Subgroups — C₅₃×#2592 #20271529

Order 6784.

520 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#397	C₂	1	2	yes	maximal, complemented
4	C₅₃×#393	C₂	1	2	yes	maximal, complemented
5	C₅₃×#391	C₂	1	2	yes	maximal, complemented
6	C₅₃×#390	C₂	1	2	yes	maximal
7	C₅₃×#388	C₂	1	4	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	#124×C₅₃	—	2	4	no	—
13	C₄×Q₈×C₅₃	—	2	4	no	—
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	2	yes	—
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	4	yes	—
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	#95×C₅₃	C₂²	1	2	yes	—
25	#95×C₅₃	C₂²	1	4	yes	—
26	#95×C₅₃	C₂²	1	4	yes	—
27	C₂⁴×C₅₃	C₂³	1	2	yes	—
28	C₂×Q₈×C₅₃	—	2	2	no	—
29	C₂×Q₈×C₅₃	—	2	2	no	—
30	C₂×Q₈×C₅₃	—	2	4	no	—
31	C₂×Q₈×C₅₃	—	2	4	no	—
32	C₂×D₄×C₅₃	—	2	2	no	—
33	C₂×D₄×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₄⋊C₄×C₅₃	—	2	4	no	—
44	C₂²⋊C₄×C₅₃	—	2	4	no	—
45	C₂²⋊C₄×C₅₃	—	2	4	no	—
46	C₂²⋊C₄×C₅₃	—	2	8	no	—
47	C₂²⋊C₄×C₅₃	—	4	4	no	—
48	C₄²×C₅₃	—	2	4	no	—
49	C₄²×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	D₄	1	4	yes	—
51	C₂³×C₅₃	—	2	2	no	—
52	C₂³×C₅₃	—	2	4	no	—
53	C₂³×C₅₃	—	2	4	no	—
54	C₂³×C₅₃	—	2	2	no	—
55	C₂³×C₅₃	—	2	2	no	—
56	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
57	Q₈×C₅₃	—	2	8	no	—
58	Q₈×C₅₃	—	4	4	no	—
59	D₄×C₅₃	—	4	4	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	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₅₃	#41	1	4	yes	—
77	C₂×C₄×C₅₃	#41	1	4	yes	—
78	C₂²×C₅₃	—	4	2	no	—
79	C₂²×C₅₃	—	4	2	no	—
80	C₂²×C₅₃	—	4	2	no	—
81	C₂²×C₅₃	—	4	4	no	—
82	C₂²×C₅₃	—	4	4	no	—
83	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×#41	1	2	yes	—
85	C₂²×C₅₃	C₂×#41	1	2	yes	—
86	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	2	no	—
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	—	4	4	no	—
94	C₄×C₅₃	#124	1	4	yes	—
95	#2592	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₅₃	—	8	2	no	—
97	C₂×C₅₃	#525	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
99	C₂×C₅₃	#537	1	1	yes	characteristic
100	C₂×C₅₃	#556	1	2	yes	—
101	C₂×C₅₃	#559	1	2	yes	—
102	C₂×#124	C₂×C₅₃	1	1	yes	characteristic
103	C₂×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
104	#397	C₂×C₅₃	1	2	yes	complemented
105	#393	C₂×C₅₃	1	2	yes	complemented
106	#391	C₂×C₅₃	1	2	yes	complemented
107	#390	C₂×C₅₃	1	2	yes	—
108	#388	C₂×C₅₃	1	4	yes	complemented
109	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
110	C₅₃	#2592	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
112	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
113	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
114	#124	—	2	4	no	—
115	C₄×Q₈	—	2	4	no	—
116	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
122	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
123	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
124	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
125	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
126	#95	C₂²×C₅₃	1	2	yes	—
127	#95	C₂²×C₅₃	1	4	yes	—
128	#95	C₂²×C₅₃	1	4	yes	—
129	C₂⁴	C₂³×C₅₃	1	2	yes	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	4	no	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	—	2	4	no	—
136	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
137	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
138	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	2	8	no	—
149	C₂²⋊C₄	—	4	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	4	2	no	—
152	C₄²	D₄×C₅₃	1	4	yes	—
153	C₂³	—	2	2	no	—
154	C₂³	—	2	4	no	—
155	C₂³	—	2	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
159	Q₈	—	2	8	no	—
160	Q₈	—	4	4	no	—
161	D₄	—	4	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
176	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
177	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
178	C₂×C₄	#41×C₅₃	1	4	yes	—
179	C₂×C₄	#41×C₅₃	1	4	yes	—
180	C₂²	—	4	2	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	4	no	—
184	C₂²	—	4	4	no	—
185	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
186	C₂²	C₂×#41×C₅₃	1	2	yes	—
187	C₂²	C₂×#41×C₅₃	1	2	yes	—
188	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
189	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	4	no	—
196	C₄	#124×C₅₃	1	4	yes	—
197	C₂	—	8	2	no	—
198	C₂	C₅₃×#525	1	1	yes	characteristic
199	C₂	C₂×#124×C₅₃	1	1	yes	characteristic
200	C₂	C₅₃×#537	1	1	yes	characteristic
201	C₂	C₅₃×#556	1	2	yes	—
202	C₂	C₅₃×#559	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×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
2. C₂×C₄×Q₈×C₅₃	C₂	semidirect
3. C₅₃×#397	C₂	semidirect
4. C₅₃×#393	C₂	semidirect
5. C₅₃×#391	C₂	semidirect
7. C₅₃×#388	C₂	semidirect
8. C₄×C₄⋊C₄×C₅₃	C₂	semidirect
95. #2592	C₅₃	direct (N × H)
103. C₂×C₄×Q₈	C₂×C₅₃	semidirect
104. #397	C₂×C₅₃	semidirect
105. #393	C₂×C₅₃	semidirect
106. #391	C₂×C₅₃	semidirect
108. #388	C₂×C₅₃	semidirect
109. C₄×C₄⋊C₄	C₂×C₅₃	semidirect
110. C₅₃	#2592	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#124×C₅₃	2
C₂×C₄×Q₈×C₅₃	2
C₅₃×#397	2
C₅₃×#393	2
C₅₃×#391	2
C₅₃×#390	2
C₅₃×#388	2
C₄×C₄⋊C₄×C₅₃	2
#2592	53

Subgroups — C53×#2592 #20271529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2592 #20271529