Subgroups — C₅₃×#9740 #27217252

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1981	C₂	1	2	yes	maximal
3	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#332	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#500	—	2	2	no	—
8	C₅₃×#500	C₄	1	4	yes	complemented
9	C₂×#107×C₅₃	—	2	1	no	—
10	C₂×#107×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#103×C₅₃	—	2	4	no	—
14	C₂×#97×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#334	—	2	2	no	—
19	C₅₃×#332	—	2	4	no	—
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	—	2	1	no	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#107×C₅₃	—	2	4	no	—
35	#107×C₅₃	—	2	4	no	—
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	1	no	—
39	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
40	#103×C₅₃	—	4	8	no	—
41	#97×C₅₃	—	4	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	#95×C₅₃	—	2	1	no	—
45	C₂×Q₈×C₅₃	—	2	4	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	4	no	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	4	1	no	—
58	C₂×C₈×C₅₃	—	4	1	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	4	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
64	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	—	4	1	no	—
71	C₄²×C₅₃	—	4	1	no	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₂³×C₅₃	#121	1	1	yes	characteristic
77	Q₈×C₅₃	—	4	8	no	—
78	Q₈×C₅₃	—	8	2	no	—
79	Q₈×C₅₃	—	8	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	8	4	no	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
95	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	—	4	2	no	—
99	C₈×C₅₃	—	8	2	no	—
100	C₈×C₅₃	#103	1	4	yes	complemented
101	#9740	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₅₃	#390	1	1	yes	characteristic
103	C₂²×C₅₃	#461	1	1	yes	characteristic
104	C₂²×C₅₃	#462	1	1	yes	characteristic
105	C₂²×C₅₃	#469	1	2	yes	—
106	C₂²×C₅₃	#473	1	2	yes	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	8	2	no	—
110	C₄×C₅₃	—	8	4	no	—
111	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
112	C₄×C₅₃	#419	1	1	yes	characteristic
113	C₄×C₅₃	#442	1	2	yes	—
114	C₄×C₅₃	#461	1	1	yes	characteristic
115	C₄×C₅₃	#462	1	1	yes	characteristic
116	C₄×C₅₃	#496	1	2	yes	—
117	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
118	#1981	C₂×C₅₃	1	2	yes	—
119	#1770	C₂×C₅₃	1	1	yes	characteristic
120	C₂×#334	C₂×C₅₃	1	1	yes	characteristic
121	C₂×#332	C₂×C₅₃	1	2	yes	—
122	C₂×C₅₃	#1693	1	2	yes	—
123	C₂×C₅₃	#1694	1	2	yes	—
124	C₂×C₅₃	#1972	1	1	yes	characteristic
125	C₂×C₅₃	#1979	1	1	yes	characteristic
126	C₂×C₅₃	#1981	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
128	#500	—	2	2	no	—
129	#500	C₄×C₅₃	1	4	yes	complemented
130	C₂×#107	—	2	1	no	—
131	C₂×#107	—	2	1	no	—
132	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
133	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
134	C₂×#103	—	2	4	no	—
135	C₂×#97	—	2	1	no	—
136	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
138	#336	—	2	1	no	—
139	#334	—	2	2	no	—
140	#332	—	2	4	no	—
141	C₅₃	#9740	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₈	—	2	2	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
156	#107	—	2	4	no	—
157	#107	—	2	4	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
162	#103	—	4	8	no	—
163	#97	—	4	2	no	—
164	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
165	C₄×C₈	D₄×C₅₃	1	2	yes	—
166	#95	—	2	1	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
184	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
185	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
186	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
197	C₄²	#41×C₅₃	1	1	yes	characteristic
198	C₂³	#121×C₅₃	1	1	yes	characteristic
199	Q₈	—	4	8	no	—
200	Q₈	—	8	2	no	—
201	Q₈	—	8	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
210	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
211	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
212	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
213	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
215	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
216	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
217	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
218	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	4	2	no	—
221	C₈	—	8	2	no	—
222	C₈	#103×C₅₃	1	4	yes	complemented
223	C₂²	C₅₃×#390	1	1	yes	characteristic
224	C₂²	C₅₃×#461	1	1	yes	characteristic
225	C₂²	C₅₃×#462	1	1	yes	characteristic
226	C₂²	C₅₃×#469	1	2	yes	—
227	C₂²	C₅₃×#473	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₂×#103×C₅₃	1	1	yes	characteristic
233	C₄	C₅₃×#419	1	1	yes	characteristic
234	C₄	C₅₃×#442	1	2	yes	—
235	C₄	C₅₃×#461	1	1	yes	characteristic
236	C₄	C₅₃×#462	1	1	yes	characteristic
237	C₄	C₅₃×#496	1	2	yes	—
238	C₂	C₅₃×#1693	1	2	yes	—
239	C₂	C₅₃×#1694	1	2	yes	—
240	C₂	C₅₃×#1972	1	1	yes	characteristic
241	C₂	C₅₃×#1979	1	1	yes	characteristic
242	C₂	C₅₃×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#121
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. C₅₃×#500	C₄	semidirect
100. C₈×C₅₃	#103	semidirect
101. #9740	C₅₃	direct (N × H)
129. #500	C₄×C₅₃	semidirect
141. C₅₃	#9740	direct (N × H)
222. C₈	#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×C₅₃×#500	2
C₅₃×#1981	2
C₅₃×#1770	2
C₂×C₅₃×#334	2
C₂×C₅₃×#332	2
#9740	53

Subgroups — C53×#9740 #27217252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9740 #27217252