Subgroups — C₂×C₅₃×#1845 #27228307

Order 13568.

1358 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₅₃	C₂	1	4	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#384	C₂	1	2	yes	maximal
4	C₅₃×#1845	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₅₃	—	2	4	no	—
7	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#107×C₅₃	—	2	16	no	—
10	C₂×#107×C₅₃	C₂²	1	16	yes	—
11	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
13	C₅₃×#384	—	2	4	no	—
14	C₅₃×#384	C₂²	1	8	yes	—
15	C₂³×C₄×C₅₃	—	2	4	no	—
16	C₂³×C₄×C₅₃	D₄	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₅₃	D₄	1	8	yes	—
20	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
21	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
22	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
25	C₂×C₄⋊C₄×C₅₃	—	4	16	no	—
26	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
31	#107×C₅₃	—	2	32	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	16	yes	complemented
33	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	8	no	—
35	C₂²×C₄×C₅₃	—	4	8	no	—
36	C₂²×C₄×C₅₃	—	4	16	no	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₅₃	D₈	1	8	yes	—
39	C₂²×C₄×C₅₃	Q₁₆	1	8	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄⋊C₄	1	32	yes	complemented
48	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
50	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
51	C₄⋊C₄×C₅₃	—	2	16	no	—
52	C₄⋊C₄×C₅₃	—	4	8	no	—
53	C₄⋊C₄×C₅₃	—	4	32	no	—
54	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
56	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
57	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂³×C₅₃	#127	1	2	yes	—
59	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
60	C₂³×C₅₃	C₂×D₈	1	4	yes	—
61	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
62	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	4	8	no	—
64	C₂×C₄×C₅₃	—	4	16	no	—
65	C₂×C₄×C₅₃	—	8	16	no	—
66	C₂×C₄×C₅₃	#107	1	32	yes	complemented
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
70	C₂×C₄×C₅₃	#127	1	2	yes	—
71	C₂×C₄×C₅₃	C₄⋊Q₈	1	8	yes	—
72	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×D₈	1	8	yes	—
74	C₂×C₄×C₅₃	C₂×Q₁₆	1	8	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	complemented
80	C₂×#1845	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	#384	1	4	yes	—
82	C₂²×C₅₃	C₂×#107	1	8	yes	—
83	C₂²×C₅₃	#493	1	4	yes	—
84	C₂²×C₅₃	#494	1	4	yes	—
85	C₂²×C₅₃	#500	1	8	yes	—
86	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
90	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
91	C₄×C₅₃	—	8	8	no	—
92	C₄×C₅₃	#384	1	4	yes	—
93	C₄×C₅₃	C₂×#107	1	16	yes	complemented
94	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
95	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
96	C₂²×#107	C₂×C₅₃	1	4	yes	—
97	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
98	C₂×#384	C₂×C₅₃	1	2	yes	—
99	#1845	C₂×C₅₃	1	8	yes	direct product factor
100	C₂×C₅₃	#1845	1	8	yes	direct product factor
101	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
102	C₂×C₅₃	C₂²×#107	1	2	yes	—
103	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
104	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
105	C₂×C₅₃	C₂×#500	1	2	yes	—
106	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
107	C₂²×C₄⋊C₄	—	2	4	no	—
108	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
109	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
110	C₂×#107	—	2	16	no	—
111	C₂×#107	C₂²×C₅₃	1	16	yes	—
112	C₂×C₄×C₈	C₄×C₅₃	1	8	yes	complemented
113	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
114	#384	—	2	4	no	—
115	#384	C₂²×C₅₃	1	8	yes	—
116	C₅₃	C₂×#1845	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×C₄	—	2	4	no	—
118	C₂³×C₄	D₄×C₅₃	1	2	yes	—
119	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
121	C₂²×C₈	D₄×C₅₃	1	8	yes	—
122	C₂²×C₈	Q₈×C₅₃	1	8	yes	—
123	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
124	C₂×C₄⋊C₄	—	2	16	no	—
125	C₂×C₄⋊C₄	—	2	16	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	4	16	no	—
128	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
129	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
130	C₂×C₄²	D₄×C₅₃	1	2	yes	—
131	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
132	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
133	#107	—	2	32	no	—
134	C₄×C₈	C₂×C₄×C₅₃	1	16	yes	complemented
135	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	4	8	no	—
138	C₂²×C₄	—	4	16	no	—
139	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
140	C₂²×C₄	D₈×C₅₃	1	8	yes	—
141	C₂²×C₄	Q₁₆×C₅₃	1	8	yes	—
142	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
145	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
146	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
147	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
148	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
149	C₂×C₈	C₄⋊C₄×C₅₃	1	32	yes	complemented
150	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
151	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
152	C₂×C₈	C₂×Q₈×C₅₃	1	8	yes	—
153	C₄⋊C₄	—	2	16	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	32	no	—
156	C₄²	C₄⋊C₄×C₅₃	1	8	yes	—
157	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
158	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
159	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
160	C₂³	#127×C₅₃	1	2	yes	—
161	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
162	C₂³	C₂×D₈×C₅₃	1	4	yes	—
163	C₂³	C₂×Q₁₆×C₅₃	1	4	yes	—
164	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	4	16	no	—
167	C₂×C₄	—	8	16	no	—
168	C₂×C₄	#107×C₅₃	1	32	yes	complemented
169	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
170	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
171	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
172	C₂×C₄	#127×C₅₃	1	2	yes	—
173	C₂×C₄	C₄⋊Q₈×C₅₃	1	8	yes	—
174	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
175	C₂×C₄	C₂×D₈×C₅₃	1	8	yes	—
176	C₂×C₄	C₂×Q₁₆×C₅₃	1	8	yes	—
177	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
179	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
180	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
181	C₈	C₂×C₄⋊C₄×C₅₃	1	16	yes	complemented
182	C₂²	C₅₃×#384	1	4	yes	—
183	C₂²	C₂×#107×C₅₃	1	8	yes	—
184	C₂²	C₅₃×#493	1	4	yes	—
185	C₂²	C₅₃×#494	1	4	yes	—
186	C₂²	C₅₃×#500	1	8	yes	—
187	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
188	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
189	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
190	C₂²	C₂²×D₈×C₅₃	1	2	yes	—
191	C₂²	C₂²×Q₁₆×C₅₃	1	2	yes	—
192	C₄	—	8	8	no	—
193	C₄	C₅₃×#384	1	4	yes	—
194	C₄	C₂×#107×C₅₃	1	16	yes	complemented
195	C₄	C₂²×C₄⋊C₄×C₅₃	1	2	yes	—
196	C₄	C₂×C₄⋊Q₈×C₅₃	1	2	yes	—
197	C₂	C₅₃×#1845	1	8	yes	direct product factor
198	C₂	C₂×C₅₃×#384	1	1	yes	characteristic
199	C₂	C₂²×#107×C₅₃	1	2	yes	—
200	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
201	C₂	C₂×C₅₃×#494	1	1	yes	characteristic
202	C₂	C₂×C₅₃×#500	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1845	C₂	direct (N × H)
11. C₂×C₄×C₈×C₅₃	C₄	semidirect
32. C₄×C₈×C₅₃	C₂×C₄	semidirect
47. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
66. C₂×C₄×C₅₃	#107	semidirect
79. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
80. C₂×#1845	C₅₃	direct (N × H)
93. C₄×C₅₃	C₂×#107	semidirect
99. #1845	C₂×C₅₃	direct (N × H)
100. C₂×C₅₃	#1845	direct (N × H)
112. C₂×C₄×C₈	C₄×C₅₃	semidirect
116. C₅₃	C₂×#1845	direct (N × H)
134. C₄×C₈	C₂×C₄×C₅₃	semidirect
149. C₂×C₈	C₄⋊C₄×C₅₃	semidirect
168. C₂×C₄	#107×C₅₃	semidirect
181. C₈	C₂×C₄⋊C₄×C₅₃	semidirect
194. C₄	C₂×#107×C₅₃	semidirect
197. C₂	C₅₃×#1845	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#107×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#384	2
C₅₃×#1845	2
C₂×#1845	53

Subgroups — C2×C53×#1845 #27228307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1845 #27228307