Subgroups — C₅₃×#22265 #27229777

Order 13568.

962 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#436	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2397	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1955	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1870	C₂	1	8	yes	maximal, complemented
5	C₅₃×#1832	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×M₁₆×C₅₃	C₂²	1	4	yes	—
8	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
10	C₅₃×#436	—	2	8	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₅₃	C₂²	1	8	yes	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#399	C₄	1	4	yes	—
17	C₅₃×#392	C₄	1	4	yes	—
18	C₅₃×#376	C₂²	1	1	yes	characteristic
19	C₅₃×#336	C₂²	1	4	yes	—
20	C₅₃×#336	C₂²	1	4	yes	—
21	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×D₄×C₅₃	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
24	C₂×M₁₆×C₅₃	—	2	8	no	—
25	C₂×M₁₆×C₅₃	—	4	8	no	—
26	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
29	C₄×D₄×C₅₃	—	2	4	no	—
30	C₄×D₄×C₅₃	—	2	4	no	—
31	C₄×D₄×C₅₃	—	4	8	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	—	2	4	no	—
39	C₂²⋊C₈×C₅₃	—	2	8	no	—
40	C₂²⋊C₈×C₅₃	—	4	8	no	—
41	#97×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	D₄	1	4	yes	complemented
43	#95×C₅₃	C₂×C₄	1	4	yes	—
44	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂×D₄×C₅₃	—	2	4	no	—
46	C₂×D₄×C₅₃	—	4	4	no	—
47	C₂×D₄×C₅₃	—	4	4	no	—
48	C₂×D₄×C₅₃	—	4	8	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	8	no	—
52	C₂²×C₄×C₅₃	—	2	8	no	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₅₃	—	4	16	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	8	no	—
63	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	4	4	no	—
66	C₂²⋊C₄×C₅₃	—	2	8	no	—
67	C₂²⋊C₄×C₅₃	—	4	8	no	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	C₂×D₄	1	2	yes	—
70	C₄²×C₅₃	#41	1	2	yes	—
71	C₂³×C₅₃	—	2	4	no	—
72	C₂³×C₅₃	—	2	8	no	—
73	C₂³×C₅₃	—	2	4	no	—
74	C₂³×C₅₃	—	2	4	no	—
75	C₂³×C₅₃	—	2	8	no	—
76	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	D₄×C₅₃	—	4	8	no	—
78	D₄×C₅₃	—	4	16	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	8	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	8	no	—
87	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
90	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
91	C₈×C₅₃	—	2	4	no	—
92	C₈×C₅₃	—	4	4	no	—
93	C₈×C₅₃	—	4	2	no	—
94	#22265	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₅₃	—	4	4	no	—
96	C₂²×C₅₃	—	4	4	no	—
97	C₂²×C₅₃	—	4	4	no	—
98	C₂²×C₅₃	—	4	8	no	—
99	C₂²×C₅₃	—	4	8	no	—
100	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
101	C₂²×C₅₃	#518	1	2	yes	—
102	C₂²×C₅₃	#568	1	4	yes	—
103	C₄×C₅₃	—	2	4	no	—
104	C₄×C₅₃	—	4	4	no	—
105	C₄×C₅₃	—	4	4	no	—
106	C₄×C₅₃	#535	1	2	yes	—
107	C₄×C₅₃	#540	1	2	yes	—
108	C₂×#436	C₂×C₅₃	1	2	yes	complemented
109	#2397	C₂×C₅₃	1	1	yes	characteristic
110	#1955	C₂×C₅₃	1	2	yes	complemented
111	#1870	C₂×C₅₃	1	8	yes	complemented
112	#1832	C₂×C₅₃	1	1	yes	characteristic, complemented
113	#1746	C₂×C₅₃	1	1	yes	characteristic, complemented
114	C₂×C₅₃	—	8	4	no	—
115	C₂×C₅₃	#2328	1	1	yes	characteristic
116	C₂×C₅₃	#2930	1	2	yes	—
117	C₂×C₅₃	#2978	1	4	yes	—
118	C₂²×M₁₆	C₂²×C₅₃	1	4	yes	—
119	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
120	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
121	#436	—	2	8	no	—
122	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
123	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
124	C₂×C₂²⋊C₈	C₂²×C₅₃	1	8	yes	—
125	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
127	#399	C₄×C₅₃	1	4	yes	—
128	#392	C₄×C₅₃	1	4	yes	—
129	#376	C₂²×C₅₃	1	1	yes	characteristic
130	#336	C₂²×C₅₃	1	4	yes	—
131	#336	C₂²×C₅₃	1	4	yes	—
132	C₅₃	#22265	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
134	C₂²×D₄	C₂×C₄×C₅₃	1	4	yes	—
135	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
136	C₂×M₁₆	—	2	8	no	—
137	C₂×M₁₆	—	4	8	no	—
138	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
139	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
140	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
141	C₄×D₄	—	2	4	no	—
142	C₄×D₄	—	2	4	no	—
143	C₄×D₄	—	4	8	no	—
144	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
146	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
147	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
148	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
149	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
150	C₄⋊C₈	—	2	4	no	—
151	C₂²⋊C₈	—	2	8	no	—
152	C₂²⋊C₈	—	4	8	no	—
153	#97	—	2	2	no	—
154	C₄×C₈	D₄×C₅₃	1	4	yes	complemented
155	#95	C₂×C₄×C₅₃	1	4	yes	—
156	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	4	4	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	—	4	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
166	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
167	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
168	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
169	M₁₆	—	4	16	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₂²⋊C₄	—	2	8	no	—
179	C₂²⋊C₄	—	4	8	no	—
180	C₄²	—	2	4	no	—
181	C₄²	C₂×D₄×C₅₃	1	2	yes	—
182	C₄²	#41×C₅₃	1	2	yes	—
183	C₂³	—	2	4	no	—
184	C₂³	—	2	8	no	—
185	C₂³	—	2	4	no	—
186	C₂³	—	2	8	no	—
187	C₂³	—	2	4	no	—
188	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
189	D₄	—	4	8	no	—
190	D₄	—	4	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
200	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
202	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
203	C₈	—	2	4	no	—
204	C₈	—	4	4	no	—
205	C₈	—	4	2	no	—
206	C₂²	—	4	8	no	—
207	C₂²	—	4	8	no	—
208	C₂²	—	4	4	no	—
209	C₂²	—	4	4	no	—
210	C₂²	—	4	4	no	—
211	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
212	C₂²	C₅₃×#518	1	2	yes	—
213	C₂²	C₅₃×#568	1	4	yes	—
214	C₄	—	2	4	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	4	no	—
217	C₄	C₅₃×#535	1	2	yes	—
218	C₄	C₅₃×#540	1	2	yes	—
219	C₂	—	8	4	no	—
220	C₂	C₅₃×#2328	1	1	yes	characteristic
221	C₂	C₅₃×#2930	1	2	yes	—
222	C₂	C₅₃×#2978	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₅₃×#2397	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#436	C₂	semidirect
3. C₅₃×#1955	C₂	semidirect
4. C₅₃×#1870	C₂	semidirect
5. C₅₃×#1832	C₂	semidirect
6. C₅₃×#1746	C₂	semidirect
42. C₄×C₈×C₅₃	D₄	semidirect
94. #22265	C₅₃	direct (N × H)
108. C₂×#436	C₂×C₅₃	semidirect
110. #1955	C₂×C₅₃	semidirect
111. #1870	C₂×C₅₃	semidirect
112. #1832	C₂×C₅₃	semidirect
113. #1746	C₂×C₅₃	semidirect
132. C₅₃	#22265	direct (N × H)
154. C₄×C₈	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#436	2
C₅₃×#2397	2
C₅₃×#1955	2
C₅₃×#1870	2
C₅₃×#1832	2
C₅₃×#1746	2
#22265	53

Subgroups — C53×#22265 #27229777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22265 #27229777