Subgroups — C₅₃×#9631 #27217143

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1770	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1570	C₂	1	2	yes	maximal
4	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1303	C₂	1	2	yes	maximal
6	C₅₃×#431	C₂²	1	2	yes	—
7	C₂×#108×C₅₃	—	2	1	no	—
8	C₂×#108×C₅₃	—	2	1	no	—
9	C₅₃×#423	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#97×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#336	—	2	1	no	—
15	C₅₃×#335	C₄	1	2	yes	complemented
16	C₅₃×#334	—	2	4	no	—
17	C₅₃×#334	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₂×C₄²×C₅₃	—	2	1	no	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	#108×C₅₃	—	4	2	no	—
26	#108×C₅₃	—	4	2	no	—
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₄⋊C₈×C₅₃	—	2	1	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
32	C₂²⋊C₈×C₅₃	—	2	2	no	—
33	#97×C₅₃	—	4	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
39	#95×C₅₃	Q₈	1	2	yes	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
44	M₁₆×C₅₃	—	4	2	no	—
45	M₁₆×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	4	1	no	—
51	C₂×C₈×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄²×C₅₃	—	4	1	no	—
61	C₄²×C₅₃	—	4	1	no	—
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₅₃	#41	1	1	yes	characteristic
66	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	1	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	4	4	no	—
85	C₈×C₅₃	—	4	2	no	—
86	C₈×C₅₃	—	8	2	no	—
87	C₈×C₅₃	#106	1	2	yes	—
88	C₈×C₅₃	#106	1	2	yes	—
89	#9631	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	#389	1	1	yes	characteristic
93	C₂²×C₅₃	#475	1	1	yes	characteristic
94	C₂²×C₅₃	#476	1	1	yes	characteristic
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	8	2	no	—
99	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
100	C₄×C₅₃	#428	1	1	yes	characteristic
101	C₄×C₅₃	#444	1	2	yes	—
102	C₄×C₅₃	#460	1	1	yes	characteristic
103	C₄×C₅₃	#461	1	1	yes	characteristic
104	#1848	C₂×C₅₃	1	1	yes	characteristic
105	#1770	C₂×C₅₃	1	1	yes	characteristic
106	#1570	C₂×C₅₃	1	2	yes	—
107	#1564	C₂×C₅₃	1	1	yes	characteristic
108	#1303	C₂×C₅₃	1	2	yes	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	#1946	1	1	yes	characteristic
112	C₂×C₅₃	#1948	1	1	yes	characteristic
113	C₂×C₅₃	#1953	1	1	yes	characteristic
114	#431	C₂²×C₅₃	1	2	yes	—
115	C₂×#108	—	2	1	no	—
116	C₂×#108	—	2	1	no	—
117	#423	C₂²×C₅₃	1	2	yes	—
118	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
119	C₂×#97	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
121	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
122	#336	—	2	1	no	—
123	#335	C₄×C₅₃	1	2	yes	complemented
124	#334	—	2	4	no	—
125	#334	C₄×C₅₃	1	2	yes	complemented
126	C₅₃	#9631	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×M₁₆	—	2	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
134	#108	—	4	2	no	—
135	#108	—	4	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	1	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
140	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
141	C₂²⋊C₈	—	2	2	no	—
142	#97	—	4	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
147	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
148	#95	Q₈×C₅₃	1	2	yes	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
152	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
163	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
166	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
167	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
173	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
174	C₄²	#41×C₅₃	1	1	yes	characteristic
175	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
183	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
184	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
185	C₂×C₄	#125×C₅₃	1	1	yes	characteristic
186	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
189	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
190	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	4	2	no	—
195	C₈	—	8	2	no	—
196	C₈	#106×C₅₃	1	2	yes	—
197	C₈	#106×C₅₃	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₅₃×#389	1	1	yes	characteristic
201	C₂²	C₅₃×#475	1	1	yes	characteristic
202	C₂²	C₅₃×#476	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	8	2	no	—
207	C₄	C₂×#106×C₅₃	1	1	yes	characteristic
208	C₄	C₅₃×#428	1	1	yes	characteristic
209	C₄	C₅₃×#444	1	2	yes	—
210	C₄	C₅₃×#460	1	1	yes	characteristic
211	C₄	C₅₃×#461	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	C₅₃×#1946	1	1	yes	characteristic
215	C₂	C₅₃×#1948	1	1	yes	characteristic
216	C₂	C₅₃×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ 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
15. C₅₃×#335	C₄	semidirect
17. C₅₃×#334	C₄	semidirect
89. #9631	C₅₃	direct (N × H)
123. #335	C₄×C₅₃	semidirect
125. #334	C₄×C₅₃	semidirect
126. C₅₃	#9631	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#1848	2
C₅₃×#1770	2
C₅₃×#1570	2
C₅₃×#1564	2
C₅₃×#1303	2
#9631	53

Subgroups — C53×#9631 #27217143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9631 #27217143