Subgroups — #27283338

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811000	C₂	1	1	yes	maximal, characteristic
2	#20275642	C₂	1	2	yes	maximal, complemented
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810366	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₂₄	—	2	1	no	—
7	C₂²×D₄₂₄	—	2	1	no	—
8	C₂×#13133214	C₂²	1	2	yes	—
9	#16811000	—	2	2	no	—
10	#16811000	C₄	1	4	yes	complemented
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×#13133137	—	2	4	no	—
13	#16810376	—	2	2	no	—
14	#16810366	—	2	4	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#336	—	2	1	no	—
20	C₂²×D₂₁₂	—	2	2	no	—
21	C₂²×D₂₁₂	—	2	2	no	—
22	C₂²×D₂₁₂	—	2	2	no	—
23	C₂×D₄₂₄	—	2	4	no	—
24	C₂×D₄₂₄	—	2	4	no	—
25	C₂×D₄₂₄	—	4	2	no	—
26	C₂×D₄₂₄	—	4	2	no	—
27	C₂×D₄₂₄	—	4	2	no	—
28	C₂×D₄₂₄	—	4	2	no	—
29	C₂²×#88322	—	2	2	no	—
30	#13133214	—	2	2	no	—
31	#13133214	—	2	2	no	—
32	#13133214	—	2	2	no	—
33	#13133214	C₂×C₄	1	4	yes	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	#13133137	—	4	8	no	—
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	1	no	—
44	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	—	4	2	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	#95×C₅₃	—	2	1	no	—
49	C₂³×D₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×D₂₁₂	—	2	4	no	—
55	C₂×D₂₁₂	—	2	4	no	—
56	C₂×D₂₁₂	—	2	4	no	—
57	C₂×D₂₁₂	—	4	4	no	—
58	C₂×D₂₁₂	—	4	2	no	—
59	C₂×D₂₁₂	—	4	2	no	—
60	C₂×D₂₁₂	—	4	4	no	—
61	C₂×D₂₁₂	—	4	2	no	—
62	C₂×D₂₁₂	—	4	2	no	—
63	C₂×D₂₁₂	—	4	2	no	—
64	C₂×D₂₁₂	—	4	2	no	—
65	C₂×D₂₁₂	—	4	2	no	—
66	C₂×D₂₁₂	—	8	4	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	D₈	1	2	yes	—
75	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	D₈	1	4	yes	complemented
78	C₄²×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	—	4	1	no	—
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	#41	1	1	yes	characteristic
84	C₂×#88322	—	4	2	no	—
85	C₂×#88322	—	4	4	no	—
86	D₄₂₄	—	4	8	no	—
87	D₄₂₄	—	4	8	no	—
88	C₂³×C₅₃	#121	1	1	yes	characteristic
89	C₂²×D₅₃	—	4	4	no	—
90	C₂²×D₅₃	—	8	2	no	—
91	C₂²×D₅₃	—	8	2	no	—
92	C₂²×D₅₃	—	8	4	no	—
93	C₂²×D₅₃	—	8	4	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#135	1	2	yes	—
108	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	D₂₁₂	—	4	4	no	—
111	D₂₁₂	—	4	4	no	—
112	D₂₁₂	—	4	8	no	—
113	D₂₁₂	—	8	4	no	—
114	D₂₁₂	—	8	4	no	—
115	D₂₁₂	—	8	2	no	—
116	D₂₁₂	—	8	2	no	—
117	D₂₁₂	—	8	2	no	—
118	D₂₁₂	—	8	2	no	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	4	2	no	—
121	C₈×C₅₃	#102	1	4	yes	complemented
122	#88322	—	8	2	no	—
123	#9734	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	#390	1	1	yes	characteristic
125	C₂²×C₅₃	#459	1	1	yes	characteristic
126	C₂²×C₅₃	#460	1	1	yes	characteristic
127	C₂²×C₅₃	#466	1	2	yes	—
128	C₂²×C₅₃	#472	1	2	yes	—
129	C₂×D₅₃	—	8	4	no	—
130	C₂×D₅₃	—	8	4	no	—
131	C₂×D₅₃	—	8	4	no	—
132	C₂×D₅₃	—	16	4	no	—
133	C₂×D₅₃	—	16	4	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	4	4	no	—
137	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
138	C₄×C₅₃	#418	1	1	yes	characteristic
139	C₄×C₅₃	C₄×D₈	1	2	yes	—
140	C₄×C₅₃	#459	1	1	yes	characteristic
141	C₄×C₅₃	#460	1	1	yes	characteristic
142	C₄×C₅₃	#493	1	1	yes	characteristic
143	C₄×C₅₃	#495	1	1	yes	characteristic
144	C₂×#493	—	53	1	no	—
145	#1982	—	53	2	no	—
146	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
147	C₂×#335	—	53	1	no	—
148	C₂×#331	—	53	2	no	—
149	C₂×C₅₃	#1667	1	2	yes	—
150	C₂×C₅₃	#1668	1	2	yes	—
151	C₂×C₅₃	#1970	1	1	yes	characteristic
152	C₂×C₅₃	#1978	1	1	yes	characteristic
153	C₂×C₅₃	#1982	1	1	yes	characteristic
154	D₅₃	—	16	4	no	—
155	C₂²×D₈	—	106	1	no	—
156	C₂²×D₈	—	106	1	no	—
157	C₂×#127	—	53	2	no	—
158	#493	—	53	4	no	—
159	#493	—	106	2	no	—
160	C₂×C₄⋊C₈	—	53	2	no	—
161	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
162	C₂×#102	—	106	4	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
166	#336	—	2	1	no	—
167	#335	—	106	2	no	—
168	#331	—	106	4	no	—
169	C₅₃	#9734	1	1	yes	characteristic, complemented, Sylow
170	C₂²×D₄	—	106	2	no	—
171	C₂²×D₄	—	106	2	no	—
172	C₂²×D₄	—	106	2	no	—
173	C₂×D₈	—	106	4	no	—
174	C₂×D₈	—	106	4	no	—
175	C₂×D₈	—	212	2	no	—
176	C₂×D₈	—	212	2	no	—
177	C₂×D₈	—	212	2	no	—
178	C₂×D₈	—	212	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	106	2	no	—
181	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
183	#127	—	53	4	no	—
184	#127	—	106	2	no	—
185	#127	—	106	2	no	—
186	#127	—	106	2	no	—
187	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	106	2	no	—
192	C₄⋊C₈	—	106	2	no	—
193	C₄⋊C₈	D₂₁₂	1	2	yes	—
194	#102	—	212	8	no	—
195	C₄×C₈	—	4	2	no	—
196	C₄×C₈	C₄×D₅₃	1	2	yes	—
197	C₄×C₈	D₂₁₂	1	2	yes	—
198	#95	—	2	1	no	—
199	C₂⁴	—	212	2	no	—
200	C₂×D₄	—	106	4	no	—
201	C₂×D₄	—	106	4	no	—
202	C₂×D₄	—	106	4	no	—
203	C₂×D₄	—	424	4	no	—
204	C₂×D₄	—	212	2	no	—
205	C₂×D₄	—	212	2	no	—
206	C₂×D₄	—	212	2	no	—
207	C₂×D₄	—	212	2	no	—
208	C₂×D₄	—	212	4	no	—
209	C₂×D₄	—	212	2	no	—
210	C₂×D₄	—	212	4	no	—
211	C₂×D₄	—	212	2	no	—
212	C₂×D₄	—	212	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
215	C₂²×C₄	#11719945	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
217	D₈	—	212	8	no	—
218	D₈	—	212	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	212	2	no	—
225	C₂×C₈	—	212	4	no	—
226	C₂×C₈	#11719921	1	2	yes	—
227	C₂×C₈	#11719921	1	2	yes	—
228	C₂×C₈	#11719922	1	2	yes	—
229	C₂×C₈	#11719924	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	D₄₂₄	1	4	yes	complemented
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
235	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
236	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
237	C₄²	#11719945	1	1	yes	characteristic
238	C₂³	—	424	2	no	—
239	C₂³	—	424	2	no	—
240	C₂³	—	424	4	no	—
241	C₂³	—	424	4	no	—
242	C₂³	—	212	4	no	—
243	C₂³	#13133191	1	1	yes	characteristic
244	D₄	—	424	4	no	—
245	D₄	—	424	4	no	—
246	D₄	—	424	2	no	—
247	D₄	—	424	2	no	—
248	D₄	—	424	2	no	—
249	D₄	—	424	2	no	—
250	D₄	—	212	4	no	—
251	D₄	—	212	4	no	—
252	D₄	—	212	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
258	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
260	C₂×C₄	#13133191	1	1	yes	characteristic
261	C₂×C₄	#13133205	1	1	yes	characteristic
262	C₂×C₄	#13133214	1	1	yes	characteristic
263	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
265	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
266	C₂×C₄	#13133239	1	2	yes	—
267	C₂×C₄	#13133242	1	1	yes	characteristic
268	C₂×C₄	#13133242	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	424	2	no	—
272	C₈	#13133137	1	4	yes	complemented
273	C₂²	—	848	4	no	—
274	C₂²	—	848	4	no	—
275	C₂²	—	424	4	no	—
276	C₂²	—	424	4	no	—
277	C₂²	—	424	4	no	—
278	C₂²	#16810525	1	1	yes	characteristic
279	C₂²	#16810818	1	1	yes	characteristic
280	C₂²	#16810823	1	1	yes	characteristic
281	C₂²	#16810859	1	2	yes	—
282	C₂²	#16810893	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	4	no	—
286	C₄	C₂×#13133137	1	1	yes	characteristic
287	C₄	#16810627	1	1	yes	characteristic
288	C₄	C₄×D₄₂₄	1	2	yes	—
289	C₄	#16810818	1	1	yes	characteristic
290	C₄	#16810823	1	1	yes	characteristic
291	C₄	#16811000	1	1	yes	characteristic
292	C₄	#16811007	1	1	yes	characteristic
293	C₂	—	848	4	no	—
294	C₂	#20274070	1	2	yes	—
295	C₂	#20274075	1	2	yes	—
296	C₂	#20275581	1	1	yes	characteristic
297	C₂	#20275625	1	1	yes	characteristic
298	C₂	#20275642	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂×#16811000	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#9734
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20275642	C₂	semidirect
3. C₂×#16810376	C₂	semidirect
4. C₂×#16810366	C₂	semidirect
5. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
10. #16811000	C₄	semidirect
77. C₄⋊C₄×C₅₃	D₈	semidirect
121. C₈×C₅₃	#102	semidirect
146. C₈×C₄⋊C₄	D₅₃	semidirect
169. C₅₃	#9734	semidirect
231. C₄⋊C₄	D₄₂₄	semidirect
272. C₈	#13133137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811000	2
#20275642	2
C₂×#16810376	2
C₂×#16810366	2
C₈×C₄⋊C₄×C₅₃	2
#9734	53