Subgroups — C₅₃×#9897 #27217409

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 190 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×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₅₃	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	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	—	2	1	no	—
9	C₅₃×#389	—	2	2	no	—
10	C₅₃×#389	C₄	1	4	yes	—
11	C₅₃×#384	C₂²	1	2	yes	—
12	C₅₃×#382	—	2	1	no	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#336	—	2	1	no	—
15	C₅₃×#336	—	2	2	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	Q₈	1	4	yes	—
31	C₄×C₈×C₅₃	—	4	2	no	—
32	#95×C₅₃	—	2	4	no	—
33	C₂²×C₄×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	4	4	no	—
44	C₄⋊C₄×C₅₃	—	4	8	no	—
45	C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄²×C₅₃	—	2	4	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	SD₁₆	1	2	yes	—
52	C₄²×C₅₃	Q₁₆	1	2	yes	—
53	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
54	C₄²×C₅₃	#41	1	2	yes	—
55	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
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	—	4	4	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	4	no	—
65	C₂×C₄×C₅₃	#103	1	4	yes	—
66	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#135	1	4	yes	—
73	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₈×C₅₃	—	4	4	no	—
75	C₈×C₅₃	—	8	2	no	—
76	#9897	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₅₃	#391	1	1	yes	characteristic
78	C₂²×C₅₃	#413	1	2	yes	—
79	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
80	C₂²×C₅₃	#416	1	2	yes	—
81	C₂²×C₅₃	#418	1	1	yes	characteristic
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	4	2	no	—
85	C₄×C₅₃	—	8	4	no	—
86	C₄×C₅₃	#476	1	2	yes	—
87	C₄×C₅₃	#477	1	2	yes	—
88	C₄×C₅₃	#488	1	2	yes	—
89	C₄×C₅₃	#499	1	2	yes	—
90	#2859	C₂×C₅₃	1	1	yes	characteristic
91	#1831	C₂×C₅₃	1	1	yes	characteristic
92	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
93	#1296	C₂×C₅₃	1	4	yes	—
94	C₂×C₅₃	#1542	1	2	yes	—
95	C₂×C₅₃	#1553	1	2	yes	—
96	C₂×C₅₃	#1987	1	1	yes	characteristic
97	C₂×C₅₃	#1991	1	1	yes	characteristic
98	C₂×C₅₃	#1998	1	1	yes	characteristic
99	C₂×#125	—	2	1	no	—
100	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
101	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
102	C₂×C₄×C₈	—	2	1	no	—
103	#389	—	2	2	no	—
104	#389	C₄×C₅₃	1	4	yes	—
105	#384	C₂²×C₅₃	1	2	yes	—
106	#382	—	2	1	no	—
107	C₄³	C₂²×C₅₃	1	1	yes	characteristic
108	#336	—	2	2	no	—
109	#336	—	2	1	no	—
110	C₅₃	#9897	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	#125	—	4	4	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
118	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
119	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
120	C₂×C₄²	D₄×C₅₃	1	2	yes	—
121	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	4	no	—
125	C₄⋊C₈	Q₈×C₅₃	1	4	yes	—
126	C₄×C₈	—	4	2	no	—
127	#95	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
130	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
131	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	SD₁₆×C₅₃	1	2	yes	—
147	C₄²	Q₁₆×C₅₃	1	2	yes	—
148	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
149	C₄²	#41×C₅₃	1	2	yes	—
150	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	#103×C₅₃	1	4	yes	—
161	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
162	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
163	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
164	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
165	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
167	C₂×C₄	#135×C₅₃	1	4	yes	—
168	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	8	2	no	—
171	C₂²	C₅₃×#391	1	1	yes	characteristic
172	C₂²	C₅₃×#413	1	2	yes	—
173	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
174	C₂²	C₅₃×#416	1	2	yes	—
175	C₂²	C₅₃×#418	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₅₃×#476	1	2	yes	—
181	C₄	C₅₃×#477	1	2	yes	—
182	C₄	C₅₃×#488	1	2	yes	—
183	C₄	C₅₃×#499	1	2	yes	—
184	C₂	C₅₃×#1542	1	2	yes	—
185	C₂	C₅₃×#1553	1	2	yes	—
186	C₂	C₅₃×#1987	1	1	yes	characteristic
187	C₂	C₅₃×#1991	1	1	yes	characteristic
188	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
76. #9897	C₅₃	direct (N × H)
110. C₅₃	#9897	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×#1831	2
C₄×C₄⋊C₈×C₅₃	2
C₅₃×#1296	2
#9897	53

Subgroups — C53×#9897 #27217409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9897 #27217409