Subgroups — #27402584

Order 13568.

846 conjugacy classes of subgroups.

The subgroup lattice has 324 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	#20283600	C₂	1	2	yes	maximal
3	#20283312	C₂	1	2	yes	maximal, complemented
4	#20275495	C₂	1	2	yes	maximal, complemented
5	#20274949	C₂	1	4	yes	maximal, complemented
6	#20274879	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	#16810736	—	2	2	no	—
14	#16810731	—	2	2	no	—
15	C₄×#11719913	—	2	4	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	#107×D₅₃	—	2	2	no	—
22	C₂×#13133149	C₂²	1	2	yes	—
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	2	yes	—
68	#97×C₅₃	—	2	2	no	—
69	C₄×C₈×C₅₃	—	2	2	no	—
70	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₂²⋊D₅₃	—	2	2	no	—
74	C₂²×Dic₅₃	—	2	4	no	—
75	#11719945	—	2	2	no	—
76	#11719945	—	2	2	no	—
77	#11719945	C₂²×C₄	1	4	yes	—
78	#11719945	C₂×Q₈	1	4	yes	—
79	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
80	C₂×C₄×D₅₃	—	2	2	no	—
81	C₂×C₄×D₅₃	—	4	4	no	—
82	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
83	M₁₆×C₅₃	—	2	2	no	—
84	M₁₆×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
93	C₂×C₈×C₅₃	#41	1	2	yes	—
94	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
95	C₂²⋊C₄×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	1	no	—
98	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
99	#11719921	—	4	4	no	—
100	C₄⋊Dic₅₃	—	2	4	no	—
101	C₄⋊Dic₅₃	—	2	4	no	—
102	#11719919	—	4	4	no	—
103	C₄×Dic₅₃	—	4	4	no	—
104	#11719917	—	2	2	no	—
105	#11719917	C₂²×C₄	1	4	yes	—
106	C₂×#88322	—	2	2	no	—
107	Dic₂₁₂	C₄⋊C₄	1	4	yes	—
108	D₄₂₄	C₄⋊C₄	1	4	yes	—
109	#11719913	C₄⋊C₄	1	8	yes	—
110	#11719912	—	2	8	no	—
111	C₈×D₅₃	—	2	8	no	—
112	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×D₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂²⋊D₅₃	—	2	4	no	—
124	C₂²⋊D₅₃	—	2	4	no	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	4	4	no	—
127	C₂×Dic₅₃	—	8	4	no	—
128	D₂₁₂	—	2	2	no	—
129	D₂₁₂	C₂×C₄⋊C₄	1	4	yes	—
130	C₄×D₅₃	—	2	4	no	—
131	C₄×D₅₃	—	2	4	no	—
132	C₄×D₅₃	—	16	4	no	—
133	Dic₁₀₆	—	2	2	no	—
134	Dic₁₀₆	C₂×C₄⋊C₄	1	4	yes	—
135	C₈×C₅₃	—	4	2	no	—
136	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₅₃	C₄×D₄	1	2	yes	—
139	C₈×C₅₃	C₄×D₄	1	2	yes	—
140	#88322	—	2	4	no	—
141	#22691	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	#520	1	1	yes	characteristic
144	C₂²×C₅₃	D₄²	1	1	yes	characteristic
145	C₂²×C₅₃	#554	1	1	yes	characteristic
146	C₂×D₅₃	—	2	4	no	—
147	C₂×D₅₃	—	4	2	no	—
148	C₂×D₅₃	—	4	2	no	—
149	C₄×C₅₃	—	4	2	no	—
150	C₄×C₅₃	—	4	4	no	—
151	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₅₃	#547	1	1	yes	characteristic
154	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
155	Dic₅₃	—	2	4	no	—
156	Dic₅₃	—	8	4	no	—
157	C₂×#443	—	53	1	no	—
158	#2952	—	53	2	no	—
159	#2913	—	53	2	no	—
160	#1951	—	53	2	no	—
161	#1860	—	53	4	no	—
162	#1846	—	53	1	no	—
163	#1816	—	53	2	no	—
164	#1772	D₅₃	1	1	yes	characteristic, complemented
165	C₂×C₅₃	—	2	1	no	—
166	C₂×C₅₃	—	2	1	no	—
167	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₅₃	#3290	1	2	yes	—
169	D₅₃	—	4	4	no	—
170	C₂×#135	—	53	1	no	—
171	C₂×#131	—	53	2	no	—
172	#520	—	53	4	no	—
173	#443	—	53	8	no	—
174	#442	—	106	2	no	—
175	#441	—	106	2	no	—
176	C₄×SD₁₆	—	106	4	no	—
177	C₂×#108	—	53	1	no	—
178	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
179	#428	—	53	4	no	—
180	#428	—	106	4	no	—
181	#427	—	106	2	no	—
182	#427	C₂×D₅₃	1	1	yes	characteristic
183	C₂×#107	—	53	2	no	—
184	C₂×#107	—	106	2	no	—
185	C₂×#106	—	53	2	no	—
186	C₂×#104	—	53	2	no	—
187	#417	—	53	4	no	—
188	#405	C₂×D₅₃	1	2	yes	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	#382	—	53	2	no	—
191	#343	—	53	4	no	—
192	#339	C₂×D₅₃	1	2	yes	—
193	C₅₃	#22691	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	53	2	no	—
195	#135	—	53	4	no	—
196	#135	—	53	4	no	—
197	C₂×Q₁₆	—	53	1	no	—
198	C₂×SD₁₆	—	53	2	no	—
199	C₂×D₈	—	53	1	no	—
200	#131	—	53	8	no	—
201	#131	—	106	4	no	—
202	C₂×M₁₆	—	53	2	no	—
203	C₂×M₁₆	—	106	2	no	—
204	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
205	C₂²×C₈	—	106	2	no	—
206	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₄×Q₈	—	106	4	no	—
209	C₄×D₄	—	106	4	no	—
210	C₄×D₄	—	212	4	no	—
211	#117	—	106	4	no	—
212	#117	C₂²×D₅₃	1	2	yes	—
213	C₂×C₄⋊C₄	—	53	4	no	—
214	C₂×C₄⋊C₄	—	106	4	no	—
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	#108	—	2	2	no	—
217	#108	—	53	4	no	—
218	#107	—	106	2	no	—
219	#107	—	106	2	no	—
220	#107	—	106	4	no	—
221	#107	D₂₁₂	1	2	yes	—
222	#106	—	106	2	no	—
223	#106	—	106	2	no	—
224	#106	—	106	4	no	—
225	#106	D₂₁₂	1	2	yes	—
226	#104	—	53	8	no	—
227	#103	—	106	4	no	—
228	#102	—	106	4	no	—
229	#97	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
232	#95	—	212	2	no	—
233	#41	—	53	4	no	—
234	#41	—	53	4	no	—
235	#41	—	106	2	no	—
236	#41	—	106	2	no	—
237	C₂×Q₈	—	53	2	no	—
238	C₂×D₄	—	53	2	no	—
239	C₂×D₄	—	106	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	106	2	no	—
242	C₂²×C₄	—	106	4	no	—
243	C₂²×C₄	—	212	4	no	—
244	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
245	Q₁₆	—	53	4	no	—
246	SD₁₆	—	53	8	no	—
247	D₈	—	53	4	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	53	4	no	—
251	M₁₆	—	106	2	no	—
252	M₁₆	—	106	8	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	—	106	8	no	—
256	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
258	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
259	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
260	C₂×C₈	#11719945	1	2	yes	—
261	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
262	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
263	C₂×C₈	Q₈×D₅₃	1	2	yes	—
264	C₄⋊C₄	—	106	4	no	—
265	C₄⋊C₄	—	106	4	no	—
266	C₄⋊C₄	—	212	4	no	—
267	C₄⋊C₄	C₂×D₂₁₂	1	4	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	212	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	212	4	no	—
273	C₄²	C₂×C₄×D₅₃	1	2	yes	—
274	C₂³	—	106	2	no	—
275	C₂³	#13133263	1	1	yes	characteristic
276	Q₈	—	53	4	no	—
277	Q₈	—	106	2	no	—
278	D₄	—	53	4	no	—
279	D₄	—	106	4	no	—
280	D₄	—	106	4	no	—
281	D₄	—	106	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	848	4	no	—
286	C₂×C₄	—	106	2	no	—
287	C₂×C₄	—	106	4	no	—
288	C₂×C₄	—	106	4	no	—
289	C₂×C₄	—	424	4	no	—
290	C₂×C₄	—	212	4	no	—
291	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
296	C₂×C₄	#13133264	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	—	106	4	no	—
299	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
300	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
301	C₈	C₄×D₂₁₂	1	2	yes	—
302	C₈	C₄×D₂₁₂	1	2	yes	—
303	C₂²	—	2	2	no	—
304	C₂²	—	106	4	no	—
305	C₂²	—	212	2	no	—
306	C₂²	—	212	2	no	—
307	C₂²	#16811103	1	1	yes	characteristic
308	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
309	C₂²	#16811333	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	106	4	no	—
313	C₄	—	424	4	no	—
314	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
315	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
316	C₄	#16811278	1	1	yes	characteristic
317	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	212	4	no	—
321	C₂	C₄⋊C₄×D₂₁₂	1	1	yes	characteristic
322	C₂	#20286503	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	#22691
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. #20275495	C₂	semidirect
5. #20274949	C₂	semidirect
6. #20274879	C₂	semidirect
7. #20274776	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
70. C₄×C₈×C₅₃	C₂×C₄	semidirect
164. #1772	D₅₃	semidirect
193. C₅₃	#22691	semidirect
231. 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
#20283600	2
#20283312	2
#20275495	2
#20274949	2
#20274879	2
#20274776	2
C₅₃×#1772	2
#22691	53