Subgroups — C₅₃×#2599 #20271536

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#397	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#393	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#388	C₂	1	2	yes	maximal
8	C₅₃×#387	C₂	1	2	yes	maximal, complemented
9	C₅₃×#387	C₂	1	2	yes	maximal, complemented
10	C₅₃×#385	C₂	1	2	yes	maximal
11	C₅₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₅₃	—	2	4	no	—
15	#117×C₅₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	#95×C₅₃	C₂²	1	2	yes	—
30	#95×C₅₃	C₂²	1	2	yes	—
31	#95×C₅₃	C₂²	1	1	yes	characteristic
32	#95×C₅₃	C₂²	1	1	yes	characteristic
33	#95×C₅₃	C₂²	1	2	yes	—
34	#95×C₅₃	C₂²	1	2	yes	—
35	#95×C₅₃	C₂²	1	2	yes	—
36	#95×C₅₃	C₂²	1	2	yes	—
37	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₂²⋊C₄×C₅₃	—	2	4	no	—
60	C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂²⋊C₄×C₅₃	—	2	4	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	4	no	—
65	C₄²×C₅₃	D₄	1	4	yes	—
66	C₂³×C₅₃	—	2	1	no	—
67	C₂³×C₅₃	—	2	1	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	#41	1	2	yes	—
71	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
72	Q₈×C₅₃	—	2	8	no	—
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	#41	1	2	yes	—
99	C₂×C₄×C₅₃	#41	1	4	yes	—
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	4	no	—
102	C₂²×C₅₃	—	4	1	no	—
103	C₂²×C₅₃	—	4	1	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×#41	1	2	yes	—
108	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	2	4	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	4	1	no	—
116	C₄×C₅₃	—	4	2	no	—
117	C₄×C₅₃	—	4	1	no	—
118	C₄×C₅₃	—	4	2	no	—
119	C₄×C₅₃	—	4	2	no	—
120	#2599	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₅₃	—	4	2	no	—
122	C₂×C₅₃	#525	1	1	yes	characteristic
123	C₂×C₅₃	#525	1	1	yes	characteristic
124	C₂×C₅₃	#536	1	1	yes	characteristic
125	C₂×C₅₃	#552	1	2	yes	—
126	C₂×C₅₃	#556	1	2	yes	—
127	C₂×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
128	C₂×#117	C₂×C₅₃	1	1	yes	characteristic
129	#398	C₂×C₅₃	1	1	yes	characteristic, complemented
130	#397	C₂×C₅₃	1	1	yes	characteristic
131	#395	C₂×C₅₃	1	1	yes	characteristic, complemented
132	#393	C₂×C₅₃	1	1	yes	characteristic
133	#388	C₂×C₅₃	1	2	yes	—
134	#387	C₂×C₅₃	1	2	yes	complemented
135	#387	C₂×C₅₃	1	2	yes	complemented
136	#385	C₂×C₅₃	1	2	yes	—
137	#383	C₂×C₅₃	1	1	yes	characteristic, complemented
138	C₅₃	#2599	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
141	C₄×Q₈	—	2	4	no	—
142	#117	—	2	4	no	—
143	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
148	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
153	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
155	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
156	#95	C₂²×C₅₃	1	2	yes	—
157	#95	C₂²×C₅₃	1	2	yes	—
158	#95	C₂²×C₅₃	1	1	yes	characteristic
159	#95	C₂²×C₅₃	1	1	yes	characteristic
160	#95	C₂²×C₅₃	1	2	yes	—
161	#95	C₂²×C₅₃	1	2	yes	—
162	#95	C₂²×C₅₃	1	2	yes	—
163	#95	C₂²×C₅₃	1	2	yes	—
164	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
171	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
172	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
175	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	4	no	—
192	C₄²	D₄×C₅₃	1	4	yes	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	2	no	—
197	C₂³	#41×C₅₃	1	2	yes	—
198	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
199	Q₈	—	2	8	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
223	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
225	C₂×C₄	#41×C₅₃	1	4	yes	—
226	C₂×C₄	#41×C₅₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
233	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
234	C₂²	C₂×#41×C₅₃	1	2	yes	—
235	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
236	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	2	no	—
247	C₂	—	4	2	no	—
248	C₂	C₅₃×#525	1	1	yes	characteristic
249	C₂	C₅₃×#525	1	1	yes	characteristic
250	C₂	C₅₃×#536	1	1	yes	characteristic
251	C₂	C₅₃×#552	1	2	yes	—
252	C₂	C₅₃×#556	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₅₃	C₂	semidirect
3. C₅₃×#398	C₂	semidirect
5. C₅₃×#395	C₂	semidirect
8. C₅₃×#387	C₂	semidirect
9. C₅₃×#387	C₂	semidirect
11. C₅₃×#383	C₂	semidirect
120. #2599	C₅₃	direct (N × H)
127. C₂×C₄×Q₈	C₂×C₅₃	semidirect
129. #398	C₂×C₅₃	semidirect
131. #395	C₂×C₅₃	semidirect
134. #387	C₂×C₅₃	semidirect
135. #387	C₂×C₅₃	semidirect
137. #383	C₂×C₅₃	semidirect
138. C₅₃	#2599	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×C₄×Q₈×C₅₃	2
C₂×#117×C₅₃	2
C₅₃×#398	2
C₅₃×#397	2
C₅₃×#395	2
C₅₃×#393	2
C₅₃×#388	2
C₅₃×#387	2
C₅₃×#385	2
C₅₃×#383	2
#2599	53

Subgroups — C53×#2599 #20271536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2599 #20271536