Subgroups — #27283063

Order 13568.

552 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133128	—	2	1	no	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	Q₈⋊#88322	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#391	C₄	1	2	yes	—
15	C₅₃×#389	—	2	1	no	—
16	C₅₃×#389	—	2	1	no	—
17	C₅₃×#384	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	D₄	1	4	yes	—
26	#13133128	—	4	2	no	—
27	C₄×#88322	—	2	4	no	—
28	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₅₃	—	2	4	no	—
30	C₄×Q₈×C₅₃	—	2	4	no	—
31	C₄×Q₈×C₅₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#95×C₅₃	—	2	2	no	—
47	#95×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	4	no	—
50	C₂×Q₈×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	4	1	no	—
73	C₄²×C₅₃	—	4	4	no	—
74	C₄²×C₅₃	SD₁₆	1	2	yes	—
75	C₄²×C₅₃	Q₁₆	1	2	yes	—
76	C₄²×C₅₃	C₂×D₄	1	2	yes	—
77	C₄²×C₅₃	#41	1	2	yes	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₅₃	—	2	8	no	—
86	Q₈×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	#103	1	2	yes	—
105	C₂×C₄×C₅₃	#103	1	2	yes	—
106	C₂×C₄×C₅₃	#104	1	2	yes	—
107	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₃	#121	1	2	yes	—
109	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₅₃	#131	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#137	1	2	yes	—
116	#88322	—	4	4	no	—
117	#88322	—	4	4	no	—
118	#9677	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	#390	1	1	yes	characteristic
120	C₂²×C₅₃	#407	1	1	yes	characteristic
121	C₂²×C₅₃	#408	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
123	C₂²×C₅₃	#419	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
125	C₂²×C₅₃	#421	1	1	yes	characteristic
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	4	4	no	—
132	C₄×C₅₃	—	4	1	no	—
133	C₄×C₅₃	—	8	2	no	—
134	C₄×C₅₃	#435	1	4	yes	—
135	C₄×C₅₃	#461	1	2	yes	—
136	C₄×C₅₃	#462	1	2	yes	—
137	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
138	#1767	—	53	1	no	—
139	C₄×C₄⋊C₈	—	53	1	no	—
140	C₂×Q₈⋊C₈	—	53	4	no	—
141	C₂×C₅₃	#1487	1	1	yes	characteristic
142	C₂×C₅₃	#1491	1	1	yes	characteristic
143	C₂×C₅₃	#1493	1	1	yes	characteristic
144	C₂×C₅₃	#1497	1	1	yes	characteristic
145	C₂×C₅₃	#1955	1	1	yes	characteristic
146	C₂×C₅₃	#1980	1	1	yes	characteristic
147	C₂×C₅₃	#1981	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
149	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₄⋊C₈	—	106	1	no	—
152	C₂×#97	—	106	1	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	#391	Dic₅₃	1	2	yes	—
155	#389	—	2	1	no	—
156	#389	—	2	1	no	—
157	#384	Dic₅₃	1	1	yes	characteristic
158	C₄×C₄⋊C₄	—	2	1	no	—
159	C₄×C₄⋊C₄	—	2	1	no	—
160	C₄³	C₂×D₅₃	1	1	yes	characteristic
161	#336	—	106	1	no	—
162	#336	—	106	1	no	—
163	Q₈⋊C₈	—	106	8	no	—
164	C₅₃	#9677	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
166	C₂²×C₈	—	106	2	no	—
167	C₂²×C₈	—	106	2	no	—
168	C₄⋊Q₈	—	2	4	no	—
169	C₄×Q₈	—	2	4	no	—
170	C₄×Q₈	—	2	4	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
185	C₄⋊C₈	—	53	4	no	—
186	C₄⋊C₈	—	106	4	no	—
187	C₄⋊C₈	—	106	2	no	—
188	#97	—	212	2	no	—
189	C₄×C₈	—	106	4	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	#11719917	1	4	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂×C₈	—	106	4	no	—
206	C₂×C₈	—	106	4	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₂×C₈	—	212	2	no	—
210	C₂×C₈	—	212	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	4	no	—
224	C₄²	#11719924	1	2	yes	—
225	C₄²	#11719925	1	2	yes	—
226	C₄²	D₄×D₅₃	1	2	yes	—
227	C₄²	#11719947	1	2	yes	—
228	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
229	Q₈	—	2	8	no	—
230	Q₈	—	4	4	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	#13133141	1	2	yes	—
249	C₂×C₄	#13133141	1	2	yes	—
250	C₂×C₄	#13133144	1	2	yes	—
251	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
252	C₂×C₄	#13133188	1	2	yes	—
253	C₂×C₄	#13133204	1	1	yes	characteristic
254	C₂×C₄	#13133213	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719917	1	2	yes	—
256	C₂×C₄	#13133224	1	2	yes	—
257	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
258	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
259	C₂×C₄	#13133248	1	2	yes	—
260	C₈	—	212	4	no	—
261	C₈	—	212	4	no	—
262	C₂²	#16810526	1	1	yes	characteristic
263	C₂²	#16810587	1	1	yes	characteristic
264	C₂²	#16810590	1	1	yes	characteristic
265	C₂²	C₂×#13133141	1	1	yes	characteristic
266	C₂²	#16810632	1	1	yes	characteristic
267	C₂²	C₂×#13133144	1	1	yes	characteristic
268	C₂²	#16810640	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	4	no	—
276	C₄	—	8	2	no	—
277	C₄	#16810695	1	4	yes	—
278	C₄	#16810825	1	2	yes	—
279	C₄	#16810830	1	2	yes	—
280	C₂	#20273144	1	1	yes	characteristic
281	C₂	#20273162	1	1	yes	characteristic
282	C₂	#20273170	1	1	yes	characteristic
283	C₂	#20273188	1	1	yes	characteristic
284	C₂	#20275514	1	1	yes	characteristic
285	C₂	#20275635	1	1	yes	characteristic
286	C₂	#20275639	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#9677
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
164. C₅₃	#9677	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
C₄×C₄⋊#88322	2
C₂×Q₈⋊#88322	2
C₄×C₄⋊Q₈×C₅₃	2
#9677	53