Subgroups — C₅₃×#2482 #20271419

Order 6784.

672 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#399	C₂	1	2	yes	maximal, complemented
4	C₅₃×#397	C₂	1	2	yes	maximal, complemented
5	C₅₃×#388	C₂	1	4	yes	maximal, complemented
6	C₅₃×#385	C₂	1	2	yes	maximal, complemented
7	C₅₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#379	C₂	1	2	yes	maximal
9	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	#117×C₅₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
16	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
17	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂×C₄²×C₅₃	C₂²	1	2	yes	complemented
24	#95×C₅₃	C₂²	1	2	yes	complemented
25	#95×C₅₃	C₂²	1	2	yes	complemented
26	#95×C₅₃	C₂²	1	4	yes	complemented
27	C₂⁴×C₅₃	—	2	2	no	—
28	C₂⁴×C₅₃	—	2	4	no	—
29	C₂⁴×C₅₃	—	2	4	no	—
30	C₂⁴×C₅₃	—	4	2	no	—
31	C₂⁴×C₅₃	C₂³	1	2	yes	—
32	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	D₄	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₂²⋊C₄×C₅₃	—	2	4	no	—
47	C₂²⋊C₄×C₅₃	—	2	8	no	—
48	C₂²⋊C₄×C₅₃	—	2	4	no	—
49	C₂²⋊C₄×C₅₃	—	2	4	no	—
50	C₂²⋊C₄×C₅₃	—	2	4	no	—
51	C₂²⋊C₄×C₅₃	—	4	4	no	—
52	C₄²×C₅₃	—	2	4	no	—
53	C₂³×C₅₃	—	2	2	no	—
54	C₂³×C₅₃	—	2	2	no	—
55	C₂³×C₅₃	—	2	8	no	—
56	C₂³×C₅₃	—	2	2	no	—
57	C₂³×C₅₃	—	2	2	no	—
58	C₂³×C₅₃	—	2	4	no	—
59	C₂³×C₅₃	—	2	4	no	—
60	C₂³×C₅₃	—	4	16	no	—
61	C₂³×C₅₃	—	4	4	no	—
62	C₂³×C₅₃	—	4	4	no	—
63	C₂³×C₅₃	C₂×D₄	1	2	yes	—
64	C₂³×C₅₃	#41	1	4	yes	—
65	C₂³×C₅₃	#41	1	4	yes	—
66	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	8	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	—	2	4	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	—	2	8	no	—
85	C₂²×C₅₃	—	2	4	no	—
86	C₂²×C₅₃	—	2	8	no	—
87	C₂²×C₅₃	—	4	16	no	—
88	C₂²×C₅₃	—	4	2	no	—
89	C₂²×C₅₃	—	4	2	no	—
90	C₂²×C₅₃	—	4	4	no	—
91	C₂²×C₅₃	#123	1	4	yes	—
92	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×#41	1	2	yes	—
94	C₂²×C₅₃	C₂×#41	1	2	yes	—
95	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	—	4	2	no	—
99	C₄×C₅₃	—	4	2	no	—
100	C₄×C₅₃	—	4	2	no	—
101	C₄×C₅₃	—	4	4	no	—
102	#2482	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₅₃	—	2	4	no	—
104	C₂×C₅₃	—	4	4	no	—
105	C₂×C₅₃	C₂×#123	1	1	yes	characteristic
106	C₂×C₅₃	#525	1	1	yes	characteristic
107	C₂×C₅₃	#535	1	1	yes	characteristic
108	C₂×C₅₃	#551	1	4	yes	—
109	C₂×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
110	C₂²×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
111	#399	C₂×C₅₃	1	2	yes	complemented
112	#397	C₂×C₅₃	1	2	yes	complemented
113	#388	C₂×C₅₃	1	4	yes	complemented
114	#385	C₂×C₅₃	1	2	yes	complemented
115	#381	C₂×C₅₃	1	1	yes	characteristic, complemented
116	#379	C₂×C₅₃	1	2	yes	—
117	C₅₃	#2482	1	1	yes	characteristic, direct product factor, Sylow
118	C₂⁵	C₂²×C₅₃	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
120	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
121	#117	—	2	4	no	—
122	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
123	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic, complemented
124	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
125	C₂×C₂²⋊C₄	—	2	4	no	—
126	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
132	C₂×C₄²	C₂²×C₅₃	1	2	yes	complemented
133	#95	C₂²×C₅₃	1	2	yes	complemented
134	#95	C₂²×C₅₃	1	2	yes	complemented
135	#95	C₂²×C₅₃	1	4	yes	complemented
136	C₂⁴	—	2	2	no	—
137	C₂⁴	—	2	4	no	—
138	C₂⁴	—	2	4	no	—
139	C₂⁴	—	4	2	no	—
140	C₂⁴	C₂³×C₅₃	1	2	yes	—
141	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	D₄×C₅₃	1	4	yes	—
146	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
147	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
149	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	—	2	8	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	—	2	4	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	8	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	4	16	no	—
170	C₂³	—	4	4	no	—
171	C₂³	—	4	4	no	—
172	C₂³	C₂×D₄×C₅₃	1	2	yes	—
173	C₂³	#41×C₅₃	1	4	yes	—
174	C₂³	#41×C₅₃	1	4	yes	—
175	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	—	4	16	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	4	no	—
200	C₂²	#123×C₅₃	1	4	yes	—
201	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
202	C₂²	C₂×#41×C₅₃	1	2	yes	—
203	C₂²	C₂×#41×C₅₃	1	2	yes	—
204	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	4	no	—
211	C₂	—	2	4	no	—
212	C₂	—	4	4	no	—
213	C₂	C₂×#123×C₅₃	1	1	yes	characteristic
214	C₂	C₅₃×#525	1	1	yes	characteristic
215	C₂	C₅₃×#535	1	1	yes	characteristic
216	C₂	C₅₃×#551	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#117×C₅₃	C₂	semidirect
3. C₅₃×#399	C₂	semidirect
4. C₅₃×#397	C₂	semidirect
5. C₅₃×#388	C₂	semidirect
6. C₅₃×#385	C₂	semidirect
7. C₅₃×#381	C₂	semidirect
13. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
14. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
15. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
23. C₂×C₄²×C₅₃	C₂²	semidirect
24. #95×C₅₃	C₂²	semidirect
25. #95×C₅₃	C₂²	semidirect
26. #95×C₅₃	C₂²	semidirect
102. #2482	C₅₃	direct (N × H)
109. C₂×#117	C₂×C₅₃	semidirect
111. #399	C₂×C₅₃	semidirect
112. #397	C₂×C₅₃	semidirect
113. #388	C₂×C₅₃	semidirect
114. #385	C₂×C₅₃	semidirect
115. #381	C₂×C₅₃	semidirect
117. C₅₃	#2482	direct (N × H)
122. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
123. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
124. C₂×C₄⋊C₄	C₂²×C₅₃	semidirect
132. C₂×C₄²	C₂²×C₅₃	semidirect
133. #95	C₂²×C₅₃	semidirect
134. #95	C₂²×C₅₃	semidirect
135. #95	C₂²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117×C₅₃	2
C₂²×C₂²⋊C₄×C₅₃	2
C₅₃×#399	2
C₅₃×#397	2
C₅₃×#388	2
C₅₃×#385	2
C₅₃×#381	2
C₅₃×#379	2
#2482	53

Subgroups — C53×#2482 #20271419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2482 #20271419