Subgroups — C₅₃×#3046 #20271983

Order 6784.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#464	C₂	1	2	yes	maximal, complemented
6	C₅₃×#463	C₂	1	2	yes	maximal, complemented
7	C₅₃×#462	C₂	1	2	yes	maximal, complemented
8	C₅₃×#461	C₂	1	2	yes	maximal, complemented
9	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#103×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	#137×C₅₃	—	2	4	no	—
16	#135×C₅₃	—	2	4	no	—
17	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×SD₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
26	#124×C₅₃	C₂²	1	2	yes	—
27	#123×C₅₃	—	2	2	no	—
28	C₂²⋊Q₈×C₅₃	—	2	1	no	—
29	C₂²⋊Q₈×C₅₃	—	2	1	no	—
30	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
31	C₄×D₄×C₅₃	—	2	2	no	—
32	C₄×D₄×C₅₃	C₂²	1	2	yes	—
33	#117×C₅₃	—	2	1	no	—
34	#117×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
38	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
39	#103×C₅₃	C₂²	1	2	yes	—
40	#103×C₅₃	C₂²	1	2	yes	—
41	#103×C₅₃	C₂²	1	2	yes	—
42	#102×C₅₃	C₂²	1	2	yes	—
43	#41×C₅₃	—	2	2	no	—
44	#41×C₅₃	—	2	2	no	—
45	#41×C₅₃	—	2	2	no	—
46	#41×C₅₃	D₄	1	4	yes	—
47	C₂×Q₈×C₅₃	—	2	1	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
51	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₅₃	—	2	1	no	—
54	C₂×D₄×C₅₃	—	2	1	no	—
55	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	1	no	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	Q₁₆×C₅₃	—	2	2	no	—
64	Q₁₆×C₅₃	—	2	4	no	—
65	SD₁₆×C₅₃	—	2	2	no	—
66	SD₁₆×C₅₃	—	2	2	no	—
67	SD₁₆×C₅₃	—	2	4	no	—
68	D₈×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	C₂³	1	2	yes	—
72	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₅₃	—	2	1	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
81	C₂²⋊C₄×C₅₃	—	2	2	no	—
82	C₂²⋊C₄×C₅₃	—	2	2	no	—
83	C₂²⋊C₄×C₅₃	—	2	2	no	—
84	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	C₂³	1	2	yes	—
87	C₂³×C₅₃	—	2	1	no	—
88	C₂³×C₅₃	—	2	1	no	—
89	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₅₃	—	2	1	no	—
91	Q₈×C₅₃	—	2	2	no	—
92	Q₈×C₅₃	—	2	1	no	—
93	Q₈×C₅₃	—	2	2	no	—
94	Q₈×C₅₃	—	2	1	no	—
95	Q₈×C₅₃	—	4	2	no	—
96	Q₈×C₅₃	C₂×D₄	1	2	yes	—
97	D₄×C₅₃	—	2	4	no	—
98	D₄×C₅₃	—	2	1	no	—
99	D₄×C₅₃	—	2	1	no	—
100	D₄×C₅₃	—	2	1	no	—
101	D₄×C₅₃	—	4	2	no	—
102	D₄×C₅₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	8	2	no	—
111	C₂×C₄×C₅₃	—	8	1	no	—
112	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	#41	1	2	yes	—
117	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	#121	1	2	yes	—
130	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	4	1	no	—
135	C₄×C₅₃	—	4	2	no	—
136	C₄×C₅₃	—	4	2	no	—
137	C₄×C₅₃	#121	1	2	yes	—
138	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
140	#3046	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	—	4	2	no	—
143	C₂×C₅₃	—	8	1	no	—
144	C₂×C₅₃	C₂×#121	1	1	yes	characteristic
145	C₂×C₅₃	#577	1	1	yes	characteristic
146	C₂×C₅₃	#578	1	1	yes	characteristic
147	C₂×#137	C₂×C₅₃	1	1	yes	characteristic, complemented
148	C₂×#135	C₂×C₅₃	1	1	yes	characteristic
149	#536	C₂×C₅₃	1	1	yes	characteristic, complemented
150	#520	C₂×C₅₃	1	1	yes	characteristic, complemented
151	#464	C₂×C₅₃	1	2	yes	complemented
152	#463	C₂×C₅₃	1	2	yes	complemented
153	#462	C₂×C₅₃	1	2	yes	complemented
154	#461	C₂×C₅₃	1	2	yes	complemented
155	#424	C₂×C₅₃	1	1	yes	characteristic, complemented
156	#417	C₂×C₅₃	1	1	yes	characteristic, complemented
157	C₂×#103	C₂×C₅₃	1	1	yes	characteristic, complemented
158	C₅₃	#3046	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
161	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
162	#137	—	2	4	no	—
163	#135	—	2	4	no	—
164	C₂×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
165	C₂×Q₁₆	C₂²×C₅₃	1	2	yes	—
166	C₂×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
167	C₂×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×SD₁₆	C₂²×C₅₃	1	2	yes	—
169	C₂×D₈	C₂²×C₅₃	1	1	yes	characteristic
170	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
172	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
173	#124	C₂²×C₅₃	1	2	yes	—
174	#123	—	2	2	no	—
175	C₂²⋊Q₈	—	2	1	no	—
176	C₂²⋊Q₈	—	2	1	no	—
177	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂²×C₅₃	1	2	yes	—
180	#117	—	2	1	no	—
181	#117	C₂²×C₅₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
184	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
185	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
186	#103	C₂²×C₅₃	1	2	yes	—
187	#103	C₂²×C₅₃	1	2	yes	—
188	#103	C₂²×C₅₃	1	2	yes	—
189	#102	C₂²×C₅₃	1	2	yes	—
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	D₄×C₅₃	1	4	yes	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
198	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
199	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
203	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
210	Q₁₆	—	2	2	no	—
211	Q₁₆	—	2	4	no	—
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	2	4	no	—
215	D₈	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	C₂³×C₅₃	1	2	yes	—
219	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
220	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
226	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
227	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂³×C₅₃	1	2	yes	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	2	2	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	2	1	no	—
242	Q₈	—	4	2	no	—
243	Q₈	C₂×D₄×C₅₃	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	1	no	—
248	D₄	—	4	2	no	—
249	D₄	C₂×D₄×C₅₃	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	1	no	—
259	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
261	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
263	C₂×C₄	#41×C₅₃	1	2	yes	—
264	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#121×C₅₃	1	2	yes	—
277	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	#121×C₅₃	1	2	yes	—
285	C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
286	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	8	1	no	—
290	C₂	C₂×#121×C₅₃	1	1	yes	characteristic
291	C₂	C₅₃×#577	1	1	yes	characteristic
292	C₂	C₅₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
prime radical	C₂×#135×C₅₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ #117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#137×C₅₃	C₂	semidirect
3. C₅₃×#536	C₂	semidirect
4. C₅₃×#520	C₂	semidirect
5. C₅₃×#464	C₂	semidirect
6. C₅₃×#463	C₂	semidirect
7. C₅₃×#462	C₂	semidirect
8. C₅₃×#461	C₂	semidirect
9. C₅₃×#424	C₂	semidirect
10. C₅₃×#417	C₂	semidirect
11. C₂×#103×C₅₃	C₂	semidirect
140. #3046	C₅₃	direct (N × H)
147. C₂×#137	C₂×C₅₃	semidirect
149. #536	C₂×C₅₃	semidirect
150. #520	C₂×C₅₃	semidirect
151. #464	C₂×C₅₃	semidirect
152. #463	C₂×C₅₃	semidirect
153. #462	C₂×C₅₃	semidirect
154. #461	C₂×C₅₃	semidirect
155. #424	C₂×C₅₃	semidirect
156. #417	C₂×C₅₃	semidirect
157. C₂×#103	C₂×C₅₃	semidirect
158. C₅₃	#3046	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#137×C₅₃	2
C₂×#135×C₅₃	2
C₅₃×#536	2
C₅₃×#520	2
C₅₃×#464	2
C₅₃×#463	2
C₅₃×#462	2
C₅₃×#461	2
C₅₃×#424	2
C₅₃×#417	2
C₂×#103×C₅₃	2
#3046	53

Subgroups — C53×#3046 #20271983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3046 #20271983