Subgroups — #27355592

Order 13568.

630 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	M₁₆×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20273625	C₂	1	2	yes	maximal, complemented
4	#20273588	C₂	1	4	yes	maximal, complemented
5	#20273535	C₂	1	2	yes	maximal, complemented
6	#20273471	C₂	1	1	yes	maximal, characteristic
7	#20273145	C₂	1	2	yes	maximal
8	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	#16810753	C₂²	1	4	yes	—
12	#16810751	—	2	4	no	—
13	C₈×Dic₁₀₆	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	C₂×#13133151	C₄	1	1	yes	characteristic
21	#16810647	C₂²	1	2	yes	—
22	#16810646	—	2	2	no	—
23	C₂×#13133142	C₄	1	2	yes	—
24	M₁₆×Dic₅₃	—	2	2	no	—
25	#16810374	C₂²	1	2	yes	—
26	#16810374	C₂²	1	2	yes	—
27	#16810355	C₂²	1	4	yes	—
28	#16810355	C₂²	1	4	yes	—
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₅₃×#322	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	2	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	—	2	2	no	—
53	C₄⋊#88322	C₂³	1	4	yes	—
54	#13133146	—	2	4	no	—
55	#13133146	—	2	4	no	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133142	C₂×C₄	1	2	yes	—
58	#13133142	C₂×C₄	1	2	yes	—
59	#13133142	C₂×C₄	1	2	yes	—
60	#13133127	—	2	4	no	—
61	C₈×Dic₅₃	—	2	4	no	—
62	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₅₃	—	2	1	no	—
67	C₂²⋊C₈×C₅₃	—	2	1	no	—
68	#97×C₅₃	D₄	1	2	yes	—
69	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂³	1	2	yes	—
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	2	yes	—
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	M₁₆×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	4	1	no	—
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	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
94	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	#41	1	1	yes	characteristic
97	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
100	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
101	#11719919	—	2	2	no	—
102	#11719919	—	2	2	no	—
103	#11719919	—	2	2	no	—
104	#11719919	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	#11719917	—	4	4	no	—
110	C₂×#88322	—	2	4	no	—
111	Dic₂₁₂	—	2	4	no	—
112	Dic₂₁₂	—	2	2	no	—
113	Dic₂₁₂	M₁₆	1	4	yes	complemented
114	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	4	1	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	#137	1	4	yes	—
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	4	4	no	—
130	C₂×Dic₅₃	—	8	2	no	—
131	Dic₁₀₆	—	2	2	no	—
132	Dic₁₀₆	—	2	2	no	—
133	Dic₁₀₆	—	2	2	no	—
134	Dic₁₀₆	—	4	4	no	—
135	Dic₁₀₆	C₂×M₁₆	1	4	yes	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	1	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	2	4	no	—
140	C₈×C₅₃	—	4	2	no	—
141	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
142	#88322	—	4	4	no	—
143	#18042	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	#441	1	2	yes	—
147	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	#441	1	2	yes	—
152	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
153	C₄×C₅₃	#568	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#137	1	2	yes	—
155	Dic₅₃	—	2	4	no	—
156	Dic₅₃	—	4	2	no	—
157	Dic₅₃	—	4	2	no	—
158	Dic₅₃	—	4	2	no	—
159	Q₈×M₁₆	—	53	2	no	—
160	C₂×C₄×Q₁₆	—	53	1	no	—
161	#1593	—	53	2	no	—
162	#1587	—	53	4	no	—
163	#1579	—	53	2	no	—
164	#1566	—	53	1	no	—
165	#1487	—	53	2	no	—
166	#1451	D₅₃	1	1	yes	characteristic
167	C₂×C₅₃	—	2	2	no	—
168	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
170	C₂×C₅₃	#2957	1	1	yes	characteristic
171	C₂²×Q₁₆	—	53	1	no	—
172	C₂×C₄×Q₈	—	53	2	no	—
173	#446	—	53	4	no	—
174	#446	—	106	4	no	—
175	C₈×Q₈	—	53	4	no	—
176	C₄×Q₁₆	—	53	1	no	—
177	C₄×Q₁₆	—	53	2	no	—
178	C₄×Q₁₆	—	53	1	no	—
179	C₄×Q₁₆	—	53	1	no	—
180	C₄×Q₁₆	—	53	2	no	—
181	C₄×Q₁₆	—	53	1	no	—
182	#431	C₂×D₅₃	1	1	yes	characteristic
183	C₂×#107	—	53	1	no	—
184	#423	—	53	2	no	—
185	#423	—	106	2	no	—
186	C₂×#103	—	53	2	no	—
187	C₄×M₁₆	—	106	2	no	—
188	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	#334	—	53	2	no	—
191	#334	—	53	2	no	—
192	Q₈⋊C₈	—	53	4	no	—
193	Q₈⋊C₈	—	53	4	no	—
194	#322	C₂×D₅₃	1	2	yes	—
195	#322	C₂×D₅₃	1	2	yes	—
196	C₅₃	#18042	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₈	—	53	2	no	—
198	C₂×Q₁₆	—	53	1	no	—
199	C₂×Q₁₆	—	53	2	no	—
200	C₂×Q₁₆	—	53	1	no	—
201	C₂×Q₁₆	—	106	4	no	—
202	C₂×M₁₆	—	106	2	no	—
203	C₂×M₁₆	D₂₁₂	1	2	yes	—
204	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
206	C₄×Q₈	—	53	2	no	—
207	C₄×Q₈	—	53	2	no	—
208	C₄×Q₈	—	53	2	no	—
209	C₄×Q₈	—	53	2	no	—
210	C₄×Q₈	—	106	2	no	—
211	C₄×Q₈	—	106	2	no	—
212	C₂×C₄⋊C₄	—	53	2	no	—
213	C₂×C₄⋊C₄	—	106	2	no	—
214	C₂×C₄²	—	106	2	no	—
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	#107	—	53	2	no	—
217	#107	—	53	1	no	—
218	#107	—	53	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	53	4	no	—
221	C₄⋊C₈	—	106	4	no	—
222	C₄⋊C₈	—	106	4	no	—
223	C₄⋊C₈	—	106	2	no	—
224	#103	—	53	2	no	—
225	#103	—	53	2	no	—
226	#103	—	53	2	no	—
227	#103	—	53	2	no	—
228	C₂²⋊C₈	—	2	1	no	—
229	#97	—	106	4	no	—
230	#97	D₂₁₂	1	2	yes	—
231	C₄×C₈	—	106	4	no	—
232	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
233	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₅₃	1	2	yes	—
235	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅₃	1	2	yes	—
238	C₂×Q₈	—	53	2	no	—
239	C₂×Q₈	—	53	2	no	—
240	C₂×Q₈	—	106	4	no	—
241	C₂×Q₈	—	106	2	no	—
242	C₂×Q₈	—	212	2	no	—
243	C₂²×C₄	—	106	2	no	—
244	C₂²×C₄	—	106	2	no	—
245	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
247	C₂²×C₄	#11719945	1	1	yes	characteristic
248	Q₁₆	—	53	4	no	—
249	Q₁₆	—	106	4	no	—
250	Q₁₆	—	106	2	no	—
251	M₁₆	—	2	4	no	—
252	M₁₆	—	212	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	106	4	no	—
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	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
262	C₄⋊C₄	—	53	2	no	—
263	C₄⋊C₄	—	53	2	no	—
264	C₄⋊C₄	—	106	2	no	—
265	C₄⋊C₄	—	106	2	no	—
266	C₄⋊C₄	—	106	2	no	—
267	C₄⋊C₄	—	106	2	no	—
268	C₄⋊C₄	—	106	2	no	—
269	C₄²	—	106	2	no	—
270	C₄²	—	106	2	no	—
271	C₄²	—	106	2	no	—
272	C₄²	—	106	2	no	—
273	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
275	C₄²	#11719945	1	1	yes	characteristic
276	C₄²	C₂³×D₅₃	1	1	yes	characteristic
277	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
278	Q₈	—	53	4	no	—
279	Q₈	—	106	2	no	—
280	Q₈	—	106	2	no	—
281	Q₈	—	106	2	no	—
282	Q₈	—	212	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	106	2	no	—
287	C₂×C₄	—	106	2	no	—
288	C₂×C₄	—	106	2	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	424	2	no	—
291	C₂×C₄	—	212	4	no	—
292	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
293	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
294	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
295	C₂×C₄	#13133250	1	4	yes	—
296	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
298	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	4	no	—
303	C₈	—	4	2	no	—
304	C₈	—	212	4	no	—
305	C₈	M₁₆×D₅₃	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#16810731	1	2	yes	—
309	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	106	4	no	—
314	C₄	—	212	2	no	—
315	C₄	—	212	2	no	—
316	C₄	—	212	2	no	—
317	C₄	#16810731	1	2	yes	—
318	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
319	C₄	#16811389	1	1	yes	characteristic
320	C₄	C₂×#13133250	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	M₁₆×D₂₁₂	1	1	yes	characteristic
323	C₂	C₂×#16810731	1	1	yes	characteristic
324	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#18042
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #20273625	C₂	semidirect
4. #20273588	C₂	semidirect
5. #20273535	C₂	semidirect
113. Dic₂₁₂	M₁₆	semidirect
196. C₅₃	#18042	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₀₆	2
C₂×C₄×Dic₂₁₂	2
#20273625	2
#20273588	2
#20273535	2
#20273471	2
#20273145	2
C₅₃×#1451	2
#18042	53