Subgroups — C₂×C₅₃×#1452 #27224830

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#431	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1452	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#322	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₅₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₄	1	8	yes	—
8	C₅₃×#431	C₂²	1	8	yes	—
9	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	—
10	C₂×C₂²⋊C₈×C₅₃	C₄	1	4	yes	—
11	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
16	C₅₃×#322	C₂²	1	16	yes	complemented
17	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	8	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂×C₄	1	16	yes	—
30	C₂²⋊C₈×C₅₃	C₂×C₄	1	16	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₅₃	C₂³	1	4	yes	—
35	C₄×C₈×C₅₃	C₂³	1	8	yes	—
36	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	8	no	—
38	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
40	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
41	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₅₃	—	2	16	no	—
48	C₂×C₈×C₅₃	—	2	8	no	—
49	C₂×C₈×C₅₃	—	2	16	no	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	16	yes	—
52	C₄²×C₅₃	C₄²	1	2	yes	—
53	C₄²×C₅₃	C₄²	1	4	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
58	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₂³×C₅₃	—	2	4	no	—
60	C₂³×C₅₃	#97	1	4	yes	—
61	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
62	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	4	no	—
64	C₂×C₄×C₅₃	—	2	8	no	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	#97	1	8	yes	—
67	C₂×C₄×C₅₃	#97	1	4	yes	—
68	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
70	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
74	C₂×C₄×C₅₃	#131	1	8	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
77	C₈×C₅₃	—	2	8	no	—
78	C₈×C₅₃	—	2	16	no	—
79	C₂×#1452	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₅₃	—	2	4	no	—
81	C₂²×C₅₃	—	2	8	no	—
82	C₂²×C₅₃	C₂×#97	1	4	yes	—
83	C₂²×C₅₃	C₂×#97	1	2	yes	—
84	C₂²×C₅₃	#405	1	4	yes	—
85	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	C₂×#97	1	2	yes	—
88	C₄×C₅₃	C₂×#97	1	4	yes	—
89	C₄×C₅₃	#405	1	4	yes	—
90	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
91	C₄×C₅₃	C₂×#131	1	4	yes	—
92	C₂×#431	C₂×C₅₃	1	2	yes	—
93	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
94	#1452	C₂×C₅₃	1	8	yes	direct product factor
95	C₂×#322	C₂×C₅₃	1	4	yes	complemented
96	C₂×C₅₃	—	2	4	no	—
97	C₂×C₅₃	#1452	1	4	yes	direct product factor
98	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#405	1	2	yes	—
100	C₂³×C₈	C₄×C₅₃	1	2	yes	—
101	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
102	#431	C₄×C₅₃	1	8	yes	—
103	#431	C₂²×C₅₃	1	8	yes	—
104	C₂×C₄⋊C₈	C₄×C₅₃	1	4	yes	—
105	C₂×C₂²⋊C₈	C₄×C₅₃	1	4	yes	—
106	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
107	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
110	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
111	#322	C₂²×C₅₃	1	16	yes	complemented
112	C₅₃	C₂×#1452	1	1	yes	characteristic, direct product factor, Sylow
113	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
114	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
115	C₂²×C₈	—	2	8	no	—
116	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
117	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
118	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
119	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
121	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
122	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
123	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
124	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
125	C₄⋊C₈	C₂×C₄×C₅₃	1	16	yes	—
126	C₂²⋊C₈	C₂×C₄×C₅₃	1	16	yes	—
127	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
128	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
129	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
130	C₄×C₈	C₂³×C₅₃	1	4	yes	—
131	C₄×C₈	C₂³×C₅₃	1	8	yes	—
132	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
135	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
136	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
137	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
138	C₂²×C₄	M₁₆×C₅₃	1	8	yes	—
139	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
140	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
141	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
143	C₂×C₈	—	2	16	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	16	no	—
146	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
147	C₂×C₈	C₂²×C₄×C₅₃	1	16	yes	—
148	C₄²	C₄²×C₅₃	1	2	yes	—
149	C₄²	C₄²×C₅₃	1	4	yes	—
150	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
151	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
152	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
153	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
154	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
155	C₂³	—	2	4	no	—
156	C₂³	#97×C₅₃	1	4	yes	—
157	C₂³	C₂×C₄²×C₅₃	1	2	yes	—
158	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	#97×C₅₃	1	8	yes	—
163	C₂×C₄	#97×C₅₃	1	4	yes	—
164	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
165	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
167	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
168	C₂×C₄	C₂×M₁₆×C₅₃	1	8	yes	—
169	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
170	C₂×C₄	#131×C₅₃	1	8	yes	—
171	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
172	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
173	C₈	—	2	8	no	—
174	C₈	—	2	16	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	2	4	no	—
177	C₂²	C₂×#97×C₅₃	1	4	yes	—
178	C₂²	C₂×#97×C₅₃	1	2	yes	—
179	C₂²	C₅₃×#405	1	4	yes	—
180	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	C₂×#97×C₅₃	1	2	yes	—
183	C₄	C₂×#97×C₅₃	1	4	yes	—
184	C₄	C₅₃×#405	1	4	yes	—
185	C₄	C₂²×M₁₆×C₅₃	1	2	yes	—
186	C₄	C₂×#131×C₅₃	1	4	yes	—
187	C₂	—	2	4	no	—
188	C₂	C₅₃×#1452	1	4	yes	direct product factor
189	C₂	C₂²×#97×C₅₃	1	1	yes	characteristic
190	C₂	C₂×C₅₃×#405	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅₃	C₂³
commutator	C₂²	C₂²×C₄²×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1452	C₂	direct (N × H)
4. C₂×C₅₃×#322	C₂	semidirect
16. C₅₃×#322	C₂²	semidirect
79. C₂×#1452	C₅₃	direct (N × H)
94. #1452	C₂×C₅₃	direct (N × H)
95. C₂×#322	C₂×C₅₃	semidirect
97. C₂×C₅₃	#1452	direct (N × H)
111. #322	C₂²×C₅₃	semidirect
112. C₅₃	C₂×#1452	direct (N × H)
188. C₂	C₅₃×#1452	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×#431	2
C₂²×C₄×C₈×C₅₃	2
C₅₃×#1452	2
C₂×C₅₃×#322	2
C₂×#1452	53

Subgroups — C2×C53×#1452 #27224830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1452 #27224830