Subgroups — C₅₃×#19661 #27227173

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3205	C₂	1	4	yes	maximal, complemented
2	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1721	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1712	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1709	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	—	2	2	no	—
9	C₂²×D₈×C₅₃	—	2	2	no	—
10	C₅₃×#532	C₂²	1	2	yes	—
11	C₂×#127×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#495	—	2	2	no	—
13	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
14	C₅₃×#493	—	2	2	no	—
15	C₅₃×#493	C₂²	1	2	yes	—
16	C₅₃×#493	C₂²	1	2	yes	—
17	C₅₃×#493	C₂²	1	8	yes	complemented
18	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
19	C₅₃×#492	C₂²	1	2	yes	complemented
20	C₅₃×#486	C₂²	1	8	yes	complemented
21	C₅₃×#459	—	2	8	no	—
22	C₅₃×#447	—	2	8	no	—
23	C₅₃×#431	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#322	C₂²	1	2	yes	complemented
26	C₈²×C₅₃	C₂²	1	2	yes	complemented
27	C₂×#41×C₅₃	—	2	2	no	—
28	C₂²×D₄×C₅₃	—	2	2	no	—
29	C₂²×D₄×C₅₃	—	2	4	no	—
30	#135×C₅₃	—	4	8	no	—
31	C₂×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×SD₁₆×C₅₃	—	2	4	no	—
33	C₂×D₈×C₅₃	—	2	4	no	—
34	C₂×D₈×C₅₃	—	2	4	no	—
35	C₂×D₈×C₅₃	—	2	2	no	—
36	C₂×D₈×C₅₃	—	2	16	no	—
37	C₂×D₈×C₅₃	—	4	8	no	—
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
40	#127×C₅₃	—	2	2	no	—
41	#127×C₅₃	—	2	4	no	—
42	#127×C₅₃	C₂³	1	4	yes	—
43	#127×C₅₃	C₂³	1	2	yes	—
44	#124×C₅₃	—	2	2	no	—
45	#121×C₅₃	—	2	4	no	—
46	C₄×D₄×C₅₃	—	2	4	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
49	#102×C₅₃	—	2	16	no	—
50	C₂²⋊C₈×C₅₃	D₄	1	4	yes	—
51	C₄×C₈×C₅₃	D₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	4	yes	—
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	#41×C₅₃	—	4	8	no	—
57	C₂×Q₈×C₅₃	—	2	2	no	—
58	C₂×D₄×C₅₃	—	2	8	no	—
59	C₂×D₄×C₅₃	—	2	4	no	—
60	C₂×D₄×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	2	4	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂×D₄×C₅₃	—	4	4	no	—
64	C₂×D₄×C₅₃	—	4	2	no	—
65	C₂×D₄×C₅₃	—	4	8	no	—
66	C₂×D₄×C₅₃	—	8	8	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₅₃	—	4	4	no	—
71	SD₁₆×C₅₃	—	4	8	no	—
72	D₈×C₅₃	—	4	8	no	—
73	D₈×C₅₃	—	4	8	no	—
74	D₈×C₅₃	—	4	4	no	—
75	D₈×C₅₃	—	8	16	no	—
76	C₂×C₈×C₅₃	D₈	1	8	yes	complemented
77	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
78	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₂²⋊C₄×C₅₃	—	4	4	no	—
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×D₄	1	2	yes	—
84	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	—	4	4	no	—
87	C₂³×C₅₃	—	8	8	no	—
88	C₂³×C₅₃	—	8	4	no	—
89	C₂³×C₅₃	#127	1	1	yes	characteristic
90	Q₈×C₅₃	—	4	4	no	—
91	D₄×C₅₃	—	4	8	no	—
92	D₄×C₅₃	—	4	4	no	—
93	D₄×C₅₃	—	4	16	no	—
94	D₄×C₅₃	—	4	8	no	—
95	D₄×C₅₃	—	8	8	no	—
96	D₄×C₅₃	—	8	4	no	—
97	D₄×C₅₃	—	8	4	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	8	4	no	—
101	C₂×C₄×C₅₃	#127	1	2	yes	—
102	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
106	C₈×C₅₃	—	2	8	no	—
107	C₈×C₅₃	C₂×D₈	1	4	yes	—
108	C₈×C₅₃	C₂×D₈	1	4	yes	—
109	#19661	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	8	2	no	—
112	C₂²×C₅₃	—	8	4	no	—
113	C₂²×C₅₃	—	8	8	no	—
114	C₂²×C₅₃	—	8	4	no	—
115	C₂²×C₅₃	—	16	8	no	—
116	C₂²×C₅₃	#493	1	2	yes	—
117	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	8	2	no	—
120	C₄×C₅₃	#493	1	2	yes	—
121	C₄×C₅₃	C₂²×D₈	1	2	yes	—
122	C₄×C₅₃	#576	1	4	yes	—
123	#3205	C₂×C₅₃	1	4	yes	complemented
124	#3147	C₂×C₅₃	1	1	yes	characteristic, complemented
125	C₂×#493	C₂×C₅₃	1	1	yes	characteristic, complemented
126	#1721	C₂×C₅₃	1	4	yes	complemented
127	#1712	C₂×C₅₃	1	2	yes	complemented
128	#1709	C₂×C₅₃	1	2	yes	complemented
129	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	—	16	2	no	—
132	C₂×C₅₃	—	16	4	no	—
133	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
134	C₂×C₅₃	#3155	1	2	yes	—
135	C₂×#135	—	2	2	no	—
136	C₂²×D₈	—	2	2	no	—
137	#532	C₂²×C₅₃	1	2	yes	—
138	C₂×#127	C₂²×C₅₃	1	2	yes	—
139	#495	—	2	2	no	—
140	#494	C₂²×C₅₃	1	1	yes	characteristic, complemented
141	#493	—	2	2	no	—
142	#493	C₂²×C₅₃	1	2	yes	—
143	#493	C₂²×C₅₃	1	2	yes	—
144	#493	C₂²×C₅₃	1	1	yes	characteristic, complemented
145	#493	C₂²×C₅₃	1	8	yes	complemented
146	#492	C₂²×C₅₃	1	2	yes	complemented
147	#486	C₂²×C₅₃	1	8	yes	complemented
148	#459	—	2	8	no	—
149	#447	—	2	8	no	—
150	#431	C₂²×C₅₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
152	#322	C₂²×C₅₃	1	2	yes	complemented
153	C₈²	C₂²×C₅₃	1	2	yes	complemented
154	C₅₃	#19661	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#41	—	2	2	no	—
156	C₂²×D₄	—	2	2	no	—
157	C₂²×D₄	—	2	4	no	—
158	#135	—	4	8	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×D₈	—	2	4	no	—
162	C₂×D₈	—	2	4	no	—
163	C₂×D₈	—	2	2	no	—
164	C₂×D₈	—	2	16	no	—
165	C₂×D₈	—	4	8	no	—
166	C₂²×C₈	D₄×C₅₃	1	2	yes	—
167	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
168	#127	—	2	4	no	—
169	#127	—	2	2	no	—
170	#127	C₂³×C₅₃	1	4	yes	—
171	#127	C₂³×C₅₃	1	2	yes	—
172	#124	—	2	2	no	—
173	#121	—	2	4	no	—
174	C₄×D₄	—	2	4	no	—
175	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
176	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
177	#102	—	2	16	no	—
178	C₂²⋊C₈	D₄×C₅₃	1	4	yes	—
179	C₄×C₈	D₄×C₅₃	1	2	yes	—
180	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
181	C₄×C₈	C₂³×C₅₃	1	4	yes	—
182	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
183	C₂⁴	—	4	2	no	—
184	#41	—	4	8	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	—	8	8	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
198	Q₁₆	—	4	4	no	—
199	SD₁₆	—	4	8	no	—
200	D₈	—	4	8	no	—
201	D₈	—	4	8	no	—
202	D₈	—	4	4	no	—
203	D₈	—	8	16	no	—
204	C₂×C₈	D₈×C₅₃	1	8	yes	complemented
205	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
206	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
208	C₄⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₅₃	1	2	yes	—
212	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
213	C₂³	—	4	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	8	no	—
216	C₂³	—	8	4	no	—
217	C₂³	#127×C₅₃	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	D₄	—	4	8	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	4	16	no	—
223	D₄	—	8	8	no	—
224	D₄	—	8	4	no	—
225	D₄	—	8	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	#127×C₅₃	1	2	yes	—
230	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
232	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
234	C₈	—	2	8	no	—
235	C₈	C₂×D₈×C₅₃	1	4	yes	—
236	C₈	C₂×D₈×C₅₃	1	4	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	8	no	—
241	C₂²	—	8	4	no	—
242	C₂²	—	16	8	no	—
243	C₂²	C₅₃×#493	1	2	yes	—
244	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	8	2	no	—
247	C₄	C₅₃×#493	1	2	yes	—
248	C₄	C₂²×D₈×C₅₃	1	2	yes	—
249	C₄	C₅₃×#576	1	4	yes	—
250	C₂	—	2	2	no	—
251	C₂	—	16	4	no	—
252	C₂	—	16	2	no	—
253	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
254	C₂	C₅₃×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3205	C₂	semidirect
2. C₅₃×#3147	C₂	semidirect
3. C₂×C₅₃×#493	C₂	semidirect
4. C₅₃×#1721	C₂	semidirect
5. C₅₃×#1712	C₂	semidirect
6. C₅₃×#1709	C₂	semidirect
7. C₅₃×#1450	C₂	semidirect
13. C₅₃×#494	C₂²	semidirect
17. C₅₃×#493	C₂²	semidirect
18. C₅₃×#493	C₂²	semidirect
19. C₅₃×#492	C₂²	semidirect
20. C₅₃×#486	C₂²	semidirect
25. C₅₃×#322	C₂²	semidirect
26. C₈²×C₅₃	C₂²	semidirect
76. C₂×C₈×C₅₃	D₈	semidirect
109. #19661	C₅₃	direct (N × H)
123. #3205	C₂×C₅₃	semidirect
124. #3147	C₂×C₅₃	semidirect
125. C₂×#493	C₂×C₅₃	semidirect
126. #1721	C₂×C₅₃	semidirect
127. #1712	C₂×C₅₃	semidirect
128. #1709	C₂×C₅₃	semidirect
129. #1450	C₂×C₅₃	semidirect
140. #494	C₂²×C₅₃	semidirect
144. #493	C₂²×C₅₃	semidirect
145. #493	C₂²×C₅₃	semidirect
146. #492	C₂²×C₅₃	semidirect
147. #486	C₂²×C₅₃	semidirect
152. #322	C₂²×C₅₃	semidirect
153. C₈²	C₂²×C₅₃	semidirect
154. C₅₃	#19661	direct (N × H)
204. C₂×C₈	D₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3205	2
C₅₃×#3147	2
C₂×C₅₃×#493	2
C₅₃×#1721	2
C₅₃×#1712	2
C₅₃×#1709	2
C₅₃×#1450	2
#19661	53

Subgroups — C53×#19661 #27227173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19661 #27227173