	—
223	#131	—	106	4	no	—
224	#131	—	212	2	no	—
225	C₂×M₁₆	—	2	1	no	—
226	C₂×M₁₆	—	53	1	no	—
227	C₂×M₁₆	—	212	2	no	—
228	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
229	C₄×Q₈	—	106	4	no	—
230	C₄×Q₈	—	212	2	no	—
231	#117	—	2	1	no	—
232	#117	—	53	1	no	—
233	#117	—	212	2	no	—
234	Q₃₂	—	53	8	no	—
235	SD₃₂	—	53	8	no	—
236	M₃₂	—	53	2	no	—
237	M₃₂	—	106	1	no	—
238	M₃₂	C₄×D₅₃	1	2	yes	—
239	M₃₂	C₂²×D₅₃	1	2	yes	—
240	C₂×C₁₆	—	106	1	no	—
241	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
242	#108	—	2	1	no	—
243	#108	C₂²×D₅₃	1	2	yes	—
244	#107	—	2	1	no	—
245	#106	C₂²×D₅₃	1	2	yes	—
246	#104	—	106	4	no	—
247	#103	—	106	4	no	—
248	#97	—	53	2	no	—
249	C₄×C₈	—	53	2	no	—
250	#41	—	106	2	no	—
251	#41	—	106	2	no	—
252	#41	—	212	1	no	—
253	#41	—	212	1	no	—
254	C₂×Q₈	—	106	1	no	—
255	C₂×Q₈	—	106	2	no	—
256	C₂×Q₈	—	106	2	no	—
257	C₂×Q₈	—	106	2	no	—
258	C₂×Q₈	—	212	1	no	—
259	C₂×Q₈	—	212	1	no	—
260	C₂×Q₈	—	212	1	no	—
261	C₂×D₄	—	106	1	no	—
262	C₂×D₄	—	212	1	no	—
263	C₂²×C₄	—	212	1	no	—
264	C₂²×C₄	—	212	1	no	—
265	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
266	Q₁₆	—	53	4	no	—
267	Q₁₆	—	53	2	no	—
268	Q₁₆	—	106	2	no	—
269	Q₁₆	—	106	2	no	—
270	Q₁₆	—	106	1	no	—
271	Q₁₆	—	106	2	no	—
272	SD₁₆	—	106	2	no	—
273	SD₁₆	—	106	2	no	—
274	D₈	—	53	2	no	—
275	D₈	—	106	1	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	106	1	no	—
279	M₁₆	—	106	1	no	—
280	M₁₆	—	212	4	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	53	2	no	—
283	C₂×C₈	—	212	4	no	—
284	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
287	C₂×C₈	#11719949	1	1	yes	characteristic
288	C₂×C₈	#11719949	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	106	1	no	—
292	C₄⋊C₄	—	212	4	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	106	1	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	53	2	no	—
297	C₄²	—	212	4	no	—
298	C₁₆	—	106	2	no	—
299	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
300	C₂³	—	212	1	no	—
301	C₂³	#13133244	1	1	yes	characteristic
302	Q₈	—	106	2	no	—
303	Q₈	—	106	4	no	—
304	Q₈	—	212	2	no	—
305	Q₈	—	212	2	no	—
306	Q₈	—	212	2	no	—
307	Q₈	—	212	1	no	—
308	D₄	—	106	2	no	—
309	D₄	—	212	2	no	—
310	D₄	—	212	2	no	—
311	D₄	—	212	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	106	2	no	—
314	C₂×C₄	—	212	2	no	—
315	C₂×C₄	—	212	1	no	—
316	C₂×C₄	—	212	2	no	—
317	C₂×C₄	—	212	2	no	—
318	C₂×C₄	—	212	2	no	—
319	C₂×C₄	—	212	2	no	—
320	C₂×C₄	#13133179	1	1	yes	characteristic
321	C₂×C₄	#13133179	1	1	yes	characteristic
322	C₂×C₄	#13133244	1	1	yes	characteristic
323	C₂×C₄	#13133244	1	1	yes	characteristic
324	C₂×C₄	#13133244	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
326	C₈	—	4	2	no	—
327	C₈	—	106	2	no	—
328	C₈	#13133179	1	1	yes	characteristic
329	C₈	#13133179	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
331	C₈	C₂×#11719949	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	848	1	no	—
334	C₂²	—	212	2	no	—
335	C₂²	#16810734	1	1	yes	characteristic
336	C₂²	#16810734	1	1	yes	characteristic
337	C₂²	C₂×#13133244	1	1	yes	characteristic
338	C₄	—	8	2	no	—
339	C₄	—	212	2	no	—
340	C₄	—	212	2	no	—
341	C₄	—	212	4	no	—
342	C₄	#16810734	1	1	yes	characteristic
343	C₄	#16810734	1	1	yes	characteristic
344	C₄	C₂×#13133179	1	1	yes	characteristic
345	C₄	C₂×#13133244	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	424	2	no	—
350	C₂	C₂×#16810734	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16810734
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2155	C₂
socle	C₂×C₅₃	C₂×#442
prime radical	C₂×#13133239	C₂²
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2155	D₅₃
hypocenter	C₅₃	#33470
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
3. #20283582	C₂	semidirect
4. #20276605	C₂	semidirect
5. C₂×#16810438	C₂	semidirect
7. #20276319	C₂	semidirect
8. C₅₃×#2155	C₂	semidirect
13. #16811066	C₄	semidirect
15. C₂×Dic₄₂₄	C₄	semidirect
16. C₂×#13133162	C₄	semidirect
20. #16810729	C₂²	semidirect
25. #16810438	C₂²	semidirect
27. #16810400	C₂²	semidirect
31. C₅₃×#365	C₂²	semidirect
175. #2155	D₅₃	semidirect
203. #365	C₂×D₅₃	semidirect
207. C₅₃	#33470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811066	2
#20283631	2
#20283582	2
#20276605	2
C₂×#16810438	2
#20276450	2
#20276319	2
C₅₃×#2155	2
#33470	53