Subgroups — #27293298

Order 13568.

320 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276518	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276514	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272655	C₂	1	2	yes	maximal, complemented
4	#20272639	C₂	1	2	yes	maximal
5	C₂×C₅₃×#433	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11719908	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₂²⋊#11719908	C₂²	1	2	yes	—
9	C₂²⋊#11719908	C₂²	1	2	yes	—
10	#16810399	—	2	1	no	—
11	C₄×#11719908	—	2	1	no	—
12	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#433	—	2	1	no	—
14	C₅₃×#433	—	2	1	no	—
15	C₅₃×#433	—	2	1	no	—
16	C₅₃×#433	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	#117×C₅₃	—	2	1	no	—
31	#117×C₅₃	—	2	1	no	—
32	#117×C₅₃	—	2	1	no	—
33	#117×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	1	no	—
41	C₄⋊C₈×C₅₃	—	2	1	no	—
42	C₄⋊C₈×C₅₃	—	2	1	no	—
43	C₄⋊C₈×C₅₃	—	2	1	no	—
44	C₂²⋊C₈×C₅₃	—	2	1	no	—
45	C₂²⋊C₈×C₅₃	—	2	1	no	—
46	C₂²⋊C₈×C₅₃	—	2	1	no	—
47	C₂²⋊C₈×C₅₃	—	2	1	no	—
48	#97×C₅₃	—	2	1	no	—
49	#97×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	4	1	no	—
68	C₂×C₈×C₅₃	—	4	1	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₂²⋊C₄×C₅₃	—	2	1	no	—
80	C₂²⋊C₄×C₅₃	—	2	1	no	—
81	C₂²⋊C₄×C₅₃	—	2	1	no	—
82	C₂²⋊C₄×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
88	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
89	#11719908	—	4	2	no	—
90	#11719908	—	4	2	no	—
91	C₂³×C₅₃	—	2	1	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	—	4	1	no	—
94	C₂³×C₅₃	—	4	1	no	—
95	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	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	2	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	4	1	no	—
120	C₈×C₅₃	—	4	1	no	—
121	#11461	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	8	2	no	—
128	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₂²×C₅₃	#504	1	2	yes	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	4	1	no	—
136	C₄×C₅₃	—	4	1	no	—
137	C₄×C₅₃	—	4	1	no	—
138	C₄×C₅₃	#410	1	1	yes	characteristic
139	C₄×C₅₃	#413	1	1	yes	characteristic
140	C₂×#433	D₅₃	1	1	yes	characteristic
141	#2163	—	53	1	no	—
142	#2162	—	53	1	no	—
143	#1322	—	53	2	no	—
144	#1314	—	53	2	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	—	8	1	no	—
147	C₂×C₅₃	#1458	1	1	yes	characteristic
148	C₂×C₅₃	#2113	1	1	yes	characteristic
149	C₂×C₅₃	#2115	1	1	yes	characteristic
150	C₂×#117	Dic₅₃	1	1	yes	characteristic
151	#433	—	2	1	no	—
152	#433	—	2	1	no	—
153	#433	—	2	1	no	—
154	#433	—	2	1	no	—
155	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
156	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
159	C₂×#97	Dic₅₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
161	C₄⋊C₁₆	—	106	1	no	—
162	C₄⋊C₁₆	—	106	1	no	—
163	C₂²⋊C₁₆	—	53	2	no	—
164	C₂²⋊C₁₆	—	53	2	no	—
165	#346	—	106	1	no	—
166	C₄×C₁₆	—	106	1	no	—
167	C₅₃	#11461	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
169	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	—	2	1	no	—
175	#117	—	2	1	no	—
176	#117	—	2	1	no	—
177	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
178	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	#88322	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	#88322	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
183	C₂×C₁₆	—	106	2	no	—
184	C₂×C₁₆	—	106	2	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	1	no	—
192	C₂²⋊C₈	—	2	1	no	—
193	#97	—	2	1	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
197	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
198	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂⁴	C₂×#88322	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
205	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
206	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
207	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
208	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
218	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	#11719917	1	1	yes	characteristic
231	C₄²	#11719917	1	1	yes	characteristic
232	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
234	C₁₆	—	212	2	no	—
235	C₁₆	—	212	2	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	4	1	no	—
239	C₂³	—	4	1	no	—
240	C₂³	C₂²×#88322	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
256	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
257	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	8	2	no	—
272	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
273	C₂²	#16811040	1	2	yes	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	#16810597	1	1	yes	characteristic
283	C₄	#16810607	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	8	1	no	—
286	C₂	#20273029	1	1	yes	characteristic
287	C₂	#20276329	1	1	yes	characteristic
288	C₂	#20276336	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#433	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#433	D₅₃
hypocenter	C₅₃	#11461
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
1. #20276518	C₂	semidirect
2. #20276514	C₂	semidirect
3. #20272655	C₂	semidirect
167. C₅₃	#11461	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
#20276514	2
#20272655	2
#20272639	2
C₂×C₅₃×#433	2
#11461	53