Subgroups — #27284099

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 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	#20274867	C₂	1	1	yes	maximal, characteristic
3	#20272555	C₂	1	4	yes	maximal
4	C₄²×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	—	2	1	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	#16810529	—	2	1	no	—
11	#16810521	—	2	1	no	—
12	#16810521	C₄	1	2	yes	—
13	#16810493	C₄	1	1	yes	characteristic
14	C₄×C₄⋊Dic₅₃	—	2	1	no	—
15	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
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	2	no	—
25	C₄×Dic₁₀₆	—	4	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	1	no	—
27	C₂×C₄⋊Dic₅₃	—	2	1	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×#11719919	—	2	2	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₄⋊#88322	—	2	4	no	—
34	C₄⋊#88322	—	4	2	no	—
35	C₄⋊#88322	—	4	2	no	—
36	#13133124	—	2	2	no	—
37	#13133124	—	2	2	no	—
38	C₂²×C₈×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	C₄×C₈×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	—	2	4	no	—
48	C₄×C₈×C₅₃	—	2	2	no	—
49	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	2	2	no	—
58	C₂²×Dic₅₃	—	2	2	no	—
59	C₂×Dic₁₀₆	—	2	4	no	—
60	C₂×Dic₁₀₆	—	4	2	no	—
61	C₂×Dic₁₀₆	—	4	2	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	C₂×C₈	1	2	yes	—
69	C₄²×C₅₃	M₁₆	1	2	yes	—
70	C₄²×C₅₃	D₈	1	2	yes	—
71	C₄²×C₅₃	SD₁₆	1	2	yes	—
72	C₄²×C₅₃	SD₁₆	1	2	yes	—
73	C₄²×C₅₃	Q₁₆	1	2	yes	—
74	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
75	C₄²×C₅₃	#41	1	2	yes	—
76	C₄⋊Dic₅₃	—	2	4	no	—
77	C₄⋊Dic₅₃	—	2	2	no	—
78	C₄⋊Dic₅₃	—	2	4	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄⋊Dic₅₃	—	4	2	no	—
81	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
82	#11719919	—	4	4	no	—
83	C₄×Dic₅₃	—	4	2	no	—
84	C₄×Dic₅₃	—	4	2	no	—
85	C₂×#88322	—	4	2	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	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₃	#102	1	2	yes	—
98	C₂×C₄×C₅₃	#102	1	2	yes	—
99	C₂×C₄×C₅₃	#103	1	2	yes	—
100	C₂×C₄×C₅₃	#103	1	2	yes	—
101	C₂×C₄×C₅₃	#104	1	2	yes	—
102	C₂×C₄×C₅₃	#104	1	2	yes	—
103	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#131	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	—	4	2	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	C₂×Dic₅₃	—	4	2	no	—
120	Dic₁₀₆	—	4	8	no	—
121	C₈×C₅₃	—	2	8	no	—
122	#88322	—	8	2	no	—
123	#9870	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	#391	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₂²×C₅₃	#408	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	D₄⋊C₈	1	4	yes	—
135	C₄×C₅₃	Q₈⋊C₈	1	4	yes	—
136	C₄×C₅₃	C₈×Q₈	1	2	yes	—
137	C₄×C₅₃	#446	1	2	yes	—
138	C₄×C₅₃	#486	1	1	yes	characteristic
139	C₄×C₅₃	#487	1	1	yes	characteristic
140	C₄×C₅₃	#498	1	1	yes	characteristic
141	C₄×C₅₃	#500	1	1	yes	characteristic
142	Dic₅₃	—	4	4	no	—
143	Dic₅₃	—	8	2	no	—
144	C₄×C₄⋊Q₈	—	53	1	no	—
145	#1842	—	53	1	no	—
146	C₄²×C₈	D₅₃	1	1	yes	characteristic
147	#1279	—	53	4	no	—
148	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
149	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
150	C₂×C₅₃	#1492	1	1	yes	characteristic
151	C₂×C₅₃	#1493	1	1	yes	characteristic
152	C₂×C₅₃	#1984	1	1	yes	characteristic
153	C₂×C₅₃	#1990	1	1	yes	characteristic
154	C₂×C₅₃	#1994	1	1	yes	characteristic
155	C₂×C₄⋊Q₈	—	53	1	no	—
156	C₂×C₄×Q₈	—	106	1	no	—
157	C₂×C₄⋊C₈	—	53	2	no	—
158	C₂×C₄⋊C₈	—	106	1	no	—
159	C₂×C₄⋊C₈	—	106	1	no	—
160	C₂×C₄×C₈	—	2	1	no	—
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
163	#391	—	106	1	no	—
164	#389	—	53	2	no	—
165	#389	—	106	1	no	—
166	#384	—	53	1	no	—
167	C₄×C₄⋊C₄	—	53	2	no	—
168	C₄×C₄⋊C₄	—	106	1	no	—
169	C₄³	C₂×D₅₃	1	1	yes	characteristic
170	C₅₃	#9870	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	—	106	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	106	2	no	—
174	C₄⋊Q₈	—	53	4	no	—
175	C₄⋊Q₈	—	106	2	no	—
176	C₄×Q₈	—	212	2	no	—
177	C₄×Q₈	—	212	2	no	—
178	C₂×C₄⋊C₄	—	53	2	no	—
179	C₂×C₄⋊C₄	—	53	2	no	—
180	C₂×C₄⋊C₄	—	106	1	no	—
181	C₂×C₄⋊C₄	—	106	2	no	—
182	C₂×C₄⋊C₄	—	106	1	no	—
183	C₂×C₄⋊C₄	—	106	1	no	—
184	C₂×C₄²	—	106	2	no	—
185	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
187	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
188	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
192	C₄⋊C₈	—	106	4	no	—
193	C₄⋊C₈	—	212	2	no	—
194	C₄⋊C₈	—	212	2	no	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	Dic₁₀₆	1	4	yes	complemented
199	#95	—	106	2	no	—
200	#95	—	106	2	no	—
201	C₂×Q₈	—	106	4	no	—
202	C₂×Q₈	—	212	2	no	—
203	C₂×Q₈	—	212	2	no	—
204	C₂²×C₄	—	106	2	no	—
205	C₂²×C₄	—	106	2	no	—
206	C₂²×C₄	#11719921	1	1	yes	characteristic
207	C₂²×C₄	#11719921	1	1	yes	characteristic
208	C₂²×C₄	#11719921	1	1	yes	characteristic
209	C₂²×C₄	#11719921	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	212	2	no	—
217	C₂×C₈	—	212	2	no	—
218	C₂×C₈	—	212	2	no	—
219	C₄⋊C₄	—	53	4	no	—
220	C₄⋊C₄	—	106	4	no	—
221	C₄⋊C₄	—	106	2	no	—
222	C₄⋊C₄	—	106	4	no	—
223	C₄⋊C₄	—	212	4	no	—
224	C₄⋊C₄	—	212	2	no	—
225	C₄⋊C₄	—	212	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	212	2	no	—
230	C₄²	—	212	2	no	—
231	C₄²	C₈×D₅₃	1	2	yes	—
232	C₄²	#11719912	1	2	yes	—
233	C₄²	#11719913	1	2	yes	—
234	C₄²	#11719913	1	2	yes	—
235	C₄²	D₄₂₄	1	2	yes	—
236	C₄²	Dic₂₁₂	1	2	yes	—
237	C₄²	C₂×Dic₁₀₆	1	2	yes	—
238	C₄²	#11719945	1	2	yes	—
239	C₂³	C₂×#11719921	1	1	yes	characteristic
240	Q₈	—	212	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	106	4	no	—
248	C₂×C₄	—	212	2	no	—
249	C₂×C₄	—	212	2	no	—
250	C₂×C₄	—	212	2	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	—	212	2	no	—
253	C₂×C₄	#13133129	1	2	yes	—
254	C₂×C₄	#13133129	1	2	yes	—
255	C₂×C₄	#13133139	1	2	yes	—
256	C₂×C₄	#13133139	1	2	yes	—
257	C₂×C₄	#13133142	1	2	yes	—
258	C₂×C₄	#13133142	1	2	yes	—
259	C₂×C₄	#13133145	1	2	yes	—
260	C₂×C₄	#13133145	1	2	yes	—
261	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
262	C₂×C₄	#13133196	1	2	yes	—
263	C₂×C₄	#13133205	1	1	yes	characteristic
264	C₂×C₄	#13133217	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
267	C₂×C₄	#13133223	1	2	yes	—
268	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
271	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₈	—	424	2	no	—
274	C₂²	#16810529	1	1	yes	characteristic
275	C₂²	C₂×#13133129	1	1	yes	characteristic
276	C₂²	#16810589	1	1	yes	characteristic
277	C₂²	C₂×#13133139	1	1	yes	characteristic
278	C₂²	C₂×#13133142	1	1	yes	characteristic
279	C₂²	C₂×#13133145	1	1	yes	characteristic
280	C₂²	C₂×#13133145	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	424	2	no	—
285	C₄	—	212	4	no	—
286	C₄	#16810352	1	4	yes	—
287	C₄	#16810355	1	4	yes	—
288	C₄	C₈×Dic₁₀₆	1	2	yes	—
289	C₄	#16810753	1	2	yes	—
290	C₄	#16810970	1	1	yes	characteristic
291	C₄	#16810974	1	1	yes	characteristic
292	C₄	#16811021	1	1	yes	characteristic
293	C₄	#16811028	1	1	yes	characteristic
294	C₂	C₂×#16810352	1	1	yes	characteristic
295	C₂	C₂×#16810355	1	1	yes	characteristic
296	C₂	#20273167	1	1	yes	characteristic
297	C₂	#20273171	1	1	yes	characteristic
298	C₂	#20275651	1	1	yes	characteristic
299	C₂	#20275676	1	1	yes	characteristic
300	C₂	#20275692	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#9870
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
23. #13133217	C₈	semidirect
49. C₄×C₈×C₅₃	Q₈	semidirect
170. C₅₃	#9870	semidirect
198. 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
#20274867	2
#20272555	2
C₄²×C₈×C₅₃	2
#9870	53