Subgroups — #27274563

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	2	yes	maximal, complemented
4	#20272542	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₄	1	2	yes	—
7	C₂×C₂²⋊#88322	C₄	1	2	yes	—
8	C₂×C₄×#88322	—	2	4	no	—
9	C₂×C₄×#88322	C₄	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810378	C₄	1	2	yes	—
14	#16810345	—	2	4	no	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#336	C₄	1	2	yes	—
19	C₂²×#88322	—	2	4	no	—
20	C₂²×#88322	—	2	4	no	—
21	C₂²×#88322	C₈	1	8	yes	complemented
22	C₂²×#88322	C₂×C₄	1	4	yes	—
23	C₂²×#88322	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂²⋊#88322	—	2	4	no	—
26	C₄×#88322	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	—	2	2	no	—
29	C₄×#88322	—	2	4	no	—
30	C₄×#88322	D₄	1	4	yes	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²⋊C₈×C₅₃	—	2	4	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	—	2	2	no	—
44	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
57	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	C₂×D₄	1	2	yes	—
69	C₄²×C₅₃	#41	1	2	yes	—
70	C₂×#88322	—	2	2	no	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	—	2	16	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	2	4	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	C₂²⋊C₄	1	4	yes	—
77	C₂×#88322	C₂×C₈	1	4	yes	—
78	C₂×#88322	C₂×C₈	1	4	yes	—
79	C₂×#88322	M₁₆	1	4	yes	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	C₄×C₈	1	2	yes	—
85	C₂³×C₅₃	#97	1	2	yes	—
86	C₂³×C₅₃	#97	1	2	yes	—
87	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	8	no	—
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
99	C₂×C₄×C₅₃	#97	1	2	yes	—
100	C₂×C₄×C₅₃	#97	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₅₃	#131	1	2	yes	—
110	C₂×C₄×C₅₃	#131	1	2	yes	—
111	C₈×C₅₃	—	4	4	no	—
112	#88322	—	2	4	no	—
113	#88322	—	4	4	no	—
114	#88322	C₂²⋊C₈	1	8	yes	complemented
115	#8144	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	#322	1	4	yes	complemented
123	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
127	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
128	C₂²×C₅₃	#405	1	1	yes	characteristic
129	C₂²×C₅₃	#405	1	1	yes	characteristic
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
134	C₄×C₅₃	#431	1	2	yes	—
135	C₄×C₅₃	#435	1	4	yes	—
136	C₄×C₅₃	#435	1	4	yes	—
137	C₂²×C₄×C₈	—	53	1	no	—
138	C₄×C₂²⋊C₈	—	53	1	no	—
139	C₄×C₂²⋊C₈	D₅₃	1	1	yes	characteristic
140	C₂×#322	—	53	2	no	—
141	#1273	—	53	2	no	—
142	C₂×C₅₃	—	2	4	no	—
143	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
144	C₂×C₅₃	#1450	1	1	yes	characteristic
145	C₂×C₅₃	#1451	1	1	yes	characteristic
146	C₂×C₅₃	#1451	1	1	yes	characteristic
147	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
148	C₂×C₅₃	#1752	1	1	yes	characteristic
149	C₂×C₅₃	#1752	1	1	yes	characteristic
150	C₂³×C₈	—	53	2	no	—
151	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
152	C₂×C₂²⋊C₈	—	53	2	no	—
153	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	53	2	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	2	no	—
158	C₂×C₄×C₈	—	106	4	no	—
159	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
160	#336	—	53	2	no	—
161	#336	Dic₅₃	1	2	yes	—
162	#322	—	106	4	no	—
163	C₅₃	#8144	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	—	53	2	no	—
169	C₂²×C₈	—	53	8	no	—
170	C₂²×C₈	—	53	4	no	—
171	C₂²×C₈	—	106	4	no	—
172	C₂²×C₈	—	106	4	no	—
173	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
180	C₂²⋊C₈	—	2	4	no	—
181	C₂²⋊C₈	—	106	4	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	53	4	no	—
185	C₄×C₈	—	106	4	no	—
186	C₄×C₈	—	106	2	no	—
187	C₄×C₈	—	106	2	no	—
188	C₄×C₈	—	106	2	no	—
189	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₈×D₅₃	1	4	yes	—
198	C₂²×C₄	#11719912	1	4	yes	—
199	C₂²×C₄	#11719917	1	4	yes	—
200	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	53	4	no	—
210	C₂×C₈	—	53	4	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	53	4	no	—
213	C₂×C₈	—	106	16	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	—	106	2	no	—
216	C₂×C₈	—	106	4	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	106	4	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	D₄×D₅₃	1	2	yes	—
224	C₄²	#11719947	1	2	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	C₈×Dic₅₃	1	2	yes	—
230	C₂³	#13133127	1	2	yes	—
231	C₂³	#13133128	1	2	yes	—
232	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
244	C₂×C₄	#13133127	1	2	yes	—
245	C₂×C₄	#13133128	1	2	yes	—
246	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₄	#13133173	1	1	yes	characteristic
248	C₂×C₄	#13133177	1	2	yes	—
249	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
250	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
251	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
252	C₂×C₄	C₂×#11719912	1	2	yes	—
253	C₂×C₄	C₂×#11719917	1	2	yes	—
254	C₂×C₄	#13133224	1	2	yes	—
255	C₂×C₄	#13133225	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₈	—	53	8	no	—
258	C₈	—	106	4	no	—
259	C₈	—	212	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	#16810345	1	4	yes	complemented
267	C₂²	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
268	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
269	C₂²	C₂×#13133127	1	1	yes	characteristic
270	C₂²	C₂×#13133128	1	1	yes	characteristic
271	C₂²	M₁₆×Dic₅₃	1	1	yes	characteristic
272	C₂²	#16810579	1	1	yes	characteristic
273	C₂²	#16810581	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	C₂²⋊C₈×D₅₃	1	2	yes	—
278	C₄	#16810673	1	2	yes	—
279	C₄	#16810691	1	4	yes	—
280	C₄	#16810695	1	4	yes	—
281	C₂	—	2	4	no	—
282	C₂	C₂×#16810345	1	1	yes	characteristic
283	C₂	#20272999	1	1	yes	characteristic
284	C₂	#20273002	1	1	yes	characteristic
285	C₂	#20273005	1	1	yes	characteristic
286	C₂	C₂²⋊C₈×Dic₅₃	1	1	yes	characteristic
287	C₂	#20274493	1	1	yes	characteristic
288	C₂	#20274494	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄×C₂²⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	#8144
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#16810345	C₂	semidirect
4. #20272542	C₂	semidirect
21. C₂²×#88322	C₈	semidirect
114. #88322	C₂²⋊C₈	semidirect
122. C₂²×C₅₃	#322	semidirect
163. C₅₃	#8144	semidirect
266. C₂²	#16810345	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#88322	2
C₄×C₂²⋊#88322	2
C₂×#16810345	2
#20272542	2
C₄×C₂²⋊C₈×C₅₃	2
#8144	53