Subgroups — C₅₃×#9819 #27217331

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1990	C₂	1	2	yes	maximal
3	C₅₃×#1956	C₂	1	2	yes	maximal
4	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#498	—	2	2	no	—
8	C₅₃×#498	C₄	1	4	yes	complemented
9	C₂×#106×C₅₃	—	2	2	no	—
10	C₂×#106×C₅₃	—	2	2	no	—
11	C₂×#103×C₅₃	—	2	4	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	C₂²	1	1	yes	characteristic
15	C₅₃×#384	C₂²	1	1	yes	characteristic
16	C₅₃×#340	—	2	4	no	—
17	C₈²×C₅₃	—	2	2	no	—
18	C₂²×Q₈×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	D₄	1	4	yes	—
20	C₂²×C₈×C₅₃	D₄	1	2	yes	—
21	C₄⋊Q₈×C₅₃	—	2	2	no	—
22	C₄⋊Q₈×C₅₃	—	2	2	no	—
23	C₄⋊Q₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#106×C₅₃	—	2	8	no	—
33	#106×C₅₃	—	4	4	no	—
34	#103×C₅₃	—	4	8	no	—
35	C₄×C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	—	4	2	no	—
41	C₂×Q₈×C₅₃	—	4	4	no	—
42	C₂×Q₈×C₅₃	—	4	4	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	—	4	1	no	—
45	C₂²×C₄×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₅₃	SD₁₆	1	8	yes	—
52	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
53	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	4	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	4	4	no	—
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	#41	1	2	yes	—
65	C₂³×C₅₃	#127	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	8	no	—
67	Q₈×C₅₃	—	8	4	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	8	2	no	—
70	C₂×C₄×C₅₃	—	8	2	no	—
71	C₂×C₄×C₅₃	—	8	1	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	—	8	1	no	—
74	C₂×C₄×C₅₃	—	8	2	no	—
75	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₅₃	#121	1	2	yes	—
78	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₅₃	#135	1	4	yes	—
83	C₈×C₅₃	—	2	8	no	—
84	C₈×C₅₃	#103	1	8	yes	complemented
85	#9819	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₅₃	#390	1	1	yes	characteristic
87	C₂²×C₅₃	#492	1	1	yes	characteristic
88	C₂²×C₅₃	#492	1	2	yes	—
89	C₂²×C₅₃	#494	1	1	yes	characteristic
90	C₂²×C₅₃	#495	1	2	yes	—
91	C₄×C₅₃	—	8	4	no	—
92	C₄×C₅₃	—	16	1	no	—
93	C₄×C₅₃	—	16	1	no	—
94	C₄×C₅₃	C₂×#103	1	2	yes	—
95	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
96	C₄×C₅₃	#465	1	4	yes	—
97	C₄×C₅₃	#487	1	2	yes	—
98	C₂×#498	C₂×C₅₃	1	1	yes	characteristic
99	#1990	C₂×C₅₃	1	2	yes	—
100	#1956	C₂×C₅₃	1	2	yes	—
101	#1844	C₂×C₅₃	1	1	yes	characteristic
102	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
103	C₂×C₅₃	#1705	1	1	yes	characteristic
104	C₂×C₅₃	#1707	1	2	yes	—
105	C₂×C₅₃	#1708	1	1	yes	characteristic
106	C₂×C₅₃	#1956	1	1	yes	characteristic
107	C₂×C₅₃	#1968	1	2	yes	—
108	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
109	#498	—	2	2	no	—
110	#498	C₄×C₅₃	1	4	yes	complemented
111	C₂×#106	—	2	2	no	—
112	C₂×#106	—	2	2	no	—
113	C₂×#103	—	2	4	no	—
114	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
116	#384	C₂²×C₅₃	1	1	yes	characteristic
117	#384	C₂²×C₅₃	1	1	yes	characteristic
118	#340	—	2	4	no	—
119	C₈²	—	2	2	no	—
120	C₅₃	#9819	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×C₈	D₄×C₅₃	1	4	yes	—
123	C₂²×C₈	D₄×C₅₃	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
135	#106	—	2	8	no	—
136	#106	—	4	4	no	—
137	#103	—	4	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
141	C₄×C₈	D₄×C₅₃	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
154	C₂×C₈	SD₁₆×C₅₃	1	8	yes	—
155	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
156	C₂×C₈	Q₁₆×C₅₃	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
166	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
167	C₄²	#41×C₅₃	1	2	yes	—
168	C₂³	#127×C₅₃	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
180	C₂×C₄	#121×C₅₃	1	2	yes	—
181	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
183	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
184	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
185	C₂×C₄	#135×C₅₃	1	4	yes	—
186	C₈	—	2	8	no	—
187	C₈	#103×C₅₃	1	8	yes	complemented
188	C₂²	C₅₃×#390	1	1	yes	characteristic
189	C₂²	C₅₃×#492	1	1	yes	characteristic
190	C₂²	C₅₃×#492	1	2	yes	—
191	C₂²	C₅₃×#494	1	1	yes	characteristic
192	C₂²	C₅₃×#495	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	16	1	no	—
195	C₄	—	16	1	no	—
196	C₄	C₂×#103×C₅₃	1	2	yes	—
197	C₄	C₄×SD₁₆×C₅₃	1	4	yes	—
198	C₄	C₅₃×#465	1	4	yes	—
199	C₄	C₅₃×#487	1	2	yes	—
200	C₂	C₅₃×#1705	1	1	yes	characteristic
201	C₂	C₅₃×#1707	1	2	yes	—
202	C₂	C₅₃×#1708	1	1	yes	characteristic
203	C₂	C₅₃×#1956	1	1	yes	characteristic
204	C₂	C₅₃×#1968	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₃×#498	C₄	semidirect
84. C₈×C₅₃	#103	semidirect
85. #9819	C₅₃	direct (N × H)
110. #498	C₄×C₅₃	semidirect
120. C₅₃	#9819	direct (N × H)
187. C₈	#103×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#498	2
C₅₃×#1990	2
C₅₃×#1956	2
C₅₃×#1844	2
C₂×C₈²×C₅₃	2
#9819	53

Subgroups — C53×#9819 #27217331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9819 #27217331