Subgroups — C₅₃×#33451 #27240963

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₄×#135×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2177	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2176	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2175	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2138	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₅₃	C₂²	1	2	yes	—
10	C₄×#41×C₅₃	—	2	2	no	—
11	C₅₃×#508	C₄	1	8	yes	complemented
12	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
18	C₄×SD₁₆×C₅₃	—	2	4	no	—
19	C₄×D₈×C₅₃	C₂²	1	4	yes	—
20	C₅₃×#427	C₂²	1	2	yes	—
21	C₅₃×#416	—	2	2	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#367	C₂²	1	2	yes	—
24	C₅₃×#366	C₂²	1	2	yes	—
25	C₅₃×#358	C₂²	1	4	yes	—
26	C₅₃×#357	C₂²	1	4	yes	—
27	C₅₃×#346	C₂²	1	2	yes	—
28	C₅₃×#346	C₂²	1	2	yes	—
29	C₂×#41×C₅₃	—	2	2	no	—
30	#135×C₅₃	—	2	4	no	—
31	#135×C₅₃	C₂×C₄	1	8	yes	complemented
32	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₅₃	—	2	4	no	—
38	C₄×D₄×C₅₃	—	2	4	no	—
39	C₄×D₄×C₅₃	—	4	4	no	—
40	#117×C₅₃	—	2	2	no	—
41	#117×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	—	4	2	no	—
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	Q₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
45	SD₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
46	D₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
52	#107×C₅₃	C₂³	1	4	yes	—
53	#106×C₅₃	—	2	2	no	—
54	#103×C₅₃	—	2	4	no	—
55	#102×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	D₄	1	2	yes	—
57	C₄×C₈×C₅₃	C₂³	1	2	yes	—
58	#41×C₅₃	—	2	8	no	—
59	#41×C₅₃	—	4	4	no	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	—	4	4	no	—
65	C₂²×C₄×C₅₃	D₈	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₅₃	—	2	2	no	—
68	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₅₃	—	2	8	no	—
70	D₈×C₅₃	—	2	2	no	—
71	D₈×C₅₃	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₂²⋊C₄×C₅₃	—	4	2	no	—
81	C₂²⋊C₄×C₅₃	—	4	4	no	—
82	C₄²×C₅₃	—	4	4	no	—
83	C₄²×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	D₈	1	2	yes	—
85	C₄²×C₅₃	C₂×D₄	1	2	yes	—
86	C₁₆×C₅₃	—	2	2	no	—
87	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂³×C₅₃	—	4	2	no	—
90	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₅₃	—	2	4	no	—
92	Q₈×C₅₃	—	4	2	no	—
93	D₄×C₅₃	—	2	4	no	—
94	D₄×C₅₃	—	4	8	no	—
95	D₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	8	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	8	8	no	—
101	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#135	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	C₄×D₄	1	2	yes	—
108	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
110	#33451	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	4	4	no	—
113	C₂²×C₅₃	—	8	4	no	—
114	C₂²×C₅₃	C₄×D₈	1	2	yes	—
115	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₅₃	—	2	4	no	—
117	C₄×C₅₃	—	4	4	no	—
118	C₄×C₅₃	—	8	4	no	—
119	C₄×C₅₃	C₄×D₈	1	2	yes	—
120	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
122	C₂×#508	C₂×C₅₃	1	1	yes	characteristic
123	C₄×#135	C₂×C₅₃	1	2	yes	complemented
124	#2177	C₂×C₅₃	1	2	yes	complemented
125	#2176	C₂×C₅₃	1	2	yes	complemented
126	#2175	C₂×C₅₃	1	4	yes	complemented
127	#2158	C₂×C₅₃	1	1	yes	characteristic, complemented
128	#2138	C₂×C₅₃	1	2	yes	complemented
129	C₂×#346	C₂×C₅₃	1	1	yes	characteristic, complemented
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	—	8	4	no	—
132	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
133	C₂×C₅₃	#3414	1	2	yes	—
134	C₂×#135	C₂²×C₅₃	1	2	yes	—
135	C₄×#41	—	2	2	no	—
136	#508	C₄×C₅₃	1	8	yes	complemented
137	C₂×Q₃₂	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×SD₃₂	C₂²×C₅₃	1	2	yes	—
139	C₂×D₁₆	C₂²×C₅₃	1	1	yes	characteristic
140	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
141	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
142	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	—
143	C₄×SD₁₆	—	2	4	no	—
144	C₄×D₈	C₂²×C₅₃	1	4	yes	—
145	#427	C₂²×C₅₃	1	2	yes	—
146	#416	—	2	2	no	—
147	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
148	#367	C₂²×C₅₃	1	2	yes	—
149	#366	C₂²×C₅₃	1	2	yes	—
150	#358	C₂²×C₅₃	1	4	yes	—
151	#357	C₂²×C₅₃	1	4	yes	—
152	#346	C₂²×C₅₃	1	2	yes	—
153	#346	C₂²×C₅₃	1	2	yes	—
154	C₅₃	#33451	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#41	—	2	2	no	—
156	#135	—	2	4	no	—
157	#135	C₂×C₄×C₅₃	1	8	yes	complemented
158	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	C₂³×C₅₃	1	2	yes	—
161	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
163	C₄×Q₈	—	2	4	no	—
164	C₄×D₄	—	2	4	no	—
165	C₄×D₄	—	4	4	no	—
166	#117	—	2	2	no	—
167	#117	—	4	2	no	—
168	C₂×C₄²	—	4	2	no	—
169	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
170	Q₃₂	C₂×C₄×C₅₃	1	4	yes	complemented
171	SD₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
172	D₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
175	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
176	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
178	#107	C₂³×C₅₃	1	4	yes	—
179	#106	—	2	2	no	—
180	#103	—	2	4	no	—
181	#102	—	2	4	no	—
182	C₄×C₈	D₄×C₅₃	1	2	yes	—
183	C₄×C₈	C₂³×C₅₃	1	2	yes	—
184	#41	—	2	8	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	D₈×C₅₃	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
193	Q₁₆	—	2	2	no	—
194	Q₁₆	C₂²×C₄×C₅₃	1	4	yes	—
195	SD₁₆	—	2	8	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂²×C₄×C₅₃	1	4	yes	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂×C₈	#41×C₅₃	1	2	yes	—
203	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	4	no	—
210	C₄²	D₈×C₅₃	1	2	yes	—
211	C₄²	C₂×D₄×C₅₃	1	2	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
214	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	8	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	8	8	no	—
227	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
229	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
230	C₂×C₄	#135×C₅₃	1	2	yes	—
231	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₄×D₄×C₅₃	1	2	yes	—
234	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
235	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	C₄×D₈×C₅₃	1	2	yes	—
240	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	4	4	no	—
243	C₄	—	8	4	no	—
244	C₄	C₄×D₈×C₅₃	1	2	yes	—
245	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
246	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	8	4	no	—
249	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
250	C₂	C₅₃×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×C₅₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#135×C₅₃	C₂	semidirect
3. C₅₃×#2177	C₂	semidirect
4. C₅₃×#2176	C₂	semidirect
5. C₅₃×#2175	C₂	semidirect
6. C₅₃×#2158	C₂	semidirect
7. C₅₃×#2138	C₂	semidirect
8. C₂×C₅₃×#346	C₂	semidirect
11. C₅₃×#508	C₄	semidirect
31. #135×C₅₃	C₂×C₄	semidirect
44. Q₃₂×C₅₃	C₂×C₄	semidirect
45. SD₃₂×C₅₃	C₂×C₄	semidirect
46. D₁₆×C₅₃	C₂×C₄	semidirect
48. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
110. #33451	C₅₃	direct (N × H)
123. C₄×#135	C₂×C₅₃	semidirect
124. #2177	C₂×C₅₃	semidirect
125. #2176	C₂×C₅₃	semidirect
126. #2175	C₂×C₅₃	semidirect
127. #2158	C₂×C₅₃	semidirect
128. #2138	C₂×C₅₃	semidirect
129. C₂×#346	C₂×C₅₃	semidirect
136. #508	C₄×C₅₃	semidirect
154. C₅₃	#33451	direct (N × H)
157. #135	C₂×C₄×C₅₃	semidirect
170. Q₃₂	C₂×C₄×C₅₃	semidirect
171. SD₃₂	C₂×C₄×C₅₃	semidirect
172. D₁₆	C₂×C₄×C₅₃	semidirect
174. C₂×C₁₆	C₂×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×#508	2
C₄×#135×C₅₃	2
C₅₃×#2177	2
C₅₃×#2176	2
C₅₃×#2175	2
C₅₃×#2158	2
C₅₃×#2138	2
C₂×C₅₃×#346	2
#33451	53

Subgroups — C53×#33451 #27240963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33451 #27240963