Subgroups — #27284073

Order 13568.

528 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	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272554	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₆	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810529	—	2	1	no	—
10	#16810521	—	2	1	no	—
11	#16810521	C₄	1	2	yes	—
12	#16810493	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₅₃	—	2	1	no	—
14	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
15	#16810378	—	2	1	no	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	—	2	1	no	—
18	C₄³×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#336	—	2	1	no	—
20	C₂²×Dic₁₀₆	—	2	1	no	—
21	C₂²×#88322	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	C₈	1	4	yes	complemented
24	C₄×Dic₁₀₆	—	4	4	no	—
25	C₂×C₄⋊Dic₅₃	—	2	1	no	—
26	C₂×C₄⋊Dic₅₃	—	2	1	no	—
27	C₂×C₄⋊Dic₅₃	—	2	1	no	—
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×#11719919	—	2	2	no	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	C₄⋊#88322	—	2	4	no	—
33	C₄×#88322	—	2	4	no	—
34	#13133124	—	2	2	no	—
35	#13133124	—	2	2	no	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	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	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	Q₈	1	4	yes	complemented
46	C₄×C₈×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	2	2	no	—
55	C₂²×Dic₅₃	—	2	2	no	—
56	C₂×Dic₁₀₆	—	2	4	no	—
57	C₂×Dic₁₀₆	—	4	2	no	—
58	C₂×Dic₁₀₆	—	4	2	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	4	1	no	—
66	C₄²×C₅₃	C₂×C₈	1	2	yes	—
67	C₄²×C₅₃	M₁₆	1	2	yes	—
68	C₄²×C₅₃	SD₁₆	1	2	yes	—
69	C₄²×C₅₃	Q₁₆	1	2	yes	—
70	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
71	C₄²×C₅₃	#41	1	2	yes	—
72	C₄⋊Dic₅₃	—	2	4	no	—
73	C₄⋊Dic₅₃	—	2	2	no	—
74	C₄⋊Dic₅₃	—	2	4	no	—
75	C₄⋊Dic₅₃	—	4	2	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
78	#11719919	—	4	4	no	—
79	C₄×Dic₅₃	—	4	4	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	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	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₅₃	#103	1	2	yes	—
96	C₂×C₄×C₅₃	#103	1	2	yes	—
97	C₂×C₄×C₅₃	#104	1	2	yes	—
98	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#131	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×Dic₅₃	—	2	4	no	—
109	C₂×Dic₅₃	—	4	2	no	—
110	C₂×Dic₅₃	—	4	2	no	—
111	C₂×Dic₅₃	—	4	2	no	—
112	C₂×Dic₅₃	—	4	2	no	—
113	C₂×Dic₅₃	—	4	2	no	—
114	Dic₁₀₆	—	4	8	no	—
115	C₈×C₅₃	—	4	4	no	—
116	#88322	—	4	4	no	—
117	#9864	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	#391	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₅₃	#408	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
122	C₂²×C₅₃	#418	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
124	C₂²×C₅₃	#421	1	1	yes	characteristic
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	4	1	no	—
130	C₄×C₅₃	Q₈⋊C₈	1	4	yes	—
131	C₄×C₅₃	C₈×Q₈	1	2	yes	—
132	C₄×C₅₃	#446	1	2	yes	—
133	C₄×C₅₃	#476	1	2	yes	—
134	C₄×C₅₃	#477	1	2	yes	—
135	Dic₅₃	—	4	4	no	—
136	Dic₅₃	—	8	2	no	—
137	C₄×C₄⋊Q₈	—	53	1	no	—
138	C₄×C₄⋊C₈	—	53	1	no	—
139	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
140	#1279	—	53	4	no	—
141	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
142	C₂×C₅₃	#1478	1	1	yes	characteristic
143	C₂×C₅₃	#1481	1	1	yes	characteristic
144	C₂×C₅₃	#1484	1	1	yes	characteristic
145	C₂×C₅₃	#1984	1	1	yes	characteristic
146	C₂×C₅₃	#1986	1	1	yes	characteristic
147	C₂×C₅₃	#1987	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	—	53	1	no	—
149	C₂×C₄×Q₈	—	106	1	no	—
150	C₂×C₄⋊C₈	—	106	1	no	—
151	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
152	C₂×C₄×C₈	—	2	1	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	#391	—	106	1	no	—
155	#389	—	53	2	no	—
156	#389	—	106	1	no	—
157	#384	—	53	1	no	—
158	C₄×C₄⋊C₄	—	53	2	no	—
159	C₄×C₄⋊C₄	—	106	1	no	—
160	C₄³	C₂×D₅₃	1	1	yes	characteristic
161	#336	—	2	1	no	—
162	#336	—	106	1	no	—
163	C₅₃	#9864	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	—	106	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	106	2	no	—
167	C₄⋊Q₈	—	53	4	no	—
168	C₄⋊Q₈	—	106	2	no	—
169	C₄×Q₈	—	212	4	no	—
170	C₂×C₄⋊C₄	—	53	2	no	—
171	C₂×C₄⋊C₄	—	53	2	no	—
172	C₂×C₄⋊C₄	—	106	1	no	—
173	C₂×C₄⋊C₄	—	106	2	no	—
174	C₂×C₄⋊C₄	—	106	1	no	—
175	C₂×C₄⋊C₄	—	106	1	no	—
176	C₂×C₄²	—	106	2	no	—
177	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
180	C₂×C₄²	D₂₁₂	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	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	—	106	4	no	—
186	C₄⋊C₈	Dic₁₀₆	1	4	yes	complemented
187	C₄×C₈	—	4	2	no	—
188	C₄×C₈	—	106	4	no	—
189	#95	—	106	2	no	—
190	#95	—	106	2	no	—
191	C₂×Q₈	—	106	4	no	—
192	C₂×Q₈	—	212	2	no	—
193	C₂×Q₈	—	212	2	no	—
194	C₂²×C₄	—	106	2	no	—
195	C₂²×C₄	—	106	2	no	—
196	C₂²×C₄	#11719921	1	1	yes	characteristic
197	C₂²×C₄	#11719921	1	1	yes	characteristic
198	C₂²×C₄	#11719921	1	1	yes	characteristic
199	C₂²×C₄	#11719921	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	106	4	no	—
207	C₂×C₈	—	212	2	no	—
208	C₂×C₈	—	212	2	no	—
209	C₄⋊C₄	—	53	4	no	—
210	C₄⋊C₄	—	106	4	no	—
211	C₄⋊C₄	—	106	2	no	—
212	C₄⋊C₄	—	106	4	no	—
213	C₄⋊C₄	—	212	4	no	—
214	C₄⋊C₄	—	212	2	no	—
215	C₄⋊C₄	—	212	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	212	4	no	—
221	C₄²	C₈×D₅₃	1	2	yes	—
222	C₄²	#11719912	1	2	yes	—
223	C₄²	#11719923	1	2	yes	—
224	C₄²	#11719925	1	2	yes	—
225	C₄²	C₂×Dic₁₀₆	1	2	yes	—
226	C₄²	#11719945	1	2	yes	—
227	C₂³	C₂×#11719921	1	1	yes	characteristic
228	Q₈	—	212	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	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	—	106	4	no	—
239	C₂×C₄	—	212	2	no	—
240	C₂×C₄	—	212	2	no	—
241	C₂×C₄	—	212	2	no	—
242	C₂×C₄	—	212	2	no	—
243	C₂×C₄	—	212	2	no	—
244	C₂×C₄	#13133129	1	2	yes	—
245	C₂×C₄	#13133129	1	2	yes	—
246	C₂×C₄	#13133140	1	2	yes	—
247	C₂×C₄	#13133140	1	2	yes	—
248	C₂×C₄	#13133143	1	2	yes	—
249	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
250	C₂×C₄	#13133196	1	2	yes	—
251	C₂×C₄	#13133205	1	1	yes	characteristic
252	C₂×C₄	#13133217	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
255	C₂×C₄	#13133223	1	2	yes	—
256	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
258	C₂×C₄	#13133242	1	2	yes	—
259	C₈	—	4	4	no	—
260	C₈	—	212	4	no	—
261	C₂²	#16810529	1	1	yes	characteristic
262	C₂²	C₂×#13133129	1	1	yes	characteristic
263	C₂²	#16810589	1	1	yes	characteristic
264	C₂²	C₂×#13133140	1	1	yes	characteristic
265	C₂²	#16810627	1	1	yes	characteristic
266	C₂²	C₂×#13133143	1	1	yes	characteristic
267	C₂²	#16810639	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	424	2	no	—
274	C₄	—	212	4	no	—
275	C₄	#16810353	1	4	yes	—
276	C₄	C₈×Dic₁₀₆	1	2	yes	—
277	C₄	#16810753	1	2	yes	—
278	C₄	#16810913	1	2	yes	—
279	C₄	#16810918	1	2	yes	—
280	C₂	C₂×#16810353	1	1	yes	characteristic
281	C₂	#20273105	1	1	yes	characteristic
282	C₂	#20273117	1	1	yes	characteristic
283	C₂	#20273131	1	1	yes	characteristic
284	C₂	#20275651	1	1	yes	characteristic
285	C₂	#20275659	1	1	yes	characteristic
286	C₂	#20275663	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
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	#9864
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
23. #13133217	C₈	semidirect
45. C₄⋊C₈×C₅₃	Q₈	semidirect
163. C₅₃	#9864	semidirect
186. C₄⋊C₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
C₄×C₄⋊#88322	2
#20272554	2
C₄×C₄⋊C₈×C₅₃	2
#9864	53