Subgroups — #27537095

Order 13568.

684 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287817	C₂	1	2	yes	maximal
2	#20283137	C₂	1	1	yes	maximal, characteristic
3	#20276481	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276479	C₂	1	4	yes	maximal, complemented
5	C₂×#16810443	C₂	1	2	yes	maximal, complemented
6	#364×D₅₃	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133225	C₄	1	1	yes	characteristic
9	C₂×#13133223	C₂²	1	1	yes	characteristic
10	#16811091	C₄	1	1	yes	characteristic
11	#16811041	—	2	4	no	—
12	#16811039	—	2	4	no	—
13	C₂×#13133157	C₂²	1	2	yes	—
14	M₃₂×D₅₃	C₂²	1	8	yes	—
15	#16810580	C₄	1	2	yes	—
16	#16810579	C₄	1	1	yes	characteristic
17	#16810577	C₂²	1	2	yes	—
18	#16810575	C₄	1	2	yes	—
19	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×D₅₃	C₂²	1	2	yes	—
22	#16810444	C₂²	1	4	yes	—
23	#16810443	C₂²	1	4	yes	—
24	#16810443	C₂²	1	4	yes	—
25	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#405	C₂²	1	1	yes	characteristic
27	C₅₃×#364	C₂²	1	4	yes	—
28	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
29	#13133225	C₈	1	8	yes	—
30	#13133223	D₄	1	4	yes	—
31	#13133223	Q₈	1	4	yes	—
32	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719912	C₂×C₄	1	2	yes	—
34	C₂×#11719912	C₂×C₄	1	2	yes	—
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₅₃	C₂³	1	2	yes	—
38	#13133182	—	2	1	no	—
39	#13133179	—	2	1	no	—
40	#13133177	—	2	2	no	—
41	#13133174	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₅₃	C₂×C₄	1	2	yes	—
44	#13133157	—	2	2	no	—
45	#13133157	C₂³	1	4	yes	—
46	#13133156	—	2	8	no	—
47	C₂×#11719908	—	2	2	no	—
48	C₁₆×D₅₃	—	2	8	no	—
49	#13133128	C₂×C₄	1	2	yes	—
50	#13133127	C₂×C₄	1	2	yes	—
51	C₄×#88322	C₂×C₄	1	2	yes	—
52	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
56	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
60	M₃₂×C₅₃	—	2	2	no	—
61	M₃₂×C₅₃	C₂³	1	4	yes	—
62	C₂×C₁₆×C₅₃	—	2	2	no	—
63	#97×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₃	M₁₆	1	2	yes	—
67	C₂²×Dic₅₃	—	2	1	no	—
68	#11719945	C₄⋊C₄	1	4	yes	—
69	#11719945	C₂×C₈	1	4	yes	—
70	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
71	C₂×C₄×D₅₃	—	2	2	no	—
72	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
74	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂²⋊C₄×C₅₃	—	2	1	no	—
82	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
83	#11719921	—	2	2	no	—
84	#11719919	—	2	2	no	—
85	C₄×Dic₅₃	—	2	1	no	—
86	C₄×Dic₅₃	—	2	2	no	—
87	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
89	#11719917	C₄⋊C₄	1	4	yes	—
90	C₂×#88322	C₂×C₈	1	4	yes	—
91	C₂×#88322	C₂²×C₄	1	2	yes	—
92	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
93	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
94	#11719912	C₄⋊C₄	1	8	yes	—
95	#11719912	C₂×C₈	1	8	yes	—
96	C₈×D₅₃	C₂×C₈	1	8	yes	—
97	C₈×D₅₃	C₂×D₄	1	4	yes	—
98	C₈×D₅₃	C₂×Q₈	1	4	yes	—
99	C₁₆×C₅₃	—	2	4	no	—
100	#11719908	—	2	4	no	—
101	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₂²⋊D₅₃	C₄⋊C₈	1	8	yes	—
109	C₂×Dic₅₃	—	2	1	no	—
110	C₂×Dic₅₃	—	2	1	no	—
111	C₂×Dic₅₃	—	4	2	no	—
112	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
113	D₂₁₂	C₄⋊C₈	1	4	yes	—
114	C₄×D₅₃	—	4	4	no	—
115	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
116	C₄×D₅₃	C₂²×C₈	1	4	yes	—
117	Dic₁₀₆	C₄⋊C₈	1	4	yes	—
118	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₈×C₅₃	C₂²×C₈	1	4	yes	—
120	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	#88322	C₂×C₄⋊C₄	1	4	yes	—
123	#88322	C₂²×C₈	1	2	yes	—
124	#88322	C₂²×C₈	1	2	yes	—
125	#33401	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
128	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
129	C₂×D₅₃	—	4	2	no	—
130	C₂×D₅₃	C₂×C₄⋊C₈	1	4	yes	—
131	C₄×C₅₃	—	4	2	no	—
132	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
133	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
135	Dic₅₃	—	2	2	no	—
136	Dic₅₃	—	4	1	no	—
137	Dic₅₃	C₂×C₄⋊C₈	1	4	yes	—
138	#3407	—	53	2	no	—
139	C₄×#131	—	53	1	no	—
140	#2153	—	53	1	no	—
141	#2153	—	53	4	no	—
142	#2153	D₅₃	1	1	yes	characteristic, complemented
143	C₂×#364	—	53	2	no	—
144	C₂×#364	—	53	4	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
148	D₅₃	—	2	4	no	—
149	C₂×#131	—	53	1	no	—
150	C₂×#131	—	53	1	no	—
151	C₄×#41	—	53	1	no	—
152	#504	—	106	4	no	—
153	#504	—	106	4	no	—
154	C₂×M₃₂	—	53	2	no	—
155	C₂×M₃₂	—	53	8	no	—
156	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
157	#405	—	53	1	no	—
158	#405	—	53	2	no	—
159	#405	—	53	2	no	—
160	#405	C₂×D₅₃	1	1	yes	characteristic
161	C₄×M₁₆	—	53	1	no	—
162	C₄×M₁₆	—	53	2	no	—
163	C₂×C₄×C₈	—	53	1	no	—
164	C₂×C₄×C₈	—	53	2	no	—
165	#364	—	53	4	no	—
166	#364	—	53	4	no	—
167	#364	—	53	4	no	—
168	#364	C₂×D₅₃	1	4	yes	—
169	C₅₃	#33401	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	—	53	1	no	—
171	#131	—	53	8	no	—
172	#131	—	53	4	no	—
173	#131	—	53	4	no	—
174	C₂×M₁₆	—	53	1	no	—
175	C₂×M₁₆	—	53	2	no	—
176	C₂×M₁₆	—	53	2	no	—
177	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
178	C₂²×C₈	—	53	1	no	—
179	C₂²×C₈	—	53	2	no	—
180	C₂²×C₈	—	53	2	no	—
181	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
182	C₄×Q₈	—	106	1	no	—
183	C₄×D₄	—	106	2	no	—
184	C₄×D₄	—	106	1	no	—
185	#117	—	53	2	no	—
186	#117	C₄×D₅₃	1	1	yes	characteristic
187	C₂×C₄²	—	53	1	no	—
188	C₂×C₄²	—	53	2	no	—
189	M₃₂	—	2	2	no	—
190	M₃₂	—	53	4	no	—
191	M₃₂	—	106	8	no	—
192	M₃₂	—	106	2	no	—
193	M₃₂	C₂²×D₅₃	1	4	yes	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	106	8	no	—
196	C₂×C₁₆	—	106	2	no	—
197	#97	—	53	2	no	—
198	#97	—	53	2	no	—
199	#97	C₄×D₅₃	1	2	yes	—
200	C₄×C₈	—	53	1	no	—
201	C₄×C₈	—	53	2	no	—
202	C₄×C₈	—	53	1	no	—
203	C₄×C₈	—	53	1	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	53	2	no	—
206	C₄×C₈	C₂²×D₅₃	1	2	yes	—
207	#41	—	53	4	no	—
208	#41	—	53	4	no	—
209	C₂×Q₈	—	53	1	no	—
210	C₂×D₄	—	53	2	no	—
211	C₂×D₄	—	53	1	no	—
212	C₂²×C₄	—	53	2	no	—
213	C₂²×C₄	—	106	2	no	—
214	C₂²×C₄	—	106	1	no	—
215	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
216	M₁₆	—	53	4	no	—
217	M₁₆	—	53	8	no	—
218	M₁₆	—	53	8	no	—
219	M₁₆	C₈×D₅₃	1	4	yes	—
220	C₂×C₈	—	53	4	no	—
221	C₂×C₈	—	53	1	no	—
222	C₂×C₈	—	53	2	no	—
223	C₂×C₈	—	53	1	no	—
224	C₂×C₈	—	53	8	no	—
225	C₂×C₈	—	53	4	no	—
226	C₂×C₈	—	53	4	no	—
227	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
229	C₂×C₈	D₄×D₅₃	1	2	yes	—
230	C₂×C₈	Q₈×D₅₃	1	2	yes	—
231	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	106	2	no	—
234	C₂²⋊C₄	—	2	1	no	—
235	C₂²⋊C₄	—	106	2	no	—
236	C₄²	—	53	1	no	—
237	C₄²	—	53	1	no	—
238	C₄²	—	106	2	no	—
239	C₄²	—	106	1	no	—
240	C₄²	C₂×C₄×D₅₃	1	2	yes	—
241	C₁₆	—	2	4	no	—
242	C₁₆	—	106	4	no	—
243	C₂³	—	53	2	no	—
244	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
245	Q₈	—	53	4	no	—
246	D₄	—	53	8	no	—
247	D₄	—	53	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	53	2	no	—
250	C₂×C₄	—	53	4	no	—
251	C₂×C₄	—	53	4	no	—
252	C₂×C₄	—	106	1	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	212	4	no	—
255	C₂×C₄	—	212	2	no	—
256	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
258	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
260	C₈	—	53	4	no	—
261	C₈	—	53	2	no	—
262	C₈	—	53	2	no	—
263	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
264	C₈	C₂×C₈×D₅₃	1	4	yes	—
265	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
266	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	53	4	no	—
269	C₂²	—	212	2	no	—
270	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
271	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	53	4	no	—
274	C₄	—	106	2	no	—
275	C₄	—	212	1	no	—
276	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
277	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	—	106	4	no	—
282	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2153	C₂
socle	C₂×C₅₃	C₂²×C₄⋊C₈
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2153	D₅₃
hypocenter	C₅₃	#33401
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
3. #20276481	C₂	semidirect
4. #20276479	C₂	semidirect
5. C₂×#16810443	C₂	semidirect
6. #364×D₅₃	C₂	semidirect
7. C₅₃×#2153	C₂	semidirect
142. #2153	D₅₃	semidirect
169. C₅₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#20287817	2
#20283137	2
#20276481	2
#20276479	2
C₂×#16810443	2
#364×D₅₃	2
C₅₃×#2153	2
#33401	53