Subgroups — #27301490

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810439	C₂	1	2	yes	maximal, complemented
3	#20274753	C₂	1	1	yes	maximal, characteristic
4	#20272823	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#467	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	—	2	4	no	—
7	C₂×#13133157	—	2	1	no	—
8	C₂×#13133157	C₂²	1	2	yes	—
9	C₂²×#11719908	—	2	1	no	—
10	C₂×#13133153	C₂²	1	2	yes	—
11	#16810587	—	2	1	no	—
12	#16810439	—	2	4	no	—
13	#16810388	—	2	1	no	—
14	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
17	C₅₃×#467	—	2	1	no	—
18	C₅₃×#467	—	2	1	no	—
19	C₅₃×#467	C₄	1	2	yes	—
20	C₅₃×#467	C₄	1	2	yes	—
21	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#103×C₅₃	—	2	1	no	—
23	C₂×#11719917	—	2	2	no	—
24	#13133157	—	2	4	no	—
25	#13133157	—	2	2	no	—
26	#13133157	D₄	1	4	yes	—
27	C₂×#11719908	—	2	4	no	—
28	C₂×#11719908	—	2	2	no	—
29	#13133153	—	2	2	no	—
30	#13133153	—	2	2	no	—
31	C₂²⋊#88322	—	4	1	no	—
32	C₂²⋊#88322	—	4	1	no	—
33	C₂²×Q₈×C₅₃	—	2	1	no	—
34	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₅₃	—	2	1	no	—
36	C₂×Q₁₆×C₅₃	—	2	2	no	—
37	C₂×Q₁₆×C₅₃	—	2	2	no	—
38	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²⋊Q₈×C₅₃	—	2	2	no	—
48	C₂²⋊Q₈×C₅₃	—	2	2	no	—
49	C₂²⋊Q₈×C₅₃	—	4	1	no	—
50	C₂²⋊Q₈×C₅₃	—	4	1	no	—
51	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
53	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
54	#107×C₅₃	—	2	1	no	—
55	#107×C₅₃	C₂×C₄	1	2	yes	—
56	#103×C₅₃	—	2	2	no	—
57	#103×C₅₃	—	2	2	no	—
58	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
59	C₂×Q₈×C₅₃	—	2	2	no	—
60	C₂×Q₈×C₅₃	—	4	1	no	—
61	C₂×Q₈×C₅₃	—	4	2	no	—
62	C₂×Q₈×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	—	4	1	no	—
66	C₂²×C₄×C₅₃	—	4	1	no	—
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₅₃	—	2	4	no	—
74	Q₁₆×C₅₃	—	4	1	no	—
75	Q₁₆×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	1	no	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	#41	1	2	yes	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄⋊C₄×C₅₃	—	4	2	no	—
92	C₄⋊C₄×C₅₃	—	4	1	no	—
93	C₂²⋊C₄×C₅₃	—	4	2	no	—
94	C₂²⋊C₄×C₅₃	—	4	2	no	—
95	#11719917	—	4	2	no	—
96	#11719917	—	4	2	no	—
97	C₂×#88322	—	4	2	no	—
98	#11719908	—	2	4	no	—
99	C₂³×C₅₃	—	2	2	no	—
100	C₂³×C₅₃	—	2	2	no	—
101	C₂³×C₅₃	—	4	1	no	—
102	C₂³×C₅₃	#102	1	1	yes	characteristic
103	C₂³×C₅₃	#102	1	1	yes	characteristic
104	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	Q₈×C₅₃	—	4	2	no	—
106	Q₈×C₅₃	—	4	2	no	—
107	Q₈×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	8	2	no	—
118	C₂×C₄×C₅₃	—	8	2	no	—
119	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#121	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#135	1	2	yes	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	1	no	—
127	C₈×C₅₃	C₄×D₄	1	2	yes	—
128	C₈×C₅₃	#120	1	1	yes	characteristic
129	C₈×C₅₃	#123	1	1	yes	characteristic
130	#88322	—	8	2	no	—
131	#13066	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	4	4	no	—
140	C₂²×C₅₃	#362	1	2	yes	complemented
141	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
142	C₂²×C₅₃	#465	1	2	yes	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	8	2	no	—
146	C₄×C₅₃	—	8	2	no	—
147	C₄×C₅₃	#386	1	1	yes	characteristic
148	C₄×C₅₃	#416	1	1	yes	characteristic
149	C₄×C₅₃	#466	1	2	yes	—
150	C₂²×M₃₂	—	53	1	no	—
151	C₂×#467	D₅₃	1	1	yes	characteristic
152	C₂×#362	—	53	2	no	—
153	#1810	—	53	1	no	—
154	#1387	—	53	2	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	—	4	1	no	—
158	C₂×C₅₃	#1893	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
160	C₂×C₅₃	#2148	1	1	yes	characteristic
161	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
162	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
163	C₂×C₂²⋊Q₈	—	2	1	no	—
164	C₂×M₃₂	—	53	2	no	—
165	C₂×M₃₂	—	106	4	no	—
166	C₂×M₃₂	—	106	1	no	—
167	C₂²×C₁₆	—	106	1	no	—
168	#467	—	2	1	no	—
169	#467	—	2	1	no	—
170	#467	Dic₅₃	1	2	yes	—
171	#467	Dic₅₃	1	2	yes	—
172	C₂×#108	—	53	2	no	—
173	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
174	C₂×#103	—	2	1	no	—
175	#407	—	106	1	no	—
176	#362	—	106	4	no	—
177	#341	—	106	1	no	—
178	C₅₃	#13066	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	2	1	no	—
180	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
185	C₂×M₁₆	—	106	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
194	C₂²⋊Q₈	—	2	2	no	—
195	C₂²⋊Q₈	—	2	2	no	—
196	C₂²⋊Q₈	—	4	1	no	—
197	C₂²⋊Q₈	—	4	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	4	1	no	—
200	C₂×C₂²⋊C₄	—	4	1	no	—
201	M₃₂	—	53	4	no	—
202	M₃₂	—	106	4	no	—
203	M₃₂	—	106	2	no	—
204	C₂×C₁₆	—	106	4	no	—
205	C₂×C₁₆	—	106	2	no	—
206	#108	—	106	2	no	—
207	#108	—	106	2	no	—
208	#107	—	2	1	no	—
209	#107	C₂×Dic₅₃	1	2	yes	—
210	#103	—	2	2	no	—
211	#103	—	2	2	no	—
212	C₂²⋊C₈	—	212	1	no	—
213	C₂²⋊C₈	—	212	1	no	—
214	C₂⁴	#11719923	1	1	yes	characteristic
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	#11719922	1	1	yes	characteristic
224	C₂²×C₄	#11719922	1	1	yes	characteristic
225	C₂²×C₄	#11719923	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	Q₁₆	—	2	4	no	—
230	Q₁₆	—	4	1	no	—
231	Q₁₆	—	4	1	no	—
232	M₁₆	—	212	2	no	—
233	M₁₆	—	212	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	212	2	no	—
242	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
244	C₂×C₈	D₄×D₅₃	1	2	yes	—
245	C₂×C₈	#11719947	1	2	yes	—
246	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₁₆	—	106	4	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	#13133138	1	1	yes	characteristic
259	C₂³	#13133138	1	1	yes	characteristic
260	C₂³	C₂×#11719923	1	1	yes	characteristic
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	8	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	#13133138	1	1	yes	characteristic
276	C₂×C₄	#13133138	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
278	C₂×C₄	#13133189	1	2	yes	—
279	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
280	C₂×C₄	#13133237	1	2	yes	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	424	2	no	—
284	C₈	D₄×Dic₅₃	1	2	yes	—
285	C₈	#13133185	1	1	yes	characteristic
286	C₈	#13133198	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	#16810439	1	2	yes	complemented
296	C₂²	C₂×#13133138	1	1	yes	characteristic
297	C₂²	#16810850	1	2	yes	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	#16810503	1	1	yes	characteristic
303	C₄	#16810619	1	1	yes	characteristic
304	C₄	#16810857	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	1	no	—
308	C₂	#20275144	1	1	yes	characteristic
309	C₂	C₂×#16810439	1	1	yes	characteristic
310	C₂	#20276464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#467	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#467	D₅₃
hypocenter	C₅₃	#13066
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#16810439	C₂	semidirect
4. #20272823	C₂	semidirect
140. C₂²×C₅₃	#362	semidirect
178. C₅₃	#13066	semidirect
295. C₂²	#16810439	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂×#16810439	2
#20274753	2
#20272823	2
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