Subgroups — #27356122

Order 13568.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283817	C₂	1	2	yes	maximal
2	#20273624	C₂	1	2	yes	maximal, complemented
3	#20273566	C₂	1	4	yes	maximal, complemented
4	#20273534	C₂	1	2	yes	maximal, complemented
5	#20273463	C₂	1	1	yes	maximal, characteristic
6	#20273136	C₂	1	2	yes	maximal
7	#20273004	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
9	#16810754	—	2	2	no	—
10	#16810754	C₂²	1	4	yes	—
11	Q₈×#88322	—	2	2	no	—
12	Q₈×#88322	C₂²	1	4	yes	—
13	#16810684	—	2	4	no	—
14	#16810676	C₂²	1	2	yes	—
15	#16810676	C₂²	1	1	yes	characteristic
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	2	yes	—
18	#16810374	C₂²	1	1	yes	characteristic
19	#16810374	C₂²	1	1	yes	characteristic
20	Q₈⋊#88322	C₂²	1	4	yes	—
21	Q₈⋊#88322	C₂²	1	4	yes	—
22	#16810345	C₂²	1	2	yes	—
23	#16810344	C₂²	1	2	yes	—
24	C₅₃×#533	C₂²	1	2	yes	—
25	C₅₃×#500	C₄	1	1	yes	characteristic
26	C₅₃×#494	C₄	1	1	yes	characteristic
27	C₅₃×#487	C₄	1	2	yes	—
28	C₅₃×#467	C₄	1	2	yes	—
29	C₅₃×#467	C₄	1	2	yes	—
30	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
31	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#11719917	—	2	1	no	—
34	C₄⋊#88322	—	2	4	no	—
35	C₄⋊#88322	—	2	4	no	—
36	C₄⋊#88322	D₄	1	2	yes	—
37	C₄⋊#88322	C₂³	1	2	yes	—
38	C₄⋊#88322	C₂³	1	2	yes	—
39	C₂²⋊#88322	—	2	2	no	—
40	C₂²⋊#88322	D₄	1	2	yes	—
41	#13133128	—	2	1	no	—
42	#13133128	—	2	4	no	—
43	C₄×#88322	—	2	2	no	—
44	C₄×#88322	—	2	4	no	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
53	#125×C₅₃	—	2	2	no	—
54	C₂²⋊Q₈×C₅₃	—	2	2	no	—
55	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₅₃	—	2	2	no	—
57	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
58	#117×C₅₃	—	2	2	no	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	#107×C₅₃	C₂×C₄	1	2	yes	—
61	#107×C₅₃	C₂×C₄	1	2	yes	—
62	#103×C₅₃	C₂×C₄	1	4	yes	—
63	#103×C₅₃	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₅₃	—	2	2	no	—
69	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	Q₁₆×C₅₃	—	2	4	no	—
74	Q₁₆×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	2	2	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
93	#11719917	—	4	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	C₂×D₄	1	4	yes	—
98	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₅₃	—	2	4	no	—
100	Q₈×C₅₃	—	2	4	no	—
101	Q₈×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	C₄×D₄	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	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
114	C₈×C₅₃	—	2	1	no	—
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	#131	1	2	yes	—
118	#88322	—	2	4	no	—
119	#88322	—	4	2	no	—
120	#88322	—	4	2	no	—
121	#88322	—	4	2	no	—
122	#18090	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	4	4	no	—
131	C₄×C₅₃	—	4	4	no	—
132	C₄×C₅₃	#444	1	2	yes	—
133	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
134	C₄×C₅₃	#568	1	1	yes	characteristic
135	C₄×C₅₃	#578	1	2	yes	—
136	#3170	D₅₃	1	1	yes	characteristic
137	#2986	—	53	2	no	—
138	#1593	—	53	2	no	—
139	#1584	—	53	4	no	—
140	#1579	—	53	2	no	—
141	#1564	—	53	1	no	—
142	#1485	—	53	2	no	—
143	#1451	—	53	1	no	—
144	C₂×C₅₃	—	4	1	no	—
145	C₂×C₅₃	#2935	1	1	yes	characteristic
146	C₂×C₅₃	#2950	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
148	#533	C₂×D₅₃	1	2	yes	—
149	#500	Dic₅₃	1	1	yes	characteristic
150	#494	Dic₅₃	1	1	yes	characteristic
151	#487	Dic₅₃	1	2	yes	—
152	#467	Dic₅₃	1	2	yes	—
153	#467	Dic₅₃	1	2	yes	—
154	#446	—	53	4	no	—
155	#446	—	106	2	no	—
156	C₈×Q₈	—	53	4	no	—
157	C₈×Q₈	—	106	2	no	—
158	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
159	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
160	#433	—	106	4	no	—
161	#431	—	53	1	no	—
162	#431	—	53	2	no	—
163	C₄×M₁₆	—	53	1	no	—
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	#335	—	53	2	no	—
166	#334	—	53	1	no	—
167	#334	—	53	1	no	—
168	Q₈⋊C₈	—	53	4	no	—
169	Q₈⋊C₈	—	53	4	no	—
170	#322	—	53	2	no	—
171	#322	—	53	2	no	—
172	C₅₃	#18090	1	1	yes	characteristic, complemented, Sylow
173	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
174	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
175	C₂×M₁₆	—	106	1	no	—
176	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
179	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
180	#125	—	2	2	no	—
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂²⋊Q₈	C₂×Dic₅₃	1	4	yes	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
185	#117	—	2	2	no	—
186	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	#107	C₂×Dic₅₃	1	2	yes	—
188	#107	C₂×Dic₅₃	1	2	yes	—
189	C₄⋊C₈	—	53	2	no	—
190	C₄⋊C₈	—	53	2	no	—
191	C₄⋊C₈	—	53	2	no	—
192	C₄⋊C₈	—	106	4	no	—
193	C₄⋊C₈	—	106	4	no	—
194	#103	C₂×Dic₅₃	1	4	yes	—
195	#103	C₂×Dic₅₃	1	4	yes	—
196	C₂²⋊C₈	—	53	2	no	—
197	C₂²⋊C₈	—	106	2	no	—
198	#97	—	106	4	no	—
199	#97	—	106	1	no	—
200	C₄×C₈	—	53	2	no	—
201	C₄×C₈	—	53	2	no	—
202	C₄×C₈	—	106	4	no	—
203	C₄×C₈	—	106	2	no	—
204	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
210	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
211	C₂²×C₄	#11719947	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	4	2	no	—
215	M₁₆	—	212	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	53	4	no	—
219	C₂×C₈	—	106	2	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
223	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	D₄×D₅₃	1	1	yes	characteristic
235	C₄²	#11719947	1	1	yes	characteristic
236	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
237	C₄²	C₂³×D₅₃	1	1	yes	characteristic
238	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
239	Q₈	—	2	4	no	—
240	Q₈	—	2	4	no	—
241	Q₈	—	4	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
249	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	106	4	no	—
258	C₈	—	212	2	no	—
259	C₈	—	212	2	no	—
260	C₈	—	212	2	no	—
261	C₈	#13133224	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	#16810745	1	2	yes	—
272	C₄	C₂×#13133224	1	1	yes	characteristic
273	C₄	#16811390	1	1	yes	characteristic
274	C₄	#16811436	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	#20283459	1	1	yes	characteristic
277	C₂	#20283583	1	1	yes	characteristic
278	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#18090
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #20273624	C₂	semidirect
3. #20273566	C₂	semidirect
4. #20273534	C₂	semidirect
172. C₅₃	#18090	semidirect

Maximal subgroups

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

subgroup	index
#20283817	2
#20273624	2
#20273566	2
#20273534	2
#20273463	2
#20273136	2
#20273004	2
C₅₃×#3170	2
#18090	53