Subgroups — C₅₃×#9875 #27217387

Order 13568.

434 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1279	C₂	1	4	yes	maximal
5	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₅₃×#389	C₄	1	2	yes	—
9	C₅₃×#387	—	2	2	no	—
10	C₅₃×#382	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
12	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#336	—	2	2	no	—
15	C₅₃×#336	—	2	2	no	—
16	C₂²×C₈×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	—	2	2	no	—
18	#125×C₅₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	4	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	Q₈	1	4	yes	—
32	#95×C₅₃	—	2	2	no	—
33	#95×C₅₃	—	2	2	no	—
34	#95×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	4	4	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	4	4	no	—
46	C₄⋊C₄×C₅₃	—	2	8	no	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₄⋊C₄×C₅₃	—	4	4	no	—
50	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	—	4	4	no	—
56	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
57	C₄²×C₅₃	#41	1	2	yes	—
58	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	2	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	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	#131	1	2	yes	—
78	C₂×C₄×C₅₃	#135	1	4	yes	—
79	C₈×C₅₃	—	4	4	no	—
80	C₈×C₅₃	—	8	2	no	—
81	#9875	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂²×C₅₃	#391	1	1	yes	characteristic
83	C₂²×C₅₃	#407	1	1	yes	characteristic
84	C₂²×C₅₃	#408	1	1	yes	characteristic
85	C₂²×C₅₃	#416	1	2	yes	—
86	C₂²×C₅₃	#421	1	2	yes	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	4	2	no	—
89	C₄×C₅₃	—	4	2	no	—
90	C₄×C₅₃	—	4	4	no	—
91	C₄×C₅₃	—	8	2	no	—
92	C₄×C₅₃	#446	1	4	yes	—
93	C₄×C₅₃	#488	1	2	yes	—
94	C₄×C₅₃	#499	1	2	yes	—
95	C₄×#125	C₂×C₅₃	1	1	yes	characteristic
96	#1831	C₂×C₅₃	1	1	yes	characteristic
97	#1744	C₂×C₅₃	1	1	yes	characteristic
98	#1279	C₂×C₅₃	1	4	yes	—
99	C₂×C₅₃	#1482	1	2	yes	—
100	C₂×C₅₃	#1488	1	2	yes	—
101	C₂×C₅₃	#1985	1	1	yes	characteristic
102	C₂×C₅₃	#1991	1	1	yes	characteristic
103	C₂×C₅₃	#1995	1	1	yes	characteristic
104	C₂×#125	C₄×C₅₃	1	1	yes	characteristic
105	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
107	#389	C₄×C₅₃	1	2	yes	—
108	#387	—	2	2	no	—
109	#382	C₄×C₅₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	2	2	no	—
111	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
112	C₄³	C₂²×C₅₃	1	1	yes	characteristic
113	#336	—	2	2	no	—
114	#336	—	2	2	no	—
115	C₅₃	#9875	1	1	yes	characteristic, direct product factor, Sylow
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	#125	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
126	C₂×C₄²	D₄×C₅₃	1	2	yes	—
127	C₂×C₄²	D₄×C₅₃	1	2	yes	—
128	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	4	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	Q₈×C₅₃	1	4	yes	—
132	#95	—	2	2	no	—
133	#95	—	2	2	no	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
138	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
139	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	4	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	4	4	no	—
156	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
157	C₄²	#41×C₅₃	1	2	yes	—
158	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
173	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
174	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
175	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
176	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
177	C₂×C₄	#131×C₅₃	1	2	yes	—
178	C₂×C₄	#135×C₅₃	1	4	yes	—
179	C₈	—	4	4	no	—
180	C₈	—	8	2	no	—
181	C₂²	C₅₃×#391	1	1	yes	characteristic
182	C₂²	C₅₃×#407	1	1	yes	characteristic
183	C₂²	C₅₃×#408	1	1	yes	characteristic
184	C₂²	C₅₃×#416	1	2	yes	—
185	C₂²	C₅₃×#421	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	C₅₃×#446	1	4	yes	—
192	C₄	C₅₃×#488	1	2	yes	—
193	C₄	C₅₃×#499	1	2	yes	—
194	C₂	C₅₃×#1482	1	2	yes	—
195	C₂	C₅₃×#1488	1	2	yes	—
196	C₂	C₅₃×#1985	1	1	yes	characteristic
197	C₂	C₅₃×#1991	1	1	yes	characteristic
198	C₂	C₅₃×#1995	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
81. #9875	C₅₃	direct (N × H)
115. C₅₃	#9875	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×#125×C₅₃	2
C₅₃×#1831	2
C₅₃×#1744	2
C₅₃×#1279	2
#9875	53

Subgroups — C53×#9875 #27217387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9875 #27217387