Subgroups — C₅₃×#21724 #27229236

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2331	C₂	1	2	yes	maximal
3	C₅₃×#1993	C₂	1	2	yes	maximal
4	C₅₃×#1944	C₂	1	2	yes	maximal
5	C₅₃×#1941	C₂	1	2	yes	maximal
6	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1776	C₂	1	2	yes	maximal
8	C₄×#106×C₅₃	C₂	1	2	yes	maximal
9	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#498	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#106×C₅₃	C₂²	1	2	yes	—
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×#106×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#391	—	2	4	no	—
23	C₅₃×#389	C₂²	1	4	yes	—
24	C₅₃×#387	—	2	4	no	—
25	C₅₃×#382	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₅₃×#376	C₂²	1	1	yes	characteristic
28	C₅₃×#340	C₂²	1	4	yes	—
29	C₅₃×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₃	—	2	4	no	—
36	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₅₃	—	2	4	no	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#107×C₅₃	—	2	4	no	—
46	#106×C₅₃	—	2	4	no	—
47	#106×C₅₃	C₂×C₄	1	8	yes	—
48	#106×C₅₃	C₂×C₄	1	8	yes	—
49	#97×C₅₃	—	2	2	no	—
50	#97×C₅₃	—	2	2	no	—
51	#97×C₅₃	Q₈	1	4	yes	complemented
52	C₄×C₈×C₅₃	—	2	2	no	—
53	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
55	#95×C₅₃	—	2	4	no	—
56	#95×C₅₃	—	2	2	no	—
57	#95×C₅₃	—	2	2	no	—
58	#95×C₅₃	—	2	4	no	—
59	#95×C₅₃	D₄	1	4	yes	—
60	C₂×Q₈×C₅₃	—	2	8	no	—
61	C₂×Q₈×C₅₃	—	8	4	no	—
62	C₂²×C₄×C₅₃	—	2	4	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₂×C₈×C₅₃	#41	1	2	yes	—
80	C₄⋊C₄×C₅₃	—	2	8	no	—
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	8	4	no	—
85	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
86	C₄²×C₅₃	—	2	1	no	—
87	C₄²×C₅₃	—	2	1	no	—
88	C₄²×C₅₃	—	8	4	no	—
89	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
92	C₄²×C₅₃	#41	1	2	yes	—
93	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₅₃	—	8	8	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
106	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
107	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	C₄×Q₈	1	4	yes	—
116	C₈×C₅₃	C₄×Q₈	1	4	yes	—
117	#21724	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₂²×C₅₃	#532	1	1	yes	characteristic
121	C₂²×C₅₃	#577	1	4	yes	—
122	C₄×C₅₃	—	2	4	no	—
123	C₄×C₅₃	—	2	4	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	—	4	8	no	—
126	C₄×C₅₃	—	8	4	no	—
127	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
128	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
129	C₄×C₅₃	#533	1	1	yes	characteristic
130	C₄×C₅₃	#533	1	1	yes	characteristic
131	C₂×#498	C₂×C₅₃	1	1	yes	characteristic
132	#2331	C₂×C₅₃	1	2	yes	—
133	#1993	C₂×C₅₃	1	2	yes	—
134	#1944	C₂×C₅₃	1	2	yes	—
135	#1941	C₂×C₅₃	1	2	yes	—
136	#1846	C₂×C₅₃	1	1	yes	characteristic
137	#1776	C₂×C₅₃	1	2	yes	—
138	C₄×#106	C₂×C₅₃	1	2	yes	—
139	#1726	C₂×C₅₃	1	1	yes	characteristic
140	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₅₃	#2949	1	2	yes	—
142	C₂×C₅₃	#3163	1	2	yes	—
143	C₂×C₅₃	#3173	1	2	yes	—
144	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
145	#498	C₄×C₅₃	1	8	yes	complemented
146	C₂×#107	C₂²×C₅₃	1	2	yes	—
147	C₂×#106	C₂²×C₅₃	1	2	yes	—
148	C₂×#106	C₂²×C₅₃	1	2	yes	—
149	C₂×#106	C₂²×C₅₃	1	2	yes	—
150	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
156	#391	—	2	4	no	—
157	#389	C₂²×C₅₃	1	4	yes	—
158	#387	—	2	4	no	—
159	#382	C₂²×C₅₃	1	2	yes	—
160	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
161	#376	C₂²×C₅₃	1	1	yes	characteristic
162	#340	C₂²×C₅₃	1	4	yes	—
163	#340	C₂²×C₅₃	1	4	yes	—
164	C₅₃	#21724	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
175	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
180	#107	—	2	4	no	—
181	#106	—	2	4	no	—
182	#106	C₂×C₄×C₅₃	1	8	yes	—
183	#106	C₂×C₄×C₅₃	1	8	yes	—
184	#97	—	2	2	no	—
185	#97	—	2	2	no	—
186	#97	Q₈×C₅₃	1	4	yes	complemented
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
189	C₄×C₈	Q₈×C₅₃	1	4	yes	complemented
190	#95	—	2	4	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	4	no	—
194	#95	D₄×C₅₃	1	4	yes	—
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	8	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
209	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
210	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
211	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
212	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
213	C₂×C₈	#41×C₅₃	1	2	yes	—
214	C₂×C₈	#41×C₅₃	1	2	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	8	4	no	—
220	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	8	4	no	—
224	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
225	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
226	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
227	C₄²	#41×C₅₃	1	2	yes	—
228	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
229	Q₈	—	8	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
241	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
242	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	C₄×Q₈×C₅₃	1	4	yes	—
251	C₈	C₄×Q₈×C₅₃	1	4	yes	—
252	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
253	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
254	C₂²	C₅₃×#532	1	1	yes	characteristic
255	C₂²	C₅₃×#577	1	4	yes	—
256	C₄	—	2	4	no	—
257	C₄	—	2	4	no	—
258	C₄	—	4	8	no	—
259	C₄	—	4	2	no	—
260	C₄	—	8	4	no	—
261	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
262	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
263	C₄	C₅₃×#533	1	1	yes	characteristic
264	C₄	C₅₃×#533	1	1	yes	characteristic
265	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
266	C₂	C₅₃×#2949	1	2	yes	—
267	C₂	C₅₃×#3163	1	2	yes	—
268	C₂	C₅₃×#3173	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#498	C₄	semidirect
51. #97×C₅₃	Q₈	semidirect
54. C₄×C₈×C₅₃	Q₈	semidirect
117. #21724	C₅₃	direct (N × H)
145. #498	C₄×C₅₃	semidirect
164. C₅₃	#21724	direct (N × H)
186. #97	Q₈×C₅₃	semidirect
189. C₄×C₈	Q₈×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×#498	2
C₅₃×#2331	2
C₅₃×#1993	2
C₅₃×#1944	2
C₅₃×#1941	2
C₅₃×#1846	2
C₅₃×#1776	2
C₄×#106×C₅₃	2
C₅₃×#1726	2
#21724	53

Subgroups — C53×#21724 #27229236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21724 #27229236