Subgroups — #27537184

Order 13568.

598 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	#20283137	C₂	1	1	yes	maximal, characteristic
2	#20276521	C₂	1	4	yes	maximal, complemented
3	#20276513	C₂	1	4	yes	maximal, complemented
4	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276330	C₂	1	1	yes	maximal, characteristic
6	#20276328	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133225	C₄	1	1	yes	characteristic
11	C₂×#13133223	C₂²	1	1	yes	characteristic
12	#16811091	C₄	1	1	yes	characteristic
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	C₂²×#11719908	C₂²	1	1	yes	characteristic
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	C₄⋊#11719908	C₂²	1	4	yes	—
23	#16810402	C₂²	1	4	yes	—
24	#16810402	C₂²	1	4	yes	—
25	#16810398	C₂²	1	2	yes	—
26	C₁₆×Dic₅₃	C₂²	1	2	yes	—
27	C₁₆×Dic₅₃	C₂²	1	2	yes	—
28	C₁₆×Dic₅₃	C₂²	1	2	yes	—
29	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#405	C₂²	1	1	yes	characteristic
32	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	—
33	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
34	#13133225	—	2	4	no	—
35	#13133223	—	2	2	no	—
36	#13133223	—	2	2	no	—
37	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
38	C₂×#11719912	C₂×C₄	1	2	yes	—
39	C₂×#11719912	C₂×C₄	1	2	yes	—
40	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂³	1	2	yes	—
43	#13133182	C₈	1	2	yes	—
44	#13133179	C₈	1	2	yes	—
45	#13133177	C₈	1	4	yes	—
46	#13133174	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₅₃	C₂×C₄	1	2	yes	—
49	#13133157	—	2	2	no	—
50	C₂×#11719908	—	2	2	no	—
51	C₂×#11719908	C₂³	1	2	yes	—
52	C₂×#11719908	C₂³	1	2	yes	—
53	#13133128	C₂×C₄	1	2	yes	—
54	#13133127	C₂×C₄	1	2	yes	—
55	C₄×#88322	C₂×C₄	1	2	yes	—
56	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
60	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
64	M₃₂×C₅₃	—	2	2	no	—
65	C₂×C₁₆×C₅₃	—	2	2	no	—
66	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
67	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
68	#97×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₅₃	C₂³	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
72	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
73	#11719945	—	2	2	no	—
74	#11719945	—	2	2	no	—
75	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
77	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
79	M₁₆×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	#41	1	2	yes	—
83	C₂×C₈×C₅₃	#41	1	2	yes	—
84	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
86	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	#11719921	C₂×C₈	1	4	yes	—
89	#11719919	C₂×C₈	1	4	yes	—
90	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
91	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
92	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
94	#11719917	—	2	2	no	—
95	C₂×#88322	M₁₆	1	4	yes	—
96	C₂×#88322	C₂²×C₄	1	2	yes	—
97	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
98	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
99	#11719912	—	2	4	no	—
100	#11719912	—	2	4	no	—
101	C₈×D₅₃	—	2	2	no	—
102	C₈×D₅₃	—	2	2	no	—
103	C₈×D₅₃	—	2	4	no	—
104	C₁₆×C₅₃	—	2	2	no	—
105	C₁₆×C₅₃	—	2	2	no	—
106	#11719908	—	2	2	no	—
107	#11719908	—	2	2	no	—
108	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	#117	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂²⋊D₅₃	—	2	4	no	—
116	C₂×Dic₅₃	—	2	4	no	—
117	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
118	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
119	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
120	D₂₁₂	—	2	2	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₄×D₅₃	—	2	2	no	—
123	C₄×D₅₃	—	2	8	no	—
124	Dic₁₀₆	—	2	2	no	—
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	#117	1	2	yes	—
127	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
129	#88322	—	2	2	no	—
130	#88322	C₂×M₁₆	1	2	yes	—
131	#88322	C₂×M₁₆	1	2	yes	—
132	#33420	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	#431	1	2	yes	—
135	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
136	C₂×D₅₃	—	2	4	no	—
137	C₂×D₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	4	no	—
139	C₄×C₅₃	#431	1	2	yes	—
140	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
141	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
142	Dic₅₃	—	2	2	no	—
143	Dic₅₃	—	2	2	no	—
144	Dic₅₃	#504	1	4	yes	—
145	C₄×#131	—	53	1	no	—
146	#2164	—	53	4	no	—
147	#2162	—	53	4	no	—
148	#2151	—	53	1	no	—
149	#2151	D₅₃	1	1	yes	characteristic, complemented
150	#2113	—	53	1	no	—
151	#2113	—	53	1	no	—
152	C₄×M₃₂	—	53	1	no	—
153	C₂×C₄×C₁₆	—	53	1	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
156	C₂×C₅₃	C₂×#504	1	2	yes	—
157	D₅₃	—	4	2	no	—
158	C₂×#131	—	53	1	no	—
159	C₂×#131	—	53	1	no	—
160	C₄×#41	—	53	1	no	—
161	C₂×M₃₂	—	53	1	no	—
162	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
163	C₂²×C₁₆	—	53	1	no	—
164	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
165	#405	—	53	1	no	—
166	#405	—	53	2	no	—
167	#405	—	53	2	no	—
168	#405	C₂×D₅₃	1	1	yes	characteristic
169	C₄×M₁₆	—	53	1	no	—
170	C₄×M₁₆	—	53	2	no	—
171	C₂×C₄×C₈	—	53	1	no	—
172	C₂×C₄×C₈	—	53	2	no	—
173	C₄⋊C₁₆	—	53	4	no	—
174	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	—
175	C₂²⋊C₁₆	—	53	4	no	—
176	C₂²⋊C₁₆	—	53	4	no	—
177	#346	—	53	2	no	—
178	C₄×C₁₆	—	53	2	no	—
179	C₄×C₁₆	—	53	2	no	—
180	C₄×C₁₆	—	53	2	no	—
181	C₅₃	#33420	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	53	1	no	—
183	#131	—	106	4	no	—
184	#131	—	106	2	no	—
185	#131	—	106	2	no	—
186	C₂×M₁₆	—	53	1	no	—
187	C₂×M₁₆	—	53	2	no	—
188	C₂×M₁₆	—	53	2	no	—
189	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₈	—	53	1	no	—
191	C₂²×C₈	—	53	2	no	—
192	C₂²×C₈	—	53	2	no	—
193	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
194	C₄×Q₈	—	53	2	no	—
195	C₄×D₄	—	53	2	no	—
196	C₄×D₄	—	53	4	no	—
197	#117	—	53	2	no	—
198	#117	C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄²	—	53	1	no	—
200	C₂×C₄²	—	53	2	no	—
201	M₃₂	—	2	2	no	—
202	M₃₂	—	106	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	53	2	no	—
205	C₂×C₁₆	—	53	2	no	—
206	C₂×C₁₆	—	106	2	no	—
207	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
208	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
209	#97	—	53	2	no	—
210	#97	—	53	2	no	—
211	#97	C₄×D₅₃	1	2	yes	—
212	C₄×C₈	—	53	1	no	—
213	C₄×C₈	—	53	2	no	—
214	C₄×C₈	—	53	1	no	—
215	C₄×C₈	—	53	1	no	—
216	C₄×C₈	—	53	1	no	—
217	C₄×C₈	—	53	2	no	—
218	C₄×C₈	C₂²×D₅₃	1	2	yes	—
219	#41	—	106	2	no	—
220	#41	—	106	2	no	—
221	C₂×Q₈	—	53	1	no	—
222	C₂×D₄	—	53	2	no	—
223	C₂×D₄	—	53	1	no	—
224	C₂²×C₄	—	53	2	no	—
225	C₂²×C₄	—	53	4	no	—
226	C₂²×C₄	—	53	2	no	—
227	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
228	M₁₆	—	2	2	no	—
229	M₁₆	—	106	4	no	—
230	M₁₆	—	106	2	no	—
231	M₁₆	—	106	4	no	—
232	C₂×C₈	—	53	4	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	53	1	no	—
235	C₂×C₈	—	53	2	no	—
236	C₂×C₈	—	106	4	no	—
237	C₂×C₈	—	106	2	no	—
238	C₂×C₈	—	106	2	no	—
239	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₈	#11719947	1	2	yes	—
242	C₂×C₈	#11719949	1	2	yes	—
243	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
244	C₄⋊C₄	—	53	4	no	—
245	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
246	C₂²⋊C₄	—	53	4	no	—
247	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
248	C₄²	—	53	2	no	—
249	C₄²	—	53	4	no	—
250	C₄²	—	53	1	no	—
251	C₄²	—	53	1	no	—
252	C₄²	C₂×C₄×D₅₃	1	2	yes	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	106	2	no	—
256	C₁₆	—	106	2	no	—
257	C₂³	—	53	2	no	—
258	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
259	Q₈	—	106	2	no	—
260	D₄	—	106	4	no	—
261	D₄	—	106	2	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	2	no	—
264	C₂×C₄	—	53	2	no	—
265	C₂×C₄	—	106	8	no	—
266	C₂×C₄	—	106	4	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	#13133175	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
271	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	53	2	no	—
276	C₈	—	53	2	no	—
277	C₈	—	106	2	no	—
278	C₈	#13133175	1	2	yes	—
279	C₈	C₂×#11719947	1	1	yes	characteristic
280	C₈	C₂×#11719949	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	106	4	no	—
283	C₂²	—	106	2	no	—
284	C₂²	#16810675	1	2	yes	—
285	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	53	4	no	—
288	C₄	—	106	2	no	—
289	C₄	—	106	2	no	—
290	C₄	#16810675	1	2	yes	—
291	C₄	C₂×#13133175	1	1	yes	characteristic
292	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	212	2	no	—
295	C₂	C₂×#16810675	1	1	yes	characteristic
296	C₂	#504×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2151	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2151	D₅₃
hypocenter	C₅₃	#33420
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. #20276521	C₂	semidirect
3. #20276513	C₂	semidirect
4. #20276474	C₂	semidirect
7. M₃₂×Dic₅₃	C₂	semidirect
8. C₂×C₁₆×Dic₅₃	C₂	semidirect
9. C₅₃×#2151	C₂	semidirect
149. #2151	D₅₃	semidirect
181. C₅₃	#33420	semidirect

Maximal subgroups

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

subgroup	index
#20283137	2
#20276521	2
#20276513	2
#20276474	2
#20276330	2
#20276328	2
M₃₂×Dic₅₃	2
C₂×C₁₆×Dic₅₃	2
C₅₃×#2151	2
#33420	53