Subgroups — C₅₃×#14713 #27222225

Order 13568.

3342 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#390	C₂	1	12	yes	maximal, complemented
3	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×C₅₃	C₂²	1	6	yes	—
6	C₂×#127×C₅₃	—	2	4	no	—
7	C₂×#127×C₅₃	—	2	4	no	—
8	C₂×#127×C₅₃	C₄	1	8	yes	complemented
9	C₂²×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
10	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	16	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	6	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#390	—	2	48	no	—
14	C₅₃×#384	—	2	4	no	—
15	C₄³×C₅₃	—	2	4	no	—
16	C₂⁵×C₅₃	C₂³	1	4	yes	—
17	C₂²×D₄×C₅₃	—	2	12	no	—
18	C₂²×D₄×C₅₃	—	2	12	no	—
19	C₂²×D₄×C₅₃	—	2	24	no	—
20	C₂²×D₄×C₅₃	—	2	24	no	—
21	C₂²×D₄×C₅₃	C₂×C₄	1	24	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
25	#127×C₅₃	—	2	32	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
28	C₂×C₂²⋊C₄×C₅₃	—	2	96	no	—
29	C₂×C₄²×C₅₃	—	2	12	no	—
30	C₂×C₄²×C₅₃	—	2	12	no	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅₃	D₄	1	4	yes	—
33	C₂×C₄²×C₅₃	D₄	1	4	yes	—
34	C₂×C₄²×C₅₃	D₄	1	24	yes	complemented
35	C₂⁴×C₅₃	—	2	24	no	—
36	C₂⁴×C₅₃	—	2	4	no	—
37	C₂⁴×C₅₃	—	2	4	no	—
38	C₂⁴×C₅₃	—	2	24	no	—
39	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
40	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₂×D₄×C₅₃	—	2	48	no	—
42	C₂×D₄×C₅₃	—	2	96	no	—
43	C₂×D₄×C₅₃	—	4	48	no	—
44	C₂×D₄×C₅₃	—	4	24	no	—
45	C₂×D₄×C₅₃	—	4	24	no	—
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	—	2	12	no	—
48	C₂²×C₄×C₅₃	—	2	24	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
55	C₂²×C₄×C₅₃	#41	1	12	yes	—
56	C₄⋊C₄×C₅₃	—	4	24	no	—
57	C₂²⋊C₄×C₅₃	—	4	64	no	—
58	C₄²×C₅₃	—	2	48	no	—
59	C₄²×C₅₃	C₂²⋊C₄	1	16	yes	complemented
60	C₂³×C₅₃	—	2	8	no	—
61	C₂³×C₅₃	—	2	48	no	—
62	C₂³×C₅₃	—	4	24	no	—
63	C₂³×C₅₃	—	4	12	no	—
64	C₂³×C₅₃	—	4	12	no	—
65	C₂³×C₅₃	—	4	16	no	—
66	C₂³×C₅₃	—	4	48	no	—
67	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
68	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
69	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
71	C₂³×C₅₃	C₂×#41	1	6	yes	—
72	D₄×C₅₃	—	4	96	no	—
73	C₂×C₄×C₅₃	—	2	24	no	—
74	C₂×C₄×C₅₃	—	2	24	no	—
75	C₂×C₄×C₅₃	—	4	16	no	—
76	C₂×C₄×C₅₃	—	4	12	no	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
78	C₂×C₄×C₅₃	C₄×D₄	1	24	yes	—
79	C₂×C₄×C₅₃	#121	1	24	yes	—
80	C₂×C₄×C₅₃	#124	1	12	yes	—
81	C₂×C₄×C₅₃	#127	1	12	yes	—
82	C₂×C₄×C₅₃	#127	1	16	yes	complemented
83	#14713	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₅₃	—	4	32	no	—
85	C₂²×C₅₃	—	4	24	no	—
86	C₂²×C₅₃	—	8	16	no	—
87	C₂²×C₅₃	—	8	12	no	—
88	C₂²×C₅₃	—	8	4	no	—
89	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
91	C₂²×C₅₃	C₂×#121	1	6	yes	—
92	C₂²×C₅₃	C₂×#124	1	3	yes	—
93	C₂²×C₅₃	C₂×#127	1	4	yes	—
94	C₂²×C₅₃	C₂×#127	1	3	yes	—
95	C₂²×C₅₃	#532	1	12	yes	—
96	C₄×C₅₃	—	2	16	no	—
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	#390	1	24	yes	complemented
99	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic
100	C₂×#390	C₂×C₅₃	1	12	yes	complemented
101	C₂×#384	C₂×C₅₃	1	1	yes	characteristic, complemented
102	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic, complemented
103	C₂×C₅₃	—	8	8	no	—
104	C₂×C₅₃	C₂×#390	1	3	yes	—
105	C₂×C₅₃	C₄×#127	1	4	yes	—
106	C₂×C₅₃	#2856	1	4	yes	—
107	C₂×C₅₃	#2858	1	3	yes	—
108	C₂×C₅₃	#2867	1	1	yes	characteristic
109	C₂³×D₄	C₂²×C₅₃	1	6	yes	—
110	C₂×#127	—	2	4	no	—
111	C₂×#127	—	2	4	no	—
112	C₂×#127	C₄×C₅₃	1	8	yes	complemented
113	C₂²×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
114	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	16	yes	—
115	C₂²×C₄²	C₂²×C₅₃	1	6	yes	—
116	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
117	#390	—	2	48	no	—
118	#384	—	2	4	no	—
119	C₄³	—	2	4	no	—
120	C₅₃	#14713	1	1	yes	characteristic, direct product factor, Sylow
121	C₂⁵	C₂³×C₅₃	1	4	yes	—
122	C₂²×D₄	—	2	12	no	—
123	C₂²×D₄	—	2	12	no	—
124	C₂²×D₄	—	2	24	no	—
125	C₂²×D₄	—	2	24	no	—
126	C₂²×D₄	C₂×C₄×C₅₃	1	24	yes	—
127	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
128	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
129	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
130	#127	—	2	32	no	—
131	C₂×C₄⋊C₄	—	2	12	no	—
132	C₂×C₄⋊C₄	—	2	24	no	—
133	C₂×C₂²⋊C₄	—	2	96	no	—
134	C₂×C₄²	—	2	12	no	—
135	C₂×C₄²	—	2	12	no	—
136	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
137	C₂×C₄²	D₄×C₅₃	1	4	yes	—
138	C₂×C₄²	D₄×C₅₃	1	4	yes	—
139	C₂×C₄²	D₄×C₅₃	1	24	yes	complemented
140	C₂⁴	—	2	24	no	—
141	C₂⁴	—	2	4	no	—
142	C₂⁴	—	2	4	no	—
143	C₂⁴	—	2	24	no	—
144	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	—
145	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
146	C₂×D₄	—	2	48	no	—
147	C₂×D₄	—	2	96	no	—
148	C₂×D₄	—	4	48	no	—
149	C₂×D₄	—	4	24	no	—
150	C₂×D₄	—	4	24	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	12	no	—
153	C₂²×C₄	—	2	24	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
156	C₂²×C₄	C₂×D₄×C₅₃	1	24	yes	—
157	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
158	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
159	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
160	C₂²×C₄	#41×C₅₃	1	12	yes	—
161	C₄⋊C₄	—	4	24	no	—
162	C₂²⋊C₄	—	4	64	no	—
163	C₄²	—	2	48	no	—
164	C₄²	C₂²⋊C₄×C₅₃	1	16	yes	complemented
165	C₂³	—	2	8	no	—
166	C₂³	—	2	48	no	—
167	C₂³	—	4	24	no	—
168	C₂³	—	4	12	no	—
169	C₂³	—	4	16	no	—
170	C₂³	—	4	12	no	—
171	C₂³	—	4	48	no	—
172	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
173	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
174	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
175	C₂³	C₂²×D₄×C₅₃	1	6	yes	—
176	C₂³	C₂×#41×C₅₃	1	6	yes	—
177	D₄	—	4	96	no	—
178	C₂×C₄	—	2	24	no	—
179	C₂×C₄	—	2	24	no	—
180	C₂×C₄	—	4	16	no	—
181	C₂×C₄	—	4	12	no	—
182	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	12	yes	—
183	C₂×C₄	C₄×D₄×C₅₃	1	24	yes	—
184	C₂×C₄	#121×C₅₃	1	24	yes	—
185	C₂×C₄	#124×C₅₃	1	12	yes	—
186	C₂×C₄	#127×C₅₃	1	12	yes	—
187	C₂×C₄	#127×C₅₃	1	16	yes	complemented
188	C₂²	—	4	24	no	—
189	C₂²	—	4	32	no	—
190	C₂²	—	8	16	no	—
191	C₂²	—	8	12	no	—
192	C₂²	—	8	4	no	—
193	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
194	C₂²	C₂×C₄×D₄×C₅₃	1	6	yes	—
195	C₂²	C₂×#121×C₅₃	1	6	yes	—
196	C₂²	C₂×#124×C₅₃	1	3	yes	—
197	C₂²	C₂×#127×C₅₃	1	3	yes	—
198	C₂²	C₂×#127×C₅₃	1	4	yes	—
199	C₂²	C₅₃×#532	1	12	yes	—
200	C₄	—	2	16	no	—
201	C₄	—	8	4	no	—
202	C₄	C₅₃×#390	1	24	yes	complemented
203	C₂	—	8	8	no	—
204	C₂	C₂×C₅₃×#390	1	3	yes	—
205	C₂	C₄×#127×C₅₃	1	4	yes	—
206	C₂	C₅₃×#2856	1	4	yes	—
207	C₂	C₅₃×#2858	1	3	yes	—
208	C₂	C₅₃×#2867	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
prime radical	C₂²×#127×C₅₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#390	C₂	semidirect
3. C₂×C₅₃×#384	C₂	semidirect
4. C₂×C₄³×C₅₃	C₂	semidirect
8. C₂×#127×C₅₃	C₄	semidirect
34. C₂×C₄²×C₅₃	D₄	semidirect
59. C₄²×C₅₃	C₂²⋊C₄	semidirect
82. C₂×C₄×C₅₃	#127	semidirect
83. #14713	C₅₃	direct (N × H)
98. C₄×C₅₃	#390	semidirect
100. C₂×#390	C₂×C₅₃	semidirect
101. C₂×#384	C₂×C₅₃	semidirect
102. C₂×C₄³	C₂×C₅₃	semidirect
112. C₂×#127	C₄×C₅₃	semidirect
120. C₅₃	#14713	direct (N × H)
139. C₂×C₄²	D₄×C₅₃	semidirect
164. C₄²	C₂²⋊C₄×C₅₃	semidirect
187. C₂×C₄	#127×C₅₃	semidirect
202. C₄	C₅₃×#390	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₅₃	2
C₂×C₅₃×#390	2
C₂×C₅₃×#384	2
C₂×C₄³×C₅₃	2
#14713	53

Subgroups — C53×#14713 #27222225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#14713 #27222225