Subgroups — #27355817

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283805	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20273623	C₂	1	2	yes	maximal, complemented
4	#20273564	C₂	1	4	yes	maximal, complemented
5	#20273533	C₂	1	2	yes	maximal, complemented
6	#20273161	C₂	1	2	yes	maximal
7	#20273004	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	#16810751	—	2	4	no	—
12	#16810750	C₂²	1	4	yes	—
13	#16810747	C₂²	1	4	yes	—
14	C₄×Dic₂₁₂	C₄	1	2	yes	—
15	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
16	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
17	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₂	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	#16810677	—	2	2	no	—
21	#16810676	C₂²	1	1	yes	characteristic
22	#16810673	—	2	2	no	—
23	C₂×#13133151	C₄	1	1	yes	characteristic
24	C₂×#13133142	C₄	1	2	yes	—
25	C₄×#11719917	C₂²	1	1	yes	characteristic
26	#16810353	C₂²	1	4	yes	—
27	#16810353	C₂²	1	4	yes	—
28	#16810345	C₂²	1	2	yes	—
29	#16810344	C₂²	1	2	yes	—
30	C₅₃×#432	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#334	C₂²	1	2	yes	—
34	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₁₂	—	2	4	no	—
36	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
38	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719917	—	2	1	no	—
40	C₄×Dic₁₀₆	—	2	2	no	—
41	C₄×Dic₁₀₆	—	2	2	no	—
42	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₀₆	C₂³	1	2	yes	—
45	C₄×Dic₁₀₆	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
47	C₂×#11719919	—	2	2	no	—
48	C₂×C₄×Dic₅₃	—	2	2	no	—
49	#13133151	C₂×C₄	1	2	yes	—
50	#13133151	C₂×C₄	1	1	yes	characteristic
51	#13133151	C₂×C₄	1	1	yes	characteristic
52	C₄⋊#88322	D₄	1	2	yes	—
53	#13133146	—	2	4	no	—
54	#13133146	—	2	4	no	—
55	#13133142	C₂×C₄	1	2	yes	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133142	C₂×C₄	1	2	yes	—
58	#13133142	C₂×C₄	1	2	yes	—
59	C₂²⋊#88322	D₄	1	2	yes	—
60	#13133128	—	2	1	no	—
61	#13133127	—	2	4	no	—
62	C₄×#88322	C₂³	1	2	yes	—
63	C₄×#88322	C₂³	1	2	yes	—
64	C₈×Dic₅₃	—	2	4	no	—
65	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
69	C₂²⋊C₈×C₅₃	—	2	2	no	—
70	#97×C₅₃	—	2	2	no	—
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	C₂²×Dic₅₃	—	2	2	no	—
79	C₂²×Dic₅₃	—	2	2	no	—
80	C₂×Dic₁₀₆	—	2	4	no	—
81	C₂×Dic₁₀₆	—	2	2	no	—
82	C₂×Dic₁₀₆	—	4	2	no	—
83	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	4	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₃	#41	1	1	yes	characteristic
95	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊Dic₅₃	—	2	2	no	—
97	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
98	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
99	#11719919	—	2	2	no	—
100	#11719919	—	2	2	no	—
101	#11719919	—	2	2	no	—
102	#11719919	—	2	2	no	—
103	C₄×Dic₅₃	—	2	2	no	—
104	C₄×Dic₅₃	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	#11719917	—	4	2	no	—
108	C₂×#88322	—	2	2	no	—
109	C₂×#88322	C₂×D₄	1	4	yes	—
110	Dic₂₁₂	—	2	4	no	—
111	Dic₂₁₂	—	2	2	no	—
112	Dic₂₁₂	M₁₆	1	4	yes	complemented
113	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	2	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	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	4	4	no	—
129	C₂×Dic₅₃	—	4	4	no	—
130	Dic₁₀₆	—	2	2	no	—
131	Dic₁₀₆	—	2	2	no	—
132	Dic₁₀₆	—	2	2	no	—
133	Dic₁₀₆	—	4	4	no	—
134	Dic₁₀₆	C₂×M₁₆	1	4	yes	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	1	no	—
138	C₈×C₅₃	—	4	4	no	—
139	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
140	#88322	—	2	4	no	—
141	#88322	—	4	2	no	—
142	#18062	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	#444	1	2	yes	—
146	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₅₃	#568	1	1	yes	characteristic
153	C₄×C₅₃	#578	1	2	yes	—
154	Dic₅₃	—	2	4	no	—
155	Dic₅₃	—	4	2	no	—
156	Dic₅₃	—	4	2	no	—
157	Dic₅₃	—	4	2	no	—
158	#2984	—	53	2	no	—
159	C₂×C₄×Q₁₆	—	53	1	no	—
160	#1593	—	53	2	no	—
161	#1584	—	53	4	no	—
162	#1579	—	53	2	no	—
163	#1568	D₅₃	1	1	yes	characteristic
164	#1491	—	53	2	no	—
165	#1451	—	53	1	no	—
166	C₂×C₅₃	—	2	2	no	—
167	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
168	C₂×C₅₃	#2950	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
170	C₂²×Q₁₆	—	53	1	no	—
171	C₂×C₄×Q₈	—	53	2	no	—
172	#446	—	53	4	no	—
173	#446	—	106	4	no	—
174	C₈×Q₈	—	53	4	no	—
175	C₄×Q₁₆	—	53	1	no	—
176	C₄×Q₁₆	—	53	2	no	—
177	C₄×Q₁₆	—	53	1	no	—
178	C₄×Q₁₆	—	53	1	no	—
179	C₄×Q₁₆	—	53	2	no	—
180	C₄×Q₁₆	—	53	1	no	—
181	#432	—	106	2	no	—
182	#432	C₂×D₅₃	1	2	yes	—
183	#431	—	53	1	no	—
184	#431	—	106	2	no	—
185	C₂×#107	—	53	1	no	—
186	C₂×#103	—	53	2	no	—
187	C₄×M₁₆	—	53	1	no	—
188	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
189	#335	C₂×D₅₃	1	2	yes	—
190	#334	C₂×D₅₃	1	2	yes	—
191	Q₈⋊C₈	—	53	4	no	—
192	Q₈⋊C₈	—	53	4	no	—
193	#322	—	53	2	no	—
194	#322	—	53	2	no	—
195	C₅₃	#18062	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	53	2	no	—
197	C₂×Q₁₆	—	53	1	no	—
198	C₂×Q₁₆	—	53	2	no	—
199	C₂×Q₁₆	—	53	1	no	—
200	C₂×Q₁₆	—	106	4	no	—
201	C₂×M₁₆	—	106	1	no	—
202	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
204	C₄×Q₈	—	53	2	no	—
205	C₄×Q₈	—	53	2	no	—
206	C₄×Q₈	—	53	2	no	—
207	C₄×Q₈	—	53	2	no	—
208	C₄×Q₈	—	106	2	no	—
209	C₄×Q₈	—	106	2	no	—
210	C₂×C₄⋊C₄	—	53	2	no	—
211	C₂×C₄⋊C₄	—	106	2	no	—
212	C₂×C₄²	—	106	2	no	—
213	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
214	#107	—	53	2	no	—
215	#107	—	53	1	no	—
216	#107	—	53	1	no	—
217	C₄⋊C₈	—	53	2	no	—
218	C₄⋊C₈	—	106	4	no	—
219	C₄⋊C₈	—	106	4	no	—
220	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
221	#103	—	53	2	no	—
222	#103	—	53	2	no	—
223	#103	—	53	2	no	—
224	#103	—	53	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	53	2	no	—
227	#97	—	2	2	no	—
228	#97	—	106	4	no	—
229	#97	—	106	1	no	—
230	C₄×C₈	—	53	2	no	—
231	C₄×C₈	—	53	2	no	—
232	C₄×C₈	—	106	4	no	—
233	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
234	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
237	C₂×Q₈	—	53	2	no	—
238	C₂×Q₈	—	53	2	no	—
239	C₂×Q₈	—	106	4	no	—
240	C₂×Q₈	—	106	2	no	—
241	C₂×Q₈	—	212	2	no	—
242	C₂²×C₄	—	106	2	no	—
243	C₂²×C₄	—	106	2	no	—
244	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
245	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
246	C₂²×C₄	#11719949	1	1	yes	characteristic
247	Q₁₆	—	53	4	no	—
248	Q₁₆	—	106	4	no	—
249	Q₁₆	—	106	2	no	—
250	M₁₆	—	212	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	53	4	no	—
255	C₂×C₈	—	106	2	no	—
256	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
260	C₄⋊C₄	—	53	2	no	—
261	C₄⋊C₄	—	53	2	no	—
262	C₄⋊C₄	—	106	2	no	—
263	C₄⋊C₄	—	106	2	no	—
264	C₄⋊C₄	—	106	2	no	—
265	C₄⋊C₄	—	106	2	no	—
266	C₄⋊C₄	—	106	2	no	—
267	C₄²	—	106	2	no	—
268	C₄²	—	106	2	no	—
269	C₄²	—	106	2	no	—
270	C₄²	—	106	2	no	—
271	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₄²	D₄×D₅₃	1	1	yes	characteristic
273	C₄²	#11719949	1	1	yes	characteristic
274	C₄²	C₂³×D₅₃	1	1	yes	characteristic
275	C₂³	#13133179	1	1	yes	characteristic
276	Q₈	—	53	4	no	—
277	Q₈	—	106	2	no	—
278	Q₈	—	106	2	no	—
279	Q₈	—	106	2	no	—
280	Q₈	—	212	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	106	2	no	—
286	C₂×C₄	—	106	2	no	—
287	C₂×C₄	—	106	2	no	—
288	C₂×C₄	—	106	2	no	—
289	C₂×C₄	—	212	4	no	—
290	C₂×C₄	—	212	4	no	—
291	C₂×C₄	#13133179	1	1	yes	characteristic
292	C₂×C₄	#13133179	1	1	yes	characteristic
293	C₂×C₄	#13133179	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	4	no	—
301	C₈	—	106	4	no	—
302	C₈	—	212	2	no	—
303	C₈	M₁₆×D₅₃	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	#16810744	1	2	yes	—
307	C₂²	C₂×#13133179	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	106	4	no	—
313	C₄	—	212	2	no	—
314	C₄	—	212	2	no	—
315	C₄	—	212	2	no	—
316	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
317	C₄	#16811389	1	1	yes	characteristic
318	C₄	#16811437	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	#20283450	1	1	yes	characteristic
321	C₂	#20283582	1	1	yes	characteristic
322	C₂	C₂×#16810744	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1568	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#18062
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #20273623	C₂	semidirect
4. #20273564	C₂	semidirect
5. #20273533	C₂	semidirect
112. Dic₂₁₂	M₁₆	semidirect
195. C₅₃	#18062	semidirect

Maximal subgroups

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

subgroup	index
#20283805	2
C₂×C₄×Dic₂₁₂	2
#20273623	2
#20273564	2
#20273533	2
#20273161	2
#20273004	2
C₅₃×#1568	2
#18062	53