Subgroups — C₅₃×#22257 #27229769

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3162	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2960	C₂	1	2	yes	maximal
3	C₅₃×#1995	C₂	1	2	yes	maximal
4	C₅₃×#1922	C₂	1	8	yes	maximal
5	C₅₃×#1838	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂²	1	2	yes	—
8	C₅₃×#501	C₄	1	4	yes	complemented
9	C₅₃×#501	C₄	1	4	yes	complemented
10	C₅₃×#446	—	2	2	no	—
11	C₅₃×#446	—	2	2	no	—
12	C₈×Q₈×C₅₃	—	2	2	no	—
13	C₈×Q₈×C₅₃	—	2	2	no	—
14	C₅₃×#427	C₂²	1	4	yes	—
15	C₅₃×#424	C₂²	1	4	yes	—
16	C₅₃×#423	C₂²	1	2	yes	—
17	C₅₃×#405	C₂²	1	4	yes	—
18	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#376	C₂²	1	1	yes	characteristic
21	C₅₃×#341	C₂²	1	4	yes	—
22	C₅₃×#339	C₂²	1	8	yes	—
23	C₅₃×#336	C₂²	1	4	yes	—
24	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	#125×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²⋊Q₈×C₅₃	—	2	2	no	—
31	C₂²⋊Q₈×C₅₃	—	2	2	no	—
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	#117×C₅₃	C₂³	1	4	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#107×C₅₃	C₂×C₄	1	8	yes	—
38	#106×C₅₃	C₂×C₄	1	4	yes	—
39	#106×C₅₃	C₂×C₄	1	4	yes	—
40	C₄⋊C₈×C₅₃	—	2	4	no	—
41	C₄⋊C₈×C₅₃	—	2	4	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	#97×C₅₃	—	2	4	no	—
45	#97×C₅₃	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	#95×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	#41	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	M₁₆×C₅₃	—	2	8	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
68	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂²⋊C₄×C₅₃	—	2	4	no	—
70	C₄²×C₅₃	—	2	4	no	—
71	C₄²×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
76	Q₈×C₅₃	—	4	4	no	—
77	Q₈×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
88	C₈×C₅₃	—	2	4	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	—	4	4	no	—
92	#22257	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
95	C₂²×C₅₃	#541	1	2	yes	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	—	4	4	no	—
99	C₄×C₅₃	—	4	4	no	—
100	C₄×C₅₃	—	8	2	no	—
101	C₄×C₅₃	#519	1	2	yes	—
102	C₄×C₅₃	#536	1	2	yes	—
103	#3162	C₂×C₅₃	1	1	yes	characteristic
104	#2960	C₂×C₅₃	1	2	yes	—
105	#1995	C₂×C₅₃	1	2	yes	—
106	#1922	C₂×C₅₃	1	8	yes	—
107	#1838	C₂×C₅₃	1	1	yes	characteristic
108	#1746	C₂×C₅₃	1	1	yes	characteristic
109	C₂×C₅₃	—	2	2	no	—
110	C₂×C₅₃	#2327	1	1	yes	characteristic
111	C₂×C₅₃	#2955	1	2	yes	—
112	#533	C₂²×C₅₃	1	2	yes	—
113	#501	C₄×C₅₃	1	4	yes	complemented
114	#501	C₄×C₅₃	1	4	yes	complemented
115	#446	—	2	2	no	—
116	#446	—	2	2	no	—
117	C₈×Q₈	—	2	2	no	—
118	C₈×Q₈	—	2	2	no	—
119	#427	C₂²×C₅₃	1	4	yes	—
120	#424	C₂²×C₅₃	1	4	yes	—
121	#423	C₂²×C₅₃	1	2	yes	—
122	#405	C₂²×C₅₃	1	4	yes	—
123	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
125	#376	C₂²×C₅₃	1	1	yes	characteristic
126	#341	C₂²×C₅₃	1	4	yes	—
127	#339	C₂²×C₅₃	1	8	yes	—
128	#336	C₂²×C₅₃	1	4	yes	—
129	C₅₃	#22257	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
131	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
132	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
133	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
134	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
135	#125	C₂×C₄×C₅₃	1	4	yes	—
136	C₂²⋊Q₈	—	2	2	no	—
137	C₂²⋊Q₈	—	2	2	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	#117	C₂³×C₅₃	1	4	yes	—
141	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
142	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
143	#107	C₂×C₄×C₅₃	1	8	yes	—
144	#106	C₂×C₄×C₅₃	1	4	yes	—
145	#106	C₂×C₄×C₅₃	1	4	yes	—
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	#97	—	2	4	no	—
151	#97	C₂×C₄×C₅₃	1	4	yes	—
152	C₄×C₈	—	2	4	no	—
153	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
154	C₄×C₈	D₄×C₅₃	1	2	yes	—
155	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
156	#95	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	#41×C₅₃	1	2	yes	—
161	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
162	M₁₆	—	2	8	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
167	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
168	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
174	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	4	no	—
178	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
179	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
180	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
181	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
182	Q₈	—	4	4	no	—
183	Q₈	—	4	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
190	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
191	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
193	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	4	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
200	C₂²	C₅₃×#541	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	4	no	—
204	C₄	—	4	4	no	—
205	C₄	—	8	2	no	—
206	C₄	C₅₃×#519	1	2	yes	—
207	C₄	C₅₃×#536	1	2	yes	—
208	C₂	—	2	2	no	—
209	C₂	C₅₃×#2327	1	1	yes	characteristic
210	C₂	C₅₃×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₃×#501	C₄	semidirect
9. C₅₃×#501	C₄	semidirect
92. #22257	C₅₃	direct (N × H)
113. #501	C₄×C₅₃	semidirect
114. #501	C₄×C₅₃	semidirect
129. C₅₃	#22257	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×#3162	2
C₅₃×#2960	2
C₅₃×#1995	2
C₅₃×#1922	2
C₅₃×#1838	2
C₅₃×#1746	2
#22257	53

Subgroups — C53×#22257 #27229769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22257 #27229769