Subgroups — #27273625

Order 13568.

770 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	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272556	C₂	1	4	yes	maximal, complemented
4	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	—	2	1	no	—
6	#16810676	—	2	2	no	—
7	#16810676	C₄	1	4	yes	complemented
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₄⋊#88322	C₄	1	4	yes	complemented
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₂²⋊#88322	—	2	1	no	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂×#117×C₅₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
21	C₅₃×#376	—	2	1	no	—
22	C₄³×C₅₃	—	2	1	no	—
23	C₂²×#88322	—	2	4	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	—	4	4	no	—
27	C₄⋊#88322	—	2	4	no	—
28	C₄⋊#88322	—	2	4	no	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	C₂×C₄	1	4	yes	—
31	C₂²⋊#88322	—	2	4	no	—
32	C₄×#88322	—	2	2	no	—
33	C₄×#88322	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	#117×C₅₃	—	2	4	no	—
38	#117×C₅₃	C₈	1	8	yes	complemented
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#95×C₅₃	—	2	2	no	—
53	#95×C₅₃	—	2	2	no	—
54	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
75	C₂²⋊C₄×C₅₃	—	2	8	no	—
76	C₄²×C₅₃	—	2	8	no	—
77	C₄²×C₅₃	—	4	4	no	—
78	C₄²×C₅₃	—	4	2	no	—
79	C₄²×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	C₄²	1	2	yes	—
81	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	4	4	no	—
93	C₂×#88322	—	4	8	no	—
94	C₂×#88322	—	4	4	no	—
95	C₂³×C₅₃	—	2	2	no	—
96	C₂³×C₅₃	—	2	2	no	—
97	C₂³×C₅₃	#95	1	2	yes	—
98	C₂³×C₅₃	C₂²⋊C₈	1	4	yes	—
99	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	8	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	#95	1	2	yes	—
112	C₂×C₄×C₅₃	#95	1	2	yes	—
113	C₂×C₄×C₅₃	#95	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
115	C₂×C₄×C₅₃	#97	1	4	yes	—
116	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
117	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
118	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₅₃	#137	1	2	yes	—
128	#88322	—	4	4	no	—
129	#88322	—	4	4	no	—
130	#7891	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	8	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	#336	1	4	yes	—
135	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
137	C₂²×C₅₃	#417	1	2	yes	—
138	C₂²×C₅₃	#421	1	2	yes	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	4	8	no	—
144	C₄×C₅₃	#336	1	4	yes	—
145	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
147	C₄×C₅₃	#418	1	1	yes	characteristic
148	C₄×C₅₃	#419	1	1	yes	characteristic
149	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
150	C₄×C₅₃	#428	1	2	yes	—
151	C₂×#431	—	53	1	no	—
152	C₂²×C₄⋊C₈	—	53	1	no	—
153	C₄×#117	D₅₃	1	1	yes	characteristic
154	#1279	—	53	4	no	—
155	C₂×C₅₃	—	2	4	no	—
156	C₂×C₅₃	#1477	1	4	yes	—
157	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
158	C₂×C₅₃	#1733	1	1	yes	characteristic
159	C₂×C₅₃	#1736	1	1	yes	characteristic
160	C₂³×C₈	—	106	1	no	—
161	C₂×#117	Dic₅₃	1	2	yes	—
162	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
163	#431	—	53	4	no	—
164	#431	—	106	2	no	—
165	C₂×C₄⋊C₈	—	53	4	no	—
166	C₂×C₄⋊C₈	—	53	2	no	—
167	C₂×C₄⋊C₈	—	106	2	no	—
168	C₂×C₄⋊C₈	—	106	1	no	—
169	C₂×C₄⋊C₈	—	106	1	no	—
170	C₂×C₂²⋊C₈	—	106	1	no	—
171	C₂×C₄×C₈	—	53	2	no	—
172	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
173	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
174	C₄×C₂²⋊C₄	—	2	1	no	—
175	C₄×C₂²⋊C₄	—	2	1	no	—
176	#376	—	2	1	no	—
177	C₄³	—	2	1	no	—
178	C₅₃	#7891	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
180	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	—	106	4	no	—
185	C₂²×C₈	—	212	4	no	—
186	#117	—	2	4	no	—
187	#117	#88322	1	8	yes	complemented
188	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	Dic₁₀₆	1	2	yes	—
194	C₂×C₄²	C₄×D₅₃	1	2	yes	—
195	C₂×C₄²	C₄×D₅₃	1	2	yes	—
196	C₂×C₄²	D₂₁₂	1	2	yes	—
197	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	C₄⋊C₈	—	53	4	no	—
202	C₄⋊C₈	—	106	2	no	—
203	C₄⋊C₈	—	106	4	no	—
204	C₄⋊C₈	—	106	4	no	—
205	C₂²⋊C₈	—	106	4	no	—
206	C₄×C₈	—	53	4	no	—
207	C₄×C₈	—	106	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×#88322	1	4	yes	—
216	C₂²×C₄	#11719917	1	4	yes	—
217	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
218	C₂²×C₄	#11719919	1	2	yes	—
219	C₂²×C₄	#11719919	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
221	C₂²×C₄	#11719921	1	2	yes	—
222	C₂²×C₄	#11719921	1	2	yes	—
223	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	212	8	no	—
232	C₂×C₈	—	212	4	no	—
233	C₂×C₈	—	212	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	C₂×#88322	1	8	yes	complemented
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	4	4	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₄×Dic₅₃	1	2	yes	—
242	C₄²	#11719919	1	2	yes	—
243	C₄²	#11719919	1	2	yes	—
244	C₄²	C₄⋊Dic₅₃	1	2	yes	—
245	C₄²	#11719921	1	2	yes	—
246	C₄²	#11719921	1	2	yes	—
247	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
250	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	#13133124	1	2	yes	—
254	C₂³	C₂²⋊#88322	1	4	yes	—
255	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	#13133124	1	2	yes	—
268	C₂×C₄	#13133124	1	2	yes	—
269	C₂×C₄	#13133124	1	2	yes	—
270	C₂×C₄	C₄×#88322	1	4	yes	—
271	C₂×C₄	#13133128	1	4	yes	—
272	C₂×C₄	C₂²⋊#88322	1	4	yes	—
273	C₂×C₄	C₄⋊#88322	1	8	yes	—
274	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
276	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
277	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
278	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×#88322	1	2	yes	—
281	C₂×C₄	C₂×#11719917	1	2	yes	—
282	C₂×C₄	#13133242	1	2	yes	—
283	C₂×C₄	#13133247	1	2	yes	—
284	C₈	—	212	4	no	—
285	C₈	—	212	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	8	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#16810378	1	4	yes	—
290	C₂²	C₂×#13133124	1	1	yes	characteristic
291	C₂²	C₂×C₂²⋊#88322	1	2	yes	—
292	C₂²	#16810624	1	2	yes	—
293	C₂²	#16810640	1	2	yes	—
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	8	no	—
299	C₄	#16810378	1	4	yes	—
300	C₄	C₂×C₄×#88322	1	1	yes	characteristic
301	C₄	C₂×#13133128	1	1	yes	characteristic
302	C₄	#16810627	1	1	yes	characteristic
303	C₄	#16810631	1	1	yes	characteristic
304	C₄	C₂×C₄⋊#88322	1	2	yes	—
305	C₄	#16810666	1	2	yes	—
306	C₂	—	2	4	no	—
307	C₂	#20273102	1	4	yes	—
308	C₂	C₂×#16810378	1	1	yes	characteristic
309	C₂	#20274424	1	1	yes	characteristic
310	C₂	#20274436	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#7891
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20272556	C₂	semidirect
7. #16810676	C₄	semidirect
11. C₂×C₄⋊#88322	C₄	semidirect
38. #117×C₅₃	C₈	semidirect
74. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
178. C₅₃	#7891	semidirect
187. #117	#88322	semidirect
235. C₄⋊C₄	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄⋊#88322	2
#20272556	2
C₄×#117×C₅₃	2
#7891	53