Subgroups — C₅₃×#13277 #27220789

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1380	C₂	1	2	yes	maximal
5	C₅₃×#1380	C₂	1	2	yes	maximal
6	C₂×#125×C₅₃	—	2	1	no	—
7	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₅₃×#499	—	2	2	no	—
9	C₅₃×#499	C₄	1	4	yes	complemented
10	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#106×C₅₃	—	2	1	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₅₃×C₄⋊C₁₆	—	2	2	no	—
17	C₅₃×#340	—	2	1	no	—
18	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	#125×C₅₃	—	2	4	no	—
21	#125×C₅₃	—	4	2	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
26	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
27	C₂×C₄²×C₅₃	—	4	1	no	—
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	—	2	4	no	—
31	#107×C₅₃	—	2	1	no	—
32	#107×C₅₃	—	2	1	no	—
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	C₂×C₄	1	2	yes	—
35	#107×C₅₃	C₂×C₄	1	2	yes	—
36	#107×C₅₃	Q₈	1	4	yes	—
37	#106×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	#95×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	4	4	no	—
57	C₄⋊C₄×C₅₃	—	4	4	no	—
58	C₄⋊C₄×C₅₃	—	4	1	no	—
59	C₄⋊C₄×C₅₃	—	4	1	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄²×C₅₃	—	8	2	no	—
62	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₃	SD₁₆	1	2	yes	—
64	C₁₆×C₅₃	—	4	2	no	—
65	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	—	8	1	no	—
75	C₂×C₄×C₅₃	—	8	2	no	—
76	C₂×C₄×C₅₃	—	8	1	no	—
77	C₂×C₄×C₅₃	#102	1	2	yes	—
78	C₂×C₄×C₅₃	#102	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#137	1	2	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	C₄×Q₈	1	2	yes	—
85	C₈×C₅₃	#124	1	1	yes	characteristic
86	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
87	#13277	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
89	C₂²×C₅₃	#474	1	2	yes	—
90	C₂²×C₅₃	#508	1	2	yes	—
91	C₂²×C₅₃	#509	1	1	yes	characteristic
92	C₂²×C₅₃	#510	1	1	yes	characteristic
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	8	2	no	—
96	C₄×C₅₃	—	8	2	no	—
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	#391	1	1	yes	characteristic
99	C₄×C₅₃	#419	1	1	yes	characteristic
100	C₄×C₅₃	#475	1	2	yes	—
101	C₂×#499	C₂×C₅₃	1	1	yes	characteristic
102	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
103	C₄×#107	C₂×C₅₃	1	1	yes	characteristic
104	#1380	C₂×C₅₃	1	2	yes	—
105	#1380	C₂×C₅₃	1	2	yes	—
106	C₂×C₅₃	#1988	1	1	yes	characteristic
107	C₂×C₅₃	#2138	1	1	yes	characteristic
108	C₂×C₅₃	#2139	1	1	yes	characteristic
109	C₂×C₅₃	#2228	1	2	yes	—
110	C₂×C₅₃	#2229	1	2	yes	—
111	C₂×#125	—	2	1	no	—
112	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
113	#499	—	2	2	no	—
114	#499	C₄×C₅₃	1	4	yes	complemented
115	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
116	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
117	C₂×#107	C₂²×C₅₃	1	2	yes	—
118	C₂×#106	—	2	1	no	—
119	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₁₆	—	2	2	no	—
122	#340	—	2	1	no	—
123	C₅₃	#13277	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
126	#125	—	2	4	no	—
127	#125	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	4	no	—
137	#107	—	2	1	no	—
138	#107	—	2	1	no	—
139	#107	—	2	2	no	—
140	#107	C₂×C₄×C₅₃	1	2	yes	—
141	#107	C₂×C₄×C₅₃	1	2	yes	—
142	#107	Q₈×C₅₃	1	4	yes	—
143	#106	—	2	4	no	—
144	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
145	C₄×C₈	D₄×C₅₃	1	2	yes	—
146	#95	—	4	1	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	—	4	1	no	—
150	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
155	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
156	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
157	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
158	C₂×C₈	#41×C₅₃	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	4	1	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	—	8	2	no	—
168	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
169	C₄²	SD₁₆×C₅₃	1	2	yes	—
170	C₁₆	—	4	2	no	—
171	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	1	no	—
182	C₂×C₄	—	8	1	no	—
183	C₂×C₄	#102×C₅₃	1	2	yes	—
184	C₂×C₄	#102×C₅₃	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
187	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
188	C₂×C₄	#137×C₅₃	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	C₄×Q₈×C₅₃	1	2	yes	—
191	C₈	#124×C₅₃	1	1	yes	characteristic
192	C₈	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
193	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
194	C₂²	C₅₃×#474	1	2	yes	—
195	C₂²	C₅₃×#508	1	2	yes	—
196	C₂²	C₅₃×#509	1	1	yes	characteristic
197	C₂²	C₅₃×#510	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	8	4	no	—
201	C₄	—	8	2	no	—
202	C₄	—	8	2	no	—
203	C₄	C₅₃×#391	1	1	yes	characteristic
204	C₄	C₅₃×#419	1	1	yes	characteristic
205	C₄	C₅₃×#475	1	2	yes	—
206	C₂	C₅₃×#1988	1	1	yes	characteristic
207	C₂	C₅₃×#2138	1	1	yes	characteristic
208	C₂	C₅₃×#2139	1	1	yes	characteristic
209	C₂	C₅₃×#2228	1	2	yes	—
210	C₂	C₅₃×#2229	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅₃×#499	C₄	semidirect
87. #13277	C₅₃	direct (N × H)
114. #499	C₄×C₅₃	semidirect
123. C₅₃	#13277	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×#499	2
C₂×C₅₃×C₄⋊C₁₆	2
C₄×#107×C₅₃	2
C₅₃×#1380	2
#13277	53

Subgroups — C53×#13277 #27220789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13277 #27220789