Subgroups — C₅₃×#9824 #27217336

Order 13568.

726 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	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1957	C₂	1	4	yes	maximal, complemented
3	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₅₃	—	2	2	no	—
6	C₂×#127×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#493	—	2	2	no	—
8	C₅₃×#493	C₄	1	4	yes	complemented
9	C₂×#107×C₅₃	—	2	2	no	—
10	C₂×#102×C₅₃	—	2	8	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#384	C₂²	1	2	yes	—
14	C₈²×C₅₃	—	2	2	no	—
15	C₂²×D₄×C₅₃	—	2	2	no	—
16	C₂²×D₄×C₅₃	—	2	4	no	—
17	C₂×D₈×C₅₃	—	2	8	no	—
18	C₂×D₈×C₅₃	—	4	4	no	—
19	C₂×D₈×C₅₃	—	4	4	no	—
20	C₂²×C₈×C₅₃	D₄	1	4	yes	—
21	C₂²×C₈×C₅₃	D₄	1	2	yes	—
22	#127×C₅₃	—	2	2	no	—
23	#127×C₅₃	—	2	4	no	—
24	#127×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#107×C₅₃	—	4	4	no	—
29	#102×C₅₃	—	4	16	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₂⁴×C₅₃	—	4	2	no	—
35	C₂×D₄×C₅₃	—	2	4	no	—
36	C₂×D₄×C₅₃	—	2	8	no	—
37	C₂×D₄×C₅₃	—	4	4	no	—
38	C₂×D₄×C₅₃	—	4	2	no	—
39	C₂×D₄×C₅₃	—	4	8	no	—
40	C₂×D₄×C₅₃	—	4	4	no	—
41	C₂×D₄×C₅₃	—	4	4	no	—
42	C₂×D₄×C₅₃	—	4	8	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	D₈×C₅₃	—	4	16	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
50	C₂×C₈×C₅₃	D₈	1	4	yes	—
51	C₂×C₈×C₅₃	D₈	1	8	yes	complemented
52	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	4	8	no	—
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	2	yes	—
58	C₂³×C₅₃	—	4	4	no	—
59	C₂³×C₅₃	—	8	2	no	—
60	C₂³×C₅₃	—	8	2	no	—
61	C₂³×C₅₃	—	8	4	no	—
62	C₂³×C₅₃	—	8	4	no	—
63	C₂³×C₅₃	#127	1	1	yes	characteristic
64	D₄×C₅₃	—	4	8	no	—
65	D₄×C₅₃	—	4	16	no	—
66	D₄×C₅₃	—	8	8	no	—
67	D₄×C₅₃	—	8	4	no	—
68	C₂×C₄×C₅₃	—	8	2	no	—
69	C₂×C₄×C₅₃	—	8	4	no	—
70	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₅₃	#121	1	2	yes	—
73	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	#135	1	4	yes	—
78	C₈×C₅₃	—	2	8	no	—
79	C₈×C₅₃	#102	1	8	yes	complemented
80	#9824	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	—	8	4	no	—
82	C₂²×C₅₃	—	8	8	no	—
83	C₂²×C₅₃	—	16	4	no	—
84	C₂²×C₅₃	—	16	2	no	—
85	C₂²×C₅₃	—	16	2	no	—
86	C₂²×C₅₃	#390	1	1	yes	characteristic
87	C₂²×C₅₃	#492	1	1	yes	characteristic
88	C₂²×C₅₃	#493	1	1	yes	characteristic
89	C₂²×C₅₃	#493	1	2	yes	—
90	C₂²×C₅₃	#495	1	2	yes	—
91	C₄×C₅₃	—	16	2	no	—
92	C₄×C₅₃	C₂×#102	1	2	yes	—
93	C₄×C₅₃	C₄×D₈	1	4	yes	—
94	C₄×C₅₃	#466	1	4	yes	—
95	C₄×C₅₃	#486	1	2	yes	—
96	C₂×#493	C₂×C₅₃	1	1	yes	characteristic
97	#1957	C₂×C₅₃	1	4	yes	complemented
98	#1845	C₂×C₅₃	1	1	yes	characteristic, complemented
99	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
100	C₂×C₅₃	—	16	4	no	—
101	C₂×C₅₃	#1706	1	2	yes	—
102	C₂×C₅₃	#1709	1	1	yes	characteristic
103	C₂×C₅₃	#1712	1	1	yes	characteristic
104	C₂×C₅₃	#1957	1	1	yes	characteristic
105	C₂×C₅₃	#1970	1	2	yes	—
106	C₂²×D₈	—	2	2	no	—
107	C₂×#127	C₂²×C₅₃	1	2	yes	—
108	#493	—	2	2	no	—
109	#493	C₄×C₅₃	1	4	yes	complemented
110	C₂×#107	—	2	2	no	—
111	C₂×#102	—	2	8	no	—
112	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
114	#384	C₂²×C₅₃	1	2	yes	—
115	C₈²	—	2	2	no	—
116	C₅₃	#9824	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×D₄	—	2	2	no	—
118	C₂²×D₄	—	2	4	no	—
119	C₂×D₈	—	2	8	no	—
120	C₂×D₈	—	4	4	no	—
121	C₂×D₈	—	4	4	no	—
122	C₂²×C₈	D₄×C₅₃	1	4	yes	—
123	C₂²×C₈	D₄×C₅₃	1	2	yes	—
124	#127	—	2	2	no	—
125	#127	—	2	4	no	—
126	#127	C₂×C₄×C₅₃	1	4	yes	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
130	#107	—	4	4	no	—
131	#102	—	4	16	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
135	C₄×C₈	D₄×C₅₃	1	2	yes	—
136	C₂⁴	—	4	2	no	—
137	C₂×D₄	—	2	4	no	—
138	C₂×D₄	—	2	8	no	—
139	C₂×D₄	—	4	4	no	—
140	C₂×D₄	—	4	2	no	—
141	C₂×D₄	—	4	8	no	—
142	C₂×D₄	—	4	4	no	—
143	C₂×D₄	—	4	4	no	—
144	C₂×D₄	—	4	8	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
148	D₈	—	4	16	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
152	C₂×C₈	D₈×C₅₃	1	4	yes	—
153	C₂×C₈	D₈×C₅₃	1	8	yes	complemented
154	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
158	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
159	C₄²	#41×C₅₃	1	2	yes	—
160	C₂³	—	4	4	no	—
161	C₂³	—	8	2	no	—
162	C₂³	—	8	2	no	—
163	C₂³	—	8	4	no	—
164	C₂³	—	8	4	no	—
165	C₂³	#127×C₅₃	1	1	yes	characteristic
166	D₄	—	4	8	no	—
167	D₄	—	4	16	no	—
168	D₄	—	8	4	no	—
169	D₄	—	8	8	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	4	no	—
172	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
173	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
174	C₂×C₄	#121×C₅₃	1	2	yes	—
175	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
176	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
177	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
178	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
179	C₂×C₄	#135×C₅₃	1	4	yes	—
180	C₈	—	2	8	no	—
181	C₈	#102×C₅₃	1	8	yes	complemented
182	C₂²	—	8	4	no	—
183	C₂²	—	8	8	no	—
184	C₂²	—	16	4	no	—
185	C₂²	—	16	2	no	—
186	C₂²	—	16	2	no	—
187	C₂²	C₅₃×#390	1	1	yes	characteristic
188	C₂²	C₅₃×#492	1	1	yes	characteristic
189	C₂²	C₅₃×#493	1	1	yes	characteristic
190	C₂²	C₅₃×#493	1	2	yes	—
191	C₂²	C₅₃×#495	1	2	yes	—
192	C₄	—	16	2	no	—
193	C₄	C₂×#102×C₅₃	1	2	yes	—
194	C₄	C₄×D₈×C₅₃	1	4	yes	—
195	C₄	C₅₃×#466	1	4	yes	—
196	C₄	C₅₃×#486	1	2	yes	—
197	C₂	—	16	4	no	—
198	C₂	C₅₃×#1706	1	2	yes	—
199	C₂	C₅₃×#1709	1	1	yes	characteristic
200	C₂	C₅₃×#1712	1	1	yes	characteristic
201	C₂	C₅₃×#1957	1	1	yes	characteristic
202	C₂	C₅₃×#1970	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×C₅₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1957	C₂	semidirect
3. C₅₃×#1845	C₂	semidirect
4. C₂×C₈²×C₅₃	C₂	semidirect
8. C₅₃×#493	C₄	semidirect
51. C₂×C₈×C₅₃	D₈	semidirect
79. C₈×C₅₃	#102	semidirect
80. #9824	C₅₃	direct (N × H)
97. #1957	C₂×C₅₃	semidirect
98. #1845	C₂×C₅₃	semidirect
99. C₂×C₈²	C₂×C₅₃	semidirect
109. #493	C₄×C₅₃	semidirect
116. C₅₃	#9824	direct (N × H)
153. C₂×C₈	D₈×C₅₃	semidirect
181. C₈	#102×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×#493	2
C₅₃×#1957	2
C₅₃×#1845	2
C₂×C₈²×C₅₃	2
#9824	53

Subgroups — C53×#9824 #27217336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9824 #27217336