Subgroups — C₅₃×#3585 #20272522

Order 6784.

1464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#577	C₂	1	8	yes	maximal, complemented
4	C₅₃×#576	C₂	1	8	yes	maximal, complemented
5	C₅₃×#575	C₂	1	6	yes	maximal, complemented
6	C₂×AGL(1,ℤ/8)×C₅₃	C₂	1	6	yes	maximal, complemented
7	C₅₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴×C₅₃	—	2	1	no	—
9	2₋¹⁺⁴×C₅₃	C₂²	1	4	yes	complemented
10	H(5,𝔽₂)×C₅₃	—	2	4	no	—
11	H(5,𝔽₂)×C₅₃	—	2	3	no	—
12	H(5,𝔽₂)×C₅₃	C₂²	1	8	yes	complemented
13	H(5,𝔽₂)×C₅₃	C₂²	1	4	yes	complemented
14	C₂×#41×C₅₃	—	2	4	no	—
15	C₂×#41×C₅₃	—	2	1	no	—
16	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#41×C₅₃	C₂²	1	3	yes	complemented
18	C₂×#41×C₅₃	C₂²	1	6	yes	complemented
19	C₂²×D₄×C₅₃	—	2	3	no	—
20	C₂²×D₄×C₅₃	C₂²	1	3	yes	complemented
21	#137×C₅₃	C₂²	1	4	yes	complemented
22	#137×C₅₃	C₂²	1	12	yes	complemented
23	AGL(1,ℤ/8)×C₅₃	C₂²	1	12	yes	complemented
24	AGL(1,ℤ/8)×C₅₃	C₂²	1	24	yes	complemented
25	AGL(1,ℤ/8)×C₅₃	C₂²	1	12	yes	complemented
26	#135×C₅₃	C₂²	1	24	yes	complemented
27	C₂×SD₁₆×C₅₃	C₂²	1	12	yes	complemented
28	C₂×D₈×C₅₃	C₂²	1	12	yes	complemented
29	#131×C₅₃	C₂²	1	8	yes	complemented
30	C₂×M₁₆×C₅₃	C₂²	1	6	yes	complemented
31	C₂⁴×C₅₃	—	2	3	no	—
32	#41×C₅₃	—	2	8	no	—
33	#41×C₅₃	—	2	4	no	—
34	#41×C₅₃	—	2	12	no	—
35	#41×C₅₃	—	2	6	no	—
36	#41×C₅₃	—	2	6	no	—
37	#41×C₅₃	—	2	6	no	—
38	#41×C₅₃	D₄	1	4	yes	—
39	#41×C₅₃	C₂³	1	12	yes	—
40	#41×C₅₃	C₂³	1	4	yes	—
41	#41×C₅₃	C₂³	1	4	yes	—
42	#41×C₅₃	C₂³	1	12	yes	complemented
43	C₂×Q₈×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₅₃	C₂³	1	6	yes	complemented
47	C₂×D₄×C₅₃	—	2	3	no	—
48	C₂×D₄×C₅₃	—	2	6	no	—
49	C₂×D₄×C₅₃	—	2	24	no	—
50	C₂×D₄×C₅₃	—	2	12	no	—
51	C₂×D₄×C₅₃	—	2	3	no	—
52	C₂×D₄×C₅₃	—	2	6	no	—
53	C₂×D₄×C₅₃	D₄	1	3	yes	—
54	C₂×D₄×C₅₃	C₂³	1	12	yes	—
55	C₂×D₄×C₅₃	C₂³	1	3	yes	—
56	C₂×D₄×C₅₃	C₂³	1	6	yes	complemented
57	C₂²×C₄×C₅₃	—	2	3	no	—
58	C₂²×C₄×C₅₃	—	2	6	no	—
59	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
60	Q₁₆×C₅₃	C₂³	1	8	yes	complemented
61	SD₁₆×C₅₃	C₂³	1	24	yes	complemented
62	SD₁₆×C₅₃	C₂³	1	8	yes	complemented
63	D₈×C₅₃	C₂³	1	24	yes	complemented
64	M₁₆×C₅₃	C₂³	1	4	yes	—
65	M₁₆×C₅₃	C₂³	1	12	yes	complemented
66	C₂×C₈×C₅₃	C₂³	1	12	yes	complemented
67	C₂³×C₅₃	—	2	3	no	—
68	C₂³×C₅₃	—	2	6	no	—
69	C₂³×C₅₃	—	2	12	no	—
70	C₂³×C₅₃	—	8	6	no	—
71	C₂³×C₅₃	C₂×D₄	1	3	yes	—
72	Q₈×C₅₃	—	2	6	no	—
73	Q₈×C₅₃	—	2	1	no	—
74	Q₈×C₅₃	C₂×D₄	1	4	yes	—
75	Q₈×C₅₃	C₂⁴	1	2	yes	—
76	Q₈×C₅₃	C₂⁴	1	4	yes	—
77	D₄×C₅₃	—	2	12	no	—
78	D₄×C₅₃	—	2	4	no	—
79	D₄×C₅₃	—	2	12	no	—
80	D₄×C₅₃	—	2	3	no	—
81	D₄×C₅₃	—	2	12	no	—
82	D₄×C₅₃	—	2	6	no	—
83	D₄×C₅₃	C₂×D₄	1	12	yes	—
84	D₄×C₅₃	C₂⁴	1	6	yes	—
85	D₄×C₅₃	C₂⁴	1	4	yes	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	6	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	12	no	—
90	C₂×C₄×C₅₃	—	2	12	no	—
91	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
92	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
93	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
95	C₈×C₅₃	C₂⁴	1	8	yes	—
96	C₂²×C₅₃	—	2	4	no	—
97	C₂²×C₅₃	—	2	6	no	—
98	C₂²×C₅₃	—	4	3	no	—
99	C₂²×C₅₃	—	4	6	no	—
100	C₂²×C₅₃	—	8	6	no	—
101	C₂²×C₅₃	—	8	12	no	—
102	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₅₃	C₂²×D₄	1	6	yes	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	4	no	—
106	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₅₃	C₂²×D₄	1	6	yes	—
108	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
109	#3585	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	—	2	6	no	—
112	C₂×C₅₃	—	4	6	no	—
113	C₂×C₅₃	—	4	4	no	—
114	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
115	#585	C₂×C₅₃	1	1	yes	characteristic, complemented
116	C₂×H(5,𝔽₂)	C₂×C₅₃	1	1	yes	characteristic, complemented
117	#577	C₂×C₅₃	1	8	yes	complemented
118	#576	C₂×C₅₃	1	8	yes	complemented
119	#575	C₂×C₅₃	1	6	yes	complemented
120	C₂×AGL(1,ℤ/8)	C₂×C₅₃	1	6	yes	complemented
121	#568	C₂×C₅₃	1	1	yes	characteristic, complemented
122	C₅₃	#3585	1	1	yes	characteristic, direct product factor, Sylow
123	2₋¹⁺⁴	—	2	1	no	—
124	2₋¹⁺⁴	C₂²×C₅₃	1	4	yes	complemented
125	H(5,𝔽₂)	—	2	4	no	—
126	H(5,𝔽₂)	—	2	3	no	—
127	H(5,𝔽₂)	C₂²×C₅₃	1	8	yes	complemented
128	H(5,𝔽₂)	C₂²×C₅₃	1	4	yes	complemented
129	C₂×#41	—	2	4	no	—
130	C₂×#41	—	2	1	no	—
131	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
132	C₂×#41	C₂²×C₅₃	1	3	yes	complemented
133	C₂×#41	C₂²×C₅₃	1	6	yes	complemented
134	C₂²×D₄	—	2	3	no	—
135	C₂²×D₄	C₂²×C₅₃	1	3	yes	complemented
136	#137	C₂²×C₅₃	1	4	yes	complemented
137	#137	C₂²×C₅₃	1	12	yes	complemented
138	AGL(1,ℤ/8)	C₂²×C₅₃	1	12	yes	complemented
139	AGL(1,ℤ/8)	C₂²×C₅₃	1	24	yes	complemented
140	AGL(1,ℤ/8)	C₂²×C₅₃	1	12	yes	complemented
141	#135	C₂²×C₅₃	1	24	yes	complemented
142	C₂×SD₁₆	C₂²×C₅₃	1	12	yes	complemented
143	C₂×D₈	C₂²×C₅₃	1	12	yes	complemented
144	#131	C₂²×C₅₃	1	8	yes	complemented
145	C₂×M₁₆	C₂²×C₅₃	1	6	yes	complemented
146	C₂⁴	—	2	3	no	—
147	#41	—	2	8	no	—
148	#41	—	2	4	no	—
149	#41	—	2	12	no	—
150	#41	—	2	6	no	—
151	#41	—	2	6	no	—
152	#41	—	2	6	no	—
153	#41	D₄×C₅₃	1	4	yes	—
154	#41	C₂³×C₅₃	1	12	yes	—
155	#41	C₂³×C₅₃	1	4	yes	—
156	#41	C₂³×C₅₃	1	4	yes	—
157	#41	C₂³×C₅₃	1	12	yes	complemented
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	D₄×C₅₃	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
161	C₂×Q₈	C₂³×C₅₃	1	6	yes	complemented
162	C₂×D₄	—	2	3	no	—
163	C₂×D₄	—	2	6	no	—
164	C₂×D₄	—	2	24	no	—
165	C₂×D₄	—	2	12	no	—
166	C₂×D₄	—	2	3	no	—
167	C₂×D₄	—	2	6	no	—
168	C₂×D₄	D₄×C₅₃	1	3	yes	—
169	C₂×D₄	C₂³×C₅₃	1	12	yes	—
170	C₂×D₄	C₂³×C₅₃	1	3	yes	—
171	C₂×D₄	C₂³×C₅₃	1	6	yes	complemented
172	C₂²×C₄	—	2	3	no	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
175	Q₁₆	C₂³×C₅₃	1	8	yes	complemented
176	SD₁₆	C₂³×C₅₃	1	24	yes	complemented
177	SD₁₆	C₂³×C₅₃	1	8	yes	complemented
178	D₈	C₂³×C₅₃	1	24	yes	complemented
179	M₁₆	C₂³×C₅₃	1	4	yes	—
180	M₁₆	C₂³×C₅₃	1	12	yes	complemented
181	C₂×C₈	C₂³×C₅₃	1	12	yes	complemented
182	C₂³	—	2	3	no	—
183	C₂³	—	2	6	no	—
184	C₂³	—	2	12	no	—
185	C₂³	—	8	6	no	—
186	C₂³	C₂×D₄×C₅₃	1	3	yes	—
187	Q₈	—	2	6	no	—
188	Q₈	—	2	1	no	—
189	Q₈	C₂×D₄×C₅₃	1	4	yes	—
190	Q₈	C₂⁴×C₅₃	1	4	yes	—
191	Q₈	C₂⁴×C₅₃	1	2	yes	—
192	D₄	—	2	12	no	—
193	D₄	—	2	4	no	—
194	D₄	—	2	12	no	—
195	D₄	—	2	3	no	—
196	D₄	—	2	12	no	—
197	D₄	—	2	6	no	—
198	D₄	C₂×D₄×C₅₃	1	12	yes	—
199	D₄	C₂⁴×C₅₃	1	4	yes	—
200	D₄	C₂⁴×C₅₃	1	6	yes	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	12	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	C₂×D₄×C₅₃	1	3	yes	—
207	C₂×C₄	C₂×D₄×C₅₃	1	6	yes	—
208	C₂×C₄	C₂⁴×C₅₃	1	6	yes	—
209	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
210	C₈	C₂⁴×C₅₃	1	8	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	6	no	—
213	C₂²	—	4	3	no	—
214	C₂²	—	4	6	no	—
215	C₂²	—	8	12	no	—
216	C₂²	—	8	6	no	—
217	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
218	C₂²	C₂²×D₄×C₅₃	1	6	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	2	4	no	—
221	C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
222	C₄	C₂²×D₄×C₅₃	1	6	yes	—
223	C₄	C₂⁵×C₅₃	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	6	no	—
226	C₂	—	4	6	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂³×D₄×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₅₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#585	C₂	semidirect
2. C₂×H(5,𝔽₂)×C₅₃	C₂	semidirect
3. C₅₃×#577	C₂	semidirect
4. C₅₃×#576	C₂	semidirect
5. C₅₃×#575	C₂	semidirect
6. C₂×AGL(1,ℤ/8)×C₅₃	C₂	semidirect
7. C₅₃×#568	C₂	semidirect
9. 2₋¹⁺⁴×C₅₃	C₂²	semidirect
12. H(5,𝔽₂)×C₅₃	C₂²	semidirect
13. H(5,𝔽₂)×C₅₃	C₂²	semidirect
17. C₂×#41×C₅₃	C₂²	semidirect
18. C₂×#41×C₅₃	C₂²	semidirect
20. C₂²×D₄×C₅₃	C₂²	semidirect
21. #137×C₅₃	C₂²	semidirect
22. #137×C₅₃	C₂²	semidirect
23. AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
24. AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
25. AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
26. #135×C₅₃	C₂²	semidirect
27. C₂×SD₁₆×C₅₃	C₂²	semidirect
28. C₂×D₈×C₅₃	C₂²	semidirect
29. #131×C₅₃	C₂²	semidirect
30. C₂×M₁₆×C₅₃	C₂²	semidirect
42. #41×C₅₃	C₂³	semidirect
46. C₂×Q₈×C₅₃	C₂³	semidirect
56. C₂×D₄×C₅₃	C₂³	semidirect
60. Q₁₆×C₅₃	C₂³	semidirect
61. SD₁₆×C₅₃	C₂³	semidirect
62. SD₁₆×C₅₃	C₂³	semidirect
63. D₈×C₅₃	C₂³	semidirect
65. M₁₆×C₅₃	C₂³	semidirect
66. C₂×C₈×C₅₃	C₂³	semidirect
109. #3585	C₅₃	direct (N × H)
115. #585	C₂×C₅₃	semidirect
116. C₂×H(5,𝔽₂)	C₂×C₅₃	semidirect
117. #577	C₂×C₅₃	semidirect
118. #576	C₂×C₅₃	semidirect
119. #575	C₂×C₅₃	semidirect
120. C₂×AGL(1,ℤ/8)	C₂×C₅₃	semidirect
121. #568	C₂×C₅₃	semidirect
122. C₅₃	#3585	direct (N × H)
124. 2₋¹⁺⁴	C₂²×C₅₃	semidirect
127. H(5,𝔽₂)	C₂²×C₅₃	semidirect
128. H(5,𝔽₂)	C₂²×C₅₃	semidirect
132. C₂×#41	C₂²×C₅₃	semidirect
133. C₂×#41	C₂²×C₅₃	semidirect
135. C₂²×D₄	C₂²×C₅₃	semidirect
136. #137	C₂²×C₅₃	semidirect
137. #137	C₂²×C₅₃	semidirect
138. AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
139. AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
140. AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
141. #135	C₂²×C₅₃	semidirect
142. C₂×SD₁₆	C₂²×C₅₃	semidirect
143. C₂×D₈	C₂²×C₅₃	semidirect
144. #131	C₂²×C₅₃	semidirect
145. C₂×M₁₆	C₂²×C₅₃	semidirect
157. #41	C₂³×C₅₃	semidirect
161. C₂×Q₈	C₂³×C₅₃	semidirect
171. C₂×D₄	C₂³×C₅₃	semidirect
175. Q₁₆	C₂³×C₅₃	semidirect
176. SD₁₆	C₂³×C₅₃	semidirect
177. SD₁₆	C₂³×C₅₃	semidirect
178. D₈	C₂³×C₅₃	semidirect
180. M₁₆	C₂³×C₅₃	semidirect
181. C₂×C₈	C₂³×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#585	2
C₂×H(5,𝔽₂)×C₅₃	2
C₅₃×#577	2
C₅₃×#576	2
C₅₃×#575	2
C₂×AGL(1,ℤ/8)×C₅₃	2
C₅₃×#568	2
#3585	53

Subgroups — C53×#3585 #20272522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3585 #20272522