Subgroups — C₅₃×#9658 #27217170

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 258 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×#1984	C₂	1	2	yes	maximal
3	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	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	2	no	—
9	C₄×Q₁₆×C₅₃	—	2	2	no	—
10	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#103×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	4	no	—
17	C₅₃×#334	—	2	2	no	—
18	C₅₃×Q₈⋊C₈	—	2	4	no	—
19	C₅₃×#322	—	2	2	no	—
20	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×Q₁₆×C₅₃	—	2	2	no	—
22	C₂×Q₁₆×C₅₃	—	2	2	no	—
23	C₂×Q₁₆×C₅₃	C₈	1	4	yes	complemented
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	2	yes	—
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#107×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	#103×C₅₃	—	2	4	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	Q₁₆×C₅₃	—	2	8	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	4	no	—
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	2	2	no	—
73	C₄⋊C₄×C₅₃	—	4	2	no	—
74	C₄⋊C₄×C₅₃	—	4	2	no	—
75	C₄²×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	—	4	2	no	—
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	#41	1	1	yes	characteristic
81	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
82	Q₈×C₅₃	—	2	4	no	—
83	Q₈×C₅₃	—	2	4	no	—
84	Q₈×C₅₃	—	4	2	no	—
85	Q₈×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#137	1	4	yes	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	4	2	no	—
105	C₈×C₅₃	—	4	2	no	—
106	C₈×C₅₃	—	8	2	no	—
107	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
108	#9658	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₅₃	#390	1	1	yes	characteristic
110	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
111	C₂²×C₅₃	#436	1	1	yes	characteristic
112	C₂²×C₅₃	#441	1	2	yes	—
113	C₂²×C₅₃	#444	1	2	yes	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	4	4	no	—
117	C₄×C₅₃	—	8	2	no	—
118	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₅₃	#408	1	1	yes	characteristic
120	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
121	C₄×C₅₃	#436	1	1	yes	characteristic
122	C₄×C₅₃	#470	1	2	yes	—
123	C₄×C₅₃	#497	1	2	yes	—
124	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
125	#1984	C₂×C₅₃	1	2	yes	—
126	C₂×#334	C₂×C₅₃	1	1	yes	characteristic
127	C₂×Q₈⋊C₈	C₂×C₅₃	1	2	yes	—
128	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
129	C₂×C₅₃	#1579	1	2	yes	—
130	C₂×C₅₃	#1593	1	2	yes	—
131	C₂×C₅₃	#1954	1	1	yes	characteristic
132	C₂×C₅₃	#1971	1	1	yes	characteristic
133	C₂×C₅₃	#1975	1	1	yes	characteristic
134	C₂²×Q₁₆	C₄×C₅₃	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
136	C₄×Q₁₆	—	2	2	no	—
137	C₄×Q₁₆	—	2	2	no	—
138	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
139	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
140	C₂×#103	C₄×C₅₃	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
144	#336	—	2	4	no	—
145	#334	—	2	2	no	—
146	Q₈⋊C₈	—	2	4	no	—
147	#322	—	2	2	no	—
148	C₅₃	#9658	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	C₈×C₅₃	1	4	yes	complemented
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₈	D₄×C₅₃	1	2	yes	—
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
166	#107	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	#103	—	2	4	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	D₄×C₅₃	1	2	yes	—
174	C₄×C₈	D₄×C₅₃	1	2	yes	—
175	C₄×C₈	D₄×C₅₃	1	2	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂×C₈×C₅₃	1	4	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
186	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
187	Q₁₆	—	2	8	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
197	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
198	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
199	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
209	C₄²	#41×C₅₃	1	1	yes	characteristic
210	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
211	Q₈	—	2	4	no	—
212	Q₈	—	2	4	no	—
213	Q₈	—	4	2	no	—
214	Q₈	—	4	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
224	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
225	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
226	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
229	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
230	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
231	C₂×C₄	#137×C₅₃	1	4	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	4	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	8	2	no	—
236	C₈	C₂²⋊C₈×C₅₃	1	4	yes	—
237	C₂²	C₅₃×#390	1	1	yes	characteristic
238	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
239	C₂²	C₅₃×#436	1	1	yes	characteristic
240	C₂²	C₅₃×#441	1	2	yes	—
241	C₂²	C₅₃×#444	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	4	no	—
245	C₄	—	8	2	no	—
246	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
247	C₄	C₅₃×#408	1	1	yes	characteristic
248	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
249	C₄	C₅₃×#436	1	1	yes	characteristic
250	C₄	C₅₃×#470	1	2	yes	—
251	C₄	C₅₃×#497	1	2	yes	—
252	C₂	C₅₃×#1579	1	2	yes	—
253	C₂	C₅₃×#1593	1	2	yes	—
254	C₂	C₅₃×#1954	1	1	yes	characteristic
255	C₂	C₅₃×#1971	1	1	yes	characteristic
256	C₂	C₅₃×#1975	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
23. C₂×Q₁₆×C₅₃	C₈	semidirect
108. #9658	C₅₃	direct (N × H)
148. C₅₃	#9658	direct (N × H)
152. C₂×Q₁₆	C₈×C₅₃	semidirect

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×#1984	2
C₂×C₅₃×#334	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×C₅₃×#322	2
#9658	53

Subgroups — C53×#9658 #27217170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9658 #27217170