Subgroups — #27279593

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283796	C₂	1	1	yes	maximal, characteristic
2	#20273110	C₂	1	2	yes	maximal
3	#20273039	C₂	1	1	yes	maximal, characteristic
4	#20272544	C₂	1	2	yes	maximal
5	C₅₃×#2327	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	—	2	4	no	—
7	#16810676	—	2	2	no	—
8	#16810676	C₂²	1	2	yes	—
9	#16810676	C₂²	1	2	yes	—
10	Q₈⋊#88322	—	2	4	no	—
11	#16810349	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#391	C₄	1	1	yes	characteristic
16	C₅₃×#391	C₄	1	1	yes	characteristic
17	C₅₃×#387	—	2	2	no	—
18	C₅₃×#387	—	2	2	no	—
19	C₅₃×#382	C₄	1	1	yes	characteristic
20	C₅₃×#376	C₂²	1	1	yes	characteristic
21	C₄⋊#88322	—	2	4	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₂²⋊#88322	—	2	2	no	—
25	C₂²⋊#88322	—	2	2	no	—
26	C₄×#88322	—	2	4	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	D₄	1	4	yes	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	C₈	1	4	yes	complemented
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#95×C₅₃	—	2	2	no	—
48	#95×C₅₃	—	2	2	no	—
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	1	no	—
51	#95×C₅₃	—	2	1	no	—
52	C₂×Q₈×C₅₃	—	2	2	no	—
53	C₂×Q₈×C₅₃	—	2	1	no	—
54	C₂×Q₈×C₅₃	—	2	2	no	—
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
58	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×Q₈×C₅₃	M₁₆	1	2	yes	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₅₃	—	2	8	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	—	4	1	no	—
74	C₄⋊C₄×C₅₃	—	4	1	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	C₂×C₈	1	2	yes	—
81	C₄²×C₅₃	C₂×D₄	1	2	yes	—
82	C₄²×C₅₃	#41	1	2	yes	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₅₃	—	2	4	no	—
87	Q₈×C₅₃	—	2	4	no	—
88	Q₈×C₅₃	—	2	4	no	—
89	Q₈×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	4	1	no	—
96	C₂×C₄×C₅₃	—	4	1	no	—
97	C₂×C₄×C₅₃	—	4	1	no	—
98	C₂×C₄×C₅₃	—	4	1	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#121	1	2	yes	—
111	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#131	1	2	yes	—
115	#88322	—	4	4	no	—
116	#88322	—	8	2	no	—
117	#9100	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	#386	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₅₃	#408	1	1	yes	characteristic
123	C₄×C₅₃	—	2	4	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	—	4	2	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	8	1	no	—
131	C₄×C₅₃	—	8	1	no	—
132	C₄×C₅₃	C₈×D₄	1	2	yes	—
133	C₄×C₅₃	#435	1	2	yes	—
134	#2982	—	53	1	no	—
135	#2327	D₅₃	1	1	yes	characteristic
136	#1479	—	53	2	no	—
137	#1461	—	53	1	no	—
138	#1274	—	53	2	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	#1869	1	1	yes	characteristic
142	C₂×C₅₃	#1909	1	1	yes	characteristic
143	C₂×C₅₃	#1912	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
145	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
147	C₈×Q₈	—	106	4	no	—
148	#431	—	53	2	no	—
149	#431	—	53	2	no	—
150	#431	—	106	2	no	—
151	#391	Dic₅₃	1	1	yes	characteristic
152	#391	Dic₅₃	1	1	yes	characteristic
153	#387	—	2	2	no	—
154	#387	—	2	2	no	—
155	#382	Dic₅₃	1	1	yes	characteristic
156	#376	C₂×D₅₃	1	1	yes	characteristic
157	Q₈⋊C₈	—	106	4	no	—
158	#324	—	106	2	no	—
159	C₅₃	#9100	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	#88322	1	4	yes	complemented
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	—	2	2	no	—
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
177	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
178	C₄⋊C₈	—	106	4	no	—
179	C₄⋊C₈	—	106	2	no	—
180	C₄⋊C₈	—	106	2	no	—
181	C₂²⋊C₈	—	106	2	no	—
182	C₂²⋊C₈	—	106	2	no	—
183	C₄×C₈	—	53	4	no	—
184	C₄×C₈	—	106	4	no	—
185	C₄×C₈	—	106	2	no	—
186	#95	—	2	1	no	—
187	#95	—	2	1	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	#95	—	2	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂×#88322	1	2	yes	—
197	C₂×Q₈	C₂×#88322	1	2	yes	—
198	C₂×Q₈	#11719917	1	2	yes	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
204	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	—	212	2	no	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	C₂×#88322	1	2	yes	—
222	C₄²	D₄×D₅₃	1	2	yes	—
223	C₄²	#11719947	1	2	yes	—
224	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	2	4	no	—
227	Q₈	—	2	4	no	—
228	Q₈	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	C₂²⋊#88322	1	4	yes	—
247	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
248	C₂×C₄	#13133185	1	1	yes	characteristic
249	C₂×C₄	#13133189	1	2	yes	—
250	C₂×C₄	#13133198	1	1	yes	characteristic
251	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
252	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
253	C₂×C₄	#13133224	1	2	yes	—
254	C₈	—	424	2	no	—
255	C₈	—	212	4	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	#16810503	1	1	yes	characteristic
259	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
260	C₂²	#16810590	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	1	no	—
269	C₄	—	8	1	no	—
270	C₄	D₄×#88322	1	2	yes	—
271	C₄	#16810695	1	2	yes	—
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	#20275008	1	1	yes	characteristic
275	C₂	#20275238	1	1	yes	characteristic
276	C₂	#20275253	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810503
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2327	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2327	D₅₃
hypocenter	C₅₃	#9100
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2327

Complements

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

N	H ≅ G/N	type
32. C₄⋊Q₈×C₅₃	C₈	semidirect
159. C₅₃	#9100	semidirect
163. C₄⋊Q₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20283796	2
#20273110	2
#20273039	2
#20272544	2
C₅₃×#2327	2
#9100	53