Subgroups — C₅₃×#21728 #27229240

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1991	C₂	1	2	yes	maximal
4	C₅₃×#1944	C₂	1	2	yes	maximal
5	C₅₃×#1943	C₂	1	2	yes	maximal
6	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1776	C₂	1	2	yes	maximal
8	C₅₃×#1776	C₂	1	2	yes	maximal
9	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#499	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#107×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×#97×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×#389	C₂²	1	4	yes	—
23	C₅₃×#387	—	2	4	no	—
24	C₅₃×#382	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₄³×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#340	C₂²	1	4	yes	—
29	C₅₃×#340	C₂²	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	#125×C₅₃	—	2	4	no	—
35	#125×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
40	C₂×C₄²×C₅₃	—	2	4	no	—
41	C₂×C₄²×C₅₃	D₄	1	4	yes	—
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	#107×C₅₃	C₂×C₄	1	8	yes	—
47	#106×C₅₃	—	2	4	no	—
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	—
52	#97×C₅₃	Q₈	1	4	yes	—
53	C₄×C₈×C₅₃	—	2	2	no	—
54	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
55	#95×C₅₃	—	2	4	no	—
56	#95×C₅₃	—	2	4	no	—
57	#95×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₅₃	#41	1	2	yes	—
75	C₂×C₈×C₅₃	#41	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	2	16	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	4	8	no	—
81	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	—	4	4	no	—
85	C₄²×C₅₃	—	4	8	no	—
86	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
88	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₅₃	#41	1	2	yes	—
90	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	8	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	4	4	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	—	2	2	no	—
111	C₈×C₅₃	C₄×Q₈	1	4	yes	—
112	C₈×C₅₃	C₄×Q₈	1	4	yes	—
113	#21728	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₅₃	#532	1	1	yes	characteristic
117	C₂²×C₅₃	#575	1	4	yes	—
118	C₄×C₅₃	—	2	4	no	—
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	—	4	8	no	—
122	C₄×C₅₃	—	8	4	no	—
123	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₄×C₅₃	#533	1	1	yes	characteristic
126	C₄×C₅₃	#533	1	1	yes	characteristic
127	C₂×#499	C₂×C₅₃	1	1	yes	characteristic
128	C₄×#125	C₂×C₅₃	1	2	yes	—
129	#1991	C₂×C₅₃	1	2	yes	—
130	#1944	C₂×C₅₃	1	2	yes	—
131	#1943	C₂×C₅₃	1	2	yes	—
132	#1846	C₂×C₅₃	1	1	yes	characteristic
133	#1776	C₂×C₅₃	1	2	yes	—
134	#1776	C₂×C₅₃	1	2	yes	—
135	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
136	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₅₃	#2943	1	2	yes	—
138	C₂×C₅₃	#3166	1	2	yes	—
139	C₂×C₅₃	#3183	1	2	yes	—
140	C₂×#125	C₂²×C₅₃	1	2	yes	—
141	#499	C₄×C₅₃	1	8	yes	complemented
142	C₂×#107	C₂²×C₅₃	1	2	yes	—
143	C₂×#107	C₂²×C₅₃	1	2	yes	—
144	C₂×#106	C₂²×C₅₃	1	2	yes	—
145	C₂×#106	C₂²×C₅₃	1	2	yes	—
146	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
147	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
149	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
152	#389	C₂²×C₅₃	1	4	yes	—
153	#387	—	2	4	no	—
154	#382	C₂²×C₅₃	1	2	yes	—
155	C₄×C₄⋊C₄	—	2	4	no	—
156	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
157	C₄³	C₂²×C₅₃	1	1	yes	characteristic
158	#340	C₂²×C₅₃	1	4	yes	—
159	#340	C₂²×C₅₃	1	4	yes	—
160	C₅₃	#21728	1	1	yes	characteristic, direct product factor, Sylow
161	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
165	#125	—	2	4	no	—
166	#125	C₂×C₄×C₅₃	1	8	yes	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
170	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
171	C₂×C₄²	—	2	4	no	—
172	C₂×C₄²	D₄×C₅₃	1	4	yes	—
173	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
176	#107	—	2	4	no	—
177	#107	C₂×C₄×C₅₃	1	8	yes	—
178	#106	—	2	4	no	—
179	#106	C₂×C₄×C₅₃	1	8	yes	—
180	#97	—	2	2	no	—
181	#97	—	2	2	no	—
182	#97	Q₈×C₅₃	1	4	yes	—
183	#97	Q₈×C₅₃	1	4	yes	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	#95	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
194	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
200	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
201	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
202	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
203	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
204	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
205	C₂×C₈	#41×C₅₃	1	2	yes	—
206	C₂×C₈	#41×C₅₃	1	2	yes	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	8	no	—
212	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	4	4	no	—
216	C₄²	—	4	8	no	—
217	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
218	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
219	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
220	C₄²	#41×C₅₃	1	2	yes	—
221	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
233	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
234	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₄×Q₈×C₅₃	1	4	yes	—
243	C₈	C₄×Q₈×C₅₃	1	4	yes	—
244	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
245	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
246	C₂²	C₅₃×#532	1	1	yes	characteristic
247	C₂²	C₅₃×#575	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	—	4	8	no	—
251	C₄	—	4	2	no	—
252	C₄	—	8	4	no	—
253	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
254	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
255	C₄	C₅₃×#533	1	1	yes	characteristic
256	C₄	C₅₃×#533	1	1	yes	characteristic
257	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
258	C₂	C₅₃×#2943	1	2	yes	—
259	C₂	C₅₃×#3166	1	2	yes	—
260	C₂	C₅₃×#3183	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 ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#499	C₄	semidirect
113. #21728	C₅₃	direct (N × H)
141. #499	C₄×C₅₃	semidirect
160. C₅₃	#21728	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×C₅₃×#499	2
C₄×#125×C₅₃	2
C₅₃×#1991	2
C₅₃×#1944	2
C₅₃×#1943	2
C₅₃×#1846	2
C₅₃×#1776	2
C₄×#97×C₅₃	2
#21728	53

Subgroups — C53×#21728 #27229240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21728 #27229240