Subgroups — #27272581

Order 13568.

410 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283097	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	#20272540	C₂	1	2	yes	maximal
4	#20272539	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×#13133128	—	2	1	no	—
8	#16810529	C₄	1	1	yes	characteristic
9	#16810529	C₄	1	1	yes	characteristic
10	#16810529	C₂²	1	1	yes	characteristic
11	#16810521	C₄	1	1	yes	characteristic
12	#16810521	C₄	1	1	yes	characteristic
13	#16810521	C₂²	1	1	yes	characteristic
14	#16810507	—	2	1	no	—
15	#16810507	—	2	1	no	—
16	#16810507	—	2	1	no	—
17	#16810491	—	2	1	no	—
18	#16810378	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	—	2	1	no	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#336	C₂²	1	2	yes	—
23	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719919	—	2	1	no	—
29	C₂×#11719919	—	2	1	no	—
30	C₂×#11719919	—	2	1	no	—
31	C₄⋊#88322	—	2	4	no	—
32	#13133128	—	4	2	no	—
33	#13133124	—	2	1	no	—
34	#13133124	—	2	1	no	—
35	#13133124	—	2	1	no	—
36	#13133124	—	2	1	no	—
37	#13133124	—	2	1	no	—
38	#13133124	—	2	1	no	—
39	#13133124	—	2	1	no	—
40	#13133124	—	2	1	no	—
41	#13133124	—	2	1	no	—
42	C₂²×C₈×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₅₃	—	2	4	no	—
51	C₄×C₈×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅₃	—	2	1	no	—
60	C₂²×Dic₅₃	—	2	1	no	—
61	C₂²×Dic₅₃	—	2	1	no	—
62	C₂²×Dic₅₃	—	2	1	no	—
63	C₂×Dic₁₀₆	—	2	2	no	—
64	C₂×Dic₁₀₆	—	4	1	no	—
65	C₂×Dic₁₀₆	—	4	1	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₄²×C₅₃	—	4	1	no	—
76	C₄²×C₅₃	SD₁₆	1	2	yes	—
77	C₄²×C₅₃	Q₁₆	1	2	yes	—
78	C₄⋊Dic₅₃	—	2	2	no	—
79	C₄⋊Dic₅₃	—	4	1	no	—
80	C₄⋊Dic₅₃	—	4	1	no	—
81	#11719919	—	4	2	no	—
82	#11719919	—	4	2	no	—
83	#11719919	—	4	1	no	—
84	#11719919	—	4	1	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	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	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	#103	1	2	yes	—
104	C₂×C₄×C₅₃	#103	1	2	yes	—
105	C₂×C₄×C₅₃	#104	1	2	yes	—
106	C₂×C₄×C₅₃	#104	1	2	yes	—
107	C₂×C₄×C₅₃	#131	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
112	C₂×Dic₅₃	—	4	1	no	—
113	C₂×Dic₅₃	—	4	1	no	—
114	C₂×Dic₅₃	—	4	1	no	—
115	C₂×Dic₅₃	—	4	1	no	—
116	C₂×Dic₅₃	—	4	1	no	—
117	C₂×Dic₅₃	—	4	1	no	—
118	C₂×Dic₅₃	—	4	1	no	—
119	C₂×Dic₅₃	—	4	1	no	—
120	C₂×Dic₅₃	—	4	1	no	—
121	C₂×Dic₅₃	—	4	1	no	—
122	C₂×Dic₅₃	—	4	1	no	—
123	C₂×Dic₅₃	—	4	1	no	—
124	Dic₁₀₆	—	8	2	no	—
125	C₈×C₅₃	—	4	4	no	—
126	#88322	—	4	4	no	—
127	#7689	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	#408	1	1	yes	characteristic
129	C₂²×C₅₃	#410	1	1	yes	characteristic
130	C₂²×C₅₃	#413	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
132	C₂²×C₅₃	#417	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
134	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	2	no	—
144	Dic₅₃	—	8	1	no	—
145	Dic₅₃	—	8	1	no	—
146	Dic₅₃	—	8	1	no	—
147	Dic₅₃	—	8	1	no	—
148	#2863	—	53	1	no	—
149	#1767	—	53	1	no	—
150	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
151	#1272	—	53	2	no	—
152	#1271	—	53	2	no	—
153	C₂×C₅₃	#1473	1	1	yes	characteristic
154	C₂×C₅₃	#1483	1	1	yes	characteristic
155	C₂×C₅₃	#1493	1	1	yes	characteristic
156	C₂×C₅₃	#1503	1	1	yes	characteristic
157	C₂×C₅₃	#1509	1	1	yes	characteristic
158	C₂×C₅₃	#1524	1	1	yes	characteristic
159	C₂×C₅₃	#1550	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	2	1	no	—
161	C₂×C₄⋊C₈	—	106	1	no	—
162	C₂×#97	—	106	1	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	#391	—	53	1	no	—
165	#391	—	53	1	no	—
166	#391	—	53	1	no	—
167	#389	—	53	1	no	—
168	#389	—	53	1	no	—
169	#389	—	53	1	no	—
170	#387	—	106	1	no	—
171	#387	—	106	1	no	—
172	#387	—	106	1	no	—
173	#383	—	106	1	no	—
174	C₄³	C₂×D₅₃	1	1	yes	characteristic
175	#336	—	53	2	no	—
176	#336	C₂×D₅₃	1	2	yes	—
177	C₅₃	#7689	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	53	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	106	2	no	—
181	C₂×C₄⋊C₄	—	53	1	no	—
182	C₂×C₄⋊C₄	—	53	1	no	—
183	C₂×C₄⋊C₄	—	53	1	no	—
184	C₂×C₄⋊C₄	—	106	1	no	—
185	C₂×C₄⋊C₄	—	106	1	no	—
186	C₂×C₄⋊C₄	—	106	1	no	—
187	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
189	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
190	C₂×C₄²	D₂₁₂	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	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	106	4	no	—
196	#97	—	212	2	no	—
197	C₄×C₈	—	4	2	no	—
198	#95	—	106	1	no	—
199	#95	—	106	1	no	—
200	#95	—	106	1	no	—
201	#95	—	106	1	no	—
202	#95	—	106	1	no	—
203	#95	—	106	1	no	—
204	#95	—	106	1	no	—
205	#95	—	106	1	no	—
206	#95	—	106	1	no	—
207	C₂×Q₈	—	106	2	no	—
208	C₂×Q₈	—	212	1	no	—
209	C₂×Q₈	—	212	1	no	—
210	C₂²×C₄	—	106	1	no	—
211	C₂²×C₄	—	106	1	no	—
212	C₂²×C₄	—	106	1	no	—
213	C₂²×C₄	—	106	1	no	—
214	C₂²×C₄	#11719921	1	1	yes	characteristic
215	C₂²×C₄	#11719921	1	1	yes	characteristic
216	C₂²×C₄	#11719921	1	1	yes	characteristic
217	C₂²×C₄	#11719921	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	106	4	no	—
225	C₂×C₈	—	212	2	no	—
226	C₂×C₈	—	212	2	no	—
227	C₄⋊C₄	—	106	2	no	—
228	C₄⋊C₄	—	212	1	no	—
229	C₄⋊C₄	—	212	1	no	—
230	C₄⋊C₄	—	212	2	no	—
231	C₄⋊C₄	—	212	2	no	—
232	C₄⋊C₄	—	212	1	no	—
233	C₄⋊C₄	—	212	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₄²	#11719913	1	2	yes	—
242	C₄²	Dic₂₁₂	1	2	yes	—
243	C₂³	C₂×#11719921	1	1	yes	characteristic
244	Q₈	—	424	2	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	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	212	1	no	—
260	C₂×C₄	—	212	1	no	—
261	C₂×C₄	—	212	1	no	—
262	C₂×C₄	—	212	1	no	—
263	C₂×C₄	—	212	1	no	—
264	C₂×C₄	—	212	1	no	—
265	C₂×C₄	—	212	1	no	—
266	C₂×C₄	—	212	1	no	—
267	C₂×C₄	—	212	1	no	—
268	C₂×C₄	—	212	1	no	—
269	C₂×C₄	—	212	1	no	—
270	C₂×C₄	—	212	1	no	—
271	C₂×C₄	#13133142	1	2	yes	—
272	C₂×C₄	#13133142	1	2	yes	—
273	C₂×C₄	#13133143	1	2	yes	—
274	C₂×C₄	#13133145	1	2	yes	—
275	C₂×C₄	#13133223	1	2	yes	—
276	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
277	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
278	C₂×C₄	#13133243	1	1	yes	characteristic
279	C₂×C₄	#13133248	1	1	yes	characteristic
280	C₈	—	4	4	no	—
281	C₈	—	212	4	no	—
282	C₂²	#16810589	1	1	yes	characteristic
283	C₂²	#16810596	1	1	yes	characteristic
284	C₂²	#16810605	1	1	yes	characteristic
285	C₂²	C₂×#13133142	1	1	yes	characteristic
286	C₂²	#16810622	1	1	yes	characteristic
287	C₂²	C₂×#13133143	1	1	yes	characteristic
288	C₂²	C₂×#13133145	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	424	1	no	—
299	C₄	—	424	1	no	—
300	C₄	—	424	1	no	—
301	C₄	—	424	1	no	—
302	C₂	#20273082	1	1	yes	characteristic
303	C₂	#20273126	1	1	yes	characteristic
304	C₂	#20273171	1	1	yes	characteristic
305	C₂	#20273215	1	1	yes	characteristic
306	C₂	#20273238	1	1	yes	characteristic
307	C₂	#20273304	1	1	yes	characteristic
308	C₂	#20273403	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#7689
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
177. C₅₃	#7689	semidirect

Maximal subgroups

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

subgroup	index
#20283097	2
#20274568	2
#20272540	2
#20272539	2
C₄×C₄⋊C₈×C₅₃	2
#7689	53