Subgroups — #27372058

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285660	C₂	1	2	yes	maximal
2	#20285262	C₂	1	2	yes	maximal
3	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
4	#20274395	C₂	1	4	yes	maximal, complemented
5	#20274332	C₂	1	2	yes	maximal, complemented
6	#20274309	C₂	1	2	yes	maximal, complemented
7	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133217	C₂²	1	2	yes	—
10	#16811033	—	2	4	no	—
11	#16811028	—	2	2	no	—
12	#16811028	C₂²	1	2	yes	—
13	#16811028	C₂²	1	2	yes	—
14	#16811026	C₂²	1	4	yes	—
15	#16811019	C₂²	1	4	yes	—
16	#16811002	C₂²	1	4	yes	—
17	#16810975	C₂²	1	4	yes	—
18	#16810833	—	2	4	no	—
19	#16810778	—	2	4	no	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810666	—	2	4	no	—
22	C₂×#13133151	—	2	2	no	—
23	C₄×#11719917	C₂²	1	1	yes	characteristic
24	#16810345	C₂²	1	2	yes	—
25	C₈×#88322	C₂²	1	2	yes	—
26	C₅₃×#533	C₂²	1	2	yes	—
27	C₅₃×#500	C₂²	1	1	yes	characteristic
28	C₅₃×#494	C₂²	1	1	yes	characteristic
29	C₅₃×#487	C₂²	1	2	yes	—
30	C₅₃×#467	—	2	2	no	—
31	C₄×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₀₆	—	2	2	no	—
34	C₂×#11719925	—	2	8	no	—
35	C₂×#11719917	D₄	1	2	yes	—
36	#13133217	—	2	2	no	—
37	#13133217	—	2	2	no	—
38	#13133217	—	2	2	no	—
39	#13133217	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₅₃	—	2	2	no	—
41	C₂×C₄⋊Dic₅₃	—	2	2	no	—
42	#13133151	—	2	4	no	—
43	#13133151	—	2	4	no	—
44	#13133150	—	2	8	no	—
45	#13133148	—	2	8	no	—
46	C₄⋊#88322	D₄	1	2	yes	—
47	#13133140	—	2	8	no	—
48	C₂²⋊#88322	D₄	1	2	yes	—
49	#13133128	D₄	1	2	yes	—
50	C₄×#88322	C₂³	1	2	yes	—
51	C₄×#88322	C₂³	1	2	yes	—
52	C₂×Q₁₆×C₅₃	—	2	2	no	—
53	C₂²×C₈×C₅₃	D₄	1	2	yes	—
54	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
56	#125×C₅₃	—	2	2	no	—
57	C₂²⋊Q₈×C₅₃	—	2	2	no	—
58	C₂²⋊Q₈×C₅₃	—	2	2	no	—
59	C₄×Q₈×C₅₃	—	2	2	no	—
60	C₄×Q₈×C₅₃	—	2	2	no	—
61	#117×C₅₃	—	2	2	no	—
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	#107×C₅₃	—	2	2	no	—
64	#103×C₅₃	—	2	4	no	—
65	C₄×C₈×C₅₃	D₄	1	2	yes	—
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	—	2	2	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	4	2	no	—
74	C₂×Dic₁₀₆	—	2	4	no	—
75	C₂×Dic₁₀₆	—	4	2	no	—
76	C₂×Dic₁₀₆	—	8	4	no	—
77	Q₁₆×C₅₃	—	4	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
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	2	no	—
85	C₄⋊C₄×C₅₃	—	4	2	no	—
86	C₄⋊C₄×C₅₃	—	4	2	no	—
87	C₂²⋊C₄×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	—	4	2	no	—
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
93	#11719925	—	8	8	no	—
94	C₄⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	—	2	4	no	—
96	C₄⋊Dic₅₃	—	4	2	no	—
97	C₄⋊Dic₅₃	—	4	2	no	—
98	#11719917	Q₁₆	1	8	yes	complemented
99	C₂×#88322	C₂×D₄	1	4	yes	—
100	C₂×#88322	C₂×D₄	1	4	yes	—
101	C₂³×C₅₃	#127	1	1	yes	characteristic
102	Q₈×C₅₃	—	4	4	no	—
103	Q₈×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₅₃	—	4	4	no	—
116	C₂×Dic₅₃	—	8	4	no	—
117	Dic₁₀₆	—	4	8	no	—
118	Dic₁₀₆	—	8	4	no	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	#137	1	4	yes	—
121	#88322	—	2	4	no	—
122	#88322	C₂×Q₁₆	1	8	yes	—
123	#19633	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	#494	1	2	yes	—
127	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	8	2	no	—
131	C₄×C₅₃	—	8	2	no	—
132	C₄×C₅₃	#494	1	2	yes	—
133	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
134	C₄×C₅₃	C₂×#137	1	2	yes	—
135	C₄×C₅₃	#576	1	2	yes	—
136	Dic₅₃	—	8	4	no	—
137	#3209	—	53	2	no	—
138	#3170	D₅₃	1	1	yes	characteristic
139	#3165	—	53	2	no	—
140	C₂×#500	—	53	1	no	—
141	#1723	—	53	4	no	—
142	#1713	—	53	2	no	—
143	#1707	—	53	2	no	—
144	C₈×M₁₆	—	53	1	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
147	C₂×C₅₃	#3152	1	1	yes	characteristic
148	C₂×C₅₃	#3155	1	1	yes	characteristic
149	#533	C₂×D₅₃	1	2	yes	—
150	C₂×C₄⋊Q₈	—	53	2	no	—
151	#501	—	106	4	no	—
152	#500	—	53	4	no	—
153	#500	—	53	2	no	—
154	#500	—	53	2	no	—
155	#500	—	106	2	no	—
156	#500	C₂×D₅₃	1	1	yes	characteristic
157	#498	—	53	4	no	—
158	#494	—	53	4	no	—
159	#494	C₂×D₅₃	1	1	yes	characteristic
160	#487	—	53	4	no	—
161	#487	C₂×D₅₃	1	2	yes	—
162	#467	—	2	2	no	—
163	#462	—	106	4	no	—
164	#451	—	106	4	no	—
165	C₄×Q₁₆	—	2	2	no	—
166	#431	—	53	1	no	—
167	#428	—	106	4	no	—
168	C₂×#107	—	106	2	no	—
169	C₄×M₁₆	—	53	1	no	—
170	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
171	#322	—	53	2	no	—
172	C₈²	—	53	2	no	—
173	C₅₃	#19633	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	106	2	no	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	106	8	no	—
177	C₂×M₁₆	—	53	2	no	—
178	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
179	C₄⋊Q₈	—	53	4	no	—
180	C₄⋊Q₈	—	106	2	no	—
181	C₄⋊Q₈	—	106	2	no	—
182	C₄⋊Q₈	—	106	2	no	—
183	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
184	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
185	#125	—	2	2	no	—
186	C₂²⋊Q₈	—	2	2	no	—
187	C₂²⋊Q₈	—	2	2	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×Q₈	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	—	106	2	no	—
192	C₂×C₄⋊C₄	—	106	2	no	—
193	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
194	#107	—	2	2	no	—
195	#107	—	106	8	no	—
196	#107	—	106	4	no	—
197	#107	—	106	4	no	—
198	#106	—	106	8	no	—
199	C₄⋊C₈	—	53	2	no	—
200	#103	—	2	4	no	—
201	#103	—	106	8	no	—
202	C₂²⋊C₈	—	53	2	no	—
203	#97	—	53	2	no	—
204	C₄×C₈	—	53	2	no	—
205	C₄×C₈	—	53	2	no	—
206	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
207	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	106	4	no	—
212	C₂×Q₈	—	424	4	no	—
213	C₂×Q₈	—	212	2	no	—
214	C₂²×C₄	—	212	2	no	—
215	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
218	Q₁₆	—	4	4	no	—
219	Q₁₆	—	424	8	no	—
220	M₁₆	—	53	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	53	4	no	—
223	C₂×C₈	—	53	4	no	—
224	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	106	4	no	—
233	C₄⋊C₄	—	106	4	no	—
234	C₄⋊C₄	—	212	2	no	—
235	C₄⋊C₄	—	212	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	D₄×D₅₃	1	1	yes	characteristic
239	C₄²	D₄×D₅₃	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	C₄²	C₂³×D₅₃	1	1	yes	characteristic
242	C₂³	#13133213	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	Q₈	—	4	4	no	—
245	Q₈	—	424	4	no	—
246	Q₈	—	212	8	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	424	4	no	—
252	C₂×C₄	—	212	4	no	—
253	C₂×C₄	#13133213	1	1	yes	characteristic
254	C₂×C₄	#13133213	1	1	yes	characteristic
255	C₂×C₄	#13133213	1	1	yes	characteristic
256	C₂×C₄	Q₁₆×D₅₃	1	4	yes	—
257	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	53	8	no	—
262	C₈	—	106	4	no	—
263	C₈	#13133247	1	4	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#16811001	1	2	yes	—
267	C₂²	C₂×#13133213	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	2	no	—
272	C₄	—	424	4	no	—
273	C₄	#16811001	1	2	yes	—
274	C₄	C₂×Q₁₆×D₅₃	1	2	yes	—
275	C₄	C₂×#13133247	1	2	yes	—
276	C₄	#16811425	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	C₂×#16811001	1	1	yes	characteristic
279	C₂	#20285175	1	1	yes	characteristic
280	C₂	#20285202	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133213
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	C₂³×C₅₃	#127
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#19633
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #20274395	C₂	semidirect
5. #20274332	C₂	semidirect
6. #20274309	C₂	semidirect
98. #11719917	Q₁₆	semidirect
173. C₅₃	#19633	semidirect

Maximal subgroups

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

subgroup	index
#20285660	2
#20285262	2
C₂×#16811028	2
#20274395	2
#20274332	2
#20274309	2
C₈×#11719917	2
C₅₃×#3170	2
#19633	53