Subgroups — C₅₃×#9671 #27217183

Order 13568.

434 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1479	C₂	1	2	yes	maximal
4	C₅₃×#1461	C₂	1	2	yes	maximal
5	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₄×Q₁₆×C₅₃	—	2	1	no	—
9	C₄×Q₁₆×C₅₃	—	2	1	no	—
10	C₄×Q₁₆×C₅₃	—	2	1	no	—
11	C₄×Q₁₆×C₅₃	—	2	1	no	—
12	C₅₃×#431	C₂²	1	2	yes	—
13	C₅₃×#431	C₂²	1	1	yes	characteristic
14	C₅₃×#431	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#103×C₅₃	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#335	—	2	1	no	—
19	C₅₃×#334	—	2	1	no	—
20	C₅₃×Q₈⋊C₈	—	2	4	no	—
21	C₅₃×#324	—	2	4	no	—
22	C₅₃×#322	—	2	1	no	—
23	C₅₃×#322	—	2	1	no	—
24	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₅₃	—	2	1	no	—
26	C₂×Q₁₆×C₅₃	—	2	1	no	—
27	C₂×Q₁₆×C₅₃	—	2	4	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#107×C₅₃	—	2	1	no	—
39	#107×C₅₃	—	2	1	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	1	no	—
42	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
43	#103×C₅₃	—	2	2	no	—
44	#103×C₅₃	—	2	2	no	—
45	C₂²⋊C₈×C₅₃	—	2	2	no	—
46	C₂²⋊C₈×C₅₃	—	2	1	no	—
47	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
48	C₄×C₈×C₅₃	—	2	1	no	—
49	C₄×C₈×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
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	—	2	2	no	—
60	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	Q₁₆×C₅₃	—	2	4	no	—
67	Q₁₆×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	—	4	2	no	—
82	C₄²×C₅₃	—	4	2	no	—
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
88	Q₈×C₅₃	—	2	4	no	—
89	Q₈×C₅₃	—	2	4	no	—
90	Q₈×C₅₃	—	4	2	no	—
91	Q₈×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	8	2	no	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₅₃	#131	1	2	yes	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	—	2	1	no	—
111	C₈×C₅₃	—	4	2	no	—
112	C₈×C₅₃	—	4	2	no	—
113	C₈×C₅₃	—	8	2	no	—
114	#9671	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	#390	1	1	yes	characteristic
118	C₂²×C₅₃	#435	1	2	yes	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	4	4	no	—
123	C₄×C₅₃	—	8	2	no	—
124	C₄×C₅₃	#407	1	1	yes	characteristic
125	C₄×C₅₃	#408	1	1	yes	characteristic
126	C₄×C₅₃	#435	1	2	yes	—
127	C₄×C₅₃	#473	1	2	yes	—
128	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
129	#1564	C₂×C₅₃	1	1	yes	characteristic
130	#1479	C₂×C₅₃	1	2	yes	—
131	#1461	C₂×C₅₃	1	2	yes	—
132	#1452	C₂×C₅₃	1	1	yes	characteristic
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	#1955	1	1	yes	characteristic
135	C₂×C₅₃	#1976	1	1	yes	characteristic
136	C₂×C₅₃	#1979	1	1	yes	characteristic
137	C₂²×Q₁₆	C₄×C₅₃	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
139	C₄×Q₁₆	—	2	1	no	—
140	C₄×Q₁₆	—	2	1	no	—
141	C₄×Q₁₆	—	2	1	no	—
142	C₄×Q₁₆	—	2	1	no	—
143	#431	C₂²×C₅₃	1	2	yes	—
144	#431	C₂²×C₅₃	1	1	yes	characteristic
145	#431	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
147	C₂×#103	C₄×C₅₃	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
149	#335	—	2	1	no	—
150	#334	—	2	1	no	—
151	Q₈⋊C₈	—	2	4	no	—
152	#324	—	2	4	no	—
153	#322	—	2	1	no	—
154	#322	—	2	1	no	—
155	C₅₃	#9671	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
157	C₂×Q₁₆	—	2	4	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
170	#107	—	2	1	no	—
171	#107	—	2	1	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
175	#103	—	2	2	no	—
176	#103	—	2	2	no	—
177	C₂²⋊C₈	—	2	1	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
184	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
185	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
186	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	M₁₆×C₅₃	1	4	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
197	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
198	Q₁₆	—	2	4	no	—
199	Q₁₆	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₄²	—	4	2	no	—
215	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₄²	#41×C₅₃	1	1	yes	characteristic
219	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
220	Q₈	—	2	4	no	—
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
234	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
235	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
236	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
238	C₂×C₄	#131×C₅₃	1	2	yes	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	4	2	no	—
244	C₈	—	4	2	no	—
245	C₈	—	8	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₅₃×#390	1	1	yes	characteristic
249	C₂²	C₅₃×#435	1	2	yes	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₅₃×#407	1	1	yes	characteristic
256	C₄	C₅₃×#408	1	1	yes	characteristic
257	C₄	C₅₃×#435	1	2	yes	—
258	C₄	C₅₃×#473	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	C₅₃×#1955	1	1	yes	characteristic
261	C₂	C₅₃×#1976	1	1	yes	characteristic
262	C₂	C₅₃×#1979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
114. #9671	C₅₃	direct (N × H)
155. C₅₃	#9671	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₅₃	2
C₅₃×#1564	2
C₅₃×#1479	2
C₅₃×#1461	2
C₅₃×#1452	2
#9671	53

Subgroups — C53×#9671 #27217183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9671 #27217183