Subgroups — #28311123

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719910	C₂	1	2	yes	maximal, complemented
2	C₂×#16811518	C₂	1	2	yes	maximal
3	#20289754	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	4	yes	—
6	C₄⋊#11719910	—	2	4	no	—
7	#16811518	—	2	4	no	—
8	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₅₃	—	2	4	no	—
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#11719910	—	2	4	no	—
17	C₂×#11719910	—	2	4	no	—
18	C₂×#11719910	—	2	4	no	—
19	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
20	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
21	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂³	1	1	yes	characteristic
24	C₂×C₈×D₅₃	—	2	2	no	—
25	C₂×C₈×D₅₃	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	2	no	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	—	2	2	no	—
29	C₂×C₈×D₅₃	D₄	1	2	yes	—
30	C₂×C₈×D₅₃	Q₈	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₅₃	—	2	2	no	—
35	C₄²×D₅₃	—	2	2	no	—
36	C₄×#88322	D₄	1	4	yes	—
37	C₈×Dic₅₃	—	2	2	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×D₅₃	—	2	2	no	—
51	C₂×C₄×D₅₃	—	2	2	no	—
52	C₂×C₄×D₅₃	—	2	2	no	—
53	C₂×C₄×D₅₃	—	2	2	no	—
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	—	2	1	no	—
57	C₂×C₄×D₅₃	—	2	1	no	—
58	C₂×C₄×D₅₃	—	2	2	no	—
59	C₂×C₄×D₅₃	C₁₆	1	4	yes	complemented
60	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
68	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅₃	M₁₆	1	2	yes	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₄×Dic₅₃	—	2	1	no	—
73	C₄×Dic₅₃	—	2	1	no	—
74	C₂×#88322	—	2	1	no	—
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	C₂²⋊C₄	1	2	yes	—
77	C₂×#88322	C₂×D₄	1	1	yes	characteristic
78	C₂×#88322	C₂×D₄	1	2	yes	—
79	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
80	C₂×#88322	#41	1	2	yes	—
81	C₈×D₅₃	—	2	8	no	—
82	C₈×D₅₃	—	2	4	no	—
83	C₈×D₅₃	—	2	4	no	—
84	#11719910	—	4	4	no	—
85	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂²×D₅₃	—	2	1	no	—
87	C₂²×D₅₃	—	2	1	no	—
88	C₂²×D₅₃	—	2	1	no	—
89	C₂²×D₅₃	—	2	1	no	—
90	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
91	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
92	C₂²×D₅₃	M₃₂	1	2	yes	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
98	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×Dic₅₃	—	2	1	no	—
106	C₂×Dic₅₃	—	2	1	no	—
107	C₂×Dic₅₃	—	2	1	no	—
108	C₂×Dic₅₃	—	2	1	no	—
109	C₂×Dic₅₃	—	2	1	no	—
110	C₂×Dic₅₃	—	2	1	no	—
111	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
112	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
113	C₂×Dic₅₃	M₃₂	1	2	yes	—
114	C₄×D₅₃	—	2	8	no	—
115	C₄×D₅₃	—	2	8	no	—
116	C₄×D₅₃	—	2	2	no	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	—	2	2	no	—
122	#88322	—	2	2	no	—
123	#88322	#121	1	2	yes	—
124	#88322	C₂²⋊Q₈	1	2	yes	—
125	#12783	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂²×C₅₃	#431	1	1	yes	characteristic
129	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	—	2	2	no	—
135	C₂×D₅₃	—	2	2	no	—
136	C₂×D₅₃	—	2	2	no	—
137	C₂×D₅₃	—	2	2	no	—
138	C₂×D₅₃	—	2	2	no	—
139	C₂×D₅₃	C₄⋊C₁₆	1	4	yes	complemented
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	C₂²⋊C₁₆	1	4	yes	complemented
142	C₄×C₅₃	#380	1	1	yes	characteristic
143	C₄×C₅₃	#407	1	1	yes	characteristic
144	C₄×C₅₃	#423	1	1	yes	characteristic
145	C₄×C₅₃	#432	1	1	yes	characteristic
146	Dic₅₃	—	2	2	no	—
147	Dic₅₃	—	2	2	no	—
148	Dic₅₃	—	2	2	no	—
149	C₂²×C₄×C₈	—	53	1	no	—
150	C₂×C₄⋊C₁₆	—	53	2	no	—
151	C₂×C₂²⋊C₁₆	—	53	2	no	—
152	#1374	—	53	2	no	—
153	C₂×C₅₃	#1797	1	1	yes	characteristic
154	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
155	C₂×C₅₃	#2112	1	1	yes	characteristic
156	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
157	C₂×C₅₃	#2150	1	1	yes	characteristic
158	C₂×C₅₃	#2162	1	1	yes	characteristic
159	C₂×C₅₃	#2163	1	1	yes	characteristic
160	D₅₃	—	2	4	no	—
161	C₂³×C₈	—	53	1	no	—
162	C₂³×C₈	—	53	1	no	—
163	C₂²×C₄²	—	53	1	no	—
164	C₂²×C₁₆	—	53	4	no	—
165	C₂×C₄×C₈	—	53	1	no	—
166	C₂×C₄×C₈	—	53	1	no	—
167	C₂×C₄×C₈	—	53	1	no	—
168	C₂×C₄×C₈	—	106	4	no	—
169	C₂×C₄×C₈	#6157	1	1	yes	characteristic
170	C₄⋊C₁₆	—	106	4	no	—
171	C₂²⋊C₁₆	—	106	4	no	—
172	C₅₃	#12783	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	53	1	no	—
174	C₂³×C₄	—	53	1	no	—
175	C₂³×C₄	—	53	1	no	—
176	C₂²×C₈	—	53	1	no	—
177	C₂²×C₈	—	53	1	no	—
178	C₂²×C₈	—	53	2	no	—
179	C₂²×C₈	—	53	2	no	—
180	C₂²×C₈	—	106	2	no	—
181	C₂²×C₈	—	106	2	no	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
186	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
187	C₂×C₄²	—	53	1	no	—
188	C₂×C₄²	—	53	1	no	—
189	C₂×C₄²	—	53	1	no	—
190	C₂×C₄²	—	106	2	no	—
191	C₂×C₄²	—	106	2	no	—
192	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
193	C₂×C₁₆	—	106	4	no	—
194	C₂×C₁₆	—	106	4	no	—
195	C₂×C₁₆	—	106	4	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	—	53	4	no	—
198	C₄×C₈	—	106	2	no	—
199	C₄×C₈	—	106	2	no	—
200	C₂⁴	—	53	1	no	—
201	C₂²×C₄	—	53	1	no	—
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	53	1	no	—
204	C₂²×C₄	—	53	4	no	—
205	C₂²×C₄	—	53	2	no	—
206	C₂²×C₄	—	53	2	no	—
207	C₂²×C₄	—	106	2	no	—
208	C₂²×C₄	—	106	2	no	—
209	C₂²×C₄	—	106	2	no	—
210	C₂²×C₄	—	106	2	no	—
211	C₂²×C₄	—	106	2	no	—
212	C₂²×C₄	—	106	1	no	—
213	C₂²×C₄	—	106	1	no	—
214	C₂²×C₄	—	106	1	no	—
215	C₂²×C₄	—	106	1	no	—
216	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
217	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
218	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	106	8	no	—
231	C₂×C₈	—	106	1	no	—
232	C₂×C₈	—	106	1	no	—
233	C₂×C₈	#11719936	1	2	yes	—
234	C₂×C₈	C₄⋊#6157	1	2	yes	—
235	C₄²	—	106	2	no	—
236	C₄²	—	106	2	no	—
237	C₄²	—	106	1	no	—
238	C₄²	—	106	1	no	—
239	C₄²	#11719936	1	2	yes	—
240	C₄²	C₂²⋊#6157	1	2	yes	—
241	C₁₆	—	212	4	no	—
242	C₂³	—	53	2	no	—
243	C₂³	—	53	2	no	—
244	C₂³	—	53	2	no	—
245	C₂³	—	106	1	no	—
246	C₂³	—	106	1	no	—
247	C₂³	—	106	1	no	—
248	C₂³	—	106	1	no	—
249	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	53	2	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	106	8	no	—
256	C₂×C₄	—	106	1	no	—
257	C₂×C₄	—	106	1	no	—
258	C₂×C₄	—	106	1	no	—
259	C₂×C₄	—	106	1	no	—
260	C₂×C₄	—	106	1	no	—
261	C₂×C₄	—	106	1	no	—
262	C₂×C₄	—	106	2	no	—
263	C₂×C₄	—	106	2	no	—
264	C₂×C₄	—	106	2	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	106	8	no	—
267	C₂×C₄	#13133274	1	2	yes	—
268	C₂×C₄	#13133274	1	2	yes	—
269	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
270	C₂×C₄	#13133292	1	2	yes	—
271	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
273	C₂×C₄	#13133296	1	1	yes	characteristic
274	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
275	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
276	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	53	2	no	—
280	C₈	—	53	2	no	—
281	C₈	—	106	2	no	—
282	C₂²	—	53	4	no	—
283	C₂²	—	106	2	no	—
284	C₂²	—	106	2	no	—
285	C₂²	—	106	2	no	—
286	C₂²	—	106	2	no	—
287	C₂²	—	106	2	no	—
288	C₂²	—	106	2	no	—
289	C₂²	C₂×#13133274	1	1	yes	characteristic
290	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
291	C₂²	#16811642	1	1	yes	characteristic
292	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
293	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
294	C₂²	D₅₃⋊M₃₂	1	1	yes	characteristic
295	C₂²	C₂×#13133292	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	106	2	no	—
298	C₄	—	106	2	no	—
299	C₄	—	106	2	no	—
300	C₄	#16811518	1	4	yes	complemented
301	C₄	#16811565	1	1	yes	characteristic
302	C₄	#16811594	1	1	yes	characteristic
303	C₄	#16811627	1	1	yes	characteristic
304	C₄	#16811645	1	1	yes	characteristic
305	C₂	—	106	4	no	—
306	C₂	#20290570	1	1	yes	characteristic
307	C₂	C₂×#16811518	1	1	yes	characteristic
308	C₂	#20290944	1	1	yes	characteristic
309	C₂	D₅₃⋊(C₄⋊C₁₆)	1	1	yes	characteristic
310	C₂	#20291048	1	1	yes	characteristic
311	C₂	#20291075	1	1	yes	characteristic
312	C₂	#20291079	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12783
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#11719910	C₂	semidirect
3. #20289754	C₂	semidirect
59. C₂×C₄×D₅₃	C₁₆	semidirect
139. C₂×D₅₃	C₄⋊C₁₆	semidirect
141. C₄×C₅₃	C₂²⋊C₁₆	semidirect
172. C₅₃	#12783	semidirect
300. C₄	#16811518	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719910	2
C₂×#16811518	2
#20289754	2
C₂×C₄×C₈×D₅₃	2
#12783	53