Subgroups — C₅₃×#10821 #27218333

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#340	C₂	1	2	yes	maximal
3	C₂×C₅₃×#340	C₂	1	2	yes	maximal
4	C₅₃×C₄⋊#95	C₂	1	2	yes	maximal
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#97×C₅₃	—	2	4	no	—
10	C₂×#97×C₅₃	C₄	1	4	yes	complemented
11	C₂×#95×C₅₃	—	2	4	no	—
12	C₅₃×#340	—	2	4	no	—
13	C₅₃×#340	C₄	1	8	yes	complemented
14	C₂³×C₄×C₅₃	—	2	4	no	—
15	C₂³×C₄×C₅₃	D₄	1	2	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
35	#97×C₅₃	—	2	8	no	—
36	#95×C₅₃	—	4	8	no	—
37	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	8	no	—
43	C₂²×C₄×C₅₃	—	4	4	no	—
44	C₂²×C₄×C₅₃	—	4	4	no	—
45	C₂²×C₄×C₅₃	—	4	4	no	—
46	C₂²×C₄×C₅₃	—	4	4	no	—
47	C₂²×C₄×C₅₃	—	4	4	no	—
48	C₂²×C₄×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	#41	1	2	yes	—
55	C₂²×C₄×C₅₃	#41	1	2	yes	—
56	C₂²×C₄×C₅₃	#41	1	2	yes	—
57	C₂²×C₄×C₅₃	#41	1	2	yes	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	8	no	—
63	C₂×C₈×C₅₃	—	2	8	no	—
64	C₂×C₈×C₅₃	C₄²	1	4	yes	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
66	C₄⋊C₄×C₅₃	—	2	8	no	—
67	C₄⋊C₄×C₅₃	—	2	8	no	—
68	C₄⋊C₄×C₅₃	—	4	4	no	—
69	C₄⋊C₄×C₅₃	—	4	4	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	4	no	—
73	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₅₃	C₄×D₄	1	2	yes	—
75	C₂³×C₅₃	#121	1	2	yes	—
76	C₂³×C₅₃	#124	1	2	yes	—
77	C₂³×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
78	C₂³×C₅₃	#137	1	4	yes	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	8	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	8	no	—
85	C₂×C₄×C₅₃	—	4	8	no	—
86	C₂×C₄×C₅₃	—	4	8	no	—
87	C₂×C₄×C₅₃	—	8	4	no	—
88	C₂×C₄×C₅₃	—	8	4	no	—
89	C₂×C₄×C₅₃	—	8	4	no	—
90	C₂×C₄×C₅₃	—	8	4	no	—
91	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#117	1	4	yes	—
96	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
99	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₅₃	#123	1	4	yes	—
102	C₂×C₄×C₅₃	#125	1	2	yes	—
103	C₂×C₄×C₅₃	#126	1	4	yes	—
104	C₈×C₅₃	—	2	4	no	—
105	C₈×C₅₃	—	2	4	no	—
106	#10821	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₅₃	#380	1	1	yes	characteristic
109	C₂²×C₅₃	#382	1	1	yes	characteristic
110	C₂²×C₅₃	#388	1	4	yes	—
111	C₂²×C₅₃	#428	1	4	yes	—
112	C₂²×C₅₃	#440	1	4	yes	—
113	C₂²×C₅₃	#441	1	2	yes	—
114	C₂²×C₅₃	#442	1	2	yes	—
115	C₂²×C₅₃	#468	1	4	yes	—
116	C₂²×C₅₃	#469	1	2	yes	—
117	C₂²×C₅₃	#470	1	2	yes	—
118	C₂²×C₅₃	#489	1	4	yes	—
119	C₂²×C₅₃	#490	1	2	yes	—
120	C₂²×C₅₃	#491	1	2	yes	—
121	C₄×C₅₃	—	4	4	no	—
122	C₄×C₅₃	—	8	8	no	—
123	C₄×C₅₃	#376	1	1	yes	characteristic
124	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₅₃	#381	1	1	yes	characteristic
126	C₄×C₅₃	#383	1	1	yes	characteristic
127	C₄×C₅₃	#387	1	4	yes	—
128	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
129	C₂×#340	C₂×C₅₃	1	2	yes	—
130	C₂×#340	C₂×C₅₃	1	2	yes	—
131	C₄⋊#95	C₂×C₅₃	1	2	yes	—
132	C₂×C₅₃	#1434	1	1	yes	characteristic
133	C₂×C₅₃	#1776	1	2	yes	—
134	C₂×C₅₃	#1809	1	2	yes	—
135	C₂×C₅₃	#1836	1	2	yes	—
136	C₂×C₅₃	#1935	1	4	yes	—
137	C₂×C₅₃	#1936	1	4	yes	—
138	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
141	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×#97	—	2	4	no	—
143	C₂×#97	C₄×C₅₃	1	4	yes	complemented
144	C₂×#95	—	2	4	no	—
145	#340	—	2	4	no	—
146	#340	C₄×C₅₃	1	8	yes	complemented
147	C₅₃	#10821	1	1	yes	characteristic, direct product factor, Sylow
148	C₂³×C₄	—	2	4	no	—
149	C₂³×C₄	D₄×C₅₃	1	2	yes	—
150	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
156	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
157	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
158	C₂²×C₈	D₄×C₅₃	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
168	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
169	#97	—	2	8	no	—
170	#95	—	4	8	no	—
171	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
187	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	#41×C₅₃	1	2	yes	—
189	C₂²×C₄	#41×C₅₃	1	2	yes	—
190	C₂²×C₄	#41×C₅₃	1	2	yes	—
191	C₂²×C₄	#41×C₅₃	1	2	yes	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	C₄²×C₅₃	1	4	yes	—
199	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
208	C₂³	C₄×D₄×C₅₃	1	2	yes	—
209	C₂³	#121×C₅₃	1	2	yes	—
210	C₂³	#124×C₅₃	1	2	yes	—
211	C₂³	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
212	C₂³	#137×C₅₃	1	4	yes	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
228	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
229	C₂×C₄	#117×C₅₃	1	4	yes	—
230	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
231	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
233	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
234	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
235	C₂×C₄	#123×C₅₃	1	4	yes	—
236	C₂×C₄	#125×C₅₃	1	2	yes	—
237	C₂×C₄	#126×C₅₃	1	4	yes	—
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
241	C₂²	C₅₃×#380	1	1	yes	characteristic
242	C₂²	C₅₃×#382	1	1	yes	characteristic
243	C₂²	C₅₃×#388	1	4	yes	—
244	C₂²	C₅₃×#428	1	4	yes	—
245	C₂²	C₅₃×#440	1	4	yes	—
246	C₂²	C₅₃×#441	1	2	yes	—
247	C₂²	C₅₃×#442	1	2	yes	—
248	C₂²	C₅₃×#468	1	4	yes	—
249	C₂²	C₅₃×#469	1	2	yes	—
250	C₂²	C₅₃×#470	1	2	yes	—
251	C₂²	C₅₃×#489	1	4	yes	—
252	C₂²	C₅₃×#490	1	2	yes	—
253	C₂²	C₅₃×#491	1	2	yes	—
254	C₄	—	4	4	no	—
255	C₄	—	8	8	no	—
256	C₄	C₅₃×#376	1	1	yes	characteristic
257	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
258	C₄	C₅₃×#381	1	1	yes	characteristic
259	C₄	C₅₃×#383	1	1	yes	characteristic
260	C₄	C₅₃×#387	1	4	yes	—
261	C₂	C₅₃×#1434	1	1	yes	characteristic
262	C₂	C₅₃×#1776	1	2	yes	—
263	C₂	C₅₃×#1809	1	2	yes	—
264	C₂	C₅₃×#1836	1	2	yes	—
265	C₂	C₅₃×#1935	1	4	yes	—
266	C₂	C₅₃×#1936	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₂×#97×C₅₃	C₄	semidirect
13. C₅₃×#340	C₄	semidirect
106. #10821	C₅₃	direct (N × H)
143. C₂×#97	C₄×C₅₃	semidirect
146. #340	C₄×C₅₃	semidirect
147. C₅₃	#10821	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#97×C₅₃	2
C₂×C₅₃×#340	2
C₅₃×C₄⋊#95	2
#10821	53

Subgroups — C53×#10821 #27218333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10821 #27218333