Subgroups — #27274776

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
2	#20273098	C₂	1	1	yes	maximal, characteristic
3	#20273098	C₂	1	1	yes	maximal, characteristic
4	#20273007	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272552	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272552	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₄	1	2	yes	—
9	C₄×#11719945	C₂²	1	1	yes	characteristic
10	#16810676	C₂²	1	1	yes	characteristic
11	#16810676	C₂²	1	1	yes	characteristic
12	#16810578	C₄	1	2	yes	—
13	#16810577	—	2	2	no	—
14	C₄×#11719917	C₂²	1	1	yes	characteristic
15	C₄×#11719912	—	2	1	no	—
16	C₄×C₈×D₅₃	—	2	1	no	—
17	#16810355	C₄	1	2	yes	—
18	#16810355	C₄	1	2	yes	—
19	#16810352	C₄	1	2	yes	—
20	#16810352	C₄	1	2	yes	—
21	#16810345	—	2	1	no	—
22	#16810345	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#322	—	2	2	no	—
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	#13133223	—	2	4	no	—
29	#13133223	—	2	4	no	—
30	C₂×#11719917	C₂×C₄	1	2	yes	—
31	C₂×#11719912	—	2	2	no	—
32	C₂×C₈×D₅₃	—	2	2	no	—
33	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
35	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
36	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₄×C₂²⋊D₅₃	—	2	1	no	—
38	C₄×C₂²⋊D₅₃	—	2	1	no	—
39	#13133176	C₂×C₄	1	1	yes	characteristic
40	#13133174	—	2	1	no	—
41	C₄²×D₅₃	—	2	1	no	—
42	C₄⋊#88322	C₂×C₄	1	2	yes	—
43	C₄⋊#88322	C₂×C₄	1	2	yes	—
44	C₂²⋊#88322	—	2	1	no	—
45	C₂²⋊#88322	—	2	1	no	—
46	#13133128	—	2	1	no	—
47	#13133127	—	2	2	no	—
48	C₄×#88322	—	2	1	no	—
49	C₄×#88322	—	2	1	no	—
50	C₄×#88322	—	2	1	no	—
51	C₈×Dic₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	—	2	1	no	—
53	C₂²×C₈×C₅₃	—	2	1	no	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₃	—	2	1	no	—
67	#11719945	—	2	2	no	—
68	#11719945	M₁₆	1	4	yes	—
69	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	—	2	1	no	—
73	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²⋊Dic₅₃	—	2	1	no	—
87	#11719921	—	2	1	no	—
88	#11719921	—	2	1	no	—
89	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
90	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
91	#11719919	—	2	1	no	—
92	#11719919	—	2	1	no	—
93	C₄×Dic₅₃	—	2	1	no	—
94	C₄×Dic₅₃	—	2	1	no	—
95	C₄×Dic₅₃	—	2	1	no	—
96	#11719917	—	2	2	no	—
97	#11719917	M₁₆	1	4	yes	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	2	2	no	—
101	#11719912	—	2	8	no	—
102	C₈×D₅₃	—	2	8	no	—
103	C₂³×C₅₃	#117	1	1	yes	characteristic
104	C₂²×D₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
115	C₂²⋊D₅₃	—	2	4	no	—
116	C₂×Dic₅₃	—	2	1	no	—
117	C₂×Dic₅₃	—	2	1	no	—
118	C₂×Dic₅₃	—	2	1	no	—
119	D₂₁₂	—	2	1	no	—
120	D₂₁₂	#97	1	2	yes	—
121	C₄×D₅₃	—	2	4	no	—
122	C₄×D₅₃	—	4	2	no	—
123	C₄×D₅₃	—	4	2	no	—
124	Dic₁₀₆	—	2	1	no	—
125	Dic₁₀₆	#97	1	2	yes	—
126	C₈×C₅₃	—	2	4	no	—
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	—	2	2	no	—
129	C₈×C₅₃	—	4	2	no	—
130	#88322	—	2	4	no	—
131	#88322	—	4	2	no	—
132	#88322	—	4	2	no	—
133	#8186	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₅₃	#432	1	2	yes	—
138	C₂×D₅₃	—	2	2	no	—
139	C₂×D₅₃	—	4	1	no	—
140	C₂×D₅₃	—	4	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
144	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
145	C₄×C₅₃	#407	1	2	yes	—
146	C₄×C₅₃	#443	1	2	yes	—
147	C₄×C₅₃	#444	1	2	yes	—
148	Dic₅₃	—	2	2	no	—
149	Dic₅₃	—	4	1	no	—
150	Dic₅₃	—	4	1	no	—
151	C₄×#131	—	53	1	no	—
152	#1476	—	53	1	no	—
153	#1476	—	53	1	no	—
154	#1452	—	53	1	no	—
155	C₂×#322	D₅₃	1	1	yes	characteristic, complemented
156	#1278	—	53	1	no	—
157	#1278	—	53	1	no	—
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	#1749	1	1	yes	characteristic
160	C₂×C₅₃	#1764	1	1	yes	characteristic
161	C₂×C₅₃	#1765	1	1	yes	characteristic
162	D₅₃	—	4	2	no	—
163	C₂×#131	—	53	2	no	—
164	C₄×#41	—	53	1	no	—
165	#431	—	53	1	no	—
166	#431	—	53	1	no	—
167	#405	—	53	2	no	—
168	#405	—	106	2	no	—
169	C₄×M₁₆	—	53	1	no	—
170	C₄×M₁₆	—	106	1	no	—
171	C₂×C₄×C₈	—	106	1	no	—
172	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
175	Q₈⋊C₈	—	53	2	no	—
176	Q₈⋊C₈	—	53	2	no	—
177	D₄⋊C₈	—	53	2	no	—
178	D₄⋊C₈	—	53	2	no	—
179	#322	—	2	2	no	—
180	#322	—	106	1	no	—
181	#322	—	106	1	no	—
182	C₅₃	#8186	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	—	53	1	no	—
184	#131	—	106	4	no	—
185	#131	—	106	4	no	—
186	C₂×M₁₆	—	53	2	no	—
187	C₂×M₁₆	—	106	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	106	2	no	—
191	C₂²×C₈	C₄×D₅₃	1	2	yes	—
192	C₄×Q₈	—	53	1	no	—
193	C₄×Q₈	—	53	1	no	—
194	C₄×D₄	—	53	1	no	—
195	C₄×D₄	—	53	1	no	—
196	C₄×D₄	—	106	1	no	—
197	C₄×D₄	—	106	1	no	—
198	#117	—	53	1	no	—
199	#117	—	106	1	no	—
200	C₂×C₄²	—	106	1	no	—
201	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
202	C₄⋊C₈	—	53	2	no	—
203	C₄⋊C₈	—	53	2	no	—
204	C₂²⋊C₈	—	106	1	no	—
205	C₂²⋊C₈	—	106	1	no	—
206	#97	—	106	2	no	—
207	#97	—	106	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	106	2	no	—
211	C₄×C₈	—	106	1	no	—
212	C₄×C₈	—	106	1	no	—
213	C₄×C₈	—	106	1	no	—
214	C₄×C₈	C₄×D₅₃	1	2	yes	—
215	C₄×C₈	C₄×D₅₃	1	2	yes	—
216	C₄×C₈	D₂₁₂	1	1	yes	characteristic
217	C₄×C₈	D₂₁₂	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
219	#41	—	53	4	no	—
220	#41	—	106	2	no	—
221	C₂×Q₈	—	53	1	no	—
222	C₂×D₄	—	53	1	no	—
223	C₂×D₄	—	106	1	no	—
224	C₂²×C₄	—	106	1	no	—
225	C₂²×C₄	—	106	1	no	—
226	C₂²×C₄	—	106	1	no	—
227	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₂²×C₄	#11719945	1	1	yes	characteristic
229	C₂²×C₄	#11719945	1	1	yes	characteristic
230	M₁₆	—	53	4	no	—
231	M₁₆	—	106	8	no	—
232	M₁₆	—	106	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	106	8	no	—
240	C₂×C₈	—	106	2	no	—
241	C₂×C₈	—	106	2	no	—
242	C₂×C₈	—	106	2	no	—
243	C₂×C₈	#11719921	1	2	yes	—
244	C₂×C₈	#11719921	1	2	yes	—
245	C₄⋊C₄	—	53	1	no	—
246	C₄⋊C₄	—	53	1	no	—
247	C₄⋊C₄	—	106	1	no	—
248	C₄⋊C₄	—	106	1	no	—
249	C₂²⋊C₄	—	106	1	no	—
250	C₂²⋊C₄	—	106	1	no	—
251	C₂²⋊C₄	—	106	1	no	—
252	C₄²	—	106	1	no	—
253	C₄²	—	106	1	no	—
254	C₄²	—	106	1	no	—
255	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
257	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂³	—	106	1	no	—
260	C₂³	#13133174	1	1	yes	characteristic
261	Q₈	—	53	2	no	—
262	Q₈	—	106	1	no	—
263	D₄	—	53	2	no	—
264	D₄	—	106	4	no	—
265	D₄	—	106	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	106	1	no	—
269	C₂×C₄	—	106	4	no	—
270	C₂×C₄	—	106	1	no	—
271	C₂×C₄	—	106	1	no	—
272	C₂×C₄	—	212	2	no	—
273	C₂×C₄	—	212	2	no	—
274	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
276	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
278	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
279	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
280	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
281	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	—	4	2	no	—
286	C₈	—	106	4	no	—
287	C₈	—	212	2	no	—
288	C₈	—	212	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	106	2	no	—
292	C₂²	—	212	1	no	—
293	C₂²	—	212	1	no	—
294	C₂²	C₄×#11719921	1	1	yes	characteristic
295	C₂²	#16810678	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	106	2	no	—
299	C₄	—	212	1	no	—
300	C₄	—	212	1	no	—
301	C₄	#97×D₅₃	1	1	yes	characteristic
302	C₄	#16810575	1	1	yes	characteristic
303	C₄	#16810586	1	2	yes	—
304	C₄	#16810741	1	2	yes	—
305	C₄	#16810746	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	212	2	no	—
308	C₂	#20274478	1	1	yes	characteristic
309	C₂	#20274552	1	1	yes	characteristic
310	C₂	#20274559	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#8186
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #20273007	C₂	semidirect
5. #20272552	C₂	semidirect
6. #20272552	C₂	semidirect
7. C₂×C₅₃×#322	C₂	semidirect
155. C₂×#322	D₅₃	semidirect
182. C₅₃	#8186	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133223	2
#20273098	2
#20273007	2
#20272552	2
C₂×C₅₃×#322	2
#8186	53