Subgroups — #27402592

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#20283312	C₂	1	2	yes	maximal, complemented
4	#20275500	C₂	1	2	yes	maximal, complemented
5	#20274942	C₂	1	4	yes	maximal, complemented
6	#20274875	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274776	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	#16810662	—	2	4	no	—
19	#16810660	—	2	2	no	—
20	C₂×#13133151	C₂²	1	2	yes	—
21	C₂×#13133151	C₂²	1	2	yes	—
22	#107×D₅₃	—	2	2	no	—
23	C₂×#13133145	C₂²	1	2	yes	—
24	#16810625	C₂²	1	4	yes	—
25	#16810493	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	—	4	2	no	—
48	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
49	C₄⋊C₄×D₅₃	—	2	4	no	—
50	#13133153	C₂×C₄	1	4	yes	complemented
51	#13133151	—	2	2	no	—
52	#13133151	—	2	2	no	—
53	#13133151	—	2	2	no	—
54	#13133151	—	2	2	no	—
55	#13133150	—	2	4	no	—
56	#13133148	—	2	4	no	—
57	#13133145	C₂×C₄	1	8	yes	complemented
58	#13133142	—	2	4	no	—
59	#13133139	—	2	4	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	1	yes	characteristic
67	#107×C₅₃	D₄	1	1	yes	characteristic
68	#106×C₅₃	D₄	1	2	yes	—
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	C₄⋊Dic₅₃	—	8	4	no	—
105	#11719919	—	4	4	no	—
106	C₄×Dic₅₃	—	4	4	no	—
107	#11719917	—	2	2	no	—
108	#11719917	C₂²×C₄	1	4	yes	—
109	C₂×#88322	—	2	2	no	—
110	Dic₂₁₂	C₄⋊C₄	1	4	yes	complemented
111	D₄₂₄	C₄⋊C₄	1	4	yes	complemented
112	#11719913	C₄⋊C₄	1	8	yes	complemented
113	#11719912	—	2	8	no	—
114	C₈×D₅₃	—	2	8	no	—
115	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×D₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²⋊D₅₃	—	2	4	no	—
129	C₂²⋊D₅₃	—	2	4	no	—
130	C₂×Dic₅₃	—	2	2	no	—
131	C₂×Dic₅₃	—	4	4	no	—
132	C₂×Dic₅₃	—	8	4	no	—
133	D₂₁₂	—	2	2	no	—
134	D₂₁₂	C₂×C₄⋊C₄	1	4	yes	—
135	C₄×D₅₃	—	2	4	no	—
136	C₄×D₅₃	—	2	4	no	—
137	C₄×D₅₃	—	16	4	no	—
138	Dic₁₀₆	—	2	2	no	—
139	Dic₁₀₆	C₂×C₄⋊C₄	1	4	yes	—
140	C₈×C₅₃	—	4	2	no	—
141	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₅₃	C₄×D₄	1	2	yes	—
144	C₈×C₅₃	C₄×D₄	1	2	yes	—
145	#88322	—	2	4	no	—
146	#22692	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	#520	1	1	yes	characteristic
150	C₂²×C₅₃	D₄²	1	1	yes	characteristic
151	C₂²×C₅₃	#554	1	1	yes	characteristic
152	C₂×D₅₃	—	2	4	no	—
153	C₂×D₅₃	—	4	2	no	—
154	C₂×D₅₃	—	4	2	no	—
155	C₄×C₅₃	—	2	4	no	—
156	C₄×C₅₃	—	4	2	no	—
157	C₄×C₅₃	—	4	2	no	—
158	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₅₃	#547	1	1	yes	characteristic
161	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
162	Dic₅₃	—	2	4	no	—
163	Dic₅₃	—	8	4	no	—
164	C₂×#443	—	53	1	no	—
165	#2951	—	53	2	no	—
166	#2913	—	53	2	no	—
167	#1952	—	53	2	no	—
168	#1859	—	53	4	no	—
169	#1845	—	53	1	no	—
170	#1816	—	53	2	no	—
171	#1772	D₅₃	1	1	yes	characteristic, complemented
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₅₃	#3292	1	1	yes	characteristic
176	C₂×C₅₃	#3293	1	1	yes	characteristic
177	D₅₃	—	4	4	no	—
178	C₂×#135	—	53	1	no	—
179	C₂×#131	—	53	2	no	—
180	#520	—	53	4	no	—
181	#443	—	53	8	no	—
182	#440	—	106	4	no	—
183	C₄×Q₁₆	—	106	2	no	—
184	C₄×D₈	—	106	2	no	—
185	C₂×#108	—	53	1	no	—
186	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
187	#428	—	53	4	no	—
188	#428	—	106	4	no	—
189	#427	—	106	2	no	—
190	#427	C₂×D₅₃	1	1	yes	characteristic
191	C₂×#107	—	53	2	no	—
192	C₂×#107	—	53	2	no	—
193	C₂×#107	—	106	2	no	—
194	C₂×#104	—	53	2	no	—
195	#417	—	53	4	no	—
196	#405	C₂×D₅₃	1	2	yes	—
197	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
198	#384	—	53	2	no	—
199	#343	—	53	4	no	—
200	#339	C₂×D₅₃	1	2	yes	—
201	C₅₃	#22692	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	—	53	2	no	—
203	#135	—	53	4	no	—
204	#135	—	53	4	no	—
205	C₂×Q₁₆	—	53	1	no	—
206	C₂×SD₁₆	—	53	2	no	—
207	C₂×D₈	—	53	1	no	—
208	#131	—	53	8	no	—
209	#131	—	106	4	no	—
210	C₂×M₁₆	—	53	2	no	—
211	C₂×M₁₆	—	106	2	no	—
212	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
213	C₂²×C₈	—	106	2	no	—
214	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	C₄×Q₈	—	106	4	no	—
217	C₄×D₄	—	106	4	no	—
218	C₄×D₄	—	212	4	no	—
219	#117	—	106	4	no	—
220	#117	C₂²×D₅₃	1	2	yes	—
221	C₂×C₄⋊C₄	—	53	4	no	—
222	C₂×C₄⋊C₄	—	106	4	no	—
223	C₂×C₄⋊C₄	—	212	2	no	—
224	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
225	#108	—	2	2	no	—
226	#108	—	53	4	no	—
227	#107	—	106	2	no	—
228	#107	—	106	2	no	—
229	#107	—	106	2	no	—
230	#107	—	106	2	no	—
231	#107	—	106	4	no	—
232	#107	D₂₁₂	1	1	yes	characteristic
233	#107	D₂₁₂	1	1	yes	characteristic
234	#106	—	106	4	no	—
235	#106	D₂₁₂	1	2	yes	—
236	#104	—	53	8	no	—
237	#103	—	106	4	no	—
238	#102	—	106	4	no	—
239	#97	—	2	2	no	—
240	C₄×C₈	—	2	2	no	—
241	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
242	#41	—	53	4	no	—
243	#41	—	53	4	no	—
244	#41	—	106	2	no	—
245	#41	—	106	2	no	—
246	C₂×Q₈	—	53	2	no	—
247	C₂×D₄	—	53	2	no	—
248	C₂×D₄	—	106	2	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	—	106	4	no	—
252	C₂²×C₄	—	212	4	no	—
253	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
254	Q₁₆	—	53	4	no	—
255	SD₁₆	—	53	8	no	—
256	D₈	—	53	4	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	53	4	no	—
260	M₁₆	—	106	2	no	—
261	M₁₆	—	106	8	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	106	2	no	—
264	C₂×C₈	—	106	8	no	—
265	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
267	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
268	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
269	C₂×C₈	#11719945	1	2	yes	—
270	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
271	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₅₃	1	2	yes	—
273	C₄⋊C₄	—	106	4	no	—
274	C₄⋊C₄	—	106	4	no	—
275	C₄⋊C₄	—	424	4	no	—
276	C₄⋊C₄	—	212	4	no	—
277	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
278	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	212	4	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	212	4	no	—
284	C₄²	C₂×C₄×D₅₃	1	2	yes	—
285	C₂³	—	106	2	no	—
286	C₂³	#13133263	1	1	yes	characteristic
287	Q₈	—	53	4	no	—
288	Q₈	—	106	2	no	—
289	D₄	—	53	4	no	—
290	D₄	—	106	4	no	—
291	D₄	—	106	4	no	—
292	D₄	—	106	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	848	4	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	106	4	no	—
301	C₂×C₄	—	106	4	no	—
302	C₂×C₄	—	424	4	no	—
303	C₂×C₄	—	212	4	no	—
304	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
309	C₂×C₄	#13133264	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	106	4	no	—
312	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
313	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
314	C₈	C₄×D₂₁₂	1	2	yes	—
315	C₈	C₄×D₂₁₂	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	106	4	no	—
319	C₂²	—	212	2	no	—
320	C₂²	—	212	2	no	—
321	C₂²	#16811103	1	1	yes	characteristic
322	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
323	C₂²	#16811333	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	106	4	no	—
328	C₄	—	424	4	no	—
329	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
330	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
331	C₄	#16811278	1	1	yes	characteristic
332	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	212	4	no	—
336	C₂	C₄⋊C₄×D₂₁₂	1	1	yes	characteristic
337	C₂	#20286515	1	1	yes	characteristic
338	C₂	#20286520	1	1	yes	characteristic

	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	#22692
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. #20283312	C₂	semidirect
4. #20275500	C₂	semidirect
5. #20274942	C₂	semidirect
6. #20274875	C₂	semidirect
7. #20274776	C₂	semidirect
8. C₅₃×#1772	C₂	semidirect
12. #16810741	C₄	semidirect
39. #13133223	C₂×C₄	semidirect
50. #13133153	C₂×C₄	semidirect
57. #13133145	C₂×C₄	semidirect
71. C₄×C₈×C₅₃	C₂×C₄	semidirect
110. Dic₂₁₂	C₄⋊C₄	semidirect
111. D₄₂₄	C₄⋊C₄	semidirect
112. #11719913	C₄⋊C₄	semidirect
171. #1772	D₅₃	semidirect
201. C₅₃	#22692	semidirect
241. 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
#20283312	2
#20275500	2
#20274942	2
#20274875	2
#20274776	2
C₅₃×#1772	2
#22692	53