Subgroups — C₅₃×#33704 #27241216

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2244	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2240	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2140	C₂	1	4	yes	maximal, complemented
6	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₅₃	—	2	1	no	—
8	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#533	—	2	1	no	—
10	C₂×#127×C₅₃	—	2	1	no	—
11	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#500	C₂²	1	2	yes	complemented
13	C₅₃×#499	—	2	1	no	—
14	C₅₃×#499	C₂²	1	2	yes	complemented
15	C₅₃×#498	—	2	1	no	—
16	C₅₃×#493	—	2	1	no	—
17	C₅₃×#493	—	2	1	no	—
18	C₅₃×#493	C₂²	1	2	yes	—
19	C₅₃×#493	C₂²	1	2	yes	—
20	C₅₃×#427	—	2	1	no	—
21	C₅₃×#427	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#357	C₂²	1	16	yes	complemented
24	C₅₃×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
26	C₂²×D₄×C₅₃	—	2	2	no	—
27	C₂²×D₄×C₅₃	—	4	2	no	—
28	C₂×D₈×C₅₃	—	2	2	no	—
29	C₂×D₈×C₅₃	—	2	2	no	—
30	C₂×D₈×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	—	2	8	no	—
32	C₂×D₈×C₅₃	—	4	4	no	—
33	C₂×D₈×C₅₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊Q₈×C₅₃	—	2	2	no	—
37	#127×C₅₃	—	2	2	no	—
38	#127×C₅₃	—	2	2	no	—
39	#127×C₅₃	—	4	2	no	—
40	#125×C₅₃	—	2	2	no	—
41	C₂²⋊Q₈×C₅₃	—	4	2	no	—
42	C₄×Q₈×C₅₃	—	4	2	no	—
43	#117×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	M₃₂×C₅₃	—	2	4	no	—
46	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
47	#107×C₅₃	—	2	2	no	—
48	#107×C₅₃	C₂³	1	4	yes	—
49	#106×C₅₃	—	2	2	no	—
50	#106×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	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂⁴×C₅₃	—	4	2	no	—
56	C₂×Q₈×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	2	4	no	—
58	C₂×D₄×C₅₃	—	4	4	no	—
59	C₂×D₄×C₅₃	—	4	4	no	—
60	C₂×D₄×C₅₃	—	4	4	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂×D₄×C₅₃	—	4	4	no	—
63	C₂×D₄×C₅₃	—	8	8	no	—
64	C₂²×C₄×C₅₃	D₈	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	D₈×C₅₃	—	2	8	no	—
67	D₈×C₅₃	—	4	4	no	—
68	D₈×C₅₃	—	4	4	no	—
69	D₈×C₅₃	—	4	4	no	—
70	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	4	yes	—
75	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	4	4	no	—
78	C₄⋊C₄×C₅₃	—	4	2	no	—
79	C₂²⋊C₄×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	D₈	1	2	yes	—
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₁₆×C₅₃	—	2	4	no	—
85	C₂³×C₅₃	—	4	4	no	—
86	C₂³×C₅₃	—	8	8	no	—
87	C₂³×C₅₃	—	8	4	no	—
88	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₅₃	—	8	4	no	—
90	D₄×C₅₃	—	4	4	no	—
91	D₄×C₅₃	—	4	4	no	—
92	D₄×C₅₃	—	8	8	no	—
93	D₄×C₅₃	—	8	4	no	—
94	D₄×C₅₃	—	8	8	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	#124	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
103	C₈×C₅₃	#124	1	2	yes	—
104	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
105	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
106	C₈×C₅₃	C₂×#41	1	2	yes	—
107	#33704	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	—	8	4	no	—
110	C₂²×C₅₃	—	8	8	no	—
111	C₂²×C₅₃	—	16	8	no	—
112	C₂²×C₅₃	#486	1	2	yes	—
113	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	8	4	no	—
116	C₄×C₅₃	#486	1	2	yes	—
117	C₄×C₅₃	#509	1	4	yes	—
118	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
119	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
120	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
121	C₂×#493	C₂×C₅₃	1	1	yes	characteristic
122	#2244	C₂×C₅₃	1	4	yes	complemented
123	#2240	C₂×C₅₃	1	4	yes	complemented
124	#2140	C₂×C₅₃	1	4	yes	complemented
125	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	—	16	4	no	—
128	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#509	1	2	yes	—
130	C₂²×D₈	—	2	1	no	—
131	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
132	#533	—	2	1	no	—
133	C₂×#127	—	2	1	no	—
134	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
135	#500	C₂²×C₅₃	1	2	yes	complemented
136	#499	—	2	1	no	—
137	#499	C₂²×C₅₃	1	2	yes	complemented
138	#498	—	2	1	no	—
139	#493	—	2	1	no	—
140	#493	—	2	1	no	—
141	#493	C₂²×C₅₃	1	2	yes	—
142	#493	C₂²×C₅₃	1	2	yes	—
143	#427	—	2	1	no	—
144	#427	C₂²×C₅₃	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
146	#357	C₂²×C₅₃	1	16	yes	complemented
147	#346	C₂²×C₅₃	1	2	yes	complemented
148	C₄×C₁₆	C₂²×C₅₃	1	2	yes	complemented
149	C₅₃	#33704	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×D₄	—	2	2	no	—
151	C₂²×D₄	—	4	2	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	—	2	8	no	—
156	C₂×D₈	—	4	4	no	—
157	C₂×D₈	C₂³×C₅₃	1	4	yes	—
158	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	2	no	—
161	#127	—	2	2	no	—
162	#127	—	2	2	no	—
163	#127	—	4	2	no	—
164	#125	—	2	2	no	—
165	C₂²⋊Q₈	—	4	2	no	—
166	C₄×Q₈	—	4	2	no	—
167	#117	—	2	2	no	—
168	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
169	M₃₂	—	2	4	no	—
170	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
171	#107	—	2	2	no	—
172	#107	C₂³×C₅₃	1	4	yes	—
173	#106	—	2	2	no	—
174	#106	—	2	2	no	—
175	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
176	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
179	C₂⁴	—	4	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	—	4	4	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	—	8	8	no	—
188	C₂²×C₄	D₈×C₅₃	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
190	D₈	—	2	8	no	—
191	D₈	—	4	4	no	—
192	D₈	—	4	4	no	—
193	D₈	—	4	4	no	—
194	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
195	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
198	C₂×C₈	#41×C₅₃	1	4	yes	—
199	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	D₈×C₅₃	1	2	yes	—
206	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₁₆	—	2	4	no	—
209	C₂³	—	4	4	no	—
210	C₂³	—	8	8	no	—
211	C₂³	—	8	4	no	—
212	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
213	Q₈	—	8	4	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	4	no	—
216	D₄	—	8	8	no	—
217	D₄	—	8	4	no	—
218	D₄	—	8	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	#124×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
223	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
224	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
225	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
226	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
227	C₈	#124×C₅₃	1	2	yes	—
228	C₈	C₂×SD₁₆×C₅₃	1	2	yes	—
229	C₈	C₂×SD₁₆×C₅₃	1	2	yes	—
230	C₈	C₂×#41×C₅₃	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	8	no	—
234	C₂²	—	16	8	no	—
235	C₂²	C₅₃×#486	1	2	yes	—
236	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	8	4	no	—
239	C₄	C₅₃×#486	1	2	yes	—
240	C₄	C₅₃×#509	1	4	yes	—
241	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
242	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	16	4	no	—
245	C₂	C₂×C₅₃×#486	1	1	yes	characteristic
246	C₂	C₂×C₅₃×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3160	C₂	semidirect
3. C₅₃×#2244	C₂	semidirect
4. C₅₃×#2240	C₂	semidirect
5. C₅₃×#2140	C₂	semidirect
6. C₄×M₃₂×C₅₃	C₂	semidirect
12. C₅₃×#500	C₂²	semidirect
14. C₅₃×#499	C₂²	semidirect
23. C₅₃×#357	C₂²	semidirect
24. C₅₃×#346	C₂²	semidirect
25. C₄×C₁₆×C₅₃	C₂²	semidirect
107. #33704	C₅₃	direct (N × H)
120. #3160	C₂×C₅₃	semidirect
122. #2244	C₂×C₅₃	semidirect
123. #2240	C₂×C₅₃	semidirect
124. #2140	C₂×C₅₃	semidirect
125. C₄×M₃₂	C₂×C₅₃	semidirect
135. #500	C₂²×C₅₃	semidirect
137. #499	C₂²×C₅₃	semidirect
146. #357	C₂²×C₅₃	semidirect
147. #346	C₂²×C₅₃	semidirect
148. C₄×C₁₆	C₂²×C₅₃	semidirect
149. C₅₃	#33704	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3160	2
C₂×C₅₃×#493	2
C₅₃×#2244	2
C₅₃×#2240	2
C₅₃×#2140	2
C₄×M₃₂×C₅₃	2
#33704	53

Subgroups — C53×#33704 #27241216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33704 #27241216