Subgroups — C₅₃×#8171 #27215683

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₅₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	—	2	1	no	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
12	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
13	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
14	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₄³×C₅₃	—	2	1	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	2	no	—
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×Q₈⋊C₈	C₄	1	8	yes	complemented
20	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₄×Q₈×C₅₃	—	2	4	no	—
24	C₄×Q₈×C₅₃	—	2	2	no	—
25	C₄×Q₈×C₅₃	C₈	1	8	yes	complemented
26	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	—	4	2	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
45	#95×C₅₃	—	2	2	no	—
46	#95×C₅₃	—	2	1	no	—
47	C₂×Q₈×C₅₃	—	2	1	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
50	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	4	4	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	2	8	no	—
65	C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
69	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
70	C₄²×C₅₃	—	2	8	no	—
71	C₄²×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	—	4	1	no	—
78	C₄²×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	C₂×C₈	1	4	yes	—
80	C₄²×C₅₃	M₁₆	1	4	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₅₃	#41	1	2	yes	—
83	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₅₃	—	2	2	no	—
85	Q₈×C₅₃	—	2	4	no	—
86	Q₈×C₅₃	C₄×C₈	1	4	yes	complemented
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	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	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	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	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#137	1	2	yes	—
109	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
110	C₈×C₅₃	—	4	4	no	—
111	C₈×C₅₃	—	4	4	no	—
112	#8171	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
115	C₂²×C₅₃	#417	1	1	yes	characteristic
116	C₂²×C₅₃	#419	1	1	yes	characteristic
117	C₂²×C₅₃	#421	1	2	yes	—
118	C₄×C₅₃	—	2	4	no	—
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	—	2	4	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	4	4	no	—
123	C₄×C₅₃	—	4	2	no	—
124	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
125	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
126	C₄×C₅₃	#431	1	2	yes	—
127	C₄×C₅₃	#440	1	2	yes	—
128	C₄×C₅₃	#441	1	2	yes	—
129	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic
130	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
131	#1744	C₂×C₅₃	1	1	yes	characteristic
132	C₂×Q₈⋊C₈	C₂×C₅₃	1	2	yes	—
133	#1279	C₂×C₅₃	1	2	yes	—
134	C₂×C₅₃	#1477	1	2	yes	—
135	C₂×C₅₃	#1479	1	2	yes	—
136	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₅₃	#1759	1	1	yes	characteristic
138	C₂×C₅₃	#1763	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
140	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
142	C₂×C₄×C₈	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
144	C₄×C₄⋊C₄	—	2	2	no	—
145	C₄×C₄⋊C₄	—	2	1	no	—
146	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
147	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
148	C₄³	—	2	1	no	—
149	C₄³	C₂²×C₅₃	1	1	yes	characteristic
150	#336	—	2	2	no	—
151	#336	—	2	1	no	—
152	Q₈⋊C₈	C₄×C₅₃	1	8	yes	complemented
153	C₅₃	#8171	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	—	2	4	no	—
159	C₄×Q₈	C₈×C₅₃	1	8	yes	complemented
160	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
170	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
172	C₂×C₄²	D₄×C₅₃	1	2	yes	—
173	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	4	2	no	—
178	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
179	#95	—	2	1	no	—
180	#95	—	2	2	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	C₄²×C₅₃	1	2	yes	—
184	C₂×Q₈	C₂×C₈×C₅₃	1	4	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
189	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₄⋊C₄	—	2	8	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	C₄²×C₅₃	1	2	yes	—
203	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
204	C₄²	—	2	8	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	4	4	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	C₂×C₈×C₅₃	1	4	yes	—
214	C₄²	M₁₆×C₅₃	1	4	yes	—
215	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
216	C₄²	#41×C₅₃	1	2	yes	—
217	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	Q₈	—	2	4	no	—
220	Q₈	C₄×C₈×C₅₃	1	4	yes	complemented
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
234	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
235	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
236	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
238	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
239	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
240	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
241	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
242	C₂×C₄	#137×C₅₃	1	2	yes	—
243	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
244	C₈	—	4	4	no	—
245	C₈	—	4	4	no	—
246	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
247	C₂²	C₂×C₂²⋊C₈×C₅₃	1	2	yes	—
248	C₂²	C₅₃×#417	1	1	yes	characteristic
249	C₂²	C₅₃×#419	1	1	yes	characteristic
250	C₂²	C₅₃×#421	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	4	no	—
256	C₄	—	4	2	no	—
257	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
258	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
259	C₄	C₅₃×#431	1	2	yes	—
260	C₄	C₅₃×#440	1	2	yes	—
261	C₄	C₅₃×#441	1	2	yes	—
262	C₂	C₅₃×#1477	1	2	yes	—
263	C₂	C₅₃×#1479	1	2	yes	—
264	C₂	C₄×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
265	C₂	C₅₃×#1759	1	1	yes	characteristic
266	C₂	C₅₃×#1763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
19. C₅₃×Q₈⋊C₈	C₄	semidirect
25. C₄×Q₈×C₅₃	C₈	semidirect
86. Q₈×C₅₃	C₄×C₈	semidirect
112. #8171	C₅₃	direct (N × H)
152. Q₈⋊C₈	C₄×C₅₃	semidirect
153. C₅₃	#8171	direct (N × H)
159. C₄×Q₈	C₈×C₅₃	semidirect
220. Q₈	C₄×C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Q₈×C₅₃	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1744	2
C₂×C₅₃×Q₈⋊C₈	2
C₅₃×#1279	2
#8171	53

Subgroups — C53×#8171 #27215683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8171 #27215683