Subgroups — #27538835

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276891	C₂	1	4	yes	maximal, complemented
3	#20276886	C₂	1	4	yes	maximal, complemented
4	#20276489	C₂	1	2	yes	maximal, complemented
5	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276451	C₂	1	2	yes	maximal
7	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
8	#16811180	—	2	1	no	—
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16811028	—	2	1	no	—
11	#16811025	—	2	1	no	—
12	#16811025	C₂²	1	2	yes	complemented
13	#16811021	C₂²	1	2	yes	complemented
14	#16810661	—	2	1	no	—
15	#16810661	C₂²	1	2	yes	—
16	#16810447	C₂²	1	8	yes	complemented
17	#16810444	C₂²	1	2	yes	complemented
18	#16810444	C₂²	1	2	yes	complemented
19	#16810433	C₂²	1	8	yes	—
20	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#131×C₅₃	—	2	1	no	—
22	C₅₃×#443	—	2	1	no	—
23	C₅₃×#443	—	2	1	no	—
24	C₅₃×#443	C₂²	1	2	yes	—
25	C₅₃×#443	C₂²	1	2	yes	—
26	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#104×C₅₃	—	2	1	no	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	#13133217	—	2	2	no	—
30	#13133209	—	2	2	no	—
31	#13133196	—	4	2	no	—
32	C₄×Dic₁₀₆	—	4	2	no	—
33	#13133176	—	2	2	no	—
34	#13133157	—	2	2	no	—
35	#13133157	C₂³	1	4	yes	—
36	C₂×#11719908	—	2	2	no	—
37	#13133151	—	2	2	no	—
38	#13133151	—	2	2	no	—
39	#13133149	—	2	2	no	—
40	#13133149	C₂³	1	4	yes	—
41	C₂×#41×C₅₃	—	2	1	no	—
42	#135×C₅₃	—	2	1	no	—
43	#135×C₅₃	—	2	1	no	—
44	#135×C₅₃	Q₈	1	2	yes	complemented
45	#135×C₅₃	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₅₃	Q₈	1	1	yes	characteristic, complemented
47	C₂×SD₁₆×C₅₃	—	2	1	no	—
48	C₂×D₈×C₅₃	Q₈	1	1	yes	characteristic, complemented
49	#131×C₅₃	—	2	2	no	—
50	#131×C₅₃	—	2	2	no	—
51	#131×C₅₃	—	4	1	no	—
52	#131×C₅₃	—	4	1	no	—
53	C₂×M₁₆×C₅₃	—	2	1	no	—
54	C₂×M₁₆×C₅₃	—	4	1	no	—
55	C₂²×C₈×C₅₃	—	4	1	no	—
56	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	#108×C₅₃	—	2	1	no	—
60	#108×C₅₃	C₂³	1	2	yes	—
61	#104×C₅₃	—	2	2	no	—
62	#104×C₅₃	—	2	2	no	—
63	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	#41×C₅₃	—	2	2	no	—
68	#41×C₅₃	—	2	2	no	—
69	#41×C₅₃	—	4	1	no	—
70	#41×C₅₃	—	4	1	no	—
71	C₂×Q₈×C₅₃	—	2	1	no	—
72	C₂×D₄×C₅₃	—	2	1	no	—
73	C₂×D₄×C₅₃	—	4	1	no	—
74	C₂²×C₄×C₅₃	—	2	1	no	—
75	C₂²×C₄×C₅₃	—	4	1	no	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×Dic₁₀₆	—	4	2	no	—
78	Q₁₆×C₅₃	—	2	1	no	—
79	Q₁₆×C₅₃	C₂×Q₈	1	2	yes	—
80	SD₁₆×C₅₃	—	2	2	no	—
81	SD₁₆×C₅₃	—	2	2	no	—
82	D₈×C₅₃	—	2	1	no	—
83	D₈×C₅₃	C₂×Q₈	1	2	yes	—
84	M₁₆×C₅₃	—	2	2	no	—
85	M₁₆×C₅₃	—	4	1	no	—
86	M₁₆×C₅₃	—	4	2	no	—
87	M₁₆×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	—	4	1	no	—
89	C₂×C₈×C₅₃	—	4	2	no	—
90	C₂×C₈×C₅₃	—	4	2	no	—
91	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
92	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
93	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₅₃	—	4	2	no	—
107	C₄⋊Dic₅₃	—	2	4	no	—
108	C₄⋊Dic₅₃	—	4	2	no	—
109	#11719919	—	4	4	no	—
110	C₄×Dic₅₃	—	4	2	no	—
111	#11719908	—	2	4	no	—
112	C₂³×C₅₃	—	4	1	no	—
113	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	Q₈×C₅₃	—	2	2	no	—
115	Q₈×C₅₃	—	4	1	no	—
116	D₄×C₅₃	—	2	2	no	—
117	D₄×C₅₃	—	4	2	no	—
118	D₄×C₅₃	—	4	2	no	—
119	D₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₅₃	—	4	4	no	—
133	Dic₁₀₆	—	8	4	no	—
134	C₈×C₅₃	—	4	2	no	—
135	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
139	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
140	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
141	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
142	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
143	#33637	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	—	4	1	no	—
145	C₂²×C₅₃	—	4	2	no	—
146	C₂²×C₅₃	—	8	2	no	—
147	C₂²×C₅₃	#476	1	1	yes	characteristic
148	C₂²×C₅₃	#476	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	4	1	no	—
152	C₄×C₅₃	—	4	2	no	—
153	C₄×C₅₃	#476	1	1	yes	characteristic
154	C₄×C₅₃	#476	1	1	yes	characteristic
155	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
157	Dic₅₃	—	8	4	no	—
158	#3160	—	53	1	no	—
159	C₂×#443	D₅₃	1	1	yes	characteristic
160	#2231	—	53	4	no	—
161	#2230	—	53	4	no	—
162	#2155	—	53	2	no	—
163	C₂×#364	—	53	1	no	—
164	#2145	—	53	2	no	—
165	C₂×C₅₃	—	2	1	no	—
166	C₂×C₅₃	—	2	1	no	—
167	C₂×C₅₃	—	2	1	no	—
168	C₂×C₅₃	—	8	2	no	—
169	C₂×C₅₃	C₂×#476	1	1	yes	characteristic
170	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
171	C₂×#131	—	2	1	no	—
172	#533	—	106	1	no	—
173	C₂×M₃₂	—	53	2	no	—
174	#500	—	106	1	no	—
175	#499	—	53	2	no	—
176	#499	—	106	1	no	—
177	#498	—	53	2	no	—
178	#443	—	2	1	no	—
179	#443	—	2	1	no	—
180	#443	C₂×D₅₃	1	2	yes	—
181	#443	C₂×D₅₃	1	2	yes	—
182	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
183	#427	—	53	2	no	—
184	#427	—	106	1	no	—
185	C₂×#104	—	2	1	no	—
186	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
187	#365	—	53	8	no	—
188	#364	—	53	2	no	—
189	#364	—	53	2	no	—
190	#360	—	53	8	no	—
191	C₅₃	#33637	1	1	yes	characteristic, complemented, Sylow
192	C₂×#41	—	2	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	Dic₁₀₆	1	2	yes	complemented
196	#135	C₂²×D₅₃	1	2	yes	—
197	C₂×Q₁₆	Dic₁₀₆	1	1	yes	characteristic, complemented
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	Dic₁₀₆	1	1	yes	characteristic, complemented
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	—	4	1	no	—
203	#131	—	4	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	4	1	no	—
206	C₂²×C₈	—	4	1	no	—
207	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₄⋊Q₈	—	106	2	no	—
210	#125	—	106	2	no	—
211	C₂²⋊Q₈	—	212	2	no	—
212	C₄×Q₈	—	212	2	no	—
213	#117	—	106	2	no	—
214	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
215	M₃₂	—	53	4	no	—
216	M₃₂	—	106	2	no	—
217	C₂×C₁₆	—	106	2	no	—
218	#108	—	2	1	no	—
219	#108	C₂²×D₅₃	1	2	yes	—
220	#107	—	106	2	no	—
221	#107	—	106	2	no	—
222	#106	—	53	4	no	—
223	#106	—	106	2	no	—
224	#104	—	2	2	no	—
225	#104	—	2	2	no	—
226	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
227	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	#41	—	4	1	no	—
233	#41	—	4	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	212	2	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	Q₁₆	—	2	1	no	—
242	Q₁₆	C₂×Dic₁₀₆	1	2	yes	—
243	SD₁₆	—	2	2	no	—
244	SD₁₆	—	2	2	no	—
245	D₈	—	2	1	no	—
246	D₈	C₂×Dic₁₀₆	1	2	yes	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	#11719923	1	1	yes	characteristic
255	C₂×C₈	#11719923	1	1	yes	characteristic
256	C₂×C₈	#11719923	1	1	yes	characteristic
257	C₂×C₈	#11719923	1	1	yes	characteristic
258	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
259	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
260	C₂×C₈	#11719945	1	1	yes	characteristic
261	C₂×C₈	#11719945	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
264	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
266	C₄⋊C₄	—	106	4	no	—
267	C₄⋊C₄	—	212	4	no	—
268	C₄⋊C₄	—	212	2	no	—
269	C₂²⋊C₄	—	212	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	212	2	no	—
272	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
273	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
274	C₁₆	—	106	4	no	—
275	C₂³	—	4	1	no	—
276	C₂³	#13133242	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	424	4	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	212	4	no	—
291	C₂×C₄	#13133196	1	1	yes	characteristic
292	C₂×C₄	#13133196	1	1	yes	characteristic
293	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
294	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
295	C₂×C₄	#13133242	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	#13133196	1	1	yes	characteristic
299	C₈	#13133196	1	1	yes	characteristic
300	C₈	C₂×#11719923	1	1	yes	characteristic
301	C₈	C₂×#11719923	1	1	yes	characteristic
302	C₈	C₂×#11719923	1	1	yes	characteristic
303	C₈	C₂×#11719923	1	1	yes	characteristic
304	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
305	C₈	C₂×#11719945	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	#16810913	1	1	yes	characteristic
310	C₂²	#16810913	1	1	yes	characteristic
311	C₂²	C₂×#13133242	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	424	4	no	—
316	C₄	#16810913	1	1	yes	characteristic
317	C₄	#16810913	1	1	yes	characteristic
318	C₄	C₂×#13133196	1	1	yes	characteristic
319	C₄	C₂²×#11719923	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×#16810913	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810913
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂×C₅₃	C₂×#476
prime radical	C₂×#135×C₅₃	C₂²
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#33637
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #20285229	C₂	semidirect
2. #20276891	C₂	semidirect
3. #20276886	C₂	semidirect
4. #20276489	C₂	semidirect
5. C₂×#16810444	C₂	semidirect
12. #16811025	C₂²	semidirect
13. #16811021	C₂²	semidirect
16. #16810447	C₂²	semidirect
17. #16810444	C₂²	semidirect
18. #16810444	C₂²	semidirect
44. #135×C₅₃	Q₈	semidirect
46. C₂×Q₁₆×C₅₃	Q₈	semidirect
48. C₂×D₈×C₅₃	Q₈	semidirect
191. C₅₃	#33637	semidirect
195. #135	Dic₁₀₆	semidirect
197. C₂×Q₁₆	Dic₁₀₆	semidirect
199. C₂×D₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20276891	2
#20276886	2
#20276489	2
C₂×#16810444	2
#20276451	2
C₂×C₅₃×#443	2
#33637	53