Subgroups — #27300582

Order 13568.

376 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	#20287817	C₂	1	1	yes	maximal, characteristic
2	#20276470	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276344	C₂	1	1	yes	maximal, characteristic
4	#20274554	C₂	1	1	yes	maximal, characteristic
5	#20272801	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272796	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	#16811041	—	2	2	no	—
10	#16811039	—	2	2	no	—
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	M₃₂×D₅₃	—	2	2	no	—
14	C₂×#13133143	C₄	1	1	yes	characteristic
15	#16810617	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810441	—	2	2	no	—
18	#16810408	—	2	2	no	—
19	#16810387	C₄	1	1	yes	characteristic
20	#16810382	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#405	C₂²	1	1	yes	characteristic
24	C₅₃×#364	—	2	2	no	—
25	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
26	#13133223	—	2	1	no	—
27	#13133223	—	2	1	no	—
28	#13133223	D₄	1	2	yes	—
29	#13133223	Q₈	1	2	yes	—
30	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719912	—	2	1	no	—
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅₃	—	2	1	no	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	#13133157	—	2	1	no	—
36	#13133157	—	2	1	no	—
37	#13133157	—	2	1	no	—
38	#13133157	—	2	1	no	—
39	#13133156	—	2	4	no	—
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	D₄	1	2	yes	—
42	C₁₆×D₅₃	—	2	4	no	—
43	#13133143	C₈	1	2	yes	—
44	#13133143	C₈	1	2	yes	—
45	#13133140	—	2	1	no	—
46	#13133137	—	2	1	no	—
47	C₈×Dic₅₃	—	2	1	no	—
48	C₈×Dic₅₃	—	2	1	no	—
49	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₅₃	—	2	1	no	—
53	M₃₂×C₅₃	—	2	1	no	—
54	M₃₂×C₅₃	—	2	1	no	—
55	M₃₂×C₅₃	D₄	1	2	yes	—
56	C₂×C₁₆×C₅₃	—	2	1	no	—
57	C₂×C₁₆×C₅₃	—	2	1	no	—
58	#97×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₃	—	2	1	no	—
62	C₂²×Dic₅₃	—	2	1	no	—
63	#11719945	—	2	1	no	—
64	#11719945	—	2	1	no	—
65	#11719945	C₄⋊C₄	1	2	yes	—
66	#11719945	C₂×C₈	1	2	yes	—
67	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
70	M₁₆×C₅₃	—	2	1	no	—
71	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂²⋊C₄×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	—	2	1	no	—
82	C₄×Dic₅₃	—	2	1	no	—
83	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
85	#11719917	—	2	1	no	—
86	#11719917	C₄⋊C₄	1	2	yes	—
87	C₂×#88322	—	2	1	no	—
88	C₂×#88322	—	2	1	no	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	—	2	1	no	—
91	#11719912	—	2	2	no	—
92	#11719912	—	2	2	no	—
93	C₈×D₅₃	—	2	2	no	—
94	C₈×D₅₃	—	2	2	no	—
95	C₁₆×C₅₃	—	2	2	no	—
96	C₁₆×C₅₃	—	4	1	no	—
97	#11719908	—	2	2	no	—
98	#11719908	—	4	1	no	—
99	C₂³×C₅₃	#131	1	1	yes	characteristic
100	C₂²×D₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
108	C₂²⋊D₅₃	—	2	2	no	—
109	C₂²⋊D₅₃	—	2	2	no	—
110	C₂×Dic₅₃	—	2	1	no	—
111	C₂×Dic₅₃	—	2	1	no	—
112	C₂×Dic₅₃	—	2	1	no	—
113	C₂×Dic₅₃	—	4	2	no	—
114	D₂₁₂	—	2	1	no	—
115	D₂₁₂	C₄⋊C₈	1	2	yes	—
116	C₄×D₅₃	—	2	2	no	—
117	C₄×D₅₃	—	2	2	no	—
118	Dic₁₀₆	—	2	1	no	—
119	Dic₁₀₆	C₄⋊C₈	1	2	yes	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	#120	1	1	yes	characteristic
122	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₅₃	#123	1	1	yes	characteristic
125	#88322	—	2	2	no	—
126	#88322	—	4	1	no	—
127	#12918	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	#424	1	1	yes	characteristic
130	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₅₃	#435	1	1	yes	characteristic
132	C₂×D₅₃	—	2	2	no	—
133	C₂×D₅₃	—	8	1	no	—
134	C₄×C₅₃	—	8	1	no	—
135	C₄×C₅₃	#385	1	1	yes	characteristic
136	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
138	C₄×C₅₃	#435	1	1	yes	characteristic
139	Dic₅₃	—	2	2	no	—
140	Dic₅₃	—	2	2	no	—
141	Dic₅₃	—	4	1	no	—
142	#3407	—	53	1	no	—
143	#2153	D₅₃	1	1	yes	characteristic, complemented
144	#2150	—	53	1	no	—
145	#2117	—	53	1	no	—
146	#1764	—	53	1	no	—
147	#1378	—	53	1	no	—
148	#1376	—	53	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	#1850	1	1	yes	characteristic
153	D₅₃	—	4	2	no	—
154	C₂×#131	—	53	1	no	—
155	#504	—	106	2	no	—
156	#504	—	106	2	no	—
157	C₂×M₃₂	—	53	1	no	—
158	C₂×M₃₂	—	53	1	no	—
159	C₂×M₃₂	—	106	2	no	—
160	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
161	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
162	C₂×#104	—	53	1	no	—
163	#416	—	53	1	no	—
164	#405	C₂×D₅₃	1	1	yes	characteristic
165	C₂×C₄×C₈	—	53	1	no	—
166	#364	—	2	2	no	—
167	C₄⋊C₁₆	—	106	2	no	—
168	#350	—	106	2	no	—
169	#341	—	53	1	no	—
170	#339	—	53	1	no	—
171	C₅₃	#12918	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	53	1	no	—
173	#131	—	53	2	no	—
174	#131	—	53	2	no	—
175	#131	—	106	1	no	—
176	#131	—	106	1	no	—
177	C₂×M₁₆	—	53	1	no	—
178	C₂×M₁₆	—	106	1	no	—
179	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
180	C₂²×C₈	—	53	1	no	—
181	C₂²×C₈	—	106	1	no	—
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	#117	C₄×D₅₃	1	1	yes	characteristic
184	C₂×C₄²	—	53	1	no	—
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	106	4	no	—
189	M₃₂	—	106	1	no	—
190	M₃₂	—	106	1	no	—
191	M₃₂	—	106	1	no	—
192	M₃₂	—	106	1	no	—
193	M₃₂	C₂²⋊D₅₃	1	2	yes	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	53	2	no	—
197	C₂×C₁₆	—	106	4	no	—
198	C₂×C₁₆	—	106	1	no	—
199	#104	—	53	2	no	—
200	#104	—	53	2	no	—
201	#103	—	106	1	no	—
202	#102	—	106	1	no	—
203	#97	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	106	1	no	—
206	C₄×C₈	—	106	1	no	—
207	#41	—	53	2	no	—
208	#41	—	53	2	no	—
209	#41	—	106	1	no	—
210	#41	—	106	1	no	—
211	C₂×Q₈	—	53	1	no	—
212	C₂×D₄	—	53	1	no	—
213	C₂×D₄	—	106	1	no	—
214	C₂²×C₄	—	106	1	no	—
215	C₂²×C₄	—	106	1	no	—
216	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
217	M₁₆	—	2	1	no	—
218	M₁₆	—	53	2	no	—
219	M₁₆	—	106	1	no	—
220	M₁₆	—	106	2	no	—
221	M₁₆	—	106	2	no	—
222	M₁₆	C₈×D₅₃	1	2	yes	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	106	1	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	2	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	#11719945	1	1	yes	characteristic
231	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
232	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
233	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
234	C₂×C₈	#11719949	1	1	yes	characteristic
235	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	53	1	no	—
240	C₄²	—	53	1	no	—
241	C₄²	—	106	1	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	4	1	no	—
244	C₁₆	—	106	2	no	—
245	C₁₆	—	212	1	no	—
246	C₂³	—	106	1	no	—
247	C₂³	#13133225	1	1	yes	characteristic
248	Q₈	—	53	2	no	—
249	Q₈	—	106	1	no	—
250	D₄	—	53	2	no	—
251	D₄	—	106	2	no	—
252	D₄	—	106	2	no	—
253	D₄	—	106	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	106	1	no	—
256	C₂×C₄	—	106	1	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	106	1	no	—
260	C₂×C₄	—	212	2	no	—
261	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₂×C₄	#13133179	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
265	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
266	C₂×C₄	#13133223	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	106	2	no	—
269	C₈	—	212	1	no	—
270	C₈	#13133185	1	1	yes	characteristic
271	C₈	#13133193	1	1	yes	characteristic
272	C₈	#13133195	1	1	yes	characteristic
273	C₈	#13133202	1	1	yes	characteristic
274	C₂²	—	4	1	no	—
275	C₂²	—	106	2	no	—
276	C₂²	—	424	1	no	—
277	C₂²	#16810648	1	1	yes	characteristic
278	C₂²	#16810688	1	1	yes	characteristic
279	C₂²	#16810692	1	1	yes	characteristic
280	C₄	—	8	1	no	—
281	C₄	—	106	2	no	—
282	C₄	—	106	2	no	—
283	C₄	—	212	1	no	—
284	C₄	#16810494	1	1	yes	characteristic
285	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
286	C₄	C₈×C₂²⋊D₅₃	1	1	yes	characteristic
287	C₄	#16810693	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	212	2	no	—
292	C₂	#20274891	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133185
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2153	C₂
socle	C₂×C₅₃	#1850
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2153	D₅₃
hypocenter	C₅₃	#12918
LCS	G ▷ C₂×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
2. #20276470	C₂	semidirect
5. #20272801	C₂	semidirect
6. #20272796	C₂	semidirect
7. C₅₃×#2153	C₂	semidirect
143. #2153	D₅₃	semidirect
171. C₅₃	#12918	semidirect

Maximal subgroups

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

subgroup	index
#20287817	2
#20276470	2
#20276344	2
#20274554	2
#20272801	2
#20272796	2
C₅₃×#2153	2
#12918	53