Subgroups — C₅₃×#2336 #20271273

Order 6784.

428 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#389	C₂	1	2	yes	maximal
3	C₅₃×#387	C₂	1	2	yes	maximal
4	C₅₃×#387	C₂	1	2	yes	maximal
5	C₅₃×#387	C₂	1	2	yes	maximal
6	C₅₃×#387	C₂	1	2	yes	maximal
7	C₅₃×#383	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
10	#125×C₅₃	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	#95×C₅₃	C₂²	1	2	yes	—
24	#95×C₅₃	C₂²	1	2	yes	—
25	#95×C₅₃	C₂²	1	2	yes	—
26	#95×C₅₃	C₂²	1	1	yes	characteristic
27	#95×C₅₃	C₂²	1	2	yes	—
28	#95×C₅₃	C₂²	1	1	yes	characteristic
29	#95×C₅₃	C₂²	1	2	yes	—
30	#95×C₅₃	C₂²	1	2	yes	—
31	#95×C₅₃	C₂²	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
45	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
46	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
47	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
48	C₄²×C₅₃	—	2	4	no	—
49	C₄²×C₅₃	—	4	1	no	—
50	C₄²×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	—	4	1	no	—
52	C₄²×C₅₃	C₂×C₄	1	4	yes	—
53	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	1	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	1	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	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₅₃	#41	1	4	yes	—
79	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
80	C₂²×C₅₃	C₂×#41	1	2	yes	—
81	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
83	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	2	4	no	—
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	—	4	2	no	—
92	C₄×C₅₃	—	4	2	no	—
93	#2336	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
95	C₂×C₅₃	#519	1	1	yes	characteristic
96	C₂×C₅₃	#520	1	1	yes	characteristic
97	C₂×C₅₃	#539	1	2	yes	—
98	C₂×C₅₃	#541	1	2	yes	—
99	C₂×#125	C₂×C₅₃	1	1	yes	characteristic
100	#389	C₂×C₅₃	1	2	yes	—
101	#387	C₂×C₅₃	1	2	yes	—
102	#387	C₂×C₅₃	1	2	yes	—
103	#387	C₂×C₅₃	1	2	yes	—
104	#387	C₂×C₅₃	1	2	yes	—
105	#383	C₂×C₅₃	1	1	yes	characteristic
106	C₄×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
107	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
108	C₅₃	#2336	1	1	yes	characteristic, direct product factor, Sylow
109	#125	C₄×C₅₃	1	8	yes	complemented
110	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
117	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
118	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
121	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
122	#95	C₂²×C₅₃	1	2	yes	—
123	#95	C₂²×C₅₃	1	2	yes	—
124	#95	C₂²×C₅₃	1	2	yes	—
125	#95	C₂²×C₅₃	1	1	yes	characteristic
126	#95	C₂²×C₅₃	1	2	yes	—
127	#95	C₂²×C₅₃	1	1	yes	characteristic
128	#95	C₂²×C₅₃	1	2	yes	—
129	#95	C₂²×C₅₃	1	2	yes	—
130	#95	C₂²×C₅₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
132	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
137	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
144	C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
145	C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
146	C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
147	C₄²	—	2	4	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	1	no	—
151	C₄²	C₂×C₄×C₅₃	1	4	yes	—
152	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	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	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
173	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
174	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
175	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
176	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
177	C₂×C₄	#41×C₅₃	1	4	yes	—
178	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
179	C₂²	C₂×#41×C₅₃	1	2	yes	—
180	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
181	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
182	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₂	C₄×#41×C₅₃	1	1	yes	characteristic
193	C₂	C₅₃×#519	1	1	yes	characteristic
194	C₂	C₅₃×#520	1	1	yes	characteristic
195	C₂	C₅₃×#539	1	2	yes	—
196	C₂	C₅₃×#541	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
10. #125×C₅₃	C₄	semidirect
93. #2336	C₅₃	direct (N × H)
108. C₅₃	#2336	direct (N × H)
109. #125	C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125×C₅₃	2
C₅₃×#389	2
C₅₃×#387	2
C₅₃×#383	2
C₄×C₄⋊C₄×C₅₃	2
#2336	53

Subgroups — C53×#2336 #20271273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2336 #20271273