Subgroups — #27355869

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283664	C₂	1	2	yes	maximal
2	#20273612	C₂	1	2	yes	maximal, complemented
3	#20273566	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#88322	C₂	1	2	yes	maximal, complemented
5	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
6	#20273118	C₂	1	2	yes	maximal
7	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
9	#16810754	—	2	4	no	—
10	#16810754	C₂²	1	4	yes	—
11	Q₈×#88322	C₂²	1	4	yes	—
12	#16810676	C₂²	1	1	yes	characteristic
13	#16810650	—	2	2	no	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	#16810579	—	2	2	no	—
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	1	yes	characteristic
18	#16810376	C₂²	1	2	yes	—
19	#16810376	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	C₈×#88322	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	D₄	1	1	yes	characteristic
34	C₂×#11719917	D₄	1	1	yes	characteristic
35	C₂²×#88322	—	2	2	no	—
36	C₄⋊#88322	—	2	2	no	—
37	C₄⋊#88322	—	2	1	no	—
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	—	2	4	no	—
40	C₄⋊#88322	C₂³	1	2	yes	—
41	C₄⋊#88322	C₂³	1	2	yes	—
42	C₂²⋊#88322	—	2	1	no	—
43	#13133128	—	2	4	no	—
44	#13133128	D₄	1	2	yes	—
45	C₄×#88322	—	2	4	no	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₄×#88322	C₂³	1	2	yes	—
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
54	#125×C₅₃	—	2	2	no	—
55	C₂²⋊Q₈×C₅₃	—	2	2	no	—
56	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
57	C₄×Q₈×C₅₃	—	2	2	no	—
58	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
59	#117×C₅₃	—	2	2	no	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#107×C₅₃	C₂×C₄	1	2	yes	—
62	#107×C₅₃	C₂×C₄	1	2	yes	—
63	#103×C₅₃	C₂×C₄	1	4	yes	—
64	#103×C₅₃	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×Q₈×C₅₃	—	2	2	no	—
70	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
74	Q₁₆×C₅₃	—	2	4	no	—
75	Q₁₆×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
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	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
88	C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	#41	1	1	yes	characteristic
93	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
94	#11719917	—	2	2	no	—
95	#11719917	Q₁₆	1	4	yes	complemented
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	4	4	no	—
100	C₂×#88322	C₂×D₄	1	2	yes	—
101	C₂×#88322	C₂×D₄	1	2	yes	—
102	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
103	Q₈×C₅₃	—	2	4	no	—
104	Q₈×C₅₃	—	2	4	no	—
105	Q₈×C₅₃	—	8	2	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	4	1	no	—
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄×C₅₃	#135	1	2	yes	—
116	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	#131	1	2	yes	—
123	#88322	—	2	2	no	—
124	#88322	—	4	2	no	—
125	#88322	—	4	2	no	—
126	#88322	—	4	2	no	—
127	#88322	C₂×Q₁₆	1	4	yes	—
128	#18068	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
132	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	4	4	no	—
137	C₄×C₅₃	—	4	4	no	—
138	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
139	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
140	C₄×C₅₃	#568	1	1	yes	characteristic
141	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
143	#3170	D₅₃	1	1	yes	characteristic
144	#2962	—	53	2	no	—
145	#1591	—	53	2	no	—
146	#1584	—	53	4	no	—
147	C₈×Q₁₆	—	53	2	no	—
148	C₂×#335	—	53	1	no	—
149	#1481	—	53	2	no	—
150	C₈×M₁₆	—	53	1	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	#2935	1	1	yes	characteristic
153	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
154	C₂×C₅₃	#2957	1	1	yes	characteristic
155	#533	C₂×D₅₃	1	2	yes	—
156	#500	Dic₅₃	1	1	yes	characteristic
157	#494	Dic₅₃	1	1	yes	characteristic
158	#487	Dic₅₃	1	2	yes	—
159	#467	Dic₅₃	1	2	yes	—
160	#467	Dic₅₃	1	2	yes	—
161	#446	—	53	4	no	—
162	#446	—	106	4	no	—
163	C₈×Q₈	—	53	4	no	—
164	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
165	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
166	#431	—	53	1	no	—
167	#424	—	106	2	no	—
168	C₂×C₄⋊C₈	—	53	2	no	—
169	#405	—	106	2	no	—
170	C₄×M₁₆	—	53	1	no	—
171	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
172	#335	—	53	1	no	—
173	#335	—	53	2	no	—
174	#335	—	53	1	no	—
175	Q₈⋊C₈	—	53	4	no	—
176	Q₈⋊C₈	—	53	4	no	—
177	#322	—	53	2	no	—
178	C₈²	—	53	2	no	—
179	C₅₃	#18068	1	1	yes	characteristic, complemented, Sylow
180	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
181	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
182	C₂×M₁₆	—	53	1	no	—
183	C₂×M₁₆	—	53	1	no	—
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
188	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
189	#125	—	2	2	no	—
190	C₂²⋊Q₈	—	2	2	no	—
191	C₂²⋊Q₈	C₂×Dic₅₃	1	4	yes	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
194	#117	—	2	2	no	—
195	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
196	#107	C₂×Dic₅₃	1	2	yes	—
197	#107	C₂×Dic₅₃	1	2	yes	—
198	C₄⋊C₈	—	53	2	no	—
199	C₄⋊C₈	—	53	2	no	—
200	C₄⋊C₈	—	106	4	no	—
201	C₄⋊C₈	—	106	4	no	—
202	C₄⋊C₈	—	106	2	no	—
203	C₄⋊C₈	—	106	1	no	—
204	#103	C₂×Dic₅₃	1	4	yes	—
205	#103	C₂×Dic₅₃	1	4	yes	—
206	C₂²⋊C₈	—	106	1	no	—
207	#97	—	53	2	no	—
208	#97	—	106	4	no	—
209	C₄×C₈	—	53	2	no	—
210	C₄×C₈	—	53	2	no	—
211	C₄×C₈	—	106	4	no	—
212	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
213	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
218	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
219	C₂²×C₄	#11719947	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	4	2	no	—
223	M₁₆	—	53	4	no	—
224	M₁₆	—	106	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	53	2	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	—	212	4	no	—
233	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	D₄×D₅₃	1	1	yes	characteristic
246	C₄²	#11719947	1	1	yes	characteristic
247	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
248	C₄²	C₂³×D₅₃	1	1	yes	characteristic
249	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
259	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
260	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
261	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
262	C₂×C₄	#13133240	1	2	yes	—
263	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	53	4	no	—
270	C₈	—	106	2	no	—
271	C₈	—	212	2	no	—
272	C₈	—	212	2	no	—
273	C₈	—	212	2	no	—
274	C₈	#13133224	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	Q₁₆×Dic₅₃	1	2	yes	—
278	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	Q₁₆×Dic₅₃	1	2	yes	—
285	C₄	C₂×#13133224	1	1	yes	characteristic
286	C₄	#16811390	1	1	yes	characteristic
287	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
288	C₄	C₂×#13133240	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#20283459	1	1	yes	characteristic
291	C₂	C₂×Q₁₆×Dic₅₃	1	1	yes	characteristic
292	C₂	#20283632	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	#18068
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. #20273612	C₂	semidirect
3. #20273566	C₂	semidirect
4. Q₁₆×#88322	C₂	semidirect
95. #11719917	Q₁₆	semidirect
179. C₅₃	#18068	semidirect

Maximal subgroups

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

subgroup	index
#20283664	2
#20273612	2
#20273566	2
Q₁₆×#88322	2
C₂×#16810376	2
#20273118	2
C₈×#11719917	2
C₅₃×#3170	2
#18068	53