Subgroups — C₅₃×#3094 #20272031

Order 6784.

462 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×#121×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#482	C₂	1	4	yes	maximal, complemented
4	C₅₃×#481	C₂	1	2	yes	maximal, complemented
5	C₅₃×#480	C₂	1	2	yes	maximal, complemented
6	C₅₃×#428	C₂	1	2	yes	maximal, complemented
7	C₅₃×#418	C₂	1	2	yes	maximal, complemented
8	C₅₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₅₃	—	2	1	no	—
10	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
13	#121×C₅₃	—	2	1	no	—
14	#121×C₅₃	—	2	1	no	—
15	#121×C₅₃	C₂²	1	2	yes	—
16	#121×C₅₃	C₂²	1	2	yes	—
17	#117×C₅₃	—	2	1	no	—
18	#117×C₅₃	—	2	1	no	—
19	#117×C₅₃	C₂²	1	2	yes	—
20	#117×C₅₃	C₂²	1	2	yes	complemented
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
26	C₂×C₄²×C₅₃	—	2	1	no	—
27	#107×C₅₃	C₂²	1	4	yes	complemented
28	#106×C₅₃	C₂²	1	4	yes	complemented
29	#102×C₅₃	C₂²	1	4	yes	complemented
30	#102×C₅₃	C₂²	1	4	yes	complemented
31	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	complemented
32	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	complemented
33	C₂⁴×C₅₃	—	2	1	no	—
34	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	2	2	no	—
37	C₂×D₄×C₅₃	—	2	1	no	—
38	C₂×D₄×C₅₃	—	2	4	no	—
39	C₂×D₄×C₅₃	—	4	4	no	—
40	C₂×D₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
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	M₁₆×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	C₂³	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
58	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₅₃	—	2	2	no	—
60	C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₂³×C₅₃	—	2	2	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	—	4	1	no	—
69	C₂³×C₅₃	—	4	2	no	—
70	C₂³×C₅₃	—	4	2	no	—
71	C₂³×C₅₃	—	4	2	no	—
72	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
75	D₄×C₅₃	—	2	4	no	—
76	D₄×C₅₃	—	4	4	no	—
77	D₄×C₅₃	—	4	4	no	—
78	D₄×C₅₃	—	4	2	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	2	no	—
83	C₂×C₄×C₅₃	—	2	2	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	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#41	1	4	yes	—
94	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₅₃	—	2	4	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	4	4	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	—	4	2	no	—
106	C₂²×C₅₃	—	4	2	no	—
107	C₂²×C₅₃	—	8	4	no	—
108	C₂²×C₅₃	#123	1	2	yes	—
109	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
110	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	—	4	2	no	—
116	C₄×C₅₃	#123	1	2	yes	—
117	C₄×C₅₃	C₂×#41	1	2	yes	—
118	#3094	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	—	2	2	no	—
121	C₂×C₅₃	—	4	1	no	—
122	C₂×C₅₃	—	8	2	no	—
123	C₂×C₅₃	C₂×#123	1	1	yes	characteristic
124	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₅₃	#575	1	1	yes	characteristic
126	C₂×#121	C₂×C₅₃	1	1	yes	characteristic
127	C₂×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
128	#482	C₂×C₅₃	1	4	yes	complemented
129	#481	C₂×C₅₃	1	2	yes	complemented
130	#480	C₂×C₅₃	1	2	yes	complemented
131	#428	C₂×C₅₃	1	2	yes	complemented
132	#418	C₂×C₅₃	1	2	yes	complemented
133	#407	C₂×C₅₃	1	1	yes	characteristic, complemented
134	C₅₃	#3094	1	1	yes	characteristic, direct product factor, Sylow
135	C₂²×D₄	—	2	1	no	—
136	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
137	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
139	#121	—	2	1	no	—
140	#121	—	2	1	no	—
141	#121	C₂²×C₅₃	1	2	yes	—
142	#121	C₂²×C₅₃	1	2	yes	—
143	#117	—	2	1	no	—
144	#117	—	2	1	no	—
145	#117	C₂²×C₅₃	1	2	yes	—
146	#117	C₂²×C₅₃	1	2	yes	complemented
147	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
149	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₂²⋊C₄	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	#107	C₂²×C₅₃	1	4	yes	complemented
154	#106	C₂²×C₅₃	1	4	yes	complemented
155	#102	C₂²×C₅₃	1	4	yes	complemented
156	#102	C₂²×C₅₃	1	4	yes	complemented
157	C₂²⋊C₈	C₂²×C₅₃	1	2	yes	complemented
158	C₂²⋊C₈	C₂²×C₅₃	1	2	yes	complemented
159	C₂⁴	—	2	1	no	—
160	C₂⁴	D₄×C₅₃	1	1	yes	characteristic
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	4	4	no	—
166	C₂×D₄	C₂³×C₅₃	1	2	yes	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
173	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
177	M₁₆	—	2	4	no	—
178	C₂×C₈	C₂³×C₅₃	1	4	yes	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
183	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
184	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	4	1	no	—
195	C₂³	—	4	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
199	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
200	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
201	D₄	—	2	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄	#41×C₅₃	1	4	yes	—
220	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	4	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	8	4	no	—
234	C₂²	#123×C₅₃	1	2	yes	—
235	C₂²	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
236	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	#123×C₅₃	1	2	yes	—
243	C₄	C₂×#41×C₅₃	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	2	2	no	—
246	C₂	—	4	1	no	—
247	C₂	—	8	2	no	—
248	C₂	C₂×#123×C₅₃	1	1	yes	characteristic
249	C₂	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
250	C₂	C₅₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
prime radical	C₂×#121×C₅₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#117×C₅₃	C₂	semidirect
3. C₅₃×#482	C₂	semidirect
4. C₅₃×#481	C₂	semidirect
5. C₅₃×#480	C₂	semidirect
6. C₅₃×#428	C₂	semidirect
7. C₅₃×#418	C₂	semidirect
8. C₅₃×#407	C₂	semidirect
20. #117×C₅₃	C₂²	semidirect
22. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
23. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
27. #107×C₅₃	C₂²	semidirect
28. #106×C₅₃	C₂²	semidirect
29. #102×C₅₃	C₂²	semidirect
30. #102×C₅₃	C₂²	semidirect
31. C₂²⋊C₈×C₅₃	C₂²	semidirect
32. C₂²⋊C₈×C₅₃	C₂²	semidirect
118. #3094	C₅₃	direct (N × H)
127. C₂×#117	C₂×C₅₃	semidirect
128. #482	C₂×C₅₃	semidirect
129. #481	C₂×C₅₃	semidirect
130. #480	C₂×C₅₃	semidirect
131. #428	C₂×C₅₃	semidirect
132. #418	C₂×C₅₃	semidirect
133. #407	C₂×C₅₃	semidirect
134. C₅₃	#3094	direct (N × H)
146. #117	C₂²×C₅₃	semidirect
148. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
149. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
153. #107	C₂²×C₅₃	semidirect
154. #106	C₂²×C₅₃	semidirect
155. #102	C₂²×C₅₃	semidirect
156. #102	C₂²×C₅₃	semidirect
157. C₂²⋊C₈	C₂²×C₅₃	semidirect
158. C₂²⋊C₈	C₂²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#121×C₅₃	2
C₂×#117×C₅₃	2
C₅₃×#482	2
C₅₃×#481	2
C₅₃×#480	2
C₅₃×#428	2
C₅₃×#418	2
C₅₃×#407	2
#3094	53

Subgroups — C53×#3094 #20272031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3094 #20272031