Subgroups — C₅₃×#22253 #27229765

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2331	C₂	1	2	yes	maximal
3	C₅₃×#1993	C₂	1	2	yes	maximal
4	C₅₃×#1920	C₂	1	4	yes	maximal
5	C₅₃×#1919	C₂	1	4	yes	maximal
6	C₅₃×#1835	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#498	C₄	1	8	yes	complemented
10	C₂×#106×C₅₃	C₂²	1	4	yes	—
11	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#391	—	2	4	no	—
14	C₅₃×#389	C₂²	1	4	yes	—
15	C₅₃×#387	—	2	4	no	—
16	C₅₃×#382	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
18	C₅₃×#376	C₂²	1	1	yes	characteristic
19	C₅₃×#340	C₂²	1	4	yes	—
20	C₅₃×#340	C₂²	1	4	yes	—
21	C₅₃×#340	C₂²	1	4	yes	—
22	C₅₃×#336	C₂²	1	2	yes	—
23	C₅₃×#336	C₂²	1	2	yes	—
24	C₂²×Q₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₄⋊Q₈×C₅₃	—	2	4	no	—
29	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
34	C₂×C₄²×C₅₃	—	2	4	no	—
35	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#106×C₅₃	C₂×C₄	1	16	yes	—
38	#97×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
40	#95×C₅₃	—	2	4	no	—
41	#95×C₅₃	—	2	2	no	—
42	#95×C₅₃	—	2	2	no	—
43	#95×C₅₃	—	2	4	no	—
44	#95×C₅₃	D₄	1	4	yes	—
45	C₂×Q₈×C₅₃	—	2	8	no	—
46	C₂×Q₈×C₅₃	—	8	4	no	—
47	C₂²×C₄×C₅₃	—	2	4	no	—
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
59	C₄⋊C₄×C₅₃	—	2	8	no	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	—	8	4	no	—
64	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	8	4	no	—
68	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₅₃	#41	1	2	yes	—
70	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
71	Q₈×C₅₃	—	8	8	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	—	4	4	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
86	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
87	C₈×C₅₃	—	2	4	no	—
88	C₈×C₅₃	—	4	1	no	—
89	C₈×C₅₃	—	4	1	no	—
90	#22253	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
92	C₂²×C₅₃	#536	1	2	yes	—
93	C₂²×C₅₃	#577	1	4	yes	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	4	8	no	—
96	C₄×C₅₃	—	4	4	no	—
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	#519	1	2	yes	—
99	C₄×C₅₃	#541	1	1	yes	characteristic
100	C₄×C₅₃	#541	1	1	yes	characteristic
101	C₂×#498	C₂×C₅₃	1	1	yes	characteristic
102	#2331	C₂×C₅₃	1	2	yes	—
103	#1993	C₂×C₅₃	1	2	yes	—
104	#1920	C₂×C₅₃	1	4	yes	—
105	#1919	C₂×C₅₃	1	4	yes	—
106	#1835	C₂×C₅₃	1	1	yes	characteristic
107	#1746	C₂×C₅₃	1	1	yes	characteristic
108	C₂×C₅₃	#2327	1	1	yes	characteristic
109	C₂×C₅₃	#2949	1	2	yes	—
110	C₂×C₅₃	#3244	1	4	yes	—
111	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
112	#498	C₄×C₅₃	1	8	yes	complemented
113	C₂×#106	C₂²×C₅₃	1	4	yes	—
114	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
116	#391	—	2	4	no	—
117	#389	C₂²×C₅₃	1	4	yes	—
118	#387	—	2	4	no	—
119	#382	C₂²×C₅₃	1	2	yes	—
120	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
121	#376	C₂²×C₅₃	1	1	yes	characteristic
122	#340	C₂²×C₅₃	1	4	yes	—
123	#340	C₂²×C₅₃	1	4	yes	—
124	#340	C₂²×C₅₃	1	4	yes	—
125	#336	C₂²×C₅₃	1	2	yes	—
126	#336	C₂²×C₅₃	1	2	yes	—
127	C₅₃	#22253	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×Q₈	—	2	2	no	—
129	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
132	C₄⋊Q₈	—	2	4	no	—
133	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
137	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
138	C₂×C₄²	—	2	4	no	—
139	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
140	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
141	#106	C₂×C₄×C₅₃	1	16	yes	—
142	#97	—	2	2	no	—
143	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
144	#95	—	2	4	no	—
145	#95	—	2	2	no	—
146	#95	—	2	2	no	—
147	#95	—	2	4	no	—
148	#95	D₄×C₅₃	1	4	yes	—
149	C₂×Q₈	—	2	8	no	—
150	C₂×Q₈	—	8	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
154	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
155	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	8	4	no	—
168	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
169	C₄²	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	8	4	no	—
172	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
173	C₄²	#41×C₅₃	1	2	yes	—
174	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
175	Q₈	—	8	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
187	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
189	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
190	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	4	1	no	—
193	C₈	—	4	1	no	—
194	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
195	C₂²	C₅₃×#536	1	2	yes	—
196	C₂²	C₅₃×#577	1	4	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	4	8	no	—
199	C₄	—	4	4	no	—
200	C₄	—	8	4	no	—
201	C₄	C₅₃×#519	1	2	yes	—
202	C₄	C₅₃×#541	1	1	yes	characteristic
203	C₄	C₅₃×#541	1	1	yes	characteristic
204	C₂	C₅₃×#2327	1	1	yes	characteristic
205	C₂	C₅₃×#2949	1	2	yes	—
206	C₂	C₅₃×#3244	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅₃×#498	C₄	semidirect
90. #22253	C₅₃	direct (N × H)
112. #498	C₄×C₅₃	semidirect
127. C₅₃	#22253	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×#498	2
C₅₃×#2331	2
C₅₃×#1993	2
C₅₃×#1920	2
C₅₃×#1919	2
C₅₃×#1835	2
C₅₃×#1746	2
#22253	53

Subgroups — C53×#22253 #27229765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22253 #27229765