Subgroups — #27367970

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20284845	C₂	1	2	yes	maximal
3	#20274145	C₂	1	4	yes	maximal, complemented
4	#20274109	C₂	1	2	yes	maximal, complemented
5	#20274085	C₂	1	2	yes	maximal, complemented
6	#20273478	C₂	1	1	yes	maximal, characteristic
7	#20273413	C₂	1	2	yes	maximal
8	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	#16811028	C₂²	1	1	yes	characteristic
11	#16811003	C₂²	1	1	yes	characteristic
12	#16810974	C₂²	1	2	yes	—
13	#16810953	—	2	2	no	—
14	#16810908	C₂²	1	4	yes	—
15	#16810864	—	2	1	no	—
16	#16810864	—	2	1	no	—
17	#16810833	C₂²	1	4	yes	—
18	#16810779	—	2	2	no	—
19	#16810728	—	2	4	no	—
20	C₄×Dic₂₁₂	—	2	1	no	—
21	C₄×Dic₂₁₂	—	2	1	no	—
22	#16810680	C₂²	1	2	yes	—
23	#16810376	C₂²	1	2	yes	—
24	#16810374	C₂²	1	2	yes	—
25	#16810370	C₂²	1	4	yes	—
26	#16810368	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₅₃	—	2	1	no	—
29	C₈×Q₈×C₅₃	—	2	1	no	—
30	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
31	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₅₃	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×Dic₂₁₂	—	2	1	no	—
37	C₂×Dic₂₁₂	—	2	1	no	—
38	C₂×#11719925	—	2	4	no	—
39	#13133217	C₂³	1	2	yes	—
40	#13133217	C₂³	1	2	yes	—
41	#13133209	—	2	2	no	—
42	#13133196	—	2	2	no	—
43	#13133196	—	2	2	no	—
44	C₄×Dic₁₀₆	—	2	2	no	—
45	C₄×Dic₁₀₆	—	2	2	no	—
46	#13133176	—	2	2	no	—
47	#13133151	—	2	1	no	—
48	#13133151	—	2	1	no	—
49	#13133148	—	2	4	no	—
50	C₄⋊#88322	C₂³	1	4	yes	—
51	#13133142	—	2	2	no	—
52	#13133142	—	2	2	no	—
53	#13133141	—	2	4	no	—
54	#13133140	—	2	4	no	—
55	C₂²⋊#88322	—	2	2	no	—
56	#13133128	—	2	2	no	—
57	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	—	2	1	no	—
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₅₃	—	2	1	no	—
62	C₄×Q₈×C₅₃	—	2	1	no	—
63	C₄×Q₈×C₅₃	D₄	1	2	yes	—
64	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	—	2	1	no	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	—	2	1	no	—
72	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	—	2	2	no	—
74	C₄×C₈×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	4	no	—
80	C₂×Q₈×C₅₃	—	2	1	no	—
81	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₅₃	—	2	1	no	—
83	C₂²×C₄×C₅₃	—	2	1	no	—
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
87	C₂×Dic₁₀₆	—	2	2	no	—
88	C₂×Dic₁₀₆	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	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	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₃	#41	1	1	yes	characteristic
106	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₅₃	—	4	2	no	—
108	#11719925	—	4	8	no	—
109	C₄⋊Dic₅₃	—	2	2	no	—
110	C₄⋊Dic₅₃	—	2	2	no	—
111	C₄⋊Dic₅₃	—	2	2	no	—
112	C₄⋊Dic₅₃	—	2	2	no	—
113	#11719919	—	4	2	no	—
114	#11719919	—	4	2	no	—
115	C₄×Dic₅₃	—	4	2	no	—
116	C₂×#88322	—	2	4	no	—
117	Dic₂₁₂	—	4	2	no	—
118	Dic₂₁₂	—	4	2	no	—
119	C₂³×C₅₃	#121	1	1	yes	characteristic
120	Q₈×C₅₃	—	2	4	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	Q₈×C₅₃	#135	1	4	yes	—
123	C₂×C₄×C₅₃	—	2	4	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	4	2	no	—
131	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
137	C₂×Dic₅₃	—	4	2	no	—
138	C₂×Dic₅₃	—	4	2	no	—
139	Dic₁₀₆	—	4	4	no	—
140	Dic₁₀₆	—	4	2	no	—
141	Dic₁₀₆	—	4	2	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	4	2	no	—
146	C₈×C₅₃	#137	1	2	yes	—
147	#88322	—	4	4	no	—
148	#19265	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	#473	1	2	yes	—
152	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	4	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	4	2	no	—
159	C₄×C₅₃	C₂×#135	1	2	yes	—
160	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
161	C₄×C₅₃	#575	1	1	yes	characteristic
162	Dic₅₃	—	8	2	no	—
163	Dic₅₃	—	8	2	no	—
164	#3170	—	53	1	no	—
165	#3113	—	53	2	no	—
166	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
167	#1680	—	53	4	no	—
168	#1674	—	53	2	no	—
169	#1670	—	53	2	no	—
170	#1568	—	53	1	no	—
171	#1552	—	53	2	no	—
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	#3037	1	1	yes	characteristic
174	C₂×C₅₃	#3050	1	1	yes	characteristic
175	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
176	#533	—	53	2	no	—
177	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
178	#500	—	53	1	no	—
179	#494	—	53	1	no	—
180	#487	—	53	2	no	—
181	#483	—	106	2	no	—
182	#475	—	53	4	no	—
183	#467	—	106	1	no	—
184	#467	—	106	1	no	—
185	#462	—	53	4	no	—
186	#451	—	106	2	no	—
187	C₈×Q₈	—	2	1	no	—
188	C₈×Q₈	—	2	1	no	—
189	C₈×Q₈	C₂×D₅₃	1	2	yes	—
190	C₈×Q₈	C₂×D₅₃	1	2	yes	—
191	#441	—	106	4	no	—
192	C₄×Q₁₆	—	106	1	no	—
193	C₄×Q₁₆	—	106	1	no	—
194	#432	—	53	2	no	—
195	C₂×C₄⋊C₈	—	2	1	no	—
196	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
197	C₂×C₄×C₈	—	2	1	no	—
198	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
199	#335	—	53	2	no	—
200	#334	—	53	2	no	—
201	#332	—	53	4	no	—
202	#332	—	53	4	no	—
203	C₅₃	#19265	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
205	C₂×Q₁₆	—	106	4	no	—
206	C₂×Q₁₆	—	106	1	no	—
207	C₂×Q₁₆	—	106	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
211	C₄⋊Q₈	—	53	2	no	—
212	C₄⋊Q₈	—	53	2	no	—
213	#125	—	106	2	no	—
214	C₂²⋊Q₈	—	106	2	no	—
215	C₂²⋊Q₈	—	106	2	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×Q₈	—	106	2	no	—
219	C₄×Q₈	—	106	2	no	—
220	C₄×Q₈	D₂₁₂	1	2	yes	—
221	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
222	#117	—	106	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
227	#107	—	106	1	no	—
228	#107	—	106	1	no	—
229	#106	—	106	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	—	53	4	no	—
234	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
235	#103	—	106	4	no	—
236	#103	—	106	2	no	—
237	#103	—	106	2	no	—
238	#103	—	106	4	no	—
239	C₂²⋊C₈	—	106	2	no	—
240	#97	—	106	2	no	—
241	C₄×C₈	—	2	2	no	—
242	C₄×C₈	—	2	2	no	—
243	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	106	2	no	—
250	C₂×Q₈	—	106	2	no	—
251	C₂×Q₈	C₂×D₂₁₂	1	2	yes	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
255	C₂²×C₄	#11719945	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
257	Q₁₆	—	212	2	no	—
258	Q₁₆	—	212	8	no	—
259	Q₁₆	—	212	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	106	4	no	—
265	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₄⋊C₄	—	106	2	no	—
276	C₄⋊C₄	—	212	2	no	—
277	C₄⋊C₄	—	212	2	no	—
278	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
279	C₂²⋊C₄	—	212	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	212	2	no	—
283	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
284	C₄²	#11719945	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
286	C₄²	C₂³×D₅₃	1	1	yes	characteristic
287	C₂³	#13133191	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	212	4	no	—
291	Q₈	—	212	2	no	—
292	Q₈	—	212	2	no	—
293	Q₈	#13133239	1	4	yes	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	212	2	no	—
303	C₂×C₄	—	212	2	no	—
304	C₂×C₄	#13133191	1	1	yes	characteristic
305	C₂×C₄	#13133191	1	1	yes	characteristic
306	C₂×C₄	#13133191	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
309	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	4	2	no	—
314	C₈	—	212	4	no	—
315	C₈	#13133248	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	#16810898	1	2	yes	—
319	C₂²	C₂×#13133191	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	4	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	424	2	no	—
327	C₄	—	424	2	no	—
328	C₄	C₂×#13133239	1	2	yes	—
329	C₄	C₂×#13133248	1	1	yes	characteristic
330	C₄	#16811421	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	#20284187	1	1	yes	characteristic
333	C₂	#20284296	1	1	yes	characteristic
334	C₂	C₂×#16810898	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	#121
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#19265
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #20274145	C₂	semidirect
4. #20274109	C₂	semidirect
5. #20274085	C₂	semidirect
203. C₅₃	#19265	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20284845	2
#20274145	2
#20274109	2
#20274085	2
#20273478	2
#20273413	2
C₂×C₈×Q₈×C₅₃	2
#19265	53