Subgroups — C₅₃×#2409 #20271346

Order 6784.

840 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	C₂³×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#395	C₂	1	2	yes	maximal, complemented
5	C₅₃×#393	C₂	1	2	yes	maximal, complemented
6	C₅₃×#391	C₂	1	2	yes	maximal, complemented
7	C₅₃×#386	C₂	1	2	yes	maximal
8	C₅₃×#385	C₂	1	2	yes	maximal, complemented
9	C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₅₃	—	2	8	no	—
11	C₂²×Q₈×C₅₃	—	2	4	no	—
12	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
13	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²⋊Q₈×C₅₃	—	2	8	no	—
19	C₄×D₄×C₅₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
26	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	#95×C₅₃	C₂²	1	2	yes	—
29	#95×C₅₃	C₂²	1	4	yes	—
30	#95×C₅₃	C₂²	1	4	yes	—
31	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
32	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	—	2	32	no	—
34	C₂×Q₈×C₅₃	—	2	8	no	—
35	C₂×Q₈×C₅₃	—	2	4	no	—
36	C₂×Q₈×C₅₃	—	2	4	no	—
37	C₂×Q₈×C₅₃	—	2	4	no	—
38	C₂×Q₈×C₅₃	D₄	1	8	yes	complemented
39	C₂×D₄×C₅₃	—	2	2	no	—
40	C₂×D₄×C₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	Q₈	1	4	yes	complemented
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	8	no	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	D₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	2	8	no	—
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₂²⋊C₄×C₅₃	—	2	2	no	—
59	C₂²⋊C₄×C₅₃	—	2	4	no	—
60	C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	2	8	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	—	2	2	no	—
64	C₂³×C₅₃	—	2	2	no	—
65	C₂³×C₅₃	—	2	1	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	1	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	C₂×D₄	1	2	yes	—
70	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂³×C₅₃	C₂×Q₈	1	2	yes	complemented
72	C₂³×C₅₃	#41	1	2	yes	—
73	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
74	Q₈×C₅₃	—	2	16	no	—
75	Q₈×C₅₃	—	2	16	no	—
76	D₄×C₅₃	—	2	8	no	—
77	C₂×C₄×C₅₃	—	2	16	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	C₂×D₄	1	8	yes	—
93	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
95	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	complemented
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	2	4	no	—
100	C₂²×C₅₃	—	2	4	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	4	no	—
103	C₂²×C₅₃	—	4	1	no	—
104	C₂²×C₅₃	—	4	1	no	—
105	C₂²×C₅₃	—	4	2	no	—
106	C₂²×C₅₃	C₂²⋊Q₈	1	4	yes	complemented
107	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
109	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
112	C₄×C₅₃	—	2	8	no	—
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	4	1	no	—
116	C₄×C₅₃	—	4	1	no	—
117	C₄×C₅₃	—	4	4	no	—
118	C₄×C₅₃	#120	1	4	yes	—
119	#2409	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₅₃	—	2	4	no	—
121	C₂×C₅₃	—	4	2	no	—
122	C₂×C₅₃	C₂×#120	1	1	yes	characteristic
123	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₅₃	#536	1	1	yes	characteristic
125	C₂×C₅₃	#548	1	2	yes	—
126	C₂×C₅₃	D₄×Q₈	1	2	yes	—
127	C₂³×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
128	C₂×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	complemented
129	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
130	#395	C₂×C₅₃	1	2	yes	complemented
131	#393	C₂×C₅₃	1	2	yes	complemented
132	#391	C₂×C₅₃	1	2	yes	complemented
133	#386	C₂×C₅₃	1	2	yes	—
134	#385	C₂×C₅₃	1	2	yes	complemented
135	#380	C₂×C₅₃	1	1	yes	characteristic, complemented
136	C₅₃	#2409	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×Q₈	—	2	8	no	—
138	C₂²×Q₈	—	2	4	no	—
139	C₂²×Q₈	C₂²×C₅₃	1	2	yes	—
140	C₂²×Q₈	C₂²×C₅₃	1	2	yes	—
141	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
143	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
144	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
145	C₂²⋊Q₈	—	2	8	no	—
146	C₄×D₄	—	2	4	no	—
147	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
154	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
155	#95	C₂²×C₅₃	1	2	yes	—
156	#95	C₂²×C₅₃	1	4	yes	—
157	#95	C₂²×C₅₃	1	4	yes	—
158	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
159	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
160	C₂×Q₈	—	2	32	no	—
161	C₂×Q₈	—	2	8	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	D₄×C₅₃	1	8	yes	complemented
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	Q₈×C₅₃	1	4	yes	complemented
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	D₄×C₅₃	1	4	yes	—
175	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
177	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
179	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	8	no	—
189	C₄²	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	2	no	—
196	C₂³	C₂×D₄×C₅₃	1	2	yes	—
197	C₂³	C₂×Q₈×C₅₃	1	2	yes	—
198	C₂³	C₂×Q₈×C₅₃	1	2	yes	complemented
199	C₂³	#41×C₅₃	1	2	yes	—
200	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
201	Q₈	—	2	16	no	—
202	Q₈	—	2	16	no	—
203	D₄	—	2	8	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	C₂×D₄×C₅₃	1	8	yes	—
220	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
221	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
223	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	complemented
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	C₂²⋊Q₈×C₅₃	1	4	yes	complemented
234	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
235	C₂²	C₂²×D₄×C₅₃	1	2	yes	—
236	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
237	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
238	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
239	C₄	—	2	8	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	4	no	—
245	C₄	#120×C₅₃	1	4	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	4	2	no	—
248	C₂	C₂×#120×C₅₃	1	1	yes	characteristic
249	C₂	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
250	C₂	C₅₃×#536	1	1	yes	characteristic
251	C₂	C₅₃×#548	1	2	yes	—
252	C₂	D₄×Q₈×C₅₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
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. C₂³×Q₈×C₅₃	C₂	semidirect
2. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
4. C₅₃×#395	C₂	semidirect
5. C₅₃×#393	C₂	semidirect
6. C₅₃×#391	C₂	semidirect
8. C₅₃×#385	C₂	semidirect
9. C₅₃×#380	C₂	semidirect
38. C₂×Q₈×C₅₃	D₄	semidirect
41. C₂×D₄×C₅₃	Q₈	semidirect
71. C₂³×C₅₃	C₂×Q₈	semidirect
96. C₂×C₄×C₅₃	C₂×Q₈	semidirect
106. C₂²×C₅₃	C₂²⋊Q₈	semidirect
119. #2409	C₅₃	direct (N × H)
127. C₂³×Q₈	C₂×C₅₃	semidirect
128. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
130. #395	C₂×C₅₃	semidirect
131. #393	C₂×C₅₃	semidirect
132. #391	C₂×C₅₃	semidirect
134. #385	C₂×C₅₃	semidirect
135. #380	C₂×C₅₃	semidirect
136. C₅₃	#2409	direct (N × H)
165. C₂×Q₈	D₄×C₅₃	semidirect
168. C₂×D₄	Q₈×C₅₃	semidirect
198. C₂³	C₂×Q₈×C₅₃	semidirect
223. C₂×C₄	C₂×Q₈×C₅₃	semidirect
233. C₂²	C₂²⋊Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₅₃	2
C₂×C₂²⋊Q₈×C₅₃	2
C₂×C₄×D₄×C₅₃	2
C₅₃×#395	2
C₅₃×#393	2
C₅₃×#391	2
C₅₃×#386	2
C₅₃×#385	2
C₅₃×#380	2
#2409	53

Subgroups — C53×#2409 #20271346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2409 #20271346