Subgroups — C₅₃×#61883 #27269395

Order 13568.

1958 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3497	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3392	C₂	1	8	yes	maximal, complemented
4	C₅₃×#3359	C₂	1	8	yes	maximal, complemented
5	C₅₃×#3206	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3103	C₂	1	6	yes	maximal, complemented
7	C₅₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
9	C₅₃×#557	—	2	4	no	—
10	C₅₃×#557	C₂²	1	8	yes	—
11	C₅₃×#550	C₂²	1	4	yes	—
12	D₄×Q₈×C₅₃	C₂²	1	4	yes	—
13	C₅₃×#548	—	2	4	no	—
14	C₅₃×#537	—	2	3	no	—
15	C₅₃×#532	C₂²	1	6	yes	—
16	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
17	C₂×#125×C₅₃	—	2	3	no	—
18	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#492	C₂²	1	12	yes	—
22	C₅₃×#487	C₂²	1	12	yes	—
23	C₅₃×#481	C₂²	1	4	yes	—
24	C₅₃×#475	C₂²	1	12	yes	—
25	C₅₃×#461	C₂²	1	24	yes	—
26	C₅₃×#460	C₂²	1	12	yes	—
27	C₅₃×#448	C₂²	1	12	yes	—
28	C₈×Q₈×C₅₃	C₂²	1	8	yes	—
29	C₄×SD₁₆×C₅₃	C₂²	1	24	yes	—
30	C₅₃×#431	C₂²	1	6	yes	—
31	2₋¹⁺⁴×C₅₃	—	4	8	no	—
32	C₂×#41×C₅₃	—	2	4	no	—
33	C₂×#41×C₅₃	—	2	6	no	—
34	C₂²×Q₈×C₅₃	—	2	4	no	—
35	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
37	C₂×SD₁₆×C₅₃	C₂³	1	24	yes	—
38	C₄⋊Q₈×C₅₃	—	2	6	no	—
39	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
40	C₄⋊Q₈×C₅₃	C₂³	1	12	yes	—
41	#127×C₅₃	C₂³	1	6	yes	—
42	#125×C₅₃	—	2	24	no	—
43	#124×C₅₃	—	2	6	no	—
44	C₂²⋊Q₈×C₅₃	—	2	12	no	—
45	#121×C₅₃	—	2	12	no	—
46	#121×C₅₃	—	2	6	no	—
47	#121×C₅₃	C₂³	1	4	yes	—
48	C₄×Q₈×C₅₃	—	2	6	no	—
49	C₄×Q₈×C₅₃	—	2	8	no	—
50	C₄×Q₈×C₅₃	D₄	1	4	yes	—
51	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
52	C₄×Q₈×C₅₃	C₂³	1	12	yes	—
53	C₄×D₄×C₅₃	—	2	6	no	—
54	C₄×D₄×C₅₃	C₂³	1	12	yes	—
55	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
58	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	—	2	3	no	—
61	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
62	#106×C₅₃	C₂³	1	8	yes	—
63	C₄⋊C₈×C₅₃	C₂³	1	12	yes	—
64	#103×C₅₃	C₂³	1	24	yes	—
65	#102×C₅₃	C₂³	1	8	yes	—
66	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
67	C₄×C₈×C₅₃	C₂³	1	12	yes	—
68	#41×C₅₃	—	4	16	no	—
69	#41×C₅₃	—	4	24	no	—
70	C₂×Q₈×C₅₃	—	2	3	no	—
71	C₂×Q₈×C₅₃	—	2	6	no	—
72	C₂×Q₈×C₅₃	—	4	6	no	—
73	C₂×Q₈×C₅₃	—	4	16	no	—
74	C₂×Q₈×C₅₃	SD₁₆	1	8	yes	complemented
75	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
76	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
77	C₂×D₄×C₅₃	—	2	4	no	—
78	C₂×D₄×C₅₃	—	2	6	no	—
79	C₂×D₄×C₅₃	—	2	6	no	—
80	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
81	C₂²×C₄×C₅₃	—	2	1	no	—
82	C₂²×C₄×C₅₃	—	2	3	no	—
83	C₂²×C₄×C₅₃	—	2	12	no	—
84	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	SD₁₆×C₅₃	—	4	24	no	—
88	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
89	C₄⋊C₄×C₅₃	—	2	12	no	—
90	C₄⋊C₄×C₅₃	—	2	12	no	—
91	C₄⋊C₄×C₅₃	—	2	12	no	—
92	C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₄⋊C₄×C₅₃	—	2	6	no	—
94	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
95	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
96	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
97	C₂²⋊C₄×C₅₃	—	2	4	no	—
98	C₂²⋊C₄×C₅₃	—	2	12	no	—
99	C₄²×C₅₃	—	2	12	no	—
100	C₄²×C₅₃	C₂×D₄	1	6	yes	—
101	C₄²×C₅₃	C₂⁴	1	6	yes	—
102	C₂³×C₅₃	—	2	4	no	—
103	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₅₃	—	2	12	no	—
105	Q₈×C₅₃	—	4	6	no	—
106	Q₈×C₅₃	—	4	12	no	—
107	Q₈×C₅₃	C₂×SD₁₆	1	16	yes	—
108	D₄×C₅₃	—	2	8	no	—
109	D₄×C₅₃	—	4	8	no	—
110	D₄×C₅₃	—	4	12	no	—
111	D₄×C₅₃	—	8	6	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	2	6	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	4	3	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	6	no	—
119	C₂×C₄×C₅₃	—	4	24	no	—
120	C₂×C₄×C₅₃	C₂×SD₁₆	1	12	yes	—
121	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
124	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
125	C₈×C₅₃	—	2	8	no	—
126	#61883	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	4	4	no	—
130	C₂²×C₅₃	—	4	4	no	—
131	C₂²×C₅₃	—	4	4	no	—
132	C₂²×C₅₃	C₂²×SD₁₆	1	2	yes	—
133	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
134	C₄×C₅₃	—	2	6	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	4	2	no	—
137	C₄×C₅₃	—	4	6	no	—
138	C₄×C₅₃	—	4	4	no	—
139	C₄×C₅₃	C₂²×SD₁₆	1	12	yes	—
140	C₄×C₅₃	#585	1	2	yes	—
141	#3497	C₂×C₅₃	1	1	yes	characteristic
142	C₂×#557	C₂×C₅₃	1	1	yes	characteristic, complemented
143	#3392	C₂×C₅₃	1	8	yes	complemented
144	#3359	C₂×C₅₃	1	8	yes	complemented
145	#3206	C₂×C₅₃	1	6	yes	complemented
146	#3103	C₂×C₅₃	1	6	yes	complemented
147	#2982	C₂×C₅₃	1	1	yes	characteristic, complemented
148	C₂×C₅₃	—	2	2	no	—
149	C₂×C₅₃	—	8	4	no	—
150	C₂×C₅₃	#3487	1	1	yes	characteristic
151	C₂×C₅₃	C₂³×SD₁₆	1	1	yes	characteristic
152	C₂×C₅₃	#3586	1	1	yes	characteristic
153	C₂×2₋¹⁺⁴	—	2	1	no	—
154	#557	—	2	4	no	—
155	#557	C₂²×C₅₃	1	8	yes	—
156	#550	C₂²×C₅₃	1	4	yes	—
157	D₄×Q₈	C₂²×C₅₃	1	4	yes	—
158	#548	—	2	4	no	—
159	#537	—	2	3	no	—
160	#532	C₂²×C₅₃	1	6	yes	—
161	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
162	C₂×#125	—	2	3	no	—
163	C₂×C₄×Q₈	—	2	1	no	—
164	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	—
165	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
166	#492	C₂²×C₅₃	1	12	yes	—
167	#487	C₂²×C₅₃	1	12	yes	—
168	#481	C₂²×C₅₃	1	4	yes	—
169	#475	C₂²×C₅₃	1	12	yes	—
170	#461	C₂²×C₅₃	1	24	yes	—
171	#460	C₂²×C₅₃	1	12	yes	—
172	#448	C₂²×C₅₃	1	12	yes	—
173	C₈×Q₈	C₂²×C₅₃	1	8	yes	—
174	C₄×SD₁₆	C₂²×C₅₃	1	24	yes	—
175	#431	C₂²×C₅₃	1	6	yes	—
176	C₅₃	#61883	1	1	yes	characteristic, direct product factor, Sylow
177	2₋¹⁺⁴	—	4	8	no	—
178	C₂×#41	—	2	4	no	—
179	C₂×#41	—	2	6	no	—
180	C₂²×Q₈	—	2	4	no	—
181	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
182	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
183	C₂×SD₁₆	C₂³×C₅₃	1	24	yes	—
184	C₄⋊Q₈	—	2	6	no	—
185	C₄⋊Q₈	C₂³×C₅₃	1	12	yes	—
186	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
187	#127	C₂³×C₅₃	1	6	yes	—
188	#125	—	2	24	no	—
189	#124	—	2	6	no	—
190	C₂²⋊Q₈	—	2	12	no	—
191	#121	—	2	6	no	—
192	#121	—	2	12	no	—
193	#121	C₂³×C₅₃	1	4	yes	—
194	C₄×Q₈	—	2	8	no	—
195	C₄×Q₈	—	2	6	no	—
196	C₄×Q₈	D₄×C₅₃	1	4	yes	—
197	C₄×Q₈	C₂³×C₅₃	1	12	yes	—
198	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
199	C₄×D₄	—	2	6	no	—
200	C₄×D₄	C₂³×C₅₃	1	12	yes	—
201	C₂×C₄⋊C₄	—	2	3	no	—
202	C₂×C₄⋊C₄	—	2	3	no	—
203	C₂×C₄⋊C₄	—	2	3	no	—
204	C₂×C₄⋊C₄	D₄×C₅₃	1	3	yes	—
205	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
206	C₂×C₄²	—	2	3	no	—
207	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
208	#106	C₂³×C₅₃	1	8	yes	—
209	C₄⋊C₈	C₂³×C₅₃	1	12	yes	—
210	#103	C₂³×C₅₃	1	24	yes	—
211	#102	C₂³×C₅₃	1	8	yes	—
212	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
213	C₄×C₈	C₂³×C₅₃	1	12	yes	—
214	#41	—	4	16	no	—
215	#41	—	4	24	no	—
216	C₂×Q₈	—	2	3	no	—
217	C₂×Q₈	—	2	6	no	—
218	C₂×Q₈	—	4	6	no	—
219	C₂×Q₈	—	4	16	no	—
220	C₂×Q₈	SD₁₆×C₅₃	1	8	yes	complemented
221	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
222	C₂×Q₈	C₂⁴×C₅₃	1	6	yes	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	6	no	—
225	C₂×D₄	—	2	6	no	—
226	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	3	no	—
229	C₂²×C₄	—	2	12	no	—
230	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
231	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
232	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
233	SD₁₆	—	4	24	no	—
234	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
235	C₄⋊C₄	—	2	12	no	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	2	12	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	6	no	—
240	C₄⋊C₄	C₂×D₄×C₅₃	1	12	yes	—
241	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
242	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	2	12	no	—
245	C₄²	—	2	12	no	—
246	C₄²	C₂×D₄×C₅₃	1	6	yes	—
247	C₄²	C₂⁴×C₅₃	1	6	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
250	Q₈	—	2	12	no	—
251	Q₈	—	4	6	no	—
252	Q₈	—	4	12	no	—
253	Q₈	C₂×SD₁₆×C₅₃	1	16	yes	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	12	no	—
257	D₄	—	8	6	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	3	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	6	no	—
265	C₂×C₄	—	4	24	no	—
266	C₂×C₄	C₂×SD₁₆×C₅₃	1	12	yes	—
267	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
269	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
270	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
271	C₈	—	2	8	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	C₂²×SD₁₆×C₅₃	1	2	yes	—
278	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	6	no	—
281	C₄	—	4	6	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	2	no	—
284	C₄	C₂²×SD₁₆×C₅₃	1	12	yes	—
285	C₄	C₅₃×#585	1	2	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	8	4	no	—
288	C₂	C₅₃×#3487	1	1	yes	characteristic
289	C₂	C₂³×SD₁₆×C₅₃	1	1	yes	characteristic
290	C₂	C₅₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3497	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#557	C₂	semidirect
3. C₅₃×#3392	C₂	semidirect
4. C₅₃×#3359	C₂	semidirect
5. C₅₃×#3206	C₂	semidirect
6. C₅₃×#3103	C₂	semidirect
7. C₅₃×#2982	C₂	semidirect
74. C₂×Q₈×C₅₃	SD₁₆	semidirect
126. #61883	C₅₃	direct (N × H)
142. C₂×#557	C₂×C₅₃	semidirect
143. #3392	C₂×C₅₃	semidirect
144. #3359	C₂×C₅₃	semidirect
145. #3206	C₂×C₅₃	semidirect
146. #3103	C₂×C₅₃	semidirect
147. #2982	C₂×C₅₃	semidirect
176. C₅₃	#61883	direct (N × H)
220. C₂×Q₈	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3497	2
C₂×C₅₃×#557	2
C₅₃×#3392	2
C₅₃×#3359	2
C₅₃×#3206	2
C₅₃×#3103	2
C₅₃×#2982	2
#61883	53

Subgroups — C53×#61883 #27269395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61883 #27269395