Subgroups — C₅₃×#19608 #27227120

Order 13568.

798 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3167	C₂	1	4	yes	maximal, complemented
2	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1716	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1715	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1714	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	—	2	2	no	—
9	C₅₃×#533	C₂²	1	2	yes	—
10	C₅₃×#532	C₂²	1	2	yes	—
11	C₅₃×#500	C₂²	1	2	yes	—
12	C₅₃×#499	—	2	2	no	—
13	C₅₃×#498	C₂²	1	4	yes	—
14	C₅₃×#498	C₂²	1	2	yes	—
15	C₅₃×#495	—	2	2	no	—
16	C₅₃×#494	C₂²	1	1	yes	characteristic
17	C₅₃×#493	C₂²	1	1	yes	characteristic
18	C₅₃×#492	C₂²	1	2	yes	—
19	C₅₃×#492	C₂²	1	4	yes	—
20	C₅₃×#487	C₂²	1	4	yes	—
21	C₅₃×#486	C₂²	1	4	yes	—
22	C₅₃×#465	—	2	8	no	—
23	C₄×SD₁₆×C₅₃	—	2	8	no	—
24	C₅₃×#427	—	2	2	no	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#322	C₂²	1	2	yes	—
28	C₅₃×#322	C₂²	1	2	yes	—
29	C₂×#41×C₅₃	—	2	2	no	—
30	#135×C₅₃	—	4	8	no	—
31	C₂×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×SD₁₆×C₅₃	—	2	4	no	—
33	C₂×SD₁₆×C₅₃	—	2	8	no	—
34	C₂×D₈×C₅₃	—	2	2	no	—
35	C₂²×C₈×C₅₃	D₄	1	4	yes	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
39	#127×C₅₃	C₂³	1	2	yes	—
40	#125×C₅₃	—	2	2	no	—
41	#124×C₅₃	—	2	2	no	—
42	C₂²⋊Q₈×C₅₃	—	2	4	no	—
43	#121×C₅₃	—	2	4	no	—
44	C₄×Q₈×C₅₃	—	2	4	no	—
45	C₄×D₄×C₅₃	—	2	4	no	—
46	#117×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#107×C₅₃	—	2	4	no	—
49	#106×C₅₃	—	2	4	no	—
50	#106×C₅₃	—	2	8	no	—
51	#103×C₅₃	—	2	8	no	—
52	#102×C₅₃	—	2	8	no	—
53	C₄×C₈×C₅₃	D₄	1	2	yes	—
54	C₄×C₈×C₅₃	D₄	1	4	yes	—
55	C₄×C₈×C₅₃	C₂³	1	4	yes	—
56	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	#41×C₅₃	—	4	8	no	—
59	C₂×Q₈×C₅₃	—	2	4	no	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	2	4	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₅₃	—	4	4	no	—
68	SD₁₆×C₅₃	—	4	16	no	—
69	SD₁₆×C₅₃	—	4	8	no	—
70	D₈×C₅₃	—	4	4	no	—
71	C₂×C₈×C₅₃	—	4	4	no	—
72	C₂×C₈×C₅₃	SD₁₆	1	8	yes	complemented
73	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₂²⋊C₄×C₅₃	—	4	2	no	—
81	C₂²⋊C₄×C₅₃	—	4	4	no	—
82	C₄²×C₅₃	—	4	2	no	—
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	2	yes	—
85	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂³×C₅₃	—	4	2	no	—
87	C₂³×C₅₃	#127	1	1	yes	characteristic
88	Q₈×C₅₃	—	4	4	no	—
89	Q₈×C₅₃	—	4	8	no	—
90	D₄×C₅₃	—	4	4	no	—
91	D₄×C₅₃	—	4	8	no	—
92	D₄×C₅₃	—	8	4	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	8	4	no	—
97	C₂×C₄×C₅₃	#127	1	2	yes	—
98	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
102	C₈×C₅₃	—	2	8	no	—
103	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
104	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
105	#19608	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	8	2	no	—
108	C₂²×C₅₃	—	8	4	no	—
109	C₂²×C₅₃	#492	1	2	yes	—
110	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	8	2	no	—
113	C₄×C₅₃	—	8	4	no	—
114	C₄×C₅₃	#492	1	2	yes	—
115	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
116	C₄×C₅₃	#575	1	4	yes	—
117	#3167	C₂×C₅₃	1	4	yes	complemented
118	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
119	#3147	C₂×C₅₃	1	1	yes	characteristic
120	#1716	C₂×C₅₃	1	2	yes	complemented
121	#1715	C₂×C₅₃	1	2	yes	complemented
122	#1714	C₂×C₅₃	1	4	yes	complemented
123	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
124	C₂×C₅₃	—	2	2	no	—
125	C₂×C₅₃	—	16	2	no	—
126	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
127	C₂×C₅₃	#3150	1	2	yes	—
128	C₂×#135	—	2	2	no	—
129	#533	C₂²×C₅₃	1	2	yes	—
130	#532	C₂²×C₅₃	1	2	yes	—
131	#500	C₂²×C₅₃	1	2	yes	—
132	#499	—	2	2	no	—
133	#498	C₂²×C₅₃	1	2	yes	—
134	#498	C₂²×C₅₃	1	4	yes	—
135	#495	—	2	2	no	—
136	#494	C₂²×C₅₃	1	1	yes	characteristic
137	#493	C₂²×C₅₃	1	1	yes	characteristic
138	#492	C₂²×C₅₃	1	2	yes	—
139	#492	C₂²×C₅₃	1	4	yes	—
140	#487	C₂²×C₅₃	1	4	yes	—
141	#486	C₂²×C₅₃	1	4	yes	—
142	#465	—	2	8	no	—
143	C₄×SD₁₆	—	2	8	no	—
144	#427	—	2	2	no	—
145	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
147	#322	C₂²×C₅₃	1	2	yes	—
148	#322	C₂²×C₅₃	1	2	yes	—
149	C₅₃	#19608	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×#41	—	2	2	no	—
151	#135	—	4	8	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×SD₁₆	—	2	4	no	—
154	C₂×SD₁₆	—	2	8	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂²×C₈	D₄×C₅₃	1	4	yes	—
157	C₂²×C₈	D₄×C₅₃	1	2	yes	—
158	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
159	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
160	#127	C₂³×C₅₃	1	2	yes	—
161	#125	—	2	2	no	—
162	#124	—	2	2	no	—
163	C₂²⋊Q₈	—	2	4	no	—
164	#121	—	2	4	no	—
165	C₄×Q₈	—	2	4	no	—
166	C₄×D₄	—	2	4	no	—
167	#117	—	2	2	no	—
168	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
169	#107	—	2	4	no	—
170	#106	—	2	4	no	—
171	#106	—	2	8	no	—
172	#103	—	2	8	no	—
173	#102	—	2	8	no	—
174	C₄×C₈	D₄×C₅₃	1	4	yes	—
175	C₄×C₈	D₄×C₅₃	1	2	yes	—
176	C₄×C₈	C₂³×C₅₃	1	4	yes	—
177	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
179	#41	—	4	8	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
188	Q₁₆	—	4	4	no	—
189	SD₁₆	—	4	16	no	—
190	SD₁₆	—	4	8	no	—
191	D₈	—	4	4	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	SD₁₆×C₅₃	1	8	yes	complemented
194	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
195	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	4	no	—
203	C₄²	—	4	2	no	—
204	C₄²	C₂×D₄×C₅₃	1	2	yes	—
205	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
206	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
207	C₂³	—	4	2	no	—
208	C₂³	#127×C₅₃	1	1	yes	characteristic
209	Q₈	—	4	4	no	—
210	Q₈	—	4	8	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	8	no	—
213	D₄	—	8	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	#127×C₅₃	1	2	yes	—
219	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
220	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
221	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
223	C₈	—	2	8	no	—
224	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
225	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	8	2	no	—
228	C₂²	—	8	4	no	—
229	C₂²	C₅₃×#492	1	2	yes	—
230	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	8	2	no	—
233	C₄	—	8	4	no	—
234	C₄	C₅₃×#492	1	2	yes	—
235	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
236	C₄	C₅₃×#575	1	4	yes	—
237	C₂	—	2	2	no	—
238	C₂	—	16	2	no	—
239	C₂	C₂×C₅₃×#492	1	1	yes	characteristic
240	C₂	C₅₃×#3150	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×#127
prime radical	C₅₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3167	C₂	semidirect
2. C₅₃×#3160	C₂	semidirect
4. C₅₃×#1716	C₂	semidirect
5. C₅₃×#1715	C₂	semidirect
6. C₅₃×#1714	C₂	semidirect
7. C₂×C₅₃×#322	C₂	semidirect
72. C₂×C₈×C₅₃	SD₁₆	semidirect
105. #19608	C₅₃	direct (N × H)
117. #3167	C₂×C₅₃	semidirect
118. #3160	C₂×C₅₃	semidirect
120. #1716	C₂×C₅₃	semidirect
121. #1715	C₂×C₅₃	semidirect
122. #1714	C₂×C₅₃	semidirect
123. C₂×#322	C₂×C₅₃	semidirect
149. C₅₃	#19608	direct (N × H)
193. C₂×C₈	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3167	2
C₅₃×#3160	2
C₅₃×#3147	2
C₅₃×#1716	2
C₅₃×#1715	2
C₅₃×#1714	2
C₂×C₅₃×#322	2
#19608	53

Subgroups — C53×#19608 #27227120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19608 #27227120