Subgroups — #27384271

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	1	yes	maximal, characteristic
2	#20283305	C₂	1	1	yes	maximal, characteristic
3	#20274839	C₂	1	8	yes	maximal, complemented
4	C₂×#16810394	C₂	1	2	yes	maximal
5	C₂×#16810379	C₂	1	2	yes	maximal
6	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂×#13133176	C₂²	1	2	yes	—
9	#16810672	—	2	4	no	—
10	C₂×#13133153	C₂²	1	4	yes	—
11	#16810665	—	2	4	no	—
12	#16810661	C₂²	1	4	yes	—
13	#16810394	C₂²	1	8	yes	—
14	#16810379	C₂²	1	8	yes	—
15	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#405	C₄	1	4	yes	complemented
19	C₅₃×#405	C₂²	1	4	yes	—
20	C₄×M₁₆×C₅₃	C₄	1	4	yes	complemented
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	C₂³	1	4	yes	—
27	C₂²×#88322	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	#13133176	—	2	2	no	—
30	#13133176	—	2	2	no	—
31	#13133176	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
34	C₂×C₄×Dic₅₃	—	2	2	no	—
35	#13133153	—	2	8	no	—
36	#13133151	—	2	4	no	—
37	#13133149	—	2	4	no	—
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
45	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
46	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	#117×C₅₃	—	2	1	no	—
50	#117×C₅₃	—	2	1	no	—
51	#117×C₅₃	C₂×C₄	1	2	yes	—
52	#117×C₅₃	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
56	#97×C₅₃	C₂×C₄	1	8	yes	complemented
57	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
58	C₂⁴×C₅₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	2	yes	—
66	C₂²×C₄×C₅₃	#41	1	2	yes	—
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₅₃	—	4	2	no	—
71	M₁₆×C₅₃	—	2	4	no	—
72	M₁₆×C₅₃	—	2	2	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
75	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₂²⋊C₄×C₅₃	—	2	2	no	—
84	C₂²⋊C₄×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊Dic₅₃	—	2	4	no	—
88	C₂²⋊Dic₅₃	—	4	2	no	—
89	C₄⋊Dic₅₃	—	2	4	no	—
90	C₄⋊Dic₅₃	—	4	2	no	—
91	C₄×Dic₅₃	—	2	4	no	—
92	C₄×Dic₅₃	—	4	2	no	—
93	#11719917	—	2	4	no	—
94	#11719917	—	2	4	no	—
95	#11719917	—	4	2	no	—
96	#11719917	—	4	2	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	—	4	2	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂³×C₅₃	—	8	1	no	—
101	C₂³×C₅₃	#117	1	1	yes	characteristic
102	C₂³×C₅₃	#117	1	1	yes	characteristic
103	C₂³×C₅₃	#124	1	2	yes	—
104	C₂³×C₅₃	#125	1	2	yes	—
105	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₃	#117	1	2	yes	—
111	C₂×C₄×C₅₃	#117	1	2	yes	—
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#124	1	2	yes	—
115	C₂×C₄×C₅₃	#124	1	2	yes	—
116	C₂×C₄×C₅₃	#124	1	2	yes	—
117	C₂×C₄×C₅₃	#125	1	2	yes	—
118	C₂×C₄×C₅₃	#125	1	2	yes	—
119	C₂×C₄×C₅₃	#125	1	2	yes	—
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×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₅₃	—	4	4	no	—
127	C₂×Dic₅₃	—	8	4	no	—
128	C₈×C₅₃	—	2	4	no	—
129	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
130	#88322	—	4	4	no	—
131	#20757	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	4	1	no	—
134	C₂²×C₅₃	—	4	1	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	—	4	1	no	—
137	C₂²×C₅₃	#382	1	2	yes	—
138	C₂²×C₅₃	#382	1	2	yes	—
139	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
142	C₄×C₅₃	—	4	4	no	—
143	C₄×C₅₃	#382	1	2	yes	—
144	C₄×C₅₃	#382	1	2	yes	—
145	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
148	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
149	Dic₅₃	—	8	4	no	—
150	C₂×#430	—	53	1	no	—
151	#2911	—	53	1	no	—
152	#2873	D₅₃	1	1	yes	characteristic
153	#1833	—	53	8	no	—
154	C₂×#344	—	53	2	no	—
155	C₂×H(3,ℤ/4)	—	53	2	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	2	2	no	—
160	C₂×C₅₃	—	2	2	no	—
161	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
162	C₂²×M₁₆	—	53	2	no	—
163	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
164	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
165	C₂×#117	—	53	2	no	—
166	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
167	#430	—	106	4	no	—
168	C₂×#108	—	53	4	no	—
169	#428	—	106	4	no	—
170	#427	—	53	4	no	—
171	#405	Dic₅₃	1	4	yes	complemented
172	#405	C₂×D₅₃	1	4	yes	—
173	C₄×M₁₆	Dic₅₃	1	4	yes	complemented
174	#344	—	53	8	no	—
175	H(3,ℤ/4)	—	53	8	no	—
176	C₅₃	#20757	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	53	4	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	—	106	2	no	—
186	C₂×M₁₆	Dic₁₀₆	1	4	yes	—
187	C₂×M₁₆	D₂₁₂	1	4	yes	—
188	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
189	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
190	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
191	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
194	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
195	#117	—	2	1	no	—
196	#117	—	2	1	no	—
197	#117	—	53	4	no	—
198	#117	—	106	2	no	—
199	#117	—	106	2	no	—
200	#117	—	106	2	no	—
201	#117	C₂×Dic₅₃	1	2	yes	—
202	#117	C₂²×D₅₃	1	2	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	106	2	no	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₂²⋊C₄	—	106	2	no	—
207	C₂×C₄²	—	106	2	no	—
208	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
209	#108	—	106	8	no	—
210	#107	—	106	4	no	—
211	#106	—	106	4	no	—
212	#97	C₂×Dic₅₃	1	8	yes	complemented
213	C₄×C₈	C₂×Dic₅₃	1	8	yes	complemented
214	C₂⁴	#11719945	1	1	yes	characteristic
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	212	2	no	—
217	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
218	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
219	C₂²×C₄	#11719945	1	1	yes	characteristic
220	C₂²×C₄	#11719945	1	2	yes	—
221	C₂²×C₄	#11719945	1	2	yes	—
222	C₂²×C₄	#11719945	1	1	yes	characteristic
223	C₂²×C₄	#11719945	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	106	4	no	—
231	M₁₆	—	106	4	no	—
232	M₁₆	—	212	2	no	—
233	M₁₆	—	212	2	no	—
234	M₁₆	C₄⋊Dic₅₃	1	8	yes	—
235	M₁₆	C₄⋊Dic₅₃	1	8	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	106	4	no	—
238	C₂×C₈	—	212	2	no	—
239	C₂×C₈	—	212	2	no	—
240	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
241	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
242	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
243	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	106	4	no	—
247	C₄⋊C₄	—	212	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	106	4	no	—
251	C₂²⋊C₄	—	212	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	106	4	no	—
254	C₄²	—	212	2	no	—
255	C₄²	C₂²×Dic₅₃	1	4	yes	—
256	C₂³	—	8	1	no	—
257	C₂³	#13133176	1	1	yes	characteristic
258	C₂³	#13133176	1	1	yes	characteristic
259	C₂³	#13133205	1	2	yes	—
260	C₂³	#13133209	1	2	yes	—
261	C₂³	C₂×#11719945	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	—	424	4	no	—
265	C₂×C₄	—	212	4	no	—
266	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
268	C₂×C₄	#13133176	1	2	yes	—
269	C₂×C₄	#13133176	1	2	yes	—
270	C₂×C₄	#13133176	1	1	yes	characteristic
271	C₂×C₄	#13133176	1	1	yes	characteristic
272	C₂×C₄	#13133205	1	2	yes	—
273	C₂×C₄	#13133205	1	2	yes	—
274	C₂×C₄	#13133205	1	2	yes	—
275	C₂×C₄	#13133209	1	2	yes	—
276	C₂×C₄	#13133209	1	2	yes	—
277	C₂×C₄	#13133209	1	2	yes	—
278	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
280	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
281	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
282	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
283	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	212	4	no	—
286	C₈	C₂×C₄⋊Dic₅₃	1	8	yes	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	1	no	—
292	C₂²	#16810489	1	2	yes	—
293	C₂²	#16810489	1	2	yes	—
294	C₂²	C₂×#13133176	1	1	yes	characteristic
295	C₂²	C₂×#13133205	1	1	yes	characteristic
296	C₂²	C₂×#13133209	1	1	yes	characteristic
297	C₄	—	4	4	no	—
298	C₄	—	424	4	no	—
299	C₄	#16810489	1	2	yes	—
300	C₄	#16810489	1	2	yes	—
301	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
302	C₄	C₂×#13133176	1	1	yes	characteristic
303	C₄	C₂×#13133205	1	1	yes	characteristic
304	C₄	C₂×#13133209	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₂×#16810489	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133176
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2873	C₂
socle	C₂×C₅₃	C₂×#382
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2873	D₅₃
hypocenter	C₅₃	#20757
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ #2873

Complements

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

N	H ≅ G/N	type
3. #20274839	C₂	semidirect
18. C₅₃×#405	C₄	semidirect
20. C₄×M₁₆×C₅₃	C₄	semidirect
56. #97×C₅₃	C₂×C₄	semidirect
57. C₄×C₈×C₅₃	C₂×C₄	semidirect
171. #405	Dic₅₃	semidirect
173. C₄×M₁₆	Dic₅₃	semidirect
176. C₅₃	#20757	semidirect
212. #97	C₂×Dic₅₃	semidirect
213. C₄×C₈	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
#20283305	2
#20274839	2
C₂×#16810394	2
C₂×#16810379	2
C₅₃×#2873	2
#20757	53