Subgroups — C₅₃×#9902 #27217414

Order 13568.

438 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	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1296	C₂	1	4	yes	maximal
5	C₂×#125×C₅₃	—	2	1	no	—
6	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
7	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×#97×C₅₃	—	2	1	no	—
10	C₅₃×#389	—	2	2	no	—
11	C₅₃×#389	C₄	1	4	yes	—
12	C₅₃×#384	C₂²	1	2	yes	—
13	C₅₃×#382	—	2	1	no	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#336	—	2	1	no	—
16	C₂²×C₈×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	—	2	2	no	—
18	#125×C₅₃	—	4	4	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	2	yes	—
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	—	2	4	no	—
30	C₄⋊C₈×C₅₃	—	4	4	no	—
31	C₄⋊C₈×C₅₃	Q₈	1	4	yes	—
32	#97×C₅₃	—	4	2	no	—
33	#95×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	4	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	4	4	no	—
45	C₄⋊C₄×C₅₃	—	4	8	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊C₄×C₅₃	—	4	4	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	SD₁₆	1	2	yes	—
55	C₄²×C₅₃	Q₁₆	1	2	yes	—
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	2	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	—	4	4	no	—
66	C₂×C₄×C₅₃	—	4	4	no	—
67	C₂×C₄×C₅₃	—	4	4	no	—
68	C₂×C₄×C₅₃	#103	1	4	yes	—
69	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₅₃	#137	1	2	yes	—
78	C₈×C₅₃	—	4	4	no	—
79	C₈×C₅₃	—	8	2	no	—
80	#9902	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	#391	1	1	yes	characteristic
82	C₂²×C₅₃	#413	1	2	yes	—
83	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
84	C₂²×C₅₃	#417	1	2	yes	—
85	C₂²×C₅₃	#418	1	1	yes	characteristic
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	4	2	no	—
89	C₄×C₅₃	—	8	4	no	—
90	C₄×C₅₃	#476	1	2	yes	—
91	C₄×C₅₃	#477	1	2	yes	—
92	C₄×C₅₃	#490	1	1	yes	characteristic
93	C₄×C₅₃	#491	1	1	yes	characteristic
94	C₄×C₅₃	#501	1	2	yes	—
95	#2859	C₂×C₅₃	1	1	yes	characteristic
96	#1842	C₂×C₅₃	1	1	yes	characteristic
97	#1767	C₂×C₅₃	1	1	yes	characteristic
98	#1296	C₂×C₅₃	1	4	yes	—
99	C₂×C₅₃	#1550	1	2	yes	—
100	C₂×C₅₃	#1557	1	2	yes	—
101	C₂×C₅₃	#1987	1	1	yes	characteristic
102	C₂×C₅₃	#1992	1	1	yes	characteristic
103	C₂×C₅₃	#1998	1	1	yes	characteristic
104	C₂×#125	—	2	1	no	—
105	C₂×C₄⋊C₈	—	2	2	no	—
106	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
107	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
108	C₂×#97	—	2	1	no	—
109	#389	—	2	2	no	—
110	#389	C₄×C₅₃	1	4	yes	—
111	#384	C₂²×C₅₃	1	2	yes	—
112	#382	—	2	1	no	—
113	C₄³	C₂²×C₅₃	1	1	yes	characteristic
114	#336	—	2	1	no	—
115	C₅₃	#9902	1	1	yes	characteristic, direct product factor, Sylow
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	2	2	no	—
118	#125	—	4	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
123	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
124	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
125	C₂×C₄²	D₄×C₅₃	1	2	yes	—
126	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
127	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	4	no	—
130	C₄⋊C₈	—	4	4	no	—
131	C₄⋊C₈	Q₈×C₅₃	1	4	yes	—
132	#97	—	4	2	no	—
133	#95	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
136	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
137	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	8	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	SD₁₆×C₅₃	1	2	yes	—
155	C₄²	Q₁₆×C₅₃	1	2	yes	—
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	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#103×C₅₃	1	4	yes	—
169	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
170	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
171	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
172	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
173	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
175	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
176	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
177	C₂×C₄	#137×C₅₃	1	2	yes	—
178	C₈	—	4	4	no	—
179	C₈	—	8	2	no	—
180	C₂²	C₅₃×#391	1	1	yes	characteristic
181	C₂²	C₅₃×#413	1	2	yes	—
182	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
183	C₂²	C₅₃×#417	1	2	yes	—
184	C₂²	C₅₃×#418	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	4	no	—
189	C₄	C₅₃×#476	1	2	yes	—
190	C₄	C₅₃×#477	1	2	yes	—
191	C₄	C₅₃×#490	1	1	yes	characteristic
192	C₄	C₅₃×#491	1	1	yes	characteristic
193	C₄	C₅₃×#501	1	2	yes	—
194	C₂	C₅₃×#1550	1	2	yes	—
195	C₂	C₅₃×#1557	1	2	yes	—
196	C₂	C₅₃×#1987	1	1	yes	characteristic
197	C₂	C₅₃×#1992	1	1	yes	characteristic
198	C₂	C₅₃×#1998	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
80. #9902	C₅₃	direct (N × H)
115. C₅₃	#9902	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×#2859	2
C₅₃×#1842	2
C₅₃×#1767	2
C₅₃×#1296	2
#9902	53

Subgroups — C53×#9902 #27217414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9902 #27217414