Subgroups — C₂²×C₅₃×#496 #27268145

Order 13568.

4038 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂³×D₈×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#496	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₅₃	—	2	2	no	—
9	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₅₃	—	2	16	no	—
12	C₂²×SD₁₆×C₅₃	C₂²	1	24	yes	complemented
13	C₂²×D₈×C₅₃	—	2	16	no	—
14	C₂²×D₈×C₅₃	C₂²	1	24	yes	complemented
15	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#127×C₅₃	—	2	6	no	—
17	C₂×#127×C₅₃	C₂²	1	12	yes	complemented
18	C₂×#124×C₅₃	—	2	6	no	—
19	C₂×#124×C₅₃	C₂²	1	12	yes	complemented
20	C₂²×C₂²⋊C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#496	C₂²	1	64	yes	direct product factor
23	C₂×#97×C₅₃	C₂²	1	12	yes	complemented
24	C₂⁵×C₅₃	—	2	1	no	—
25	C₂⁵×C₅₃	—	2	2	no	—
26	C₂²×Q₈×C₅₃	—	2	3	no	—
27	C₂²×Q₈×C₅₃	—	2	8	no	—
28	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
29	C₂²×D₄×C₅₃	—	2	8	no	—
30	C₂²×D₄×C₅₃	—	2	8	no	—
31	C₂²×D₄×C₅₃	—	2	8	no	—
32	C₂²×D₄×C₅₃	—	2	3	no	—
33	C₂²×D₄×C₅₃	—	2	6	no	—
34	C₂²×D₄×C₅₃	—	2	24	no	—
35	C₂²×D₄×C₅₃	—	4	16	no	—
36	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
37	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
38	C₂³×C₄×C₅₃	—	2	1	no	—
39	C₂³×C₄×C₅₃	D₄	1	2	yes	—
40	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×SD₁₆×C₅₃	—	2	96	no	—
42	C₂×SD₁₆×C₅₃	C₂³	1	32	yes	complemented
43	C₂×D₈×C₅₃	—	2	96	no	—
44	C₂×D₈×C₅₃	C₂³	1	32	yes	complemented
45	C₂²×C₈×C₅₃	D₄	1	16	yes	complemented
46	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
47	#127×C₅₃	—	2	24	no	—
48	#127×C₅₃	C₂³	1	16	yes	complemented
49	#124×C₅₃	—	2	24	no	—
50	#124×C₅₃	C₂³	1	16	yes	complemented
51	C₂×C₂²⋊C₄×C₅₃	—	2	24	no	—
52	C₂×C₄²×C₅₃	D₄	1	6	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
54	#97×C₅₃	C₂³	1	16	yes	complemented
55	C₂⁴×C₅₃	—	2	6	no	—
56	C₂⁴×C₅₃	—	2	12	no	—
57	C₂⁴×C₅₃	—	4	4	no	—
58	C₂⁴×C₅₃	—	4	8	no	—
59	C₂⁴×C₅₃	—	4	16	no	—
60	C₂⁴×C₅₃	—	4	8	no	—
61	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	—	2	6	no	—
63	C₂×Q₈×C₅₃	—	2	24	no	—
64	C₂×Q₈×C₅₃	—	4	12	no	—
65	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
66	C₂×D₄×C₅₃	—	2	24	no	—
67	C₂×D₄×C₅₃	—	2	24	no	—
68	C₂×D₄×C₅₃	—	2	24	no	—
69	C₂×D₄×C₅₃	—	2	6	no	—
70	C₂×D₄×C₅₃	—	2	12	no	—
71	C₂×D₄×C₅₃	—	2	32	no	—
72	C₂×D₄×C₅₃	—	4	96	no	—
73	C₂×D₄×C₅₃	—	4	12	no	—
74	C₂×D₄×C₅₃	—	4	24	no	—
75	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
76	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
77	C₂²×C₄×C₅₃	—	2	8	no	—
78	C₂²×C₄×C₅₃	—	2	6	no	—
79	C₂²×C₄×C₅₃	—	4	4	no	—
80	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
84	SD₁₆×C₅₃	—	2	64	no	—
85	D₈×C₅₃	—	2	64	no	—
86	C₂×C₈×C₅₃	C₂×D₄	1	48	yes	complemented
87	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
88	C₂²⋊C₄×C₅₃	—	2	32	no	—
89	C₄²×C₅₃	C₂×D₄	1	12	yes	—
90	C₄²×C₅₃	C₂⁴	1	4	yes	—
91	C₂³×C₅₃	—	2	8	no	—
92	C₂³×C₅₃	—	2	4	no	—
93	C₂³×C₅₃	—	4	12	no	—
94	C₂³×C₅₃	—	4	24	no	—
95	C₂³×C₅₃	—	4	48	no	—
96	C₂³×C₅₃	—	4	24	no	—
97	C₂³×C₅₃	—	8	32	no	—
98	C₂³×C₅₃	—	8	16	no	—
99	C₂³×C₅₃	#127	1	4	yes	—
100	C₂³×C₅₃	AGL(1,ℤ/8)	1	8	yes	—
101	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
102	Q₈×C₅₃	—	2	8	no	—
103	Q₈×C₅₃	—	4	12	no	—
104	D₄×C₅₃	—	2	8	no	—
105	D₄×C₅₃	—	2	8	no	—
106	D₄×C₅₃	—	2	8	no	—
107	D₄×C₅₃	—	4	64	no	—
108	D₄×C₅₃	—	4	12	no	—
109	D₄×C₅₃	—	4	24	no	—
110	C₂×C₄×C₅₃	—	2	24	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	4	12	no	—
113	C₂×C₄×C₅₃	#127	1	12	yes	—
114	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
115	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
116	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
117	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
118	C₈×C₅₃	C₂²×D₄	1	16	yes	complemented
119	C₂²×#496	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₅₃	—	4	4	no	—
121	C₂²×C₅₃	—	4	8	no	—
122	C₂²×C₅₃	—	4	16	no	—
123	C₂²×C₅₃	—	4	8	no	—
124	C₂²×C₅₃	—	8	48	no	—
125	C₂²×C₅₃	—	8	24	no	—
126	C₂²×C₅₃	#496	1	16	yes	direct product factor
127	C₂²×C₅₃	C₂×#127	1	6	yes	—
128	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	12	yes	—
129	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
130	C₄×C₅₃	—	2	8	no	—
131	C₄×C₅₃	—	4	4	no	—
132	C₄×C₅₃	C₂×#127	1	6	yes	—
133	C₄×C₅₃	C₂³×D₄	1	2	yes	—
134	C₂³×SD₁₆	C₂×C₅₃	1	2	yes	complemented
135	C₂³×D₈	C₂×C₅₃	1	2	yes	complemented
136	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic, complemented
137	C₂²×#124	C₂×C₅₃	1	1	yes	characteristic, complemented
138	C₂×#496	C₂×C₅₃	1	24	yes	direct product factor
139	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
140	C₂×C₅₃	—	8	8	no	—
141	C₂×C₅₃	—	8	4	no	—
142	C₂×C₅₃	C₂×#496	1	12	yes	direct product factor
143	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
144	C₂×C₅₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
145	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
146	C₂³×D₄	—	2	2	no	—
147	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
148	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
149	C₂²×SD₁₆	—	2	16	no	—
150	C₂²×SD₁₆	C₂²×C₅₃	1	24	yes	complemented
151	C₂²×D₈	—	2	16	no	—
152	C₂²×D₈	C₂²×C₅₃	1	24	yes	complemented
153	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
154	C₂×#127	—	2	6	no	—
155	C₂×#127	C₂²×C₅₃	1	12	yes	complemented
156	C₂×#124	—	2	6	no	—
157	C₂×#124	C₂²×C₅₃	1	12	yes	complemented
158	C₂²×C₂²⋊C₄	—	2	2	no	—
159	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
160	#496	C₂²×C₅₃	1	64	yes	direct product factor
161	C₂×#97	C₂²×C₅₃	1	12	yes	complemented
162	C₅₃	C₂²×#496	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁵	—	2	1	no	—
164	C₂⁵	—	2	2	no	—
165	C₂²×Q₈	—	2	3	no	—
166	C₂²×Q₈	—	2	8	no	—
167	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
168	C₂²×D₄	—	2	8	no	—
169	C₂²×D₄	—	2	8	no	—
170	C₂²×D₄	—	2	8	no	—
171	C₂²×D₄	—	2	3	no	—
172	C₂²×D₄	—	2	6	no	—
173	C₂²×D₄	—	2	24	no	—
174	C₂²×D₄	—	4	16	no	—
175	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
176	C₂²×D₄	C₂³×C₅₃	1	12	yes	—
177	C₂³×C₄	—	2	1	no	—
178	C₂³×C₄	D₄×C₅₃	1	2	yes	—
179	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
180	C₂×SD₁₆	—	2	96	no	—
181	C₂×SD₁₆	C₂³×C₅₃	1	32	yes	complemented
182	C₂×D₈	—	2	96	no	—
183	C₂×D₈	C₂³×C₅₃	1	32	yes	complemented
184	C₂²×C₈	D₄×C₅₃	1	16	yes	complemented
185	C₂²×C₈	C₂³×C₅₃	1	12	yes	—
186	#127	—	2	24	no	—
187	#127	C₂³×C₅₃	1	16	yes	complemented
188	#124	—	2	24	no	—
189	#124	C₂³×C₅₃	1	16	yes	complemented
190	C₂×C₂²⋊C₄	—	2	24	no	—
191	C₂×C₄²	D₄×C₅₃	1	6	yes	—
192	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
193	#97	C₂³×C₅₃	1	16	yes	complemented
194	C₂⁴	—	2	6	no	—
195	C₂⁴	—	2	12	no	—
196	C₂⁴	—	4	4	no	—
197	C₂⁴	—	4	8	no	—
198	C₂⁴	—	4	16	no	—
199	C₂⁴	—	4	8	no	—
200	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₂×Q₈	—	2	6	no	—
202	C₂×Q₈	—	2	24	no	—
203	C₂×Q₈	—	4	12	no	—
204	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
205	C₂×D₄	—	2	24	no	—
206	C₂×D₄	—	2	24	no	—
207	C₂×D₄	—	2	24	no	—
208	C₂×D₄	—	2	6	no	—
209	C₂×D₄	—	2	12	no	—
210	C₂×D₄	—	2	32	no	—
211	C₂×D₄	—	4	96	no	—
212	C₂×D₄	—	4	12	no	—
213	C₂×D₄	—	4	24	no	—
214	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
215	C₂×D₄	C₂⁴×C₅₃	1	8	yes	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	6	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
220	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
221	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
222	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
223	SD₁₆	—	2	64	no	—
224	D₈	—	2	64	no	—
225	C₂×C₈	C₂×D₄×C₅₃	1	48	yes	complemented
226	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
227	C₂²⋊C₄	—	2	32	no	—
228	C₄²	C₂×D₄×C₅₃	1	12	yes	—
229	C₄²	C₂⁴×C₅₃	1	4	yes	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	8	no	—
232	C₂³	—	4	12	no	—
233	C₂³	—	4	24	no	—
234	C₂³	—	4	48	no	—
235	C₂³	—	4	24	no	—
236	C₂³	—	8	32	no	—
237	C₂³	—	8	16	no	—
238	C₂³	#127×C₅₃	1	4	yes	—
239	C₂³	AGL(1,ℤ/8)×C₅₃	1	8	yes	—
240	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
241	Q₈	—	2	8	no	—
242	Q₈	—	4	12	no	—
243	D₄	—	2	8	no	—
244	D₄	—	2	8	no	—
245	D₄	—	2	8	no	—
246	D₄	—	4	64	no	—
247	D₄	—	4	12	no	—
248	D₄	—	4	24	no	—
249	C₂×C₄	—	2	24	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	12	no	—
252	C₂×C₄	#127×C₅₃	1	12	yes	—
253	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
254	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
255	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
256	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
257	C₈	C₂²×D₄×C₅₃	1	16	yes	complemented
258	C₂²	—	4	16	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	8	48	no	—
263	C₂²	—	8	24	no	—
264	C₂²	C₅₃×#496	1	16	yes	direct product factor
265	C₂²	C₂×#127×C₅₃	1	6	yes	—
266	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	12	yes	—
267	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	4	4	no	—
270	C₄	C₂×#127×C₅₃	1	6	yes	—
271	C₄	C₂³×D₄×C₅₃	1	2	yes	—
272	C₂	—	8	8	no	—
273	C₂	—	8	4	no	—
274	C₂	C₂×C₅₃×#496	1	12	yes	direct product factor
275	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
276	C₂	C₂²×AGL(1,ℤ/8)×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×SD₁₆×C₅₃	C₂	semidirect
2. C₂³×D₈×C₅₃	C₂	semidirect
3. C₂²×#127×C₅₃	C₂	semidirect
4. C₂²×#124×C₅₃	C₂	semidirect
5. C₂×C₅₃×#496	C₂	direct (N × H)
6. C₂²×#97×C₅₃	C₂	semidirect
12. C₂²×SD₁₆×C₅₃	C₂²	semidirect
14. C₂²×D₈×C₅₃	C₂²	semidirect
17. C₂×#127×C₅₃	C₂²	semidirect
19. C₂×#124×C₅₃	C₂²	semidirect
22. C₅₃×#496	C₂²	direct (N × H)
23. C₂×#97×C₅₃	C₂²	semidirect
42. C₂×SD₁₆×C₅₃	C₂³	semidirect
44. C₂×D₈×C₅₃	C₂³	semidirect
45. C₂²×C₈×C₅₃	D₄	semidirect
48. #127×C₅₃	C₂³	semidirect
50. #124×C₅₃	C₂³	semidirect
54. #97×C₅₃	C₂³	semidirect
86. C₂×C₈×C₅₃	C₂×D₄	semidirect
118. C₈×C₅₃	C₂²×D₄	semidirect
119. C₂²×#496	C₅₃	direct (N × H)
126. C₂²×C₅₃	#496	direct (N × H)
134. C₂³×SD₁₆	C₂×C₅₃	semidirect
135. C₂³×D₈	C₂×C₅₃	semidirect
136. C₂²×#127	C₂×C₅₃	semidirect
137. C₂²×#124	C₂×C₅₃	semidirect
138. C₂×#496	C₂×C₅₃	direct (N × H)
139. C₂²×#97	C₂×C₅₃	semidirect
142. C₂×C₅₃	C₂×#496	direct (N × H)
150. C₂²×SD₁₆	C₂²×C₅₃	semidirect
152. C₂²×D₈	C₂²×C₅₃	semidirect
155. C₂×#127	C₂²×C₅₃	semidirect
157. C₂×#124	C₂²×C₅₃	semidirect
160. #496	C₂²×C₅₃	direct (N × H)
161. C₂×#97	C₂²×C₅₃	semidirect
162. C₅₃	C₂²×#496	direct (N × H)
181. C₂×SD₁₆	C₂³×C₅₃	semidirect
183. C₂×D₈	C₂³×C₅₃	semidirect
184. C₂²×C₈	D₄×C₅₃	semidirect
187. #127	C₂³×C₅₃	semidirect
189. #124	C₂³×C₅₃	semidirect
193. #97	C₂³×C₅₃	semidirect
225. C₂×C₈	C₂×D₄×C₅₃	semidirect
257. C₈	C₂²×D₄×C₅₃	semidirect
264. C₂²	C₅₃×#496	direct (N × H)
274. C₂	C₂×C₅₃×#496	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×SD₁₆×C₅₃	2
C₂³×D₈×C₅₃	2
C₂²×#127×C₅₃	2
C₂²×#124×C₅₃	2
C₂×C₅₃×#496	2
C₂²×#97×C₅₃	2
C₂²×#496	53

Subgroups — C22×C53×#496 #27268145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#496 #27268145