Subgroups — C₅₃×#12611 #27220123

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#346	C₂	1	2	yes	maximal, complemented
4	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1374	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	—	2	4	no	—
8	C₂²×C₁₆×C₅₃	C₄	1	4	yes	complemented
9	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
15	C₅₃×C₂²⋊C₁₆	C₄	1	4	yes	complemented
16	C₅₃×#346	—	2	4	no	—
17	C₅₃×#336	C₄	1	2	yes	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₅₃	—	2	2	no	—
29	C₂×C₁₆×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	—	2	8	no	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
36	C₂²⋊C₈×C₅₃	C₈	1	4	yes	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	#95×C₅₃	C₈	1	2	yes	—
40	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	C₄²	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₂²⋊C₄×C₅₃	—	2	4	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₁₆×C₅₃	—	2	2	no	—
64	C₁₆×C₅₃	—	2	4	no	—
65	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	complemented
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	C₄×C₈	1	2	yes	—
69	C₂³×C₅₃	M₃₂	1	4	yes	—
70	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	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	—	2	4	no	—
78	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#131	1	2	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	2	4	no	—
85	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₅₃	#117	1	1	yes	characteristic
87	C₈×C₅₃	C₄×D₄	1	2	yes	—
88	#12611	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	—	2	8	no	—
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	#346	1	4	yes	complemented
93	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²×C₅₃	C₈×D₄	1	2	yes	—
95	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
96	C₂²×C₅₃	#504	1	2	yes	—
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	—	4	4	no	—
99	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₅₃	#405	1	1	yes	characteristic
101	C₄×C₅₃	C₈×D₄	1	2	yes	—
102	C₂³×C₁₆	C₂×C₅₃	1	1	yes	characteristic
103	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
104	C₂×#346	C₂×C₅₃	1	2	yes	complemented
105	C₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
106	#1374	C₂×C₅₃	1	2	yes	complemented
107	C₂×C₅₃	—	2	4	no	—
108	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
110	C₂×C₅₃	#2108	1	1	yes	characteristic
111	C₂×C₅₃	#2168	1	4	yes	—
112	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
113	C₂²×C₁₆	—	2	4	no	—
114	C₂²×C₁₆	C₄×C₅₃	1	4	yes	complemented
115	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
116	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
117	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
118	C₂×C₂²⋊C₈	C₄×C₅₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
120	C₄×C₂²⋊C₄	C₄×C₅₃	1	1	yes	characteristic
121	C₂²⋊C₁₆	C₄×C₅₃	1	4	yes	complemented
122	#346	—	2	4	no	—
123	#336	C₄×C₅₃	1	2	yes	—
124	C₅₃	#12611	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
129	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
132	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₈×C₅₃	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	8	no	—
139	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
140	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
141	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
142	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
143	C₂²⋊C₈	C₈×C₅₃	1	4	yes	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	#95	C₈×C₅₃	1	2	yes	—
147	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
151	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
152	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	C₄²×C₅₃	1	2	yes	—
162	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
163	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
166	C₂×C₈	#41×C₅₃	1	2	yes	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₁₆	—	2	2	no	—
171	C₁₆	—	2	4	no	—
172	C₁₆	C₂²⋊C₄×C₅₃	1	4	yes	complemented
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₄×C₈×C₅₃	1	2	yes	—
176	C₂³	M₃₂×C₅₃	1	4	yes	—
177	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
186	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
187	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
188	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
189	C₂×C₄	#131×C₅₃	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	—	2	4	no	—
192	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
193	C₈	#117×C₅₃	1	1	yes	characteristic
194	C₈	C₄×D₄×C₅₃	1	2	yes	—
195	C₂²	—	2	8	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₅₃×#346	1	4	yes	complemented
199	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
200	C₂²	C₈×D₄×C₅₃	1	2	yes	—
201	C₂²	C₂×M₃₂×C₅₃	1	2	yes	—
202	C₂²	C₅₃×#504	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	4	4	no	—
205	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
206	C₄	C₅₃×#405	1	1	yes	characteristic
207	C₄	C₈×D₄×C₅₃	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	C₈×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₂	C₂×C₅₃×#346	1	1	yes	characteristic
211	C₂	C₅₃×#2108	1	1	yes	characteristic
212	C₂	C₅₃×#2168	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#346	C₂	semidirect
5. C₅₃×#1374	C₂	semidirect
8. C₂²×C₁₆×C₅₃	C₄	semidirect
15. C₅₃×C₂²⋊C₁₆	C₄	semidirect
65. C₁₆×C₅₃	C₂²⋊C₄	semidirect
88. #12611	C₅₃	direct (N × H)
92. C₂²×C₅₃	#346	semidirect
104. C₂×#346	C₂×C₅₃	semidirect
106. #1374	C₂×C₅₃	semidirect
114. C₂²×C₁₆	C₄×C₅₃	semidirect
121. C₂²⋊C₁₆	C₄×C₅₃	semidirect
124. C₅₃	#12611	direct (N × H)
172. C₁₆	C₂²⋊C₄×C₅₃	semidirect
198. C₂²	C₅₃×#346	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₅₃	2
C₂×C₅₃×C₂²⋊C₁₆	2
C₂×C₅₃×#346	2
C₈×C₂²⋊C₄×C₅₃	2
C₅₃×#1374	2
#12611	53

Subgroups — C53×#12611 #27220123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12611 #27220123