Subgroups — C₅₃×#3141 #20272078

Order 6784.

424 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#489	C₂	1	4	yes	maximal, complemented
4	C₅₃×#487	C₂	1	2	yes	maximal, complemented
5	C₅₃×#486	C₂	1	2	yes	maximal, complemented
6	C₅₃×#417	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₅₃	C₂²	1	2	yes	—
9	C₂²×Q₈×C₅₃	—	2	1	no	—
10	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	#127×C₅₃	C₂²	1	1	yes	characteristic
17	#124×C₅₃	C₂²	1	2	yes	—
18	#121×C₅₃	—	2	1	no	—
19	#121×C₅₃	—	2	1	no	—
20	C₄×D₄×C₅₃	—	2	1	no	—
21	C₄×D₄×C₅₃	—	2	1	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	#103×C₅₃	C₂²	1	8	yes	—
26	#102×C₅₃	C₂²	1	8	yes	—
27	#97×C₅₃	C₂²	1	2	yes	—
28	C₄×C₈×C₅₃	C₂²	1	2	yes	—
29	#41×C₅₃	—	2	8	no	—
30	C₂×Q₈×C₅₃	—	2	2	no	—
31	C₂×Q₈×C₅₃	—	2	2	no	—
32	C₂×Q₈×C₅₃	—	4	4	no	—
33	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
34	C₂×D₄×C₅₃	—	2	2	no	—
35	C₂×D₄×C₅₃	—	2	1	no	—
36	C₂×D₄×C₅₃	—	2	1	no	—
37	C₂×D₄×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	M₁₆×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	C₂³	1	4	yes	—
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	2	1	no	—
49	C₄⋊C₄×C₅₃	—	2	1	no	—
50	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
51	C₂²⋊C₄×C₅₃	—	2	2	no	—
52	C₂²⋊C₄×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂³×C₅₃	—	2	2	no	—
58	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
59	Q₈×C₅₃	—	2	4	no	—
60	Q₈×C₅₃	—	4	4	no	—
61	Q₈×C₅₃	—	4	4	no	—
62	D₄×C₅₃	—	2	4	no	—
63	D₄×C₅₃	—	4	4	no	—
64	D₄×C₅₃	—	4	2	no	—
65	D₄×C₅₃	—	4	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	4	no	—
70	C₂×C₄×C₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#41	1	4	yes	—
78	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₈×C₅₃	—	2	4	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	—	4	2	no	—
83	C₂²×C₅₃	—	4	2	no	—
84	C₂²×C₅₃	—	4	2	no	—
85	C₂²×C₅₃	#124	1	2	yes	—
86	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	4	2	no	—
90	C₄×C₅₃	—	4	4	no	—
91	C₄×C₅₃	#124	1	2	yes	—
92	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
93	C₄×C₅₃	#137	1	2	yes	—
94	C₄×C₅₃	C₂×#41	1	2	yes	—
95	#3141	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	—	8	2	no	—
98	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
99	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
101	#532	C₂×C₅₃	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
103	#489	C₂×C₅₃	1	4	yes	complemented
104	#487	C₂×C₅₃	1	2	yes	complemented
105	#486	C₂×C₅₃	1	2	yes	complemented
106	#417	C₂×C₅₃	1	4	yes	complemented
107	C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
108	C₅₃	#3141	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×#41	C₂²×C₅₃	1	2	yes	—
110	C₂²×Q₈	—	2	1	no	—
111	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
115	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
116	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
117	#127	C₂²×C₅₃	1	1	yes	characteristic
118	#124	C₂²×C₅₃	1	2	yes	—
119	#121	—	2	1	no	—
120	#121	—	2	1	no	—
121	C₄×D₄	—	2	1	no	—
122	C₄×D₄	—	2	1	no	—
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
125	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
126	#103	C₂²×C₅₃	1	8	yes	—
127	#102	C₂²×C₅₃	1	8	yes	—
128	#97	C₂²×C₅₃	1	2	yes	—
129	C₄×C₈	C₂²×C₅₃	1	2	yes	—
130	#41	—	2	8	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	C₂³×C₅₃	1	2	yes	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	—	2	1	no	—
137	C₂×D₄	—	2	1	no	—
138	C₂×D₄	C₂³×C₅₃	1	2	yes	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
142	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
144	M₁₆	—	2	4	no	—
145	C₂×C₈	C₂³×C₅₃	1	4	yes	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	D₄×C₅₃	1	1	yes	characteristic
155	C₄²	D₄×C₅₃	1	1	yes	characteristic
156	C₄²	C₂³×C₅₃	1	1	yes	characteristic
157	C₄²	C₂³×C₅₃	1	1	yes	characteristic
158	C₂³	—	2	2	no	—
159	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
160	Q₈	—	2	4	no	—
161	Q₈	—	4	4	no	—
162	Q₈	—	4	4	no	—
163	D₄	—	2	4	no	—
164	D₄	—	4	4	no	—
165	D₄	—	4	2	no	—
166	D₄	—	4	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
175	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
176	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
178	C₂×C₄	#41×C₅₃	1	4	yes	—
179	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	#124×C₅₃	1	2	yes	—
187	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	4	no	—
192	C₄	#124×C₅₃	1	2	yes	—
193	C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
194	C₄	#137×C₅₃	1	2	yes	—
195	C₄	C₂×#41×C₅₃	1	2	yes	—
196	C₂	—	2	2	no	—
197	C₂	—	8	2	no	—
198	C₂	C₂×#124×C₅₃	1	1	yes	characteristic
199	C₂	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
200	C₂	C₂×#137×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
prime radical	C₅₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×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×#489	C₂	semidirect
4. C₅₃×#487	C₂	semidirect
5. C₅₃×#486	C₂	semidirect
6. C₅₃×#417	C₂	semidirect
7. C₄×M₁₆×C₅₃	C₂	semidirect
95. #3141	C₅₃	direct (N × H)
102. C₂×C₄⋊Q₈	C₂×C₅₃	semidirect
103. #489	C₂×C₅₃	semidirect
104. #487	C₂×C₅₃	semidirect
105. #486	C₂×C₅₃	semidirect
106. #417	C₂×C₅₃	semidirect
107. C₄×M₁₆	C₂×C₅₃	semidirect
108. C₅₃	#3141	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×#532	2
C₂×C₄⋊Q₈×C₅₃	2
C₅₃×#489	2
C₅₃×#487	2
C₅₃×#486	2
C₅₃×#417	2
C₄×M₁₆×C₅₃	2
#3141	53

Subgroups — C53×#3141 #20272078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3141 #20272078