Subgroups — #27275068

Order 13568.

528 conjugacy classes of subgroups.

The subgroup lattice has 304 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×#16810369	C₂	1	2	yes	maximal
4	#20272598	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133128	—	2	1	no	—
9	C₂×C₄×#88322	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	#16810369	C₄	1	8	yes	complemented
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
15	C₅₃×#391	—	2	1	no	—
16	C₅₃×#389	—	2	1	no	—
17	C₅₃×#389	C₄	1	2	yes	—
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	C₂×C₄	1	4	yes	—
27	C₄⋊#88322	C₂×C₄	1	4	yes	—
28	#13133128	—	4	2	no	—
29	C₄×#88322	—	4	2	no	—
30	C₂²×Q₈×C₅₃	—	2	1	no	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₄×Q₈×C₅₃	—	4	2	no	—
34	C₄×Q₈×C₅₃	—	4	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	2	4	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	1	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	4	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	4	1	no	—
70	C₄⋊C₄×C₅₃	—	4	1	no	—
71	C₄⋊C₄×C₅₃	—	4	1	no	—
72	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	—	2	2	no	—
78	C₄²×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	—	4	4	no	—
80	C₄²×C₅₃	D₈	1	2	yes	—
81	C₄²×C₅₃	SD₁₆	1	2	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₅₃	#41	1	2	yes	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₅₃	—	4	4	no	—
92	Q₈×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	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	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	#101	1	2	yes	—
111	C₂×C₄×C₅₃	#102	1	2	yes	—
112	C₂×C₄×C₅₃	#102	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#135	1	2	yes	—
120	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#137	1	2	yes	—
123	#88322	—	4	4	no	—
124	#88322	—	4	4	no	—
125	#8236	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
128	C₂²×C₅₃	#413	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
130	C₂²×C₅₃	#416	1	1	yes	characteristic
131	C₂²×C₅₃	#417	1	1	yes	characteristic
132	C₂²×C₅₃	#419	1	1	yes	characteristic
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	4	1	no	—
139	C₄×C₅₃	—	4	4	no	—
140	C₄×C₅₃	—	8	2	no	—
141	C₄×C₅₃	C₄×D₈	1	2	yes	—
142	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
143	C₄×C₅₃	#440	1	2	yes	—
144	C₄×C₅₃	#441	1	2	yes	—
145	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
146	#1767	—	53	1	no	—
147	C₄×C₄⋊C₈	—	53	1	no	—
148	C₂×#332	—	53	2	no	—
149	#1296	—	53	2	no	—
150	C₂×C₅₃	#1546	1	1	yes	characteristic
151	C₂×C₅₃	#1547	1	1	yes	characteristic
152	C₂×C₅₃	#1552	1	1	yes	characteristic
153	C₂×C₅₃	#1554	1	1	yes	characteristic
154	C₂×C₅₃	C₄×#101	1	1	yes	characteristic
155	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
156	C₂×C₅₃	#1763	1	1	yes	characteristic
157	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	2	1	no	—
159	C₂×C₄⋊C₈	—	53	2	no	—
160	C₂×C₄⋊C₈	—	53	2	no	—
161	C₂×#97	—	106	1	no	—
162	C₂×C₄×C₈	—	106	1	no	—
163	#391	—	2	1	no	—
164	#389	—	2	1	no	—
165	#389	Dic₅₃	1	2	yes	—
166	#384	C₂×D₅₃	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	2	1	no	—
168	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
169	C₄³	C₂×D₅₃	1	1	yes	characteristic
170	#336	—	106	1	no	—
171	#336	—	106	1	no	—
172	#332	—	53	8	no	—
173	C₅₃	#8236	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	2	1	no	—
175	C₂²×C₈	—	106	2	no	—
176	C₂²×C₈	—	106	2	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	C₄×D₅₃	1	4	yes	—
179	C₄×Q₈	—	4	2	no	—
180	C₄×Q₈	—	4	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
195	C₄⋊C₈	—	53	4	no	—
196	C₄⋊C₈	—	53	4	no	—
197	C₄⋊C₈	—	106	2	no	—
198	C₄⋊C₈	—	106	2	no	—
199	#97	—	212	2	no	—
200	C₄×C₈	—	212	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂×C₈	—	106	4	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	212	2	no	—
219	C₂×C₈	—	212	2	no	—
220	C₂×C₈	—	212	2	no	—
221	C₂×C₈	—	212	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	4	no	—
238	C₄²	#11719922	1	2	yes	—
239	C₄²	#11719923	1	2	yes	—
240	C₄²	C₂×C₄×D₅₃	1	2	yes	—
241	C₄²	#11719945	1	2	yes	—
242	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	#13133136	1	2	yes	—
263	C₂×C₄	#13133138	1	2	yes	—
264	C₂×C₄	#13133138	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
266	C₂×C₄	#13133176	1	1	yes	characteristic
267	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
268	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
269	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
271	C₂×C₄	#13133237	1	2	yes	—
272	C₂×C₄	#13133242	1	1	yes	characteristic
273	C₂×C₄	#13133247	1	1	yes	characteristic
274	C₂×C₄	#13133248	1	2	yes	—
275	C₈	—	212	4	no	—
276	C₈	—	212	4	no	—
277	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
278	C₂²	C₂×#13133136	1	1	yes	characteristic
279	C₂²	#16810607	1	1	yes	characteristic
280	C₂²	C₂×#13133138	1	1	yes	characteristic
281	C₂²	#16810619	1	1	yes	characteristic
282	C₂²	#16810624	1	1	yes	characteristic
283	C₂²	#16810632	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	2	no	—
292	C₄	C₄×#11719922	1	2	yes	—
293	C₄	C₄×#11719923	1	2	yes	—
294	C₄	#16810721	1	2	yes	—
295	C₄	#16810728	1	2	yes	—
296	C₂	#20273387	1	1	yes	characteristic
297	C₂	#20273392	1	1	yes	characteristic
298	C₂	#20273415	1	1	yes	characteristic
299	C₂	#20273425	1	1	yes	characteristic
300	C₂	C₄×#13133136	1	1	yes	characteristic
301	C₂	C₄×#13133138	1	1	yes	characteristic
302	C₂	#20274545	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
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	#8236
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
12. #16810369	C₄	semidirect
173. C₅₃	#8236	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×#16810369	2
#20272598	2
C₄×C₄⋊Q₈×C₅₃	2
#8236	53