Subgroups — #27402576

Order 13568.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810741	C₂	1	1	yes	maximal, characteristic
2	#20283592	C₂	1	2	yes	maximal
3	#20283307	C₂	1	2	yes	maximal, complemented
4	#20275500	C₂	1	2	yes	maximal, complemented
5	#20274949	C₂	1	4	yes	maximal, complemented
6	#20274879	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274771	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	C₂×#13133223	C₂²	1	2	yes	—
11	#16811103	C₂²	1	4	yes	—
12	#16810741	C₄	1	8	yes	complemented
13	#16810725	—	2	4	no	—
14	C₄×Dic₂₁₂	—	2	2	no	—
15	C₄×D₄₂₄	—	2	2	no	—
16	C₂×#13133153	C₂²	1	1	yes	characteristic
17	#16810666	C₂²	1	4	yes	—
18	#16810664	—	2	4	no	—
19	#16810658	—	2	2	no	—
20	C₂×#13133151	C₂²	1	2	yes	—
21	C₂×#13133149	C₂²	1	2	yes	—
22	#106×D₅₃	—	2	2	no	—
23	C₂×#13133145	C₂²	1	2	yes	—
24	#16810625	C₂²	1	4	yes	—
25	#16810489	C₂²	1	2	yes	—
26	#16810391	C₂²	1	4	yes	—
27	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#427	C₂²	1	1	yes	characteristic
29	C₅₃×#405	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#339	C₂²	1	2	yes	—
32	C₂×#11719945	C₂³	1	2	yes	—
33	#13133239	C₂×C₄	1	4	yes	—
34	#13133239	Q₈	1	4	yes	—
35	C₂×Dic₂₁₂	D₄	1	1	yes	characteristic
36	C₂×#11719913	D₄	1	2	yes	—
37	C₂×D₄₂₄	D₄	1	1	yes	characteristic
38	#13133223	—	2	4	no	—
39	#13133223	C₂×C₄	1	8	yes	complemented
40	C₂×#11719917	C₂³	1	2	yes	—
41	C₂×#11719912	—	2	2	no	—
42	C₂×C₈×D₅₃	—	2	2	no	—
43	C₄×Dic₁₀₆	—	2	4	no	—
44	C₄×D₂₁₂	—	2	4	no	—
45	#13133177	—	4	4	no	—
46	#13133175	—	2	4	no	—
47	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
48	C₄⋊C₄×D₅₃	—	2	4	no	—
49	#13133153	C₂×C₄	1	4	yes	complemented
50	#13133151	—	2	2	no	—
51	#13133151	—	2	2	no	—
52	#13133150	—	2	4	no	—
53	#13133149	—	2	2	no	—
54	#13133149	—	2	2	no	—
55	#13133148	—	2	4	no	—
56	#13133145	C₂×C₄	1	8	yes	complemented
57	#13133142	—	2	4	no	—
58	#13133139	—	2	4	no	—
59	#13133124	—	4	2	no	—
60	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	#117×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#108×C₅₃	—	2	2	no	—
66	#107×C₅₃	D₄	1	2	yes	—
67	#106×C₅₃	D₄	1	1	yes	characteristic
68	#106×C₅₃	D₄	1	1	yes	characteristic
69	#97×C₅₃	—	2	2	no	—
70	C₄×C₈×C₅₃	—	2	2	no	—
71	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₂²⋊D₅₃	—	2	2	no	—
75	C₂²×Dic₅₃	—	2	4	no	—
76	#11719945	—	2	2	no	—
77	#11719945	—	2	2	no	—
78	#11719945	C₂²×C₄	1	4	yes	—
79	#11719945	C₂×Q₈	1	4	yes	—
80	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₅₃	—	2	2	no	—
82	C₂×C₄×D₅₃	—	4	4	no	—
83	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
84	M₁₆×C₅₃	—	2	2	no	—
85	M₁₆×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₅₃	#41	1	2	yes	—
95	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
97	C₂²⋊C₄×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	1	no	—
100	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
101	#11719921	—	4	4	no	—
102	C₄⋊Dic₅₃	—	2	4	no	—
103	C₄⋊Dic₅₃	—	2	4	no	—
104	#11719919	—	4	4	no	—
105	C₄×Dic₅₃	—	4	4	no	—
106	#11719917	—	2	2	no	—
107	#11719917	C₂²×C₄	1	4	yes	—
108	C₂×#88322	—	2	2	no	—
109	Dic₂₁₂	C₄⋊C₄	1	4	yes	—
110	D₄₂₄	C₄⋊C₄	1	4	yes	—
111	#11719913	C₄⋊C₄	1	8	yes	—
112	#11719912	—	2	8	no	—
113	C₈×D₅₃	—	2	8	no	—
114	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×D₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂²⋊D₅₃	—	2	4	no	—
127	C₂²⋊D₅₃	—	2	4	no	—
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	4	4	no	—
130	C₂×Dic₅₃	—	8	4	no	—
131	D₂₁₂	—	2	2	no	—
132	D₂₁₂	C₂×C₄⋊C₄	1	4	yes	—
133	C₄×D₅₃	—	2	4	no	—
134	C₄×D₅₃	—	2	4	no	—
135	C₄×D₅₃	—	16	4	no	—
136	Dic₁₀₆	—	2	2	no	—
137	Dic₁₀₆	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₅₃	—	4	2	no	—
139	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
141	C₈×C₅₃	C₄×D₄	1	2	yes	—
142	C₈×C₅₃	C₄×D₄	1	2	yes	—
143	#88322	—	2	4	no	—
144	#22690	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	#520	1	1	yes	characteristic
147	C₂²×C₅₃	D₄²	1	1	yes	characteristic
148	C₂²×C₅₃	#554	1	1	yes	characteristic
149	C₂×D₅₃	—	2	4	no	—
150	C₂×D₅₃	—	4	4	no	—
151	C₄×C₅₃	—	4	2	no	—
152	C₄×C₅₃	—	4	2	no	—
153	C₄×C₅₃	—	4	2	no	—
154	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₅₃	#547	1	1	yes	characteristic
157	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
158	Dic₅₃	—	2	4	no	—
159	Dic₅₃	—	8	4	no	—
160	C₂×#443	—	53	1	no	—
161	#2951	—	53	2	no	—
162	#2912	—	53	2	no	—
163	#1952	—	53	2	no	—
164	#1860	—	53	4	no	—
165	#1846	—	53	1	no	—
166	#1815	—	53	2	no	—
167	#1772	D₅₃	1	1	yes	characteristic, complemented
168	C₂×C₅₃	—	2	1	no	—
169	C₂×C₅₃	—	2	1	no	—
170	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₅₃	#3290	1	2	yes	—
172	D₅₃	—	4	4	no	—
173	C₂×#135	—	53	1	no	—
174	C₂×#131	—	53	2	no	—
175	#520	—	53	4	no	—
176	#443	—	53	8	no	—
177	#440	—	106	4	no	—
178	C₄×Q₁₆	—	106	2	no	—
179	C₄×D₈	—	106	2	no	—
180	C₂×#108	—	53	1	no	—
181	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
182	#428	—	53	4	no	—
183	#428	—	106	4	no	—
184	#427	—	106	2	no	—
185	#427	C₂×D₅₃	1	1	yes	characteristic
186	C₂×#107	—	53	2	no	—
187	C₂×#106	—	53	2	no	—
188	C₂×#106	—	106	2	no	—
189	C₂×#104	—	53	2	no	—
190	#417	—	53	4	no	—
191	#405	C₂×D₅₃	1	2	yes	—
192	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
193	#382	—	53	2	no	—
194	#343	—	53	4	no	—
195	#339	C₂×D₅₃	1	2	yes	—
196	C₅₃	#22690	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	53	2	no	—
198	#135	—	53	4	no	—
199	#135	—	53	4	no	—
200	C₂×Q₁₆	—	53	1	no	—
201	C₂×SD₁₆	—	53	2	no	—
202	C₂×D₈	—	53	1	no	—
203	#131	—	53	8	no	—
204	#131	—	106	4	no	—
205	C₂×M₁₆	—	53	2	no	—
206	C₂×M₁₆	—	106	2	no	—
207	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
208	C₂²×C₈	—	106	2	no	—
209	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
211	C₄×Q₈	—	106	4	no	—
212	C₄×D₄	—	106	4	no	—
213	C₄×D₄	—	212	4	no	—
214	#117	—	106	4	no	—
215	#117	C₂²×D₅₃	1	2	yes	—
216	C₂×C₄⋊C₄	—	53	4	no	—
217	C₂×C₄⋊C₄	—	106	4	no	—
218	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
219	#108	—	2	2	no	—
220	#108	—	53	4	no	—
221	#107	—	106	2	no	—
222	#107	—	106	2	no	—
223	#107	—	106	4	no	—
224	#107	D₂₁₂	1	2	yes	—
225	#106	—	106	2	no	—
226	#106	—	106	2	no	—
227	#106	—	106	4	no	—
228	#106	D₂₁₂	1	1	yes	characteristic
229	#106	D₂₁₂	1	1	yes	characteristic
230	#104	—	53	8	no	—
231	#103	—	106	4	no	—
232	#102	—	106	4	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
236	#95	—	212	2	no	—
237	#41	—	53	4	no	—
238	#41	—	53	4	no	—
239	#41	—	106	2	no	—
240	#41	—	106	2	no	—
241	C₂×Q₈	—	53	2	no	—
242	C₂×D₄	—	53	2	no	—
243	C₂×D₄	—	106	2	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	106	2	no	—
246	C₂²×C₄	—	106	4	no	—
247	C₂²×C₄	—	212	4	no	—
248	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
249	Q₁₆	—	53	4	no	—
250	SD₁₆	—	53	8	no	—
251	D₈	—	53	4	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	53	4	no	—
255	M₁₆	—	106	2	no	—
256	M₁₆	—	106	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	8	no	—
260	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
262	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
263	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
264	C₂×C₈	#11719945	1	2	yes	—
265	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	Q₈×D₅₃	1	2	yes	—
268	C₄⋊C₄	—	106	4	no	—
269	C₄⋊C₄	—	106	4	no	—
270	C₄⋊C₄	—	212	4	no	—
271	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
272	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	212	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	212	4	no	—
278	C₄²	C₂×C₄×D₅₃	1	2	yes	—
279	C₂³	—	106	2	no	—
280	C₂³	#13133263	1	1	yes	characteristic
281	Q₈	—	53	4	no	—
282	Q₈	—	106	2	no	—
283	D₄	—	53	4	no	—
284	D₄	—	106	4	no	—
285	D₄	—	106	4	no	—
286	D₄	—	106	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	848	4	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	106	4	no	—
294	C₂×C₄	—	106	4	no	—
295	C₂×C₄	—	424	4	no	—
296	C₂×C₄	—	212	4	no	—
297	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
301	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
302	C₂×C₄	#13133264	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	—	106	4	no	—
305	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
306	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
307	C₈	C₄×D₂₁₂	1	2	yes	—
308	C₈	C₄×D₂₁₂	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	106	4	no	—
311	C₂²	—	212	4	no	—
312	C₂²	#16811103	1	1	yes	characteristic
313	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
314	C₂²	#16811333	1	1	yes	characteristic
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	106	4	no	—
319	C₄	—	424	4	no	—
320	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
321	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
322	C₄	#16811278	1	1	yes	characteristic
323	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	212	4	no	—
327	C₂	C₄⋊C₄×D₂₁₂	1	1	yes	characteristic
328	C₂	#20286502	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#22690
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #20283307	C₂	semidirect
4. #20275500	C₂	semidirect
5. #20274949	C₂	semidirect
6. #20274879	C₂	semidirect
7. #20274771	C₂	semidirect
8. C₅₃×#1772	C₂	semidirect
12. #16810741	C₄	semidirect
39. #13133223	C₂×C₄	semidirect
49. #13133153	C₂×C₄	semidirect
56. #13133145	C₂×C₄	semidirect
71. C₄×C₈×C₅₃	C₂×C₄	semidirect
167. #1772	D₅₃	semidirect
196. C₅₃	#22690	semidirect
235. C₄×C₈	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810741	2
#20283592	2
#20283307	2
#20275500	2
#20274949	2
#20274879	2
#20274771	2
C₅₃×#1772	2
#22690	53