Subgroups — #27299145

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810407	C₂	1	2	yes	maximal
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272804	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	#16810407	C₄	1	8	yes	complemented
9	#16810399	—	2	1	no	—
10	#16810399	—	2	1	no	—
11	#16810399	—	2	1	no	—
12	C₄×#11719908	—	2	1	no	—
13	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
16	C₅₃×#405	—	2	1	no	—
17	C₅₃×#405	—	2	1	no	—
18	C₅₃×#405	C₄	1	1	yes	characteristic
19	C₅₃×#405	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₅₃	—	2	1	no	—
21	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#13133157	—	2	2	no	—
25	#13133157	C₂×C₄	1	4	yes	complemented
26	C₂×#11719908	—	2	2	no	—
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	C₂×C₄	1	4	yes	complemented
30	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#131×C₅₃	—	2	2	no	—
32	#131×C₅₃	—	2	2	no	—
33	#131×C₅₃	C₈	1	4	yes	—
34	#131×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	—	2	1	no	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	—	2	1	no	—
40	C₂²×C₈×C₅₃	—	2	1	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₅₃	C₈	1	2	yes	—
44	C₄×D₄×C₅₃	—	2	2	no	—
45	C₄×D₄×C₅₃	C₈	1	2	yes	—
46	#117×C₅₃	—	2	1	no	—
47	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	#97×C₅₃	—	2	1	no	—
51	#97×C₅₃	—	2	1	no	—
52	#97×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	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	#41×C₅₃	—	2	1	no	—
61	#41×C₅₃	—	2	1	no	—
62	#41×C₅₃	C₄²	1	2	yes	—
63	#41×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₅₃	—	2	1	no	—
66	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	M₁₆×C₅₃	—	2	1	no	—
72	M₁₆×C₅₃	—	2	1	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	C₄²	1	2	yes	—
76	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
84	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₅₃	—	2	1	no	—
92	C₂²⋊C₄×C₅₃	—	2	2	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×C₈	1	2	yes	—
98	#11719908	—	2	4	no	—
99	#11719908	—	2	4	no	—
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	1	no	—
103	Q₈×C₅₃	C₄×C₈	1	2	yes	—
104	D₄×C₅₃	—	2	2	no	—
105	D₄×C₅₃	—	2	2	no	—
106	D₄×C₅₃	—	2	1	no	—
107	D₄×C₅₃	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	—	2	2	no	—
124	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
125	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
126	C₈×C₅₃	C₄×D₄	1	2	yes	—
127	C₈×C₅₃	C₄×D₄	1	2	yes	—
128	#12604	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₅₃	#431	1	1	yes	characteristic
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	4	2	no	—
139	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₄×C₅₃	#431	1	1	yes	characteristic
143	C₄×#131	D₅₃	1	1	yes	characteristic
144	C₂×#350	—	53	2	no	—
145	C₄×M₃₂	—	53	1	no	—
146	C₂×#346	—	53	1	no	—
147	#1379	—	53	2	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	4	2	no	—
151	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
152	C₂×C₅₃	#2117	1	2	yes	—
153	C₂×#131	Dic₅₃	1	1	yes	characteristic
154	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
155	C₄×#41	Dic₅₃	1	1	yes	characteristic
156	C₂×M₃₂	—	53	2	no	—
157	C₂²×C₁₆	—	53	2	no	—
158	#405	—	2	1	no	—
159	#405	—	2	1	no	—
160	#405	Dic₅₃	1	1	yes	characteristic
161	#405	C₂×D₅₃	1	1	yes	characteristic
162	C₄×M₁₆	—	2	1	no	—
163	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
164	C₂×C₄×C₈	—	2	1	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#350	—	53	8	no	—
167	#346	—	106	1	no	—
168	#346	—	106	1	no	—
169	#346	—	106	1	no	—
170	C₄×C₁₆	—	106	1	no	—
171	C₅₃	#12604	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
173	#131	—	2	2	no	—
174	#131	—	2	2	no	—
175	#131	#88322	1	4	yes	—
176	#131	C₄×D₅₃	1	4	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
180	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₄×Q₈	#88322	1	2	yes	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	#88322	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	C₂×Dic₅₃	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
192	M₃₂	—	53	4	no	—
193	M₃₂	—	106	2	no	—
194	C₂×C₁₆	—	53	4	no	—
195	C₂×C₁₆	—	106	2	no	—
196	C₂×C₁₆	—	106	2	no	—
197	C₂×C₁₆	—	106	2	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	C₂×#88322	1	2	yes	—
211	#41	C₄×Dic₅₃	1	2	yes	—
212	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	C₂×#88322	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	#11719917	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	C₂×#88322	1	2	yes	—
224	M₁₆	C₄×Dic₅₃	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	C₂×#88322	1	2	yes	—
230	C₂×C₈	#11719917	1	2	yes	—
231	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
234	C₂×C₈	#11719945	1	2	yes	—
235	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×#88322	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂×#88322	1	2	yes	—
244	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
245	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
246	C₁₆	—	106	4	no	—
247	C₁₆	—	106	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₂×#11719917	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×#88322	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×#88322	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊#88322	1	2	yes	—
263	C₂×C₄	C₂²⋊#88322	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
266	C₂×C₄	#13133176	1	1	yes	characteristic
267	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
268	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	C₂²⋊#88322	1	2	yes	—
273	C₈	C₂²⋊#88322	1	2	yes	—
274	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
275	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×#11719917	1	1	yes	characteristic
280	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
281	C₂²	#16810676	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	2	no	—
286	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
287	C₄	C₂×C₄×#88322	1	1	yes	characteristic
288	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
289	C₄	#16810676	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊#88322	1	1	yes	characteristic
294	C₂	#20276343	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#12604
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. C₄×#13133157	C₂	semidirect
3. C₂×#16810399	C₂	semidirect
4. #20272804	C₂	semidirect
8. #16810407	C₄	semidirect
25. #13133157	C₂×C₄	semidirect
29. C₂×#11719908	C₂×C₄	semidirect
171. C₅₃	#12604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810407	2
C₄×#13133157	2
C₂×#16810399	2
#20272804	2
C₄×#131×C₅₃	2
#12604	53