Subgroups — C₅₃×#36701 #27244213

Order 13568.

1304 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2845	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2839	C₂	1	12	yes	maximal
3	C₅₃×#2822	C₂	1	3	yes	maximal
4	C₅₃×#2402	C₂	1	12	yes	maximal
5	C₄×#125×C₅₃	C₂	1	3	yes	maximal
6	C₂×#125×C₅₃	C₂²	1	3	yes	—
7	C₅₃×#400	C₂²	1	12	yes	—
8	C₅₃×#400	C₂²	1	12	yes	—
9	C₅₃×#400	C₂²	1	12	yes	—
10	C₅₃×#400	C₂²	1	12	yes	—
11	C₅₃×#398	C₂²	1	12	yes	—
12	C₅₃×#398	C₂²	1	12	yes	—
13	C₅₃×#398	C₂²	1	12	yes	—
14	C₅₃×#398	C₂²	1	12	yes	—
15	C₅₃×#389	C₂²	1	6	yes	—
16	C₅₃×#387	C₂²	1	12	yes	—
17	C₅₃×#382	C₂²	1	3	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
19	C₄×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
21	C₅₃×#376	C₂²	1	4	yes	—
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	#125×C₅₃	—	2	12	no	—
24	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
26	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
27	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
28	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
29	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
34	#95×C₅₃	C₂³	1	12	yes	—
35	#95×C₅₃	C₂³	1	4	yes	—
36	#95×C₅₃	C₂³	1	12	yes	—
37	#95×C₅₃	C₂³	1	12	yes	—
38	#95×C₅₃	C₂³	1	12	yes	—
39	#95×C₅₃	C₂³	1	12	yes	—
40	#95×C₅₃	C₂³	1	12	yes	—
41	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
43	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
44	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
45	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	2	24	no	—
47	C₄⋊C₄×C₅₃	—	2	12	no	—
48	C₄⋊C₄×C₅₃	—	4	6	no	—
49	C₄⋊C₄×C₅₃	—	4	12	no	—
50	C₄⋊C₄×C₅₃	—	4	12	no	—
51	C₄⋊C₄×C₅₃	—	4	12	no	—
52	C₄²×C₅₃	—	2	6	no	—
53	C₄²×C₅₃	—	2	24	no	—
54	C₄²×C₅₃	—	4	3	no	—
55	C₄²×C₅₃	#41	1	4	yes	—
56	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	3	no	—
58	C₂×C₄×C₅₃	—	2	3	no	—
59	C₂×C₄×C₅₃	—	2	12	no	—
60	C₂×C₄×C₅₃	—	2	12	no	—
61	C₂×C₄×C₅₃	—	2	3	no	—
62	C₂×C₄×C₅₃	—	2	12	no	—
63	C₂×C₄×C₅₃	—	2	6	no	—
64	C₂×C₄×C₅₃	—	2	12	no	—
65	C₂×C₄×C₅₃	—	2	12	no	—
66	C₂×C₄×C₅₃	—	2	12	no	—
67	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
68	C₂×C₄×C₅₃	2₋¹⁺⁴	1	6	yes	—
69	#36701	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
71	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	3	yes	—
72	C₂²×C₅₃	#585	1	3	yes	—
73	C₄×C₅₃	—	2	6	no	—
74	C₄×C₅₃	—	4	12	no	—
75	C₄×C₅₃	—	4	3	no	—
76	C₄×C₅₃	—	4	12	no	—
77	C₄×C₅₃	#564	1	4	yes	—
78	#2845	C₂×C₅₃	1	1	yes	characteristic
79	#2839	C₂×C₅₃	1	12	yes	—
80	#2822	C₂×C₅₃	1	3	yes	—
81	#2402	C₂×C₅₃	1	12	yes	—
82	C₄×#125	C₂×C₅₃	1	3	yes	—
83	C₂×C₅₃	#3461	1	3	yes	—
84	C₂×C₅₃	C₂×#564	1	1	yes	characteristic
85	C₂×C₅₃	#3531	1	3	yes	—
86	C₂×#125	C₂²×C₅₃	1	3	yes	—
87	#400	C₂²×C₅₃	1	12	yes	—
88	#400	C₂²×C₅₃	1	12	yes	—
89	#400	C₂²×C₅₃	1	12	yes	—
90	#400	C₂²×C₅₃	1	12	yes	—
91	#398	C₂²×C₅₃	1	12	yes	—
92	#398	C₂²×C₅₃	1	12	yes	—
93	#398	C₂²×C₅₃	1	12	yes	—
94	#398	C₂²×C₅₃	1	12	yes	—
95	#389	C₂²×C₅₃	1	6	yes	—
96	#387	C₂²×C₅₃	1	12	yes	—
97	#382	C₂²×C₅₃	1	3	yes	—
98	C₄×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
99	C₄×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
100	C₄×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
101	#376	C₂²×C₅₃	1	4	yes	—
102	C₄³	C₂²×C₅₃	1	1	yes	characteristic
103	C₅₃	#36701	1	1	yes	characteristic, direct product factor, Sylow
104	#125	—	2	12	no	—
105	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
106	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
107	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
108	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
109	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
110	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
111	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
112	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
113	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
114	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
115	#95	C₂³×C₅₃	1	12	yes	—
116	#95	C₂³×C₅₃	1	4	yes	—
117	#95	C₂³×C₅₃	1	12	yes	—
118	#95	C₂³×C₅₃	1	12	yes	—
119	#95	C₂³×C₅₃	1	12	yes	—
120	#95	C₂³×C₅₃	1	12	yes	—
121	#95	C₂³×C₅₃	1	12	yes	—
122	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
123	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
124	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
125	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
126	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	24	no	—
128	C₄⋊C₄	—	2	12	no	—
129	C₄⋊C₄	—	4	6	no	—
130	C₄⋊C₄	—	4	12	no	—
131	C₄⋊C₄	—	4	12	no	—
132	C₄⋊C₄	—	4	12	no	—
133	C₄²	—	2	6	no	—
134	C₄²	—	2	24	no	—
135	C₄²	—	4	3	no	—
136	C₄²	#41×C₅₃	1	4	yes	—
137	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	3	no	—
140	C₂×C₄	—	2	12	no	—
141	C₂×C₄	—	2	12	no	—
142	C₂×C₄	—	2	3	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	2	6	no	—
145	C₂×C₄	—	2	12	no	—
146	C₂×C₄	—	2	12	no	—
147	C₂×C₄	—	2	12	no	—
148	C₂×C₄	C₂×#41×C₅₃	1	6	yes	—
149	C₂×C₄	2₋¹⁺⁴×C₅₃	1	6	yes	—
150	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
151	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	3	yes	—
152	C₂²	C₅₃×#585	1	3	yes	—
153	C₄	—	2	6	no	—
154	C₄	—	4	12	no	—
155	C₄	—	4	3	no	—
156	C₄	—	4	12	no	—
157	C₄	C₅₃×#564	1	4	yes	—
158	C₂	C₅₃×#3461	1	3	yes	—
159	C₂	C₂×C₅₃×#564	1	1	yes	characteristic
160	C₂	C₅₃×#3531	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle, prime radical	C₂³×C₅₃	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
69. #36701	C₅₃	direct (N × H)
103. C₅₃	#36701	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×#2845	2
C₅₃×#2839	2
C₅₃×#2822	2
C₅₃×#2402	2
C₄×#125×C₅₃	2
#36701	53

Subgroups — C53×#36701 #27244213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36701 #27244213