Subgroups — C₅₃×#8451 #27215963

Order 13568.

804 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	C₂×Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1803	C₂	1	2	yes	maximal
4	C₂×C₅₃×#333	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×Q₈⋊C₈	C₂	1	2	yes	maximal
6	Q₈²×C₅₃	—	2	2	no	—
7	Q₈²×C₅₃	—	2	4	no	—
8	Q₈²×C₅₃	C₄	1	4	yes	complemented
9	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×#103×C₅₃	—	2	2	no	—
16	C₂×#103×C₅₃	—	2	2	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₅₃×#340	—	2	2	no	—
20	C₅₃×#333	—	2	2	no	—
21	C₅₃×Q₈⋊C₈	—	2	4	no	—
22	C₂²×Q₈×C₅₃	—	2	2	no	—
23	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
24	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	8	no	—
28	C₄⋊Q₈×C₅₃	—	2	8	no	—
29	C₄⋊Q₈×C₅₃	—	2	8	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	2	8	no	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
34	C₄×Q₈×C₅₃	—	2	8	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	C₄×Q₈×C₅₃	—	2	8	no	—
38	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₄×Q₈×C₅₃	D₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₄⋊C₈×C₅₃	—	2	2	no	—
51	#103×C₅₃	—	4	4	no	—
52	#103×C₅₃	—	4	4	no	—
53	C₄×C₈×C₅₃	—	2	2	no	—
54	C₂×Q₈×C₅₃	—	2	8	no	—
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	—	2	2	no	—
58	C₂×Q₈×C₅₃	—	2	2	no	—
59	C₂×Q₈×C₅₃	—	4	2	no	—
60	C₂×Q₈×C₅₃	—	4	2	no	—
61	C₂×Q₈×C₅₃	—	8	2	no	—
62	C₂×Q₈×C₅₃	—	8	2	no	—
63	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
64	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
65	C₂×Q₈×C₅₃	SD₁₆	1	4	yes	—
66	C₂×Q₈×C₅₃	Q₁₆	1	4	yes	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	—	4	1	no	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	2	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₄⋊C₄×C₅₃	—	2	16	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	—	4	2	no	—
86	C₄⋊C₄×C₅₃	—	4	2	no	—
87	C₄⋊C₄×C₅₃	—	4	2	no	—
88	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
89	C₄²×C₅₃	—	2	8	no	—
90	C₄²×C₅₃	—	2	4	no	—
91	C₄²×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×D₄	1	2	yes	—
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₅₃	#120	1	1	yes	characteristic
96	Q₈×C₅₃	—	2	4	no	—
97	Q₈×C₅₃	—	2	4	no	—
98	Q₈×C₅₃	—	2	4	no	—
99	Q₈×C₅₃	—	4	4	no	—
100	Q₈×C₅₃	—	8	2	no	—
101	Q₈×C₅₃	—	8	8	no	—
102	Q₈×C₅₃	#103	1	8	yes	complemented
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	8	2	no	—
115	C₂×C₄×C₅₃	—	8	1	no	—
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#120	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
122	C₂×C₄×C₅₃	#137	1	2	yes	—
123	C₂×C₄×C₅₃	#137	1	2	yes	—
124	C₈×C₅₃	—	4	2	no	—
125	C₈×C₅₃	—	8	2	no	—
126	#8451	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₃	#379	1	1	yes	characteristic
128	C₂²×C₅₃	#448	1	2	yes	—
129	C₂²×C₅₃	#451	1	2	yes	—
130	C₂²×C₅₃	#453	1	1	yes	characteristic
131	C₂²×C₅₃	#455	1	1	yes	characteristic
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	8	no	—
135	C₄×C₅₃	—	4	4	no	—
136	C₄×C₅₃	—	4	4	no	—
137	C₄×C₅₃	—	16	1	no	—
138	C₄×C₅₃	C₂×#103	1	2	yes	—
139	C₄×C₅₃	#419	1	2	yes	—
140	C₄×C₅₃	#448	1	2	yes	—
141	C₄×C₅₃	#451	1	2	yes	—
142	C₂×Q₈²	C₂×C₅₃	1	1	yes	characteristic
143	#1956	C₂×C₅₃	1	1	yes	characteristic
144	#1803	C₂×C₅₃	1	2	yes	—
145	C₂×#333	C₂×C₅₃	1	1	yes	characteristic
146	C₂×Q₈⋊C₈	C₂×C₅₃	1	2	yes	—
147	C₂×C₅₃	#1623	1	1	yes	characteristic
148	C₂×C₅₃	#1633	1	1	yes	characteristic
149	C₂×C₅₃	#1645	1	2	yes	—
150	C₂×C₅₃	#1787	1	2	yes	—
151	C₂×C₅₃	#1792	1	1	yes	characteristic
152	Q₈²	—	2	2	no	—
153	Q₈²	—	2	4	no	—
154	Q₈²	C₄×C₅₃	1	4	yes	complemented
155	C₂×C₄⋊Q₈	—	2	2	no	—
156	C₂×C₄⋊Q₈	—	2	2	no	—
157	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	2	2	no	—
159	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
161	C₂×#103	—	2	2	no	—
162	C₂×#103	—	2	2	no	—
163	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
164	#384	C₂²×C₅₃	1	1	yes	characteristic
165	#340	—	2	2	no	—
166	#333	—	2	2	no	—
167	Q₈⋊C₈	—	2	4	no	—
168	C₅₃	#8451	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
171	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	8	no	—
175	C₄⋊Q₈	—	2	8	no	—
176	C₄⋊Q₈	—	2	8	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
180	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
181	C₄×Q₈	—	2	8	no	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
186	C₄×Q₈	D₄×C₅₃	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	#103	—	4	4	no	—
199	#103	—	4	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₂×Q₈	—	2	8	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	8	2	no	—
209	C₂×Q₈	—	8	2	no	—
210	C₂×Q₈	C₂²⋊C₄×C₅₃	1	4	yes	—
211	C₂×Q₈	C₂²⋊C₄×C₅₃	1	4	yes	—
212	C₂×Q₈	SD₁₆×C₅₃	1	4	yes	—
213	C₂×Q₈	Q₁₆×C₅₃	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	2	16	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
236	C₄²	—	2	8	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₅₃	1	2	yes	—
241	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
242	C₂³	#120×C₅₃	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	8	2	no	—
248	Q₈	—	8	8	no	—
249	Q₈	#103×C₅₃	1	8	yes	complemented
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
265	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
266	C₂×C₄	#120×C₅₃	1	2	yes	—
267	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
268	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
269	C₂×C₄	#137×C₅₃	1	2	yes	—
270	C₂×C₄	#137×C₅₃	1	2	yes	—
271	C₈	—	4	2	no	—
272	C₈	—	8	2	no	—
273	C₂²	C₅₃×#379	1	1	yes	characteristic
274	C₂²	C₅₃×#448	1	2	yes	—
275	C₂²	C₅₃×#451	1	2	yes	—
276	C₂²	C₅₃×#453	1	1	yes	characteristic
277	C₂²	C₅₃×#455	1	1	yes	characteristic
278	C₄	—	2	8	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	16	1	no	—
284	C₄	C₂×#103×C₅₃	1	2	yes	—
285	C₄	C₅₃×#419	1	2	yes	—
286	C₄	C₅₃×#448	1	2	yes	—
287	C₄	C₅₃×#451	1	2	yes	—
288	C₂	C₅₃×#1623	1	1	yes	characteristic
289	C₂	C₅₃×#1633	1	1	yes	characteristic
290	C₂	C₅₃×#1645	1	2	yes	—
291	C₂	C₅₃×#1787	1	2	yes	—
292	C₂	C₅₃×#1792	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. Q₈²×C₅₃	C₄	semidirect
102. Q₈×C₅₃	#103	semidirect
126. #8451	C₅₃	direct (N × H)
154. Q₈²	C₄×C₅₃	semidirect
168. C₅₃	#8451	direct (N × H)
249. Q₈	#103×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₅₃	2
C₅₃×#1956	2
C₅₃×#1803	2
C₂×C₅₃×#333	2
C₂×C₅₃×Q₈⋊C₈	2
#8451	53

Subgroups — C53×#8451 #27215963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8451 #27215963