Subgroups — C₅₃×#20760 #27228272

Order 13568.

950 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2285	C₂	1	2	yes	maximal
3	C₅₃×#1835	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1834	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1735	C₂	1	4	yes	maximal
6	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×#117×C₅₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#384	C₂²	1	4	yes	—
13	C₅₃×#382	C₂²	1	4	yes	—
14	C₅₃×#380	—	2	4	no	—
15	C₄×C₄⋊C₄×C₅₃	—	2	4	no	—
16	C₅₃×#340	C₂²	1	16	yes	—
17	C₂³×C₄×C₅₃	D₄	1	2	yes	—
18	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	8	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
21	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
22	#117×C₅₃	—	2	16	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
24	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
25	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
26	C₂×C₄²×C₅₃	—	2	4	no	—
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	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
34	#95×C₅₃	—	2	8	no	—
35	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	8	no	—
37	C₂²×C₄×C₅₃	—	2	8	no	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₅₃	#41	1	8	yes	—
45	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂×C₈×C₅₃	—	2	16	no	—
47	C₂×C₈×C₅₃	—	2	8	no	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
49	C₄⋊C₄×C₅₃	—	2	16	no	—
50	C₄⋊C₄×C₅₃	—	4	16	no	—
51	C₂²⋊C₄×C₅₃	—	4	8	no	—
52	C₄²×C₅₃	—	4	8	no	—
53	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
56	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
57	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂³×C₅₃	—	2	4	no	—
59	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₅₃	#124	1	4	yes	—
61	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	—	2	16	no	—
64	C₂×C₄×C₅₃	—	4	8	no	—
65	C₂×C₄×C₅₃	—	4	16	no	—
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₅₃	#117	1	8	yes	—
70	C₂×C₄×C₅₃	#124	1	4	yes	—
71	C₂×C₄×C₅₃	#125	1	8	yes	—
72	C₂×C₄×C₅₃	#135	1	8	yes	—
73	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
76	C₈×C₅₃	—	2	8	no	—
77	#20760	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×C₅₃	—	2	8	no	—
79	C₂²×C₅₃	—	2	4	no	—
80	C₂²×C₅₃	#382	1	4	yes	—
81	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×#124	1	2	yes	—
83	C₂²×C₅₃	C₂×#135	1	4	yes	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	8	8	no	—
86	C₄×C₅₃	#382	1	4	yes	—
87	C₄×C₅₃	#427	1	8	yes	—
88	C₄×C₅₃	C₂×#117	1	2	yes	—
89	C₄×C₅₃	C₂×#125	1	2	yes	—
90	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
91	#2285	C₂×C₅₃	1	2	yes	—
92	#1835	C₂×C₅₃	1	4	yes	complemented
93	#1834	C₂×C₅₃	1	4	yes	complemented
94	#1735	C₂×C₅₃	1	4	yes	—
95	C₂×C₅₃	—	2	4	no	—
96	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
97	C₂×C₅₃	C₂×#427	1	2	yes	—
98	C₂×C₅₃	#3131	1	4	yes	—
99	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
100	C₂×#117	C₂²×C₅₃	1	4	yes	—
101	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₄×C₅₃	1	8	yes	complemented
103	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
105	#384	C₂²×C₅₃	1	4	yes	—
106	#382	C₂²×C₅₃	1	4	yes	—
107	#380	—	2	4	no	—
108	C₄×C₄⋊C₄	—	2	4	no	—
109	#340	C₂²×C₅₃	1	16	yes	—
110	C₅₃	#20760	1	1	yes	characteristic, direct product factor, Sylow
111	C₂³×C₄	D₄×C₅₃	1	2	yes	—
112	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
113	C₂²×C₈	—	2	8	no	—
114	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
115	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
116	#117	—	2	16	no	—
117	C₂×C₄⋊C₄	—	2	8	no	—
118	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
119	C₂×C₂²⋊C₄	—	2	4	no	—
120	C₂×C₄²	—	2	4	no	—
121	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
122	C₂×C₄²	D₄×C₅₃	1	2	yes	—
123	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
124	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
126	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
127	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
128	#95	—	2	8	no	—
129	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
133	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
135	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
137	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
138	C₂²×C₄	#41×C₅₃	1	8	yes	—
139	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
140	C₂×C₈	—	2	16	no	—
141	C₂×C₈	—	2	8	no	—
142	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
143	C₄⋊C₄	—	2	16	no	—
144	C₄⋊C₄	—	4	16	no	—
145	C₂²⋊C₄	—	4	8	no	—
146	C₄²	—	4	8	no	—
147	C₄²	C₄⋊C₄×C₅₃	1	8	yes	—
148	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
149	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
150	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
151	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
152	C₂³	—	2	4	no	—
153	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
154	C₂³	#124×C₅₃	1	4	yes	—
155	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	16	no	—
158	C₂×C₄	—	4	8	no	—
159	C₂×C₄	—	4	16	no	—
160	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
161	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
163	C₂×C₄	#117×C₅₃	1	8	yes	—
164	C₂×C₄	#124×C₅₃	1	4	yes	—
165	C₂×C₄	#125×C₅₃	1	8	yes	—
166	C₂×C₄	#135×C₅₃	1	8	yes	—
167	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
168	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
169	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
170	C₈	—	2	8	no	—
171	C₂²	—	2	8	no	—
172	C₂²	—	2	4	no	—
173	C₂²	C₅₃×#382	1	4	yes	—
174	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
175	C₂²	C₂×#124×C₅₃	1	2	yes	—
176	C₂²	C₂×#135×C₅₃	1	4	yes	—
177	C₄	—	2	4	no	—
178	C₄	—	8	8	no	—
179	C₄	C₅₃×#382	1	4	yes	—
180	C₄	C₅₃×#427	1	8	yes	—
181	C₄	C₂×#117×C₅₃	1	2	yes	—
182	C₄	C₂×#125×C₅₃	1	2	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₂×C₅₃×#382	1	1	yes	characteristic
185	C₂	C₂×C₅₃×#427	1	2	yes	—
186	C₂	C₅₃×#3131	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1835	C₂	semidirect
4. C₅₃×#1834	C₂	semidirect
9. C₂×C₄×C₈×C₅₃	C₄	semidirect
77. #20760	C₅₃	direct (N × H)
92. #1835	C₂×C₅₃	semidirect
93. #1834	C₂×C₅₃	semidirect
102. C₂×C₄×C₈	C₄×C₅₃	semidirect
110. C₅₃	#20760	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×C₄×C₈×C₅₃	2
C₅₃×#2285	2
C₅₃×#1835	2
C₅₃×#1834	2
C₅₃×#1735	2
#20760	53

Subgroups — C53×#20760 #27228272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20760 #27228272