Subgroups — C₅₃×#21154 #27228666

Order 13568.

998 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	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2296	C₂	1	2	yes	maximal
5	C₅₃×#1944	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1943	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1735	C₂	1	2	yes	maximal
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₃	C₂²	1	2	yes	—
11	C₂×#117×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#427	—	2	4	no	—
14	C₅₃×#427	C₄	1	8	yes	complemented
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₂×#106×C₅₃	C₂²	1	2	yes	—
19	C₂×#97×C₅₃	—	2	4	no	—
20	C₂×#97×C₅₃	C₂²	1	2	yes	—
21	C₂×#97×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#389	C₂²	1	8	yes	—
23	C₅₃×#385	—	2	4	no	—
24	C₄×C₄⋊C₄×C₅₃	—	2	4	no	—
25	C₅₃×#340	C₂²	1	8	yes	—
26	C₂³×C₄×C₅₃	D₄	1	2	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	D₄	1	4	yes	—
32	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
33	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	#117×C₅₃	—	2	4	no	—
38	#117×C₅₃	—	2	4	no	—
39	#117×C₅₃	—	2	4	no	—
40	#117×C₅₃	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	#107×C₅₃	—	2	4	no	—
53	#107×C₅₃	C₂×C₄	1	8	yes	complemented
54	#106×C₅₃	—	2	4	no	—
55	#106×C₅₃	C₂×C₄	1	8	yes	complemented
56	#97×C₅₃	—	2	4	no	—
57	#97×C₅₃	—	2	4	no	—
58	#95×C₅₃	—	2	8	no	—
59	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	—	2	4	no	—
62	C₂²×C₄×C₅₃	—	2	4	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	2	yes	—
71	C₂²×C₄×C₅₃	#41	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₅₃	—	2	8	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	4	16	no	—
88	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
89	C₂²⋊C₄×C₅₃	—	2	8	no	—
90	C₂²⋊C₄×C₅₃	—	4	4	no	—
91	C₄²×C₅₃	—	2	8	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	—	4	4	no	—
96	C₂³×C₅₃	—	2	4	no	—
97	C₂³×C₅₃	C₄×D₄	1	2	yes	—
98	C₂³×C₅₃	#121	1	2	yes	—
99	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
100	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	8	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	8	no	—
104	C₂×C₄×C₅₃	—	4	8	no	—
105	C₂×C₄×C₅₃	—	4	8	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	8	no	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₅₃	#121	1	2	yes	—
112	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
113	C₂×C₄×C₅₃	#125	1	4	yes	—
114	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
121	C₈×C₅₃	—	2	4	no	—
122	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
123	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
124	#21154	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₅₃	—	2	4	no	—
126	C₂²×C₅₃	—	2	4	no	—
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	#389	1	4	yes	—
129	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₅₃	#575	1	4	yes	—
133	C₄×C₅₃	—	4	8	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	—	8	4	no	—
136	C₄×C₅₃	#389	1	4	yes	—
137	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
141	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
142	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
143	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
144	#2296	C₂×C₅₃	1	2	yes	—
145	#1944	C₂×C₅₃	1	4	yes	complemented
146	#1943	C₂×C₅₃	1	4	yes	complemented
147	#1735	C₂×C₅₃	1	2	yes	—
148	C₂×C₅₃	—	2	4	no	—
149	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
150	C₂×C₅₃	#2943	1	2	yes	—
151	C₂×C₅₃	#3054	1	2	yes	—
152	C₂×C₅₃	#3162	1	2	yes	—
153	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
155	C₂×#117	C₂²×C₅₃	1	2	yes	—
156	C₂×#117	C₂²×C₅₃	1	2	yes	—
157	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
158	#427	—	2	4	no	—
159	#427	C₄×C₅₃	1	8	yes	complemented
160	C₂×#107	C₂²×C₅₃	1	2	yes	—
161	C₂×#107	C₂²×C₅₃	1	2	yes	—
162	C₂×#106	C₂²×C₅₃	1	2	yes	—
163	C₂×#106	C₂²×C₅₃	1	2	yes	—
164	C₂×#97	—	2	4	no	—
165	C₂×#97	C₂²×C₅₃	1	2	yes	—
166	C₂×#97	C₂²×C₅₃	1	2	yes	—
167	#389	C₂²×C₅₃	1	8	yes	—
168	#385	—	2	4	no	—
169	C₄×C₄⋊C₄	—	2	4	no	—
170	#340	C₂²×C₅₃	1	8	yes	—
171	C₅₃	#21154	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³×C₄	D₄×C₅₃	1	2	yes	—
173	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
177	C₂²×C₈	D₄×C₅₃	1	4	yes	—
178	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
179	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
180	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
181	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
183	#117	—	2	4	no	—
184	#117	—	2	4	no	—
185	#117	—	2	4	no	—
186	#117	C₂×C₄×C₅₃	1	8	yes	—
187	C₂×C₄⋊C₄	—	2	8	no	—
188	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
189	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	D₄×C₅₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
198	#107	—	2	4	no	—
199	#107	C₂×C₄×C₅₃	1	8	yes	complemented
200	#106	—	2	4	no	—
201	#106	C₂×C₄×C₅₃	1	8	yes	complemented
202	#97	—	2	4	no	—
203	#97	—	2	4	no	—
204	#95	—	2	8	no	—
205	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
212	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
215	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
216	C₂²×C₄	#41×C₅₃	1	2	yes	—
217	C₂²×C₄	#41×C₅₃	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₄⋊C₄×C₅₃	1	16	yes	complemented
223	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
224	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
227	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
228	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
229	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	16	no	—
234	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
235	C₂²⋊C₄	—	2	8	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	C₄×D₄×C₅₃	1	2	yes	—
244	C₂³	#121×C₅₃	1	2	yes	—
245	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
246	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
255	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
256	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
257	C₂×C₄	#121×C₅₃	1	2	yes	—
258	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
259	C₂×C₄	#125×C₅₃	1	4	yes	—
260	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
261	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
269	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	C₅₃×#389	1	4	yes	—
274	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
275	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
276	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
277	C₂²	C₅₃×#575	1	4	yes	—
278	C₄	—	4	8	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₅₃×#389	1	4	yes	—
282	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
283	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
284	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
285	C₄	C₂×#125×C₅₃	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	C₂×C₅₃×#389	1	1	yes	characteristic
288	C₂	C₅₃×#2943	1	2	yes	—
289	C₂	C₅₃×#3054	1	2	yes	—
290	C₂	C₅₃×#3162	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅₃×#1944	C₂	semidirect
6. C₅₃×#1943	C₂	semidirect
14. C₅₃×#427	C₄	semidirect
53. #107×C₅₃	C₂×C₄	semidirect
55. #106×C₅₃	C₂×C₄	semidirect
76. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
124. #21154	C₅₃	direct (N × H)
145. #1944	C₂×C₅₃	semidirect
146. #1943	C₂×C₅₃	semidirect
159. #427	C₄×C₅₃	semidirect
171. C₅₃	#21154	direct (N × H)
199. #107	C₂×C₄×C₅₃	semidirect
201. #106	C₂×C₄×C₅₃	semidirect
222. C₂×C₈	C₄⋊C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#427	2
C₂²×#97×C₅₃	2
C₅₃×#2296	2
C₅₃×#1944	2
C₅₃×#1943	2
C₅₃×#1735	2
#21154	53

Subgroups — C53×#21154 #27228666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21154 #27228666