Subgroups — C₅₃×#17835 #27225347

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2991	C₂	1	2	yes	maximal
2	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1574	C₂	1	6	yes	maximal, complemented
4	C₅₃×#1566	C₂	1	3	yes	maximal, complemented
5	C₅₃×#1564	C₂	1	3	yes	maximal, complemented
6	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₅₃	—	2	2	no	—
8	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#436	—	2	6	no	—
11	C₈×D₄×C₅₃	C₄	1	3	yes	—
12	C₈×D₄×C₅₃	C₄	1	3	yes	—
13	C₅₃×#431	C₂²	1	6	yes	—
14	C₅₃×#431	C₂²	1	3	yes	—
15	C₅₃×#423	C₂²	1	6	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
17	C₅₃×#335	C₂²	1	6	yes	—
18	C₅₃×#334	C₂²	1	1	yes	characteristic
19	C₅₃×#334	C₂²	1	3	yes	—
20	C₅₃×#334	C₂²	1	6	yes	—
21	C₂×#41×C₅₃	Q₈	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	6	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×D₄×C₅₃	C₂×C₄	1	3	yes	—
28	C₄×D₄×C₅₃	D₄	1	3	yes	—
29	#117×C₅₃	Q₈	1	3	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
31	C₄⋊C₈×C₅₃	—	2	6	no	—
32	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
33	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
34	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
35	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
36	C₂²⋊C₈×C₅₃	—	2	6	no	—
37	C₂²⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
38	C₄×C₈×C₅₃	—	2	6	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂³	1	3	yes	—
43	#41×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×D₄×C₅₃	C₄⋊C₄	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
49	M₁₆×C₅₃	—	8	6	no	—
50	C₂×C₈×C₅₃	—	2	6	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	6	no	—
53	C₂×C₈×C₅₃	—	4	3	no	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
59	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
60	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	6	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
62	C₄²×C₅₃	C₂×D₄	1	3	yes	—
63	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
64	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₅₃	#108	1	4	yes	—
67	D₄×C₅₃	—	2	6	no	—
68	C₂×C₄×C₅₃	—	2	3	no	—
69	C₂×C₄×C₅₃	—	2	3	no	—
70	C₂×C₄×C₅₃	—	2	6	no	—
71	C₂×C₄×C₅₃	—	2	6	no	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₅₃	—	2	3	no	—
79	C₈×C₅₃	—	2	1	no	—
80	C₈×C₅₃	—	2	3	no	—
81	C₈×C₅₃	—	4	2	no	—
82	C₈×C₅₃	—	4	6	no	—
83	C₈×C₅₃	#131	1	2	yes	—
84	#17835	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₅₃	—	2	3	no	—
86	C₂²×C₅₃	—	2	3	no	—
87	C₂²×C₅₃	—	2	3	no	—
88	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₅₃	—	2	3	no	—
90	C₄×C₅₃	—	2	3	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	C₂×#108	1	6	yes	—
93	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
94	C₄×C₅₃	#568	1	1	yes	characteristic
95	C₄×C₅₃	#577	1	2	yes	—
96	#2991	C₂×C₅₃	1	2	yes	—
97	C₈×#41	C₂×C₅₃	1	1	yes	characteristic
98	#1574	C₂×C₅₃	1	6	yes	complemented
99	#1566	C₂×C₅₃	1	3	yes	complemented
100	#1564	C₂×C₅₃	1	3	yes	complemented
101	C₂×C₅₃	—	4	3	no	—
102	C₂×C₅₃	#2906	1	1	yes	characteristic
103	C₂×C₅₃	#2912	1	1	yes	characteristic
104	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
105	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
106	C₈×Q₈	—	2	2	no	—
107	C₈×Q₈	C₄×C₅₃	1	1	yes	characteristic
108	C₈×Q₈	C₄×C₅₃	1	1	yes	characteristic
109	#436	—	2	6	no	—
110	C₈×D₄	C₄×C₅₃	1	3	yes	—
111	C₈×D₄	C₄×C₅₃	1	3	yes	—
112	#431	C₂²×C₅₃	1	6	yes	—
113	#431	C₂²×C₅₃	1	3	yes	—
114	#423	C₂²×C₅₃	1	6	yes	—
115	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
116	#335	C₂²×C₅₃	1	6	yes	—
117	#334	C₂²×C₅₃	1	1	yes	characteristic
118	#334	C₂²×C₅₃	1	3	yes	—
119	#334	C₂²×C₅₃	1	6	yes	—
120	C₅₃	#17835	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×#41	Q₈×C₅₃	1	1	yes	characteristic
122	C₂×M₁₆	—	2	6	no	—
123	C₂²×C₈	C₂×C₄×C₅₃	1	3	yes	—
124	C₂²×C₈	C₂×C₄×C₅₃	1	3	yes	—
125	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
126	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
127	C₄×D₄	C₂×C₄×C₅₃	1	3	yes	—
128	C₄×D₄	D₄×C₅₃	1	3	yes	—
129	#117	Q₈×C₅₃	1	3	yes	—
130	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
131	C₄⋊C₈	—	2	6	no	—
132	C₄⋊C₈	C₂×C₄×C₅₃	1	3	yes	—
133	C₄⋊C₈	C₂×C₄×C₅₃	1	3	yes	—
134	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
135	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
136	C₂²⋊C₈	—	2	6	no	—
137	C₂²⋊C₈	C₂×C₄×C₅₃	1	6	yes	—
138	C₄×C₈	—	2	6	no	—
139	C₄×C₈	C₂×C₄×C₅₃	1	3	yes	—
140	C₄×C₈	C₂×C₄×C₅₃	1	3	yes	—
141	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
142	C₄×C₈	C₂³×C₅₃	1	3	yes	—
143	#41	—	2	4	no	—
144	C₂×Q₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
145	C₂×D₄	C₄⋊C₄×C₅₃	1	3	yes	—
146	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
147	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
148	C₂²×C₄	C₂×Q₈×C₅₃	1	3	yes	—
149	M₁₆	—	8	6	no	—
150	C₂×C₈	—	2	6	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	—	4	3	no	—
154	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
155	C₂×C₈	C₂²×C₄×C₅₃	1	3	yes	—
156	C₂×C₈	C₂²×C₄×C₅₃	1	3	yes	—
157	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
158	C₄⋊C₄	C₄⋊C₄×C₅₃	1	3	yes	—
159	C₄⋊C₄	C₄⋊C₄×C₅₃	1	3	yes	—
160	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	6	yes	—
161	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
162	C₄²	C₂×D₄×C₅₃	1	3	yes	—
163	C₄²	C₂×Q₈×C₅₃	1	3	yes	—
164	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
165	C₂³	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
166	Q₈	#108×C₅₃	1	4	yes	—
167	D₄	—	2	6	no	—
168	C₂×C₄	—	2	3	no	—
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	2	6	no	—
172	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
173	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
174	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
175	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
176	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
178	C₈	—	2	3	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	3	no	—
181	C₈	—	4	2	no	—
182	C₈	—	4	6	no	—
183	C₈	#131×C₅₃	1	2	yes	—
184	C₂²	—	2	3	no	—
185	C₂²	—	2	3	no	—
186	C₂²	—	2	3	no	—
187	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
188	C₄	—	2	3	no	—
189	C₄	—	2	3	no	—
190	C₄	—	2	1	no	—
191	C₄	C₂×#108×C₅₃	1	6	yes	—
192	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
193	C₄	C₅₃×#568	1	1	yes	characteristic
194	C₄	C₅₃×#577	1	2	yes	—
195	C₂	—	4	3	no	—
196	C₂	C₅₃×#2906	1	1	yes	characteristic
197	C₂	C₅₃×#2912	1	1	yes	characteristic
198	C₂	C₂²×#108×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1574	C₂	semidirect
4. C₅₃×#1566	C₂	semidirect
5. C₅₃×#1564	C₂	semidirect
84. #17835	C₅₃	direct (N × H)
98. #1574	C₂×C₅₃	semidirect
99. #1566	C₂×C₅₃	semidirect
100. #1564	C₂×C₅₃	semidirect
120. C₅₃	#17835	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×#2991	2
C₈×#41×C₅₃	2
C₅₃×#1574	2
C₅₃×#1566	2
C₅₃×#1564	2
#17835	53

Subgroups — C53×#17835 #27225347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17835 #27225347