Subgroups — C₅₃×#11968 #27219480

Order 13568.

296 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1363	C₂	1	4	yes	maximal, complemented
5	C₅₃×#533	—	2	1	no	—
6	C₅₃×#500	C₄	1	1	yes	characteristic
7	C₅₃×#500	C₄	1	1	yes	characteristic
8	C₅₃×#499	—	2	1	no	—
9	C₅₃×#499	C₂²	1	2	yes	—
10	C₅₃×#498	—	2	1	no	—
11	C₅₃×#427	—	2	1	no	—
12	C₅₃×#427	C₄	1	2	yes	—
13	C₅₃×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₁₆×C₅₃	—	2	1	no	—
18	C₅₃×#334	—	2	1	no	—
19	C₅₃×#334	C₂²	1	2	yes	—
20	C₂×M₁₆×C₅₃	—	4	1	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₅₃	—	2	1	no	—
24	C₄⋊Q₈×C₅₃	—	2	1	no	—
25	#125×C₅₃	—	2	2	no	—
26	C₂²⋊Q₈×C₅₃	—	4	1	no	—
27	C₂²⋊Q₈×C₅₃	—	4	1	no	—
28	C₄×Q₈×C₅₃	—	4	1	no	—
29	C₄×Q₈×C₅₃	—	4	1	no	—
30	#117×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	—	2	2	no	—
33	#107×C₅₃	—	2	1	no	—
34	#107×C₅₃	—	2	1	no	—
35	#107×C₅₃	C₂×C₄	1	2	yes	—
36	#107×C₅₃	C₂×C₄	1	2	yes	—
37	#106×C₅₃	—	2	2	no	—
38	#106×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	4	1	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₅₃	—	4	1	no	—
46	C₂×Q₈×C₅₃	—	4	1	no	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
50	M₁₆×C₅₃	—	8	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	2	no	—
65	C₄²×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	D₈	1	1	yes	characteristic
68	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₁₆×C₅₃	—	4	2	no	—
71	C₂³×C₅₃	#102	1	1	yes	characteristic
72	Q₈×C₅₃	—	8	2	no	—
73	Q₈×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	2	1	no	—
88	C₈×C₅₃	—	8	2	no	—
89	C₈×C₅₃	#101	1	2	yes	—
90	C₈×C₅₃	#137	1	2	yes	—
91	#11968	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	—	8	2	no	—
100	C₄×C₅₃	—	8	2	no	—
101	C₄×C₅₃	#359	1	2	yes	—
102	C₄×C₅₃	#360	1	2	yes	—
103	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
104	C₄×C₅₃	#419	1	1	yes	characteristic
105	#3160	C₂×C₅₃	1	1	yes	characteristic
106	#2162	C₂×C₅₃	1	1	yes	characteristic
107	#1566	C₂×C₅₃	1	1	yes	characteristic
108	#1363	C₂×C₅₃	1	4	yes	complemented
109	C₂×C₅₃	—	4	1	no	—
110	C₂×C₅₃	#1546	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#360	1	1	yes	characteristic
113	#533	—	2	1	no	—
114	#500	C₄×C₅₃	1	1	yes	characteristic
115	#500	C₄×C₅₃	1	1	yes	characteristic
116	#499	—	2	1	no	—
117	#499	C₂²×C₅₃	1	2	yes	—
118	#498	—	2	1	no	—
119	#427	—	2	1	no	—
120	#427	C₄×C₅₃	1	2	yes	—
121	#423	—	2	1	no	—
122	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
123	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
124	C₂²⋊C₁₆	—	2	1	no	—
125	C₄×C₁₆	—	2	1	no	—
126	#334	—	2	1	no	—
127	#334	C₂²×C₅₃	1	2	yes	—
128	C₅₃	#11968	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×M₁₆	—	4	1	no	—
130	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
131	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	1	no	—
133	C₄⋊Q₈	—	2	1	no	—
134	#125	—	2	2	no	—
135	C₂²⋊Q₈	—	4	1	no	—
136	C₂²⋊Q₈	—	4	1	no	—
137	C₄×Q₈	—	4	1	no	—
138	C₄×Q₈	—	4	1	no	—
139	#117	—	2	2	no	—
140	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
141	C₂×C₁₆	—	2	2	no	—
142	#107	—	2	1	no	—
143	#107	—	2	1	no	—
144	#107	C₂×C₄×C₅₃	1	2	yes	—
145	#107	C₂×C₄×C₅₃	1	2	yes	—
146	#106	—	2	2	no	—
147	#106	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
151	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
152	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
154	C₂×Q₈	—	4	1	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
158	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
159	M₁₆	—	8	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₄²	—	4	2	no	—
175	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
176	C₄²	D₈×C₅₃	1	1	yes	characteristic
177	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
178	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
179	C₁₆	—	4	2	no	—
180	C₂³	#102×C₅₃	1	1	yes	characteristic
181	Q₈	—	8	2	no	—
182	Q₈	—	8	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
190	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
191	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
192	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
193	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
194	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	8	2	no	—
198	C₈	#101×C₅₃	1	2	yes	—
199	C₈	#137×C₅₃	1	2	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	8	2	no	—
208	C₄	—	8	2	no	—
209	C₄	C₅₃×#359	1	2	yes	—
210	C₄	C₅₃×#360	1	2	yes	—
211	C₄	C₂×#101×C₅₃	1	1	yes	characteristic
212	C₄	C₅₃×#419	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	C₅₃×#1546	1	1	yes	characteristic
215	C₂	C₂×C₅₃×#359	1	1	yes	characteristic
216	C₂	C₂×C₅₃×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1363	C₂	semidirect
91. #11968	C₅₃	direct (N × H)
108. #1363	C₂×C₅₃	semidirect
128. C₅₃	#11968	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3160	2
C₅₃×#2162	2
C₅₃×#1566	2
C₅₃×#1363	2
#11968	53

Subgroups — C53×#11968 #27219480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11968 #27219480