Subgroups — C₅₃×#8262 #27215774

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₈×C₅₃	C₂	1	2	yes	maximal
2	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1273	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	—
7	C₂×#97×C₅₃	C₄	1	2	yes	—
8	C₂×#97×C₅₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#336	C₄	1	4	yes	—
15	C₅₃×#322	—	2	2	no	—
16	C₈²×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	—	2	8	no	—
27	#97×C₅₃	—	2	4	no	—
28	#97×C₅₃	C₈	1	8	yes	complemented
29	C₄×C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₄×C₈×C₅₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
52	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
53	C₄²×C₅₃	—	2	2	no	—
54	C₄²×C₅₃	—	2	4	no	—
55	C₄²×C₅₃	—	2	4	no	—
56	C₄²×C₅₃	C₂×C₈	1	4	yes	—
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	2	yes	—
60	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
75	C₂×C₄×C₅₃	#131	1	4	yes	—
76	C₈×C₅₃	—	2	4	no	—
77	C₈×C₅₃	—	2	8	no	—
78	C₈×C₅₃	—	4	4	no	—
79	C₈×C₅₃	C₄⋊C₈	1	8	yes	complemented
80	#8262	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
84	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₅₃	#405	1	2	yes	—
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	—	4	1	no	—
88	C₄×C₅₃	—	4	1	no	—
89	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
90	C₄×C₅₃	#431	1	2	yes	—
91	C₄×C₅₃	#435	1	4	yes	—
92	C₄×C₅₃	#446	1	4	yes	—
93	C₄×C₄⋊C₈	C₂×C₅₃	1	2	yes	—
94	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
95	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
96	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
97	#1273	C₂×C₅₃	1	2	yes	—
98	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
99	C₂×C₅₃	#1453	1	1	yes	characteristic
100	C₂×C₅₃	#1455	1	1	yes	characteristic
101	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₅₃	#1771	1	2	yes	—
103	C₂×C₄⋊C₈	C₄×C₅₃	1	4	yes	—
104	C₂×#97	C₄×C₅₃	1	2	yes	—
105	C₂×#97	C₄×C₅₃	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
110	C₄³	C₂²×C₅₃	1	1	yes	characteristic
111	#336	C₄×C₅₃	1	4	yes	—
112	#322	—	2	2	no	—
113	C₈²	—	2	2	no	—
114	C₅₃	#8262	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
116	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
117	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
118	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
119	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
120	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
122	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
124	C₄⋊C₈	—	2	8	no	—
125	#97	—	2	4	no	—
126	#97	C₈×C₅₃	1	8	yes	complemented
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	D₄×C₅₃	1	2	yes	—
133	C₄×C₈	Q₈×C₅₃	1	2	yes	—
134	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
137	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
147	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
148	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
149	C₂×C₈	M₁₆×C₅₃	1	8	yes	—
150	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	C₂×C₈×C₅₃	1	4	yes	—
155	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
156	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
157	C₄²	#41×C₅₃	1	2	yes	—
158	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	C₄×C₈×C₅₃	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
167	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
168	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
169	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
170	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
171	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
172	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
173	C₂×C₄	#131×C₅₃	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	2	8	no	—
176	C₈	—	4	4	no	—
177	C₈	C₄⋊C₈×C₅₃	1	8	yes	complemented
178	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
179	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
180	C₂²	C₄×M₁₆×C₅₃	1	2	yes	—
181	C₂²	C₄×M₁₆×C₅₃	1	1	yes	characteristic
182	C₂²	C₅₃×#405	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	C₂×C₄⋊C₈×C₅₃	1	2	yes	—
187	C₄	C₅₃×#431	1	2	yes	—
188	C₄	C₅₃×#435	1	4	yes	—
189	C₄	C₅₃×#446	1	4	yes	—
190	C₂	C₈×M₁₆×C₅₃	1	2	yes	—
191	C₂	C₅₃×#1453	1	1	yes	characteristic
192	C₂	C₅₃×#1455	1	1	yes	characteristic
193	C₂	C₄×C₄⋊C₈×C₅₃	1	1	yes	characteristic
194	C₂	C₅₃×#1771	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅₃	C₂³
commutator	C₂²	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×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
28. #97×C₅₃	C₈	semidirect
79. C₈×C₅₃	C₄⋊C₈	semidirect
80. #8262	C₅₃	direct (N × H)
114. C₅₃	#8262	direct (N × H)
126. #97	C₈×C₅₃	semidirect
177. C₈	C₄⋊C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₈×C₅₃	2
C₄×#97×C₅₃	2
C₂×C₅₃×#322	2
C₂×C₈²×C₅₃	2
C₅₃×#1273	2
#8262	53

Subgroups — C53×#8262 #27215774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8262 #27215774