Subgroups — C₂³×#121×C₅₃ #27270168

Order 13568.

19998 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂⁴×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#121×C₅₃	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
7	C₂⁶×C₅₃	C₂²	1	2	yes	—
8	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₅₃	—	2	32	no	—
10	C₂³×D₄×C₅₃	—	2	16	no	—
11	C₂³×D₄×C₅₃	C₂²	1	56	yes	complemented
12	C₂³×D₄×C₅₃	C₂²	1	28	yes	complemented
13	C₂⁴×C₄×C₅₃	—	2	16	no	—
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₅₃	C₂²	1	28	yes	complemented
18	C₂×#121×C₅₃	C₂²	1	448	yes	direct product factor
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	28	yes	complemented
20	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	56	yes	complemented
21	C₂⁵×C₅₃	—	2	8	no	—
22	C₂⁵×C₅₃	—	2	8	no	—
23	C₂⁵×C₅₃	—	2	16	no	—
24	C₂⁵×C₅₃	—	2	16	no	—
25	C₂⁵×C₅₃	—	2	16	no	—
26	C₂⁵×C₅₃	D₄	1	16	yes	complemented
27	C₂⁵×C₅₃	C₂³	1	28	yes	—
28	C₂⁵×C₅₃	C₂³	1	14	yes	—
29	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×D₄×C₅₃	—	2	448	no	—
31	C₂²×D₄×C₅₃	—	2	224	no	—
32	C₂²×D₄×C₅₃	C₂³	1	224	yes	complemented
33	C₂²×D₄×C₅₃	C₂³	1	112	yes	complemented
34	C₂³×C₄×C₅₃	—	2	8	no	—
35	C₂³×C₄×C₅₃	—	2	16	no	—
36	C₂³×C₄×C₅₃	—	2	224	no	—
37	C₂³×C₄×C₅₃	D₄	1	16	yes	complemented
38	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	28	yes	—
40	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	112	yes	complemented
42	#121×C₅₃	C₂³	1	512	yes	direct product factor
43	C₂×C₄⋊C₄×C₅₃	C₂³	1	112	yes	complemented
44	C₂×C₂²⋊C₄×C₅₃	C₂³	1	224	yes	complemented
45	C₂⁴×C₅₃	—	2	128	no	—
46	C₂⁴×C₅₃	—	2	56	no	—
47	C₂⁴×C₅₃	—	2	56	no	—
48	C₂⁴×C₅₃	—	2	112	no	—
49	C₂⁴×C₅₃	—	2	112	no	—
50	C₂⁴×C₅₃	—	2	112	no	—
51	C₂⁴×C₅₃	—	4	128	no	—
52	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
53	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
54	C₂⁴×C₅₃	C₂×D₄	1	112	yes	complemented
55	C₂⁴×C₅₃	#41	1	8	yes	—
56	C₂⁴×C₅₃	C₂⁴	1	56	yes	—
57	C₂⁴×C₅₃	C₂⁴	1	28	yes	—
58	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
59	C₂×D₄×C₅₃	—	2	896	no	—
60	C₂×D₄×C₅₃	—	2	448	no	—
61	C₂×D₄×C₅₃	C₂⁴	1	128	yes	complemented
62	C₂×D₄×C₅₃	C₂⁴	1	64	yes	complemented
63	C₂²×C₄×C₅₃	—	2	56	no	—
64	C₂²×C₄×C₅₃	—	2	112	no	—
65	C₂²×C₄×C₅₃	—	2	448	no	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	112	yes	complemented
67	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	56	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
70	C₂²×C₄×C₅₃	C₂⁴	1	64	yes	complemented
71	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	complemented
72	C₂²⋊C₄×C₅₃	C₂⁴	1	128	yes	complemented
73	C₂³×C₅₃	—	2	448	no	—
74	C₂³×C₅₃	—	2	56	no	—
75	C₂³×C₅₃	—	2	56	no	—
76	C₂³×C₅₃	—	2	112	no	—
77	C₂³×C₅₃	—	2	112	no	—
78	C₂³×C₅₃	—	2	112	no	—
79	C₂³×C₅₃	—	4	448	no	—
80	C₂³×C₅₃	#121	1	64	yes	direct product factor
81	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
82	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
83	C₂³×C₅₃	C₂²×D₄	1	112	yes	complemented
84	C₂³×C₅₃	C₂×#41	1	28	yes	—
85	C₂³×C₅₃	C₂⁵	1	16	yes	—
86	C₂³×C₅₃	C₂⁵	1	8	yes	—
87	C₂³×C₅₃	C₂⁵	1	7	yes	—
88	D₄×C₅₃	—	2	256	no	—
89	D₄×C₅₃	—	2	128	no	—
90	C₂×C₄×C₅₃	—	2	56	no	—
91	C₂×C₄×C₅₃	—	2	112	no	—
92	C₂×C₄×C₅₃	—	2	128	no	—
93	C₂×C₄×C₅₃	C₂²×D₄	1	112	yes	complemented
94	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
95	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
96	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
97	C₂³×#121	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₅₃	—	2	8	no	—
99	C₂²×C₅₃	—	2	8	no	—
100	C₂²×C₅₃	—	2	224	no	—
101	C₂²×C₅₃	—	2	16	no	—
102	C₂²×C₅₃	—	2	16	no	—
103	C₂²×C₅₃	—	2	16	no	—
104	C₂²×C₅₃	—	4	224	no	—
105	C₂²×C₅₃	C₂×#121	1	112	yes	direct product factor
106	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
107	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
108	C₂²×C₅₃	C₂³×D₄	1	16	yes	complemented
109	C₂²×C₅₃	C₂²×#41	1	14	yes	—
110	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
111	C₄×C₅₃	—	2	8	no	—
112	C₄×C₅₃	—	2	16	no	—
113	C₄×C₅₃	C₂³×D₄	1	16	yes	complemented
114	C₂⁴×D₄	C₂×C₅₃	1	2	yes	complemented
115	C₂⁴×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
116	C₂⁵×C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
117	C₂²×#121	C₂×C₅₃	1	56	yes	direct product factor
118	C₂³×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
119	C₂³×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
120	C₂×C₅₃	—	2	16	no	—
121	C₂×C₅₃	—	4	16	no	—
122	C₂×C₅₃	C₂²×#121	1	28	yes	direct product factor
123	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
124	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
125	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
126	C₂⁶	C₂²×C₅₃	1	2	yes	—
127	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
128	C₂³×D₄	—	2	32	no	—
129	C₂³×D₄	—	2	16	no	—
130	C₂³×D₄	C₂²×C₅₃	1	56	yes	complemented
131	C₂³×D₄	C₂²×C₅₃	1	28	yes	complemented
132	C₂⁴×C₄	—	2	16	no	—
133	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
134	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
135	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
136	C₂⁴×C₄	C₂²×C₅₃	1	28	yes	complemented
137	C₂×#121	C₂²×C₅₃	1	448	yes	direct product factor
138	C₂²×C₄⋊C₄	C₂²×C₅₃	1	28	yes	complemented
139	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	56	yes	complemented
140	C₅₃	C₂³×#121	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁵	—	2	8	no	—
142	C₂⁵	—	2	8	no	—
143	C₂⁵	—	2	16	no	—
144	C₂⁵	—	2	16	no	—
145	C₂⁵	—	2	16	no	—
146	C₂⁵	D₄×C₅₃	1	16	yes	complemented
147	C₂⁵	C₂³×C₅₃	1	28	yes	—
148	C₂⁵	C₂³×C₅₃	1	14	yes	—
149	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
150	C₂²×D₄	—	2	448	no	—
151	C₂²×D₄	—	2	224	no	—
152	C₂²×D₄	C₂³×C₅₃	1	224	yes	complemented
153	C₂²×D₄	C₂³×C₅₃	1	112	yes	complemented
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	16	no	—
156	C₂³×C₄	—	2	224	no	—
157	C₂³×C₄	D₄×C₅₃	1	16	yes	complemented
158	C₂³×C₄	C₂³×C₅₃	1	14	yes	—
159	C₂³×C₄	C₂³×C₅₃	1	28	yes	—
160	C₂³×C₄	C₂³×C₅₃	1	14	yes	—
161	C₂³×C₄	C₂³×C₅₃	1	112	yes	complemented
162	#121	C₂³×C₅₃	1	512	yes	direct product factor
163	C₂×C₄⋊C₄	C₂³×C₅₃	1	112	yes	complemented
164	C₂×C₂²⋊C₄	C₂³×C₅₃	1	224	yes	complemented
165	C₂⁴	—	2	128	no	—
166	C₂⁴	—	2	56	no	—
167	C₂⁴	—	2	56	no	—
168	C₂⁴	—	2	112	no	—
169	C₂⁴	—	2	112	no	—
170	C₂⁴	—	2	112	no	—
171	C₂⁴	—	4	128	no	—
172	C₂⁴	C₂×D₄×C₅₃	1	8	yes	—
173	C₂⁴	C₂×D₄×C₅₃	1	8	yes	—
174	C₂⁴	C₂×D₄×C₅₃	1	112	yes	complemented
175	C₂⁴	#41×C₅₃	1	8	yes	—
176	C₂⁴	C₂⁴×C₅₃	1	56	yes	—
177	C₂⁴	C₂⁴×C₅₃	1	28	yes	—
178	C₂⁴	C₂⁴×C₅₃	1	7	yes	—
179	C₂×D₄	—	2	896	no	—
180	C₂×D₄	—	2	448	no	—
181	C₂×D₄	C₂⁴×C₅₃	1	128	yes	complemented
182	C₂×D₄	C₂⁴×C₅₃	1	64	yes	complemented
183	C₂²×C₄	—	2	56	no	—
184	C₂²×C₄	—	2	112	no	—
185	C₂²×C₄	—	2	448	no	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	112	yes	complemented
187	C₂²×C₄	C₂⁴×C₅₃	1	28	yes	—
188	C₂²×C₄	C₂⁴×C₅₃	1	56	yes	—
189	C₂²×C₄	C₂⁴×C₅₃	1	28	yes	—
190	C₂²×C₄	C₂⁴×C₅₃	1	64	yes	complemented
191	C₄⋊C₄	C₂⁴×C₅₃	1	64	yes	complemented
192	C₂²⋊C₄	C₂⁴×C₅₃	1	128	yes	complemented
193	C₂³	—	2	448	no	—
194	C₂³	—	2	56	no	—
195	C₂³	—	2	56	no	—
196	C₂³	—	2	112	no	—
197	C₂³	—	2	112	no	—
198	C₂³	—	2	112	no	—
199	C₂³	—	4	448	no	—
200	C₂³	#121×C₅₃	1	64	yes	direct product factor
201	C₂³	C₂²×D₄×C₅₃	1	28	yes	—
202	C₂³	C₂²×D₄×C₅₃	1	28	yes	—
203	C₂³	C₂²×D₄×C₅₃	1	112	yes	complemented
204	C₂³	C₂×#41×C₅₃	1	28	yes	—
205	C₂³	C₂⁵×C₅₃	1	16	yes	—
206	C₂³	C₂⁵×C₅₃	1	8	yes	—
207	C₂³	C₂⁵×C₅₃	1	7	yes	—
208	D₄	—	2	256	no	—
209	D₄	—	2	128	no	—
210	C₂×C₄	—	2	56	no	—
211	C₂×C₄	—	2	112	no	—
212	C₂×C₄	—	2	128	no	—
213	C₂×C₄	C₂²×D₄×C₅₃	1	112	yes	complemented
214	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
215	C₂×C₄	C₂⁵×C₅₃	1	16	yes	—
216	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	224	no	—
220	C₂²	—	2	16	no	—
221	C₂²	—	2	16	no	—
222	C₂²	—	2	16	no	—
223	C₂²	—	4	224	no	—
224	C₂²	C₂×#121×C₅₃	1	112	yes	direct product factor
225	C₂²	C₂³×D₄×C₅₃	1	14	yes	—
226	C₂²	C₂³×D₄×C₅₃	1	14	yes	—
227	C₂²	C₂³×D₄×C₅₃	1	16	yes	complemented
228	C₂²	C₂²×#41×C₅₃	1	14	yes	—
229	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	—	2	16	no	—
232	C₄	C₂³×D₄×C₅₃	1	16	yes	complemented
233	C₂	—	2	16	no	—
234	C₂	—	4	16	no	—
235	C₂	C₂²×#121×C₅₃	1	28	yes	direct product factor
236	C₂	C₂⁴×D₄×C₅₃	1	1	yes	characteristic
237	C₂	C₂⁴×D₄×C₅₃	1	1	yes	characteristic
238	C₂	C₂³×#41×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₅₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₄×C₅₃	C₂	semidirect
2. C₂⁴×D₄×C₅₃	C₂	semidirect
3. C₂⁵×C₄×C₅₃	C₂	semidirect
4. C₂²×#121×C₅₃	C₂	direct (N × H)
5. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
6. C₂³×C₂²⋊C₄×C₅₃	C₂	semidirect
11. C₂³×D₄×C₅₃	C₂²	semidirect
12. C₂³×D₄×C₅₃	C₂²	semidirect
17. C₂⁴×C₄×C₅₃	C₂²	semidirect
18. C₂×#121×C₅₃	C₂²	direct (N × H)
19. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
20. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
26. C₂⁵×C₅₃	D₄	semidirect
32. C₂²×D₄×C₅₃	C₂³	semidirect
33. C₂²×D₄×C₅₃	C₂³	semidirect
37. C₂³×C₄×C₅₃	D₄	semidirect
41. C₂³×C₄×C₅₃	C₂³	semidirect
42. #121×C₅₃	C₂³	direct (N × H)
43. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
44. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
54. C₂⁴×C₅₃	C₂×D₄	semidirect
61. C₂×D₄×C₅₃	C₂⁴	semidirect
62. C₂×D₄×C₅₃	C₂⁴	semidirect
66. C₂²×C₄×C₅₃	C₂×D₄	semidirect
70. C₂²×C₄×C₅₃	C₂⁴	semidirect
71. C₄⋊C₄×C₅₃	C₂⁴	semidirect
72. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
80. C₂³×C₅₃	#121	direct (N × H)
83. C₂³×C₅₃	C₂²×D₄	semidirect
93. C₂×C₄×C₅₃	C₂²×D₄	semidirect
97. C₂³×#121	C₅₃	direct (N × H)
105. C₂²×C₅₃	C₂×#121	direct (N × H)
108. C₂²×C₅₃	C₂³×D₄	semidirect
113. C₄×C₅₃	C₂³×D₄	semidirect
114. C₂⁴×D₄	C₂×C₅₃	semidirect
115. C₂⁴×D₄	C₂×C₅₃	semidirect
116. C₂⁵×C₄	C₂×C₅₃	semidirect
117. C₂²×#121	C₂×C₅₃	direct (N × H)
118. C₂³×C₄⋊C₄	C₂×C₅₃	semidirect
119. C₂³×C₂²⋊C₄	C₂×C₅₃	semidirect
122. C₂×C₅₃	C₂²×#121	direct (N × H)
130. C₂³×D₄	C₂²×C₅₃	semidirect
131. C₂³×D₄	C₂²×C₅₃	semidirect
136. C₂⁴×C₄	C₂²×C₅₃	semidirect
137. C₂×#121	C₂²×C₅₃	direct (N × H)
138. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
139. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
140. C₅₃	C₂³×#121	direct (N × H)
146. C₂⁵	D₄×C₅₃	semidirect
152. C₂²×D₄	C₂³×C₅₃	semidirect
153. C₂²×D₄	C₂³×C₅₃	semidirect
157. C₂³×C₄	D₄×C₅₃	semidirect
161. C₂³×C₄	C₂³×C₅₃	semidirect
162. #121	C₂³×C₅₃	direct (N × H)
163. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
164. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
174. C₂⁴	C₂×D₄×C₅₃	semidirect
181. C₂×D₄	C₂⁴×C₅₃	semidirect
182. C₂×D₄	C₂⁴×C₅₃	semidirect
186. C₂²×C₄	C₂×D₄×C₅₃	semidirect
190. C₂²×C₄	C₂⁴×C₅₃	semidirect
191. C₄⋊C₄	C₂⁴×C₅₃	semidirect
192. C₂²⋊C₄	C₂⁴×C₅₃	semidirect
200. C₂³	#121×C₅₃	direct (N × H)
203. C₂³	C₂²×D₄×C₅₃	semidirect
213. C₂×C₄	C₂²×D₄×C₅₃	semidirect
224. C₂²	C₂×#121×C₅₃	direct (N × H)
227. C₂²	C₂³×D₄×C₅₃	semidirect
232. C₄	C₂³×D₄×C₅₃	semidirect
235. C₂	C₂²×#121×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₅₃	2
C₂⁵×C₄×C₅₃	2
C₂²×#121×C₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×C₂²⋊C₄×C₅₃	2
C₂³×#121	53

Subgroups — C23×#121×C53 #27270168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#121×C₅₃ #27270168