Subgroups — C₅₃×#9610 #27217122

Order 13568.

294 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1565	C₂	1	2	yes	maximal
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1283	C₂	1	2	yes	maximal
6	C₅₃×#431	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₂²	1	1	yes	characteristic
8	C₂×#108×C₅₃	C₄	1	2	yes	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#341	C₄	1	1	yes	characteristic
13	C₅₃×#341	C₄	1	1	yes	characteristic
14	C₅₃×#335	—	2	2	no	—
15	C₅₃×#335	—	2	2	no	—
16	C₅₃×#334	—	2	2	no	—
17	C₅₃×#322	—	2	2	no	—
18	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#108×C₅₃	—	2	4	no	—
25	C₄⋊C₈×C₅₃	—	2	2	no	—
26	C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₄⋊C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²⋊C₈×C₅₃	—	2	1	no	—
31	C₂²⋊C₈×C₅₃	Q₈	1	2	yes	—
32	#97×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	M₁₆×C₅₃	—	2	2	no	—
42	M₁₆×C₅₃	—	2	2	no	—
43	M₁₆×C₅₃	—	2	2	no	—
44	M₁₆×C₅₃	M₁₆	1	4	yes	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	8	2	no	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	1	yes	characteristic
60	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	4	1	no	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₅₃	#131	1	2	yes	—
72	C₈×C₅₃	—	2	2	no	—
73	C₈×C₅₃	—	2	2	no	—
74	C₈×C₅₃	—	2	4	no	—
75	C₈×C₅₃	—	4	2	no	—
76	C₈×C₅₃	—	4	2	no	—
77	C₈×C₅₃	—	4	1	no	—
78	C₈×C₅₃	—	4	1	no	—
79	C₈×C₅₃	—	8	2	no	—
80	#9610	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	—	2	1	no	—
83	C₂²×C₅₃	#389	1	1	yes	characteristic
84	C₂²×C₅₃	#446	1	2	yes	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	#423	1	1	yes	characteristic
89	C₄×C₅₃	#424	1	1	yes	characteristic
90	C₄×C₅₃	#435	1	2	yes	—
91	C₄×C₅₃	#472	1	1	yes	characteristic
92	C₄×C₅₃	#473	1	1	yes	characteristic
93	C₄×#108	C₂×C₅₃	1	1	yes	characteristic
94	#1565	C₂×C₅₃	1	2	yes	—
95	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
96	#1452	C₂×C₅₃	1	1	yes	characteristic
97	#1283	C₂×C₅₃	1	2	yes	—
98	C₂×C₅₃	—	2	2	no	—
99	C₂×C₅₃	#1940	1	1	yes	characteristic
100	C₂×C₅₃	#1952	1	1	yes	characteristic
101	C₂×C₅₃	#1953	1	1	yes	characteristic
102	#431	C₂²×C₅₃	1	1	yes	characteristic
103	#431	C₂²×C₅₃	1	1	yes	characteristic
104	C₂×#108	C₄×C₅₃	1	2	yes	—
105	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
106	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
108	#341	C₄×C₅₃	1	1	yes	characteristic
109	#341	C₄×C₅₃	1	1	yes	characteristic
110	#335	—	2	2	no	—
111	#335	—	2	2	no	—
112	#334	—	2	2	no	—
113	#322	—	2	2	no	—
114	C₅₃	#9610	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
116	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
121	#108	—	2	4	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	1	no	—
125	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
126	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
127	C₂²⋊C₈	—	2	1	no	—
128	C₂²⋊C₈	Q₈×C₅₃	1	2	yes	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	D₄×C₅₃	1	2	yes	—
134	C₄×C₈	Q₈×C₅₃	1	2	yes	—
135	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
137	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	M₁₆×C₅₃	1	4	yes	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	8	2	no	—
151	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
152	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
153	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
154	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
155	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
156	C₄²	#41×C₅₃	1	1	yes	characteristic
157	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
162	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
163	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
165	C₂×C₄	#125×C₅₃	1	1	yes	characteristic
166	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
167	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
168	C₂×C₄	#131×C₅₃	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	4	no	—
172	C₈	—	4	2	no	—
173	C₈	—	4	2	no	—
174	C₈	—	4	1	no	—
175	C₈	—	4	1	no	—
176	C₈	—	8	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	C₅₃×#389	1	1	yes	characteristic
180	C₂²	C₅₃×#446	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₅₃×#423	1	1	yes	characteristic
185	C₄	C₅₃×#424	1	1	yes	characteristic
186	C₄	C₅₃×#435	1	2	yes	—
187	C₄	C₅₃×#472	1	1	yes	characteristic
188	C₄	C₅₃×#473	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	C₅₃×#1940	1	1	yes	characteristic
191	C₂	C₅₃×#1952	1	1	yes	characteristic
192	C₂	C₅₃×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
80. #9610	C₅₃	direct (N × H)
114. C₅₃	#9610	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#108×C₅₃	2
C₅₃×#1565	2
C₂×C₅₃×#335	2
C₅₃×#1452	2
C₅₃×#1283	2
#9610	53

Subgroups — C53×#9610 #27217122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9610 #27217122