Subgroups — #27356124

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	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20283816	C₂	1	2	yes	maximal
3	#20273625	C₂	1	2	yes	maximal, complemented
4	#20273567	C₂	1	4	yes	maximal, complemented
5	#20273535	C₂	1	2	yes	maximal, complemented
6	#20273463	C₂	1	1	yes	maximal, characteristic
7	#20273137	C₂	1	2	yes	maximal
8	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	#16811028	C₄	1	1	yes	characteristic
11	#16811003	C₄	1	1	yes	characteristic
12	#16810974	C₄	1	2	yes	—
13	#16810864	C₄	1	2	yes	—
14	#16810864	C₄	1	2	yes	—
15	#16810753	C₂²	1	4	yes	—
16	#16810750	—	2	2	no	—
17	C₈×Dic₁₀₆	C₂²	1	4	yes	—
18	#16810747	—	2	2	no	—
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	C₄×Dic₂₁₂	C₂²	1	2	yes	—
21	#16810681	—	2	4	no	—
22	#16810676	C₂²	1	2	yes	—
23	#16810376	C₂²	1	2	yes	—
24	#16810374	C₂²	1	1	yes	characteristic
25	#16810374	C₂²	1	1	yes	characteristic
26	#16810355	C₂²	1	4	yes	—
27	#16810355	C₂²	1	4	yes	—
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
35	#13133217	C₂×C₄	1	2	yes	—
36	#13133217	C₂×C₄	1	2	yes	—
37	#13133209	—	2	2	no	—
38	#13133196	—	2	2	no	—
39	#13133196	C₂×C₄	1	4	yes	—
40	C₄×Dic₁₀₆	—	2	2	no	—
41	C₄×Dic₁₀₆	C₂³	1	4	yes	—
42	#13133176	—	2	2	no	—
43	#13133151	C₂×C₄	1	2	yes	—
44	#13133151	C₂×C₄	1	2	yes	—
45	C₄⋊#88322	C₂³	1	2	yes	—
46	C₄⋊#88322	C₂³	1	2	yes	—
47	#13133146	—	2	4	no	—
48	#13133146	—	2	4	no	—
49	#13133142	C₂×C₄	1	4	yes	—
50	#13133142	C₂×C₄	1	4	yes	—
51	C₂²⋊#88322	—	2	2	no	—
52	#13133127	—	2	4	no	—
53	C₄×#88322	—	2	2	no	—
54	C₈×Dic₅₃	—	2	4	no	—
55	C₂×M₁₆×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
60	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
61	#97×C₅₃	—	2	1	no	—
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂×Dic₁₀₆	—	2	2	no	—
72	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
73	M₁₆×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	2	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	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	#41	1	1	yes	characteristic
85	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₅₃	—	2	4	no	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	—	2	2	no	—
90	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
91	#11719919	—	2	4	no	—
92	#11719919	—	2	4	no	—
93	C₄×Dic₅₃	—	2	4	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	Dic₂₁₂	—	2	4	no	—
97	Dic₂₁₂	—	4	2	no	—
98	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₅₃	—	2	4	no	—
110	C₂×Dic₅₃	—	2	4	no	—
111	Dic₁₀₆	—	2	4	no	—
112	Dic₁₀₆	—	2	4	no	—
113	Dic₁₀₆	—	4	4	no	—
114	C₈×C₅₃	—	2	1	no	—
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	4	no	—
118	C₈×C₅₃	—	4	2	no	—
119	C₈×C₅₃	#131	1	2	yes	—
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	#444	1	2	yes	—
131	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
132	C₄×C₅₃	#568	1	1	yes	characteristic
133	C₄×C₅₃	#578	1	2	yes	—
134	Dic₅₃	—	4	4	no	—
135	Dic₅₃	—	4	4	no	—
136	#3170	—	53	1	no	—
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	D₅₃	1	1	yes	characteristic
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	—	53	2	no	—
149	#500	—	53	1	no	—
150	#494	—	53	1	no	—
151	#487	—	53	2	no	—
152	#467	—	53	2	no	—
153	#467	—	53	2	no	—
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	—	53	2	no	—
159	C₄×Q₁₆	—	53	2	no	—
160	#433	—	106	4	no	—
161	#431	—	53	2	no	—
162	#431	C₂×D₅₃	1	1	yes	characteristic
163	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
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	C₂×D₅₃	1	2	yes	—
171	#322	C₂×D₅₃	1	2	yes	—
172	C₅₃	#18090	1	1	yes	characteristic, complemented, Sylow
173	C₂×Q₁₆	—	53	2	no	—
174	C₂×Q₁₆	—	53	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
178	C₄⋊Q₈	—	53	2	no	—
179	C₄⋊Q₈	—	53	2	no	—
180	#125	—	106	2	no	—
181	C₂²⋊Q₈	—	53	4	no	—
182	C₂²⋊Q₈	—	106	2	no	—
183	C₄×Q₈	—	53	4	no	—
184	C₄×Q₈	—	106	2	no	—
185	#117	—	106	2	no	—
186	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	#107	—	53	2	no	—
188	#107	—	53	2	no	—
189	C₄⋊C₈	—	53	2	no	—
190	C₄⋊C₈	—	53	2	no	—
191	C₄⋊C₈	—	106	4	no	—
192	C₄⋊C₈	—	106	4	no	—
193	C₄⋊C₈	D₂₁₂	1	2	yes	—
194	#103	—	53	4	no	—
195	#103	—	53	4	no	—
196	C₂²⋊C₈	—	106	2	no	—
197	C₂²⋊C₈	D₂₁₂	1	2	yes	—
198	#97	—	2	1	no	—
199	#97	—	106	4	no	—
200	C₄×C₈	—	106	4	no	—
201	C₄×C₈	—	106	2	no	—
202	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
203	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₅₃	1	2	yes	—
205	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₃	1	2	yes	—
208	C₂×Q₈	—	53	4	no	—
209	C₂×Q₈	—	106	2	no	—
210	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
212	C₂²×C₄	#11719945	1	1	yes	characteristic
213	Q₁₆	—	106	4	no	—
214	Q₁₆	—	212	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	106	2	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
226	C₄⋊C₄	—	53	4	no	—
227	C₄⋊C₄	—	106	2	no	—
228	C₄⋊C₄	—	106	2	no	—
229	C₄⋊C₄	—	106	4	no	—
230	C₄⋊C₄	—	106	2	no	—
231	C₄⋊C₄	—	106	4	no	—
232	C₂²⋊C₄	—	106	4	no	—
233	C₄²	—	106	4	no	—
234	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
235	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
236	C₄²	#11719945	1	1	yes	characteristic
237	C₄²	C₂³×D₅₃	1	1	yes	characteristic
238	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
239	Q₈	—	106	4	no	—
240	Q₈	—	106	4	no	—
241	Q₈	—	212	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	—	106	4	no	—
247	C₂×C₄	—	106	4	no	—
248	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
249	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
250	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
253	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₈	—	212	2	no	—
260	C₈	—	212	2	no	—
261	C₈	#13133225	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	212	4	no	—
270	C₄	—	212	4	no	—
271	C₄	#16810746	1	2	yes	—
272	C₄	C₂×#13133225	1	1	yes	characteristic
273	C₄	#16811389	1	1	yes	characteristic
274	C₄	#16811440	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	#20283461	1	1	yes	characteristic
277	C₂	#20283584	1	1	yes	characteristic
278	C₂	C₂×#16810746	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#18090
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #20273625	C₂	semidirect
4. #20273567	C₂	semidirect
5. #20273535	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
#20285301	2
#20283816	2
#20273625	2
#20273567	2
#20273535	2
#20273463	2
#20273137	2
C₅₃×#1451	2
#18090	53