Subgroups — C₅₃×#12868 #27220380

Order 13568.

426 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1380	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1380	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#117×C₅₃	—	2	1	no	—
8	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#427	—	2	2	no	—
10	C₅₃×#427	C₄	1	4	yes	complemented
11	C₂×#107×C₅₃	C₂²	1	2	yes	—
12	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#106×C₅₃	—	2	1	no	—
15	C₅₃×#380	—	2	1	no	—
16	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
17	C₅₃×#340	—	2	1	no	—
18	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	4	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₅₃	D₄	1	2	yes	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	#117×C₅₃	—	2	4	no	—
25	#117×C₅₃	—	4	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
30	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
31	C₂×C₄²×C₅₃	—	4	1	no	—
32	C₂×C₁₆×C₅₃	—	2	2	no	—
33	C₂×C₁₆×C₅₃	—	2	4	no	—
34	#107×C₅₃	—	2	1	no	—
35	#107×C₅₃	—	2	1	no	—
36	#107×C₅₃	—	2	4	no	—
37	#107×C₅₃	C₂×C₄	1	2	yes	—
38	#107×C₅₃	C₂×C₄	1	2	yes	—
39	#106×C₅₃	—	2	4	no	—
40	#95×C₅₃	—	4	1	no	—
41	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	4	1	no	—
45	C₂²×C₄×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	#41	1	4	yes	—
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	4	1	no	—
61	C₄⋊C₄×C₅₃	—	4	1	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₂²⋊C₄×C₅₃	—	4	4	no	—
65	C₂²⋊C₄×C₅₃	—	8	2	no	—
66	C₄²×C₅₃	—	4	4	no	—
67	C₁₆×C₅₃	—	4	2	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	—	2	1	no	—
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	#102	1	2	yes	—
73	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	8	2	no	—
81	C₂×C₄×C₅₃	—	8	1	no	—
82	C₂×C₄×C₅₃	—	8	1	no	—
83	C₂×C₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	—	8	2	no	—
85	C₂×C₄×C₅₃	—	8	2	no	—
86	C₂×C₄×C₅₃	#102	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#123	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#137	1	2	yes	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	#117	1	2	yes	—
93	C₈×C₅₃	#123	1	2	yes	—
94	#12868	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	4	2	no	—
99	C₂²×C₅₃	—	4	2	no	—
100	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
101	C₂²×C₅₃	#480	1	2	yes	—
102	C₂²×C₅₃	#508	1	2	yes	—
103	C₂²×C₅₃	#509	1	1	yes	characteristic
104	C₂²×C₅₃	#510	1	1	yes	characteristic
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	8	2	no	—
107	C₄×C₅₃	—	8	2	no	—
108	C₄×C₅₃	—	16	2	no	—
109	C₄×C₅₃	#381	1	1	yes	characteristic
110	C₄×C₅₃	#419	1	1	yes	characteristic
111	C₄×C₅₃	#483	1	2	yes	—
112	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
113	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
114	#1808	C₂×C₅₃	1	1	yes	characteristic
115	#1380	C₂×C₅₃	1	2	yes	complemented
116	#1380	C₂×C₅₃	1	2	yes	complemented
117	C₂×C₅₃	—	4	2	no	—
118	C₂×C₅₃	#1823	1	1	yes	characteristic
119	C₂×C₅₃	#2138	1	1	yes	characteristic
120	C₂×C₅₃	#2139	1	1	yes	characteristic
121	C₂×C₅₃	#2234	1	2	yes	—
122	C₂×C₅₃	#2237	1	2	yes	—
123	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
124	C₂×#117	—	2	1	no	—
125	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
126	#427	—	2	2	no	—
127	#427	C₄×C₅₃	1	4	yes	complemented
128	C₂×#107	C₂²×C₅₃	1	2	yes	—
129	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
130	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
131	C₂×#106	—	2	1	no	—
132	#380	—	2	1	no	—
133	C₂²⋊C₁₆	—	2	2	no	—
134	#340	—	2	1	no	—
135	C₅₃	#12868	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
137	C₂²×C₈	—	2	4	no	—
138	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
139	C₂²×C₈	D₄×C₅₃	1	2	yes	—
140	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
142	#117	—	2	4	no	—
143	#117	—	4	2	no	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	4	1	no	—
148	C₂×C₂²⋊C₄	—	4	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	4	no	—
152	#107	—	2	1	no	—
153	#107	—	2	1	no	—
154	#107	—	2	4	no	—
155	#107	C₂×C₄×C₅₃	1	2	yes	—
156	#107	C₂×C₄×C₅₃	1	2	yes	—
157	#106	—	2	4	no	—
158	#95	—	4	1	no	—
159	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
166	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
167	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
168	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
173	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
175	C₂×C₈	#41×C₅₃	1	4	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	4	no	—
183	C₂²⋊C₄	—	8	2	no	—
184	C₄²	—	4	4	no	—
185	C₁₆	—	4	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	2	no	—
190	C₂³	#102×C₅₃	1	2	yes	—
191	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	1	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	#102×C₅₃	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
206	C₂×C₄	#123×C₅₃	1	2	yes	—
207	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
208	C₂×C₄	#137×C₅₃	1	2	yes	—
209	C₈	—	2	2	no	—
210	C₈	#117×C₅₃	1	2	yes	—
211	C₈	#123×C₅₃	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
218	C₂²	C₅₃×#480	1	2	yes	—
219	C₂²	C₅₃×#508	1	2	yes	—
220	C₂²	C₅₃×#509	1	1	yes	characteristic
221	C₂²	C₅₃×#510	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	8	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	16	2	no	—
226	C₄	C₅₃×#381	1	1	yes	characteristic
227	C₄	C₅₃×#419	1	1	yes	characteristic
228	C₄	C₅₃×#483	1	2	yes	—
229	C₂	—	4	2	no	—
230	C₂	C₅₃×#1823	1	1	yes	characteristic
231	C₂	C₅₃×#2138	1	1	yes	characteristic
232	C₂	C₅₃×#2139	1	1	yes	characteristic
233	C₂	C₅₃×#2234	1	2	yes	—
234	C₂	C₅₃×#2237	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1380	C₂	semidirect
5. C₅₃×#1380	C₂	semidirect
10. C₅₃×#427	C₄	semidirect
94. #12868	C₅₃	direct (N × H)
115. #1380	C₂×C₅₃	semidirect
116. #1380	C₂×C₅₃	semidirect
127. #427	C₄×C₅₃	semidirect
135. C₅₃	#12868	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×C₅₃×#427	2
C₂×C₅₃×C₂²⋊C₁₆	2
C₅₃×#1808	2
C₅₃×#1380	2
#12868	53

Subgroups — C53×#12868 #27220380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12868 #27220380