Subgroups — #27367338

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 298 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	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
3	#20274233	C₂	1	4	yes	maximal, complemented
4	#20274109	C₂	1	4	yes	maximal, complemented
5	#20273463	C₂	1	1	yes	maximal, characteristic
6	#20273401	C₂	1	2	yes	maximal
7	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133217	C₂²	1	2	yes	—
9	#16811028	—	2	1	no	—
10	#16811028	—	2	1	no	—
11	#16811028	C₂²	1	2	yes	—
12	#16811028	C₂²	1	2	yes	—
13	#16811002	—	2	2	no	—
14	#16810975	—	2	2	no	—
15	#16810833	C₂²	1	8	yes	—
16	#16810778	—	2	4	no	—
17	#16810676	C₂²	1	2	yes	—
18	C₂×#13133151	—	2	1	no	—
19	C₂×#13133151	—	2	1	no	—
20	#16810376	C₂²	1	2	yes	—
21	#16810374	C₂²	1	2	yes	—
22	#16810368	C₂²	1	8	yes	—
23	C₅₃×#533	C₂²	1	1	yes	characteristic
24	C₅₃×#446	—	2	1	no	—
25	C₅₃×#446	C₂²	1	2	yes	—
26	C₈×Q₈×C₅₃	—	2	1	no	—
27	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#424	—	2	1	no	—
29	C₅₃×#423	C₂²	1	1	yes	characteristic
30	C₅₃×#405	—	2	1	no	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×Dic₁₀₆	—	2	2	no	—
33	C₂×#11719925	—	2	8	no	—
34	#13133217	—	2	2	no	—
35	#13133217	—	2	2	no	—
36	#13133217	—	2	2	no	—
37	#13133217	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₅₃	—	2	2	no	—
39	C₂×C₄⋊Dic₅₃	—	2	2	no	—
40	#13133151	—	2	2	no	—
41	#13133151	—	2	2	no	—
42	#13133151	—	2	2	no	—
43	#13133151	—	2	2	no	—
44	C₄⋊#88322	C₂³	1	4	yes	—
45	#13133140	—	2	8	no	—
46	C₂²⋊#88322	—	2	2	no	—
47	C₄×#88322	—	2	2	no	—
48	C₂×M₁₆×C₅₃	—	2	1	no	—
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
53	#125×C₅₃	—	2	1	no	—
54	C₂²⋊Q₈×C₅₃	—	2	1	no	—
55	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
56	C₄×Q₈×C₅₃	—	2	1	no	—
57	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
58	#117×C₅₃	—	2	1	no	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	2	no	—
61	C₄⋊C₈×C₅₃	—	2	2	no	—
62	C₄⋊C₈×C₅₃	—	2	1	no	—
63	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
64	#97×C₅₃	—	2	2	no	—
65	C₄×C₈×C₅₃	—	2	2	no	—
66	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₅₃	—	2	1	no	—
71	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
75	C₂²×Dic₅₃	—	4	2	no	—
76	C₂×Dic₁₀₆	—	2	4	no	—
77	C₂×Dic₁₀₆	—	4	2	no	—
78	C₂×Dic₁₀₆	—	8	4	no	—
79	M₁₆×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
83	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₅₃	—	2	1	no	—
88	C₄⋊C₄×C₅₃	—	2	1	no	—
89	C₄⋊C₄×C₅₃	—	2	1	no	—
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
93	C₂²⋊C₄×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	#41	1	1	yes	characteristic
98	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
99	#11719925	—	8	8	no	—
100	C₄⋊Dic₅₃	—	2	4	no	—
101	C₄⋊Dic₅₃	—	2	4	no	—
102	C₄⋊Dic₅₃	—	4	2	no	—
103	C₄⋊Dic₅₃	—	4	2	no	—
104	C₂×#88322	—	2	4	no	—
105	C₂³×C₅₃	#121	1	1	yes	characteristic
106	Q₈×C₅₃	—	2	4	no	—
107	Q₈×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
116	C₂×C₄×C₅₃	#137	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₅₃	—	4	4	no	—
121	C₂×Dic₅₃	—	8	4	no	—
122	Dic₁₀₆	—	4	8	no	—
123	Dic₁₀₆	—	8	4	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	4	2	no	—
127	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
128	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
129	#88322	—	4	4	no	—
130	#19208	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	#462	1	2	yes	—
134	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	4	2	no	—
138	C₄×C₅₃	—	4	2	no	—
139	C₄×C₅₃	#462	1	2	yes	—
140	C₄×C₅₃	#473	1	2	yes	—
141	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
143	C₄×C₅₃	#576	1	2	yes	—
144	Dic₅₃	—	8	4	no	—
145	#3209	—	53	2	no	—
146	C₂×#500	—	53	1	no	—
147	#2960	D₅₃	1	1	yes	characteristic
148	#1694	—	53	4	no	—
149	#1674	—	53	4	no	—
150	#1564	—	53	1	no	—
151	#1549	—	53	2	no	—
152	C₂×C₅₃	—	2	2	no	—
153	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
154	C₂×C₅₃	#3058	1	1	yes	characteristic
155	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
156	#533	C₂×D₅₃	1	1	yes	characteristic
157	C₂×C₄⋊Q₈	—	53	2	no	—
158	#500	—	53	2	no	—
159	#500	—	53	2	no	—
160	#500	—	106	1	no	—
161	#500	—	106	1	no	—
162	#494	—	106	2	no	—
163	#487	—	106	2	no	—
164	#462	—	53	8	no	—
165	#451	—	106	4	no	—
166	#446	—	2	1	no	—
167	#446	C₂×D₅₃	1	2	yes	—
168	C₈×Q₈	—	2	1	no	—
169	C₈×Q₈	C₂×D₅₃	1	2	yes	—
170	#431	—	53	2	no	—
171	C₂×#107	—	106	1	no	—
172	C₂×#107	—	106	1	no	—
173	#424	—	2	1	no	—
174	#423	C₂×D₅₃	1	1	yes	characteristic
175	#405	—	2	1	no	—
176	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
177	#335	—	53	2	no	—
178	#334	—	53	2	no	—
179	#332	—	53	8	no	—
180	C₅₃	#19208	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	106	2	no	—
182	C₂×Q₁₆	—	106	8	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₄⋊Q₈	—	53	4	no	—
187	C₄⋊Q₈	—	106	2	no	—
188	C₄⋊Q₈	—	106	2	no	—
189	C₄⋊Q₈	—	106	2	no	—
190	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
191	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
192	#125	—	2	1	no	—
193	C₂²⋊Q₈	—	2	1	no	—
194	C₂²⋊Q₈	D₂₁₂	1	2	yes	—
195	C₄×Q₈	—	2	1	no	—
196	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
197	#117	—	2	1	no	—
198	C₂×C₄⋊C₄	—	106	2	no	—
199	C₂×C₄⋊C₄	—	106	2	no	—
200	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	#107	—	106	2	no	—
202	#107	—	106	2	no	—
203	#107	—	106	2	no	—
204	#107	—	106	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	53	4	no	—
209	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
210	#103	—	106	8	no	—
211	C₂²⋊C₈	—	106	2	no	—
212	#97	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	—	106	2	no	—
215	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	106	4	no	—
221	C₂×Q₈	—	424	4	no	—
222	C₂×Q₈	—	212	2	no	—
223	C₂×Q₈	C₂×D₂₁₂	1	2	yes	—
224	C₂²×C₄	—	212	2	no	—
225	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
226	C₂²×C₄	#11719945	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
228	Q₁₆	—	424	8	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	#11719925	1	4	yes	complemented
234	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	106	4	no	—
244	C₄⋊C₄	—	106	4	no	—
245	C₄⋊C₄	—	212	2	no	—
246	C₄⋊C₄	—	212	2	no	—
247	C₄⋊C₄	C₂×D₂₁₂	1	2	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
251	C₄²	#11719945	1	1	yes	characteristic
252	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
253	C₄²	C₂³×D₅₃	1	1	yes	characteristic
254	C₂³	#13133191	1	1	yes	characteristic
255	Q₈	—	2	4	no	—
256	Q₈	—	4	2	no	—
257	Q₈	—	424	4	no	—
258	Q₈	—	212	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	424	4	no	—
264	C₂×C₄	—	212	4	no	—
265	C₂×C₄	#13133191	1	1	yes	characteristic
266	C₂×C₄	#13133191	1	1	yes	characteristic
267	C₂×C₄	#13133191	1	1	yes	characteristic
268	C₂×C₄	C₂×#11719925	1	2	yes	—
269	C₂×C₄	#13133247	1	2	yes	—
270	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
272	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	4	2	no	—
276	C₈	—	212	4	no	—
277	C₈	C₂×#11719925	1	2	yes	—
278	C₈	C₂×#11719925	1	2	yes	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	#16810833	1	2	yes	—
282	C₂²	C₂×#13133191	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	424	4	no	—
288	C₄	#16810833	1	2	yes	—
289	C₄	#16810898	1	2	yes	—
290	C₄	C₂²×#11719925	1	1	yes	characteristic
291	C₄	C₂×#13133247	1	1	yes	characteristic
292	C₄	#16811428	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	C₂×#16810833	1	1	yes	characteristic
295	C₂	#20284363	1	1	yes	characteristic
296	C₂	C₂×#16810898	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	#121
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#19208
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #20274233	C₂	semidirect
4. #20274109	C₂	semidirect
82. C₂×C₈×C₅₃	Q₁₆	semidirect
180. C₅₃	#19208	semidirect
233. C₂×C₈	#11719925	semidirect

Maximal subgroups

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

subgroup	index
#20285660	2
C₂×#16811028	2
#20274233	2
#20274109	2
#20273463	2
#20273401	2
C₅₃×#2960	2
#19208	53