Subgroups — C₅₃×#12667 #27220179

Order 13568.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2110	C₂	1	2	yes	maximal
3	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
7	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#427	C₄	1	1	yes	characteristic
11	C₅₃×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#364	C₄	1	2	yes	complemented
14	C₅₃×#364	C₄	1	2	yes	complemented
15	C₅₃×#347	—	2	4	no	—
16	C₅₃×#346	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
18	C₅₃×#339	C₄	1	2	yes	—
19	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	#117×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	M₃₂×C₅₃	—	2	2	no	—
25	M₃₂×C₅₃	—	2	1	no	—
26	M₃₂×C₅₃	—	2	1	no	—
27	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
28	M₃₂×C₅₃	D₄	1	2	yes	—
29	M₃₂×C₅₃	Q₈	1	2	yes	—
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	#108×C₅₃	—	2	1	no	—
34	#108×C₅₃	—	2	1	no	—
35	#107×C₅₃	—	2	1	no	—
36	#106×C₅₃	—	2	1	no	—
37	#97×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	M₁₆×C₅₃	—	2	2	no	—
46	M₁₆×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	1	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	C₄²	1	1	yes	characteristic
65	C₄²×C₅₃	C₄²	1	1	yes	characteristic
66	C₁₆×C₅₃	—	2	2	no	—
67	C₁₆×C₅₃	—	2	4	no	—
68	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₂³×C₅₃	#131	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
80	C₈×C₅₃	—	4	2	no	—
81	C₈×C₅₃	#97	1	1	yes	characteristic
82	C₈×C₅₃	#97	1	1	yes	characteristic
83	C₈×C₅₃	#97	1	1	yes	characteristic
84	C₈×C₅₃	#97	1	1	yes	characteristic
85	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₈×C₅₃	#117	1	1	yes	characteristic
87	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
89	#12667	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₅₃	—	4	1	no	—
91	C₂²×C₅₃	#405	1	1	yes	characteristic
92	C₂²×C₅₃	#436	1	1	yes	characteristic
93	C₂²×C₅₃	#446	1	1	yes	characteristic
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	4	1	no	—
96	C₄×C₅₃	—	8	2	no	—
97	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
99	C₄×C₅₃	#436	1	1	yes	characteristic
100	C₄×C₅₃	#446	1	1	yes	characteristic
101	C₂×#364	C₂×C₅₃	1	1	yes	characteristic
102	#2110	C₂×C₅₃	1	2	yes	—
103	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
104	#1772	C₂×C₅₃	1	1	yes	characteristic
105	#1378	C₂×C₅₃	1	2	yes	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	#1770	1	1	yes	characteristic
110	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
111	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
112	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
113	C₂×#108	C₄×C₅₃	1	1	yes	characteristic
114	#427	C₄×C₅₃	1	1	yes	characteristic
115	#405	C₂²×C₅₃	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
117	#364	C₄×C₅₃	1	2	yes	complemented
118	#364	C₄×C₅₃	1	2	yes	complemented
119	#347	—	2	4	no	—
120	#346	C₄×C₅₃	1	2	yes	complemented
121	C₄×C₁₆	C₄×C₅₃	1	2	yes	complemented
122	#339	C₄×C₅₃	1	2	yes	—
123	C₅₃	#12667	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
125	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
127	#117	C₂×C₄×C₅₃	1	2	yes	—
128	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	2	1	no	—
131	M₃₂	—	2	1	no	—
132	M₃₂	C₂×C₄×C₅₃	1	4	yes	—
133	M₃₂	D₄×C₅₃	1	2	yes	—
134	M₃₂	Q₈×C₅₃	1	2	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
137	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
138	#108	—	2	1	no	—
139	#108	—	2	1	no	—
140	#107	—	2	1	no	—
141	#106	—	2	1	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
146	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
147	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
154	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
155	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
156	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
157	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
158	C₂×C₈	M₁₆×C₅₃	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
160	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
161	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
162	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
163	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
164	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₄²×C₅₃	1	1	yes	characteristic
170	C₄²	C₄²×C₅₃	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	4	no	—
173	C₁₆	C₄⋊C₄×C₅₃	1	4	yes	—
174	C₂³	#131×C₅₃	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
184	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
185	C₈	—	4	2	no	—
186	C₈	#97×C₅₃	1	1	yes	characteristic
187	C₈	#97×C₅₃	1	1	yes	characteristic
188	C₈	#97×C₅₃	1	1	yes	characteristic
189	C₈	#97×C₅₃	1	1	yes	characteristic
190	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
191	C₈	#117×C₅₃	1	1	yes	characteristic
192	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	C₅₃×#405	1	1	yes	characteristic
196	C₂²	C₅₃×#436	1	1	yes	characteristic
197	C₂²	C₅₃×#446	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
202	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
203	C₄	C₅₃×#436	1	1	yes	characteristic
204	C₄	C₅₃×#446	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	C₅₃×#1770	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	#117
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1770
prime radical	C₂³×C₅₃	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂×M₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅₃×#364	C₄	semidirect
14. C₅₃×#364	C₄	semidirect
16. C₅₃×#346	C₄	semidirect
17. C₄×C₁₆×C₅₃	C₄	semidirect
89. #12667	C₅₃	direct (N × H)
117. #364	C₄×C₅₃	semidirect
118. #364	C₄×C₅₃	semidirect
120. #346	C₄×C₅₃	semidirect
121. C₄×C₁₆	C₄×C₅₃	semidirect
123. C₅₃	#12667	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×C₅₃×#364	2
C₅₃×#2110	2
C₄×M₃₂×C₅₃	2
C₅₃×#1772	2
C₅₃×#1378	2
#12667	53

Subgroups — C53×#12667 #27220179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12667 #27220179