Subgroups — #27299137

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810404	C₂	1	2	yes	maximal
3	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272798	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
8	C₂×#13133174	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810577	—	2	1	no	—
11	#16810577	—	2	1	no	—
12	#16810577	—	2	1	no	—
13	#16810577	—	2	1	no	—
14	C₂×#13133128	C₄	1	1	yes	characteristic
15	C₂×#13133127	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
17	#16810404	C₄	1	8	yes	complemented
18	#16810399	—	2	1	no	—
19	C₄×#11719908	—	2	1	no	—
20	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#346	—	2	1	no	—
23	C₄×C₁₆×C₅₃	—	2	1	no	—
24	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×#11719912	—	2	1	no	—
26	C₂×#11719912	—	2	1	no	—
27	C₂×#11719912	—	2	2	no	—
28	C₂×#11719912	C₂×C₄	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	—	2	1	no	—
32	C₂×C₈×D₅₃	—	2	1	no	—
33	C₂×C₈×D₅₃	—	2	2	no	—
34	C₂×C₈×D₅₃	C₈	1	4	yes	—
35	#13133174	—	2	1	no	—
36	#13133174	—	2	1	no	—
37	#13133174	—	2	2	no	—
38	C₂×#11719919	C₈	1	2	yes	—
39	C₂×#11719921	C₈	1	2	yes	—
40	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	#13133157	—	2	2	no	—
42	#13133157	C₂×C₄	1	4	yes	—
43	C₂×#11719908	—	2	2	no	—
44	#13133128	—	2	1	no	—
45	#13133128	—	2	1	no	—
46	#13133127	—	2	1	no	—
47	#13133127	—	2	1	no	—
48	C₈×Dic₅₃	—	2	1	no	—
49	C₈×Dic₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₅₃	—	2	2	no	—
54	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×C₁₆×C₅₃	—	2	2	no	—
56	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₅₃	—	2	1	no	—
66	C₂×C₄×D₅₃	—	2	1	no	—
67	C₂×C₄×D₅₃	—	2	1	no	—
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	C₄²	1	2	yes	—
70	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₅₃	M₁₆	1	2	yes	—
82	#11719921	—	2	2	no	—
83	#11719921	—	2	2	no	—
84	#11719919	—	2	2	no	—
85	#11719919	—	2	2	no	—
86	C₄×Dic₅₃	—	2	1	no	—
87	C₄×Dic₅₃	—	2	1	no	—
88	C₂×#88322	—	2	1	no	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	C₄²	1	2	yes	—
93	C₂×#88322	C₂×C₈	1	2	yes	—
94	#11719912	—	2	4	no	—
95	#11719912	—	2	4	no	—
96	C₈×D₅₃	—	2	4	no	—
97	C₈×D₅₃	—	2	4	no	—
98	C₁₆×C₅₃	—	2	4	no	—
99	#11719908	—	2	4	no	—
100	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂²×D₅₃	—	2	1	no	—
102	C₂²×D₅₃	—	2	1	no	—
103	C₂²×D₅₃	—	4	2	no	—
104	C₂²×D₅₃	C₄×C₈	1	2	yes	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×Dic₅₃	—	2	1	no	—
115	C₂×Dic₅₃	—	2	1	no	—
116	C₂×Dic₅₃	—	2	2	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
119	C₄×D₅₃	—	2	2	no	—
120	C₄×D₅₃	—	2	2	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₄×D₅₃	—	2	2	no	—
123	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
124	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
125	C₈×C₅₃	C₄×D₄	1	2	yes	—
126	C₈×C₅₃	C₄×D₄	1	2	yes	—
127	#88322	—	2	2	no	—
128	#88322	—	2	2	no	—
129	#12602	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₅₃	#431	1	1	yes	characteristic
134	C₂×D₅₃	—	2	2	no	—
135	C₂×D₅₃	—	4	2	no	—
136	C₂×D₅₃	—	4	2	no	—
137	C₂×D₅₃	—	4	2	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
142	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₅₃	#431	1	1	yes	characteristic
144	Dic₅₃	—	2	2	no	—
145	Dic₅₃	—	4	2	no	—
146	C₂×#405	—	53	1	no	—
147	C₂×#349	—	53	2	no	—
148	C₄×M₃₂	—	53	1	no	—
149	C₄×M₃₂	D₅₃	1	1	yes	characteristic, complemented
150	#1377	—	53	2	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
154	C₂×C₅₃	#2115	1	2	yes	—
155	D₅₃	—	4	2	no	—
156	C₂²×M₁₆	—	53	1	no	—
157	C₂³×C₈	—	53	1	no	—
158	C₂×#117	—	53	1	no	—
159	C₂×M₃₂	—	53	2	no	—
160	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
161	#405	—	106	1	no	—
162	#405	—	106	1	no	—
163	#405	—	106	1	no	—
164	#405	—	106	1	no	—
165	C₂×#97	—	53	1	no	—
166	C₂×#97	—	53	1	no	—
167	C₂×C₄×C₈	—	53	1	no	—
168	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
169	#349	—	53	8	no	—
170	#346	—	2	1	no	—
171	#346	—	106	1	no	—
172	C₄×C₁₆	—	2	1	no	—
173	C₄×C₁₆	—	106	1	no	—
174	C₅₃	#12602	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	53	1	no	—
176	C₂×M₁₆	—	53	4	no	—
177	C₂×M₁₆	—	106	1	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	—	106	1	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	4	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	—	106	1	no	—
185	C₂²×C₈	—	106	1	no	—
186	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	#117	—	106	2	no	—
189	#117	—	106	1	no	—
190	#117	—	106	1	no	—
191	C₂×C₄⋊C₄	—	53	2	no	—
192	C₂×C₂²⋊C₄	—	53	2	no	—
193	C₂×C₄²	—	53	1	no	—
194	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
195	M₃₂	—	2	2	no	—
196	M₃₂	—	53	4	no	—
197	M₃₂	—	106	2	no	—
198	M₃₂	C₄×D₅₃	1	4	yes	—
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	106	2	no	—
201	#97	—	106	1	no	—
202	#97	—	106	1	no	—
203	#97	—	106	1	no	—
204	#97	—	106	1	no	—
205	C₄×C₈	—	106	1	no	—
206	C₄×C₈	—	106	1	no	—
207	C₄×C₈	D₂₁₂	1	1	yes	characteristic
208	C₄×C₈	D₂₁₂	1	1	yes	characteristic
209	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
211	C₂⁴	—	53	1	no	—
212	C₂²×C₄	—	53	1	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	106	1	no	—
217	C₂²×C₄	—	106	1	no	—
218	C₂²×C₄	—	106	1	no	—
219	C₂²×C₄	—	106	1	no	—
220	C₂²×C₄	C₈×D₅₃	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	M₁₆	—	106	4	no	—
223	M₁₆	—	106	4	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	C₈×D₅₃	1	2	yes	—
233	C₂×C₈	#11719912	1	2	yes	—
234	C₂×C₈	#11719921	1	1	yes	characteristic
235	C₂×C₈	#11719921	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
237	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
238	C₂×C₈	#11719945	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
240	C₄⋊C₄	—	106	2	no	—
241	C₄⋊C₄	—	106	2	no	—
242	C₂²⋊C₄	—	106	2	no	—
243	C₂²⋊C₄	—	106	2	no	—
244	C₄²	—	106	1	no	—
245	C₄²	—	106	1	no	—
246	C₄²	#11719912	1	2	yes	—
247	C₄²	#11719921	1	1	yes	characteristic
248	C₄²	#11719921	1	1	yes	characteristic
249	C₁₆	—	2	4	no	—
250	C₁₆	—	106	4	no	—
251	C₂³	—	53	2	no	—
252	C₂³	—	106	1	no	—
253	C₂³	—	106	1	no	—
254	C₂³	—	212	2	no	—
255	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	53	2	no	—
258	C₂×C₄	—	106	1	no	—
259	C₂×C₄	—	106	1	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	106	2	no	—
262	C₂×C₄	—	106	2	no	—
263	C₂×C₄	—	106	2	no	—
264	C₂×C₄	—	106	2	no	—
265	C₂×C₄	—	212	2	no	—
266	C₂×C₄	#13133129	1	2	yes	—
267	C₂×C₄	#13133129	1	2	yes	—
268	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
270	C₂×C₄	#13133174	1	1	yes	characteristic
271	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
273	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
274	C₈	—	106	2	no	—
275	C₈	—	106	2	no	—
276	C₈	#13133129	1	2	yes	—
277	C₈	#13133129	1	2	yes	—
278	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
279	C₈	C₄×D₂₁₂	1	2	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	106	2	no	—
282	C₂²	—	212	2	no	—
283	C₂²	—	212	2	no	—
284	C₂²	—	212	2	no	—
285	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
286	C₂²	C₂×#13133129	1	1	yes	characteristic
287	C₂²	#16810674	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	106	2	no	—
291	C₄	—	212	2	no	—
292	C₄	C₄×#11719921	1	1	yes	characteristic
293	C₄	C₄×#11719912	1	1	yes	characteristic
294	C₄	C₂×#13133129	1	1	yes	characteristic
295	C₄	#16810674	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	212	2	no	—
299	C₂	C₄×#13133129	1	1	yes	characteristic
300	C₂	#20276335	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#11719921
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12602
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×#13133157	C₂	semidirect
4. #20272798	C₂	semidirect
5. C₄×M₃₂×C₅₃	C₂	semidirect
17. #16810404	C₄	semidirect
149. C₄×M₃₂	D₅₃	semidirect
174. C₅₃	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810577	2
C₂×#16810404	2
C₄×#13133157	2
#20272798	2
C₄×M₃₂×C₅₃	2
#12602	53