Subgroups — C₅₃×#12867 #27220379

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1380	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
7	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×#107×C₅₃	—	2	1	no	—
9	C₂×#107×C₅₃	—	2	2	no	—
10	C₂×#107×C₅₃	C₄	1	4	yes	complemented
11	C₂×#107×C₅₃	C₂²	1	2	yes	—
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#380	—	2	1	no	—
14	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
15	C₅₃×#340	—	2	1	no	—
16	C₂³×C₄×C₅₃	—	4	1	no	—
17	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	—	2	4	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	D₄	1	2	yes	—
21	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
27	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
28	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
29	C₂×C₁₆×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	—	2	4	no	—
31	#107×C₅₃	—	2	4	no	—
32	#107×C₅₃	—	2	2	no	—
33	#107×C₅₃	—	2	4	no	—
34	#107×C₅₃	C₂×C₄	1	4	yes	—
35	#95×C₅₃	—	4	1	no	—
36	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	4	4	no	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	—	8	4	no	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	4	yes	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₂²⋊C₄×C₅₃	—	8	2	no	—
60	C₁₆×C₅₃	—	4	2	no	—
61	C₂³×C₅₃	—	2	2	no	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	#102	1	2	yes	—
64	C₂³×C₅₃	SD₃₂	1	2	yes	—
65	C₂³×C₅₃	Q₃₂	1	2	yes	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	8	8	no	—
73	C₂×C₄×C₅₃	—	8	4	no	—
74	C₂×C₄×C₅₃	—	8	2	no	—
75	C₂×C₄×C₅₃	—	8	2	no	—
76	C₂×C₄×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	#102	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#123	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#137	1	2	yes	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	#117	1	2	yes	—
84	C₈×C₅₃	#123	1	2	yes	—
85	#12867	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	—	2	2	no	—
88	C₂²×C₅₃	—	2	4	no	—
89	C₂²×C₅₃	—	2	4	no	—
90	C₂²×C₅₃	#358	1	4	yes	—
91	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
92	C₂²×C₅₃	#480	1	2	yes	—
93	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
95	C₂²×C₅₃	#509	1	2	yes	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	—	16	2	no	—
99	C₄×C₅₃	#381	1	1	yes	characteristic
100	C₄×C₅₃	#419	1	1	yes	characteristic
101	C₄×C₅₃	#483	1	2	yes	—
102	C₂²×#107	C₂×C₅₃	1	1	yes	characteristic
103	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
104	#1808	C₂×C₅₃	1	1	yes	characteristic
105	#1380	C₂×C₅₃	1	4	yes	complemented
106	C₂×C₅₃	—	2	4	no	—
107	C₂×C₅₃	#1823	1	1	yes	characteristic
108	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
109	C₂×C₅₃	#2140	1	1	yes	characteristic
110	C₂×C₅₃	#2233	1	2	yes	—
111	C₂×C₅₃	#2236	1	2	yes	—
112	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
113	C₂²×C₄⋊C₄	—	2	1	no	—
114	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
115	C₂×#107	—	2	2	no	—
116	C₂×#107	—	2	1	no	—
117	C₂×#107	C₄×C₅₃	1	4	yes	complemented
118	C₂×#107	C₂²×C₅₃	1	2	yes	—
119	C₂×#107	C₂²×C₅₃	1	2	yes	—
120	#380	—	2	1	no	—
121	C₂²⋊C₁₆	—	2	2	no	—
122	#340	—	2	1	no	—
123	C₅₃	#12867	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₄	—	4	1	no	—
125	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
128	C₂²×C₈	D₄×C₅₃	1	2	yes	—
129	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄⋊C₄	—	4	2	no	—
136	C₂×C₂²⋊C₄	—	4	1	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	4	no	—
139	#107	—	2	2	no	—
140	#107	—	2	4	no	—
141	#107	—	2	4	no	—
142	#107	C₂×C₄×C₅₃	1	4	yes	—
143	#95	—	4	1	no	—
144	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	8	4	no	—
151	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
152	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
154	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
159	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
161	C₂×C₈	#41×C₅₃	1	4	yes	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₂²⋊C₄	—	8	2	no	—
168	C₁₆	—	4	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	#102×C₅₃	1	2	yes	—
172	C₂³	SD₃₂×C₅₃	1	2	yes	—
173	C₂³	Q₃₂×C₅₃	1	2	yes	—
174	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	8	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	#102×C₅₃	1	2	yes	—
186	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
187	C₂×C₄	#123×C₅₃	1	2	yes	—
188	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
189	C₂×C₄	#137×C₅₃	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	#117×C₅₃	1	2	yes	—
192	C₈	#123×C₅₃	1	2	yes	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	C₅₃×#358	1	4	yes	—
198	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
199	C₂²	C₅₃×#480	1	2	yes	—
200	C₂²	C₂×SD₃₂×C₅₃	1	1	yes	characteristic
201	C₂²	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
202	C₂²	C₅₃×#509	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	8	4	no	—
205	C₄	—	16	2	no	—
206	C₄	C₅₃×#381	1	1	yes	characteristic
207	C₄	C₅₃×#419	1	1	yes	characteristic
208	C₄	C₅₃×#483	1	2	yes	—
209	C₂	—	2	4	no	—
210	C₂	C₅₃×#1823	1	1	yes	characteristic
211	C₂	C₂×C₅₃×#358	1	1	yes	characteristic
212	C₂	C₅₃×#2140	1	1	yes	characteristic
213	C₂	C₅₃×#2233	1	2	yes	—
214	C₂	C₅₃×#2236	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1380	C₂	semidirect
10. C₂×#107×C₅₃	C₄	semidirect
85. #12867	C₅₃	direct (N × H)
105. #1380	C₂×C₅₃	semidirect
117. C₂×#107	C₄×C₅₃	semidirect
123. C₅₃	#12867	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#107×C₅₃	2
C₂×C₅₃×C₂²⋊C₁₆	2
C₅₃×#1808	2
C₅₃×#1380	2
#12867	53

Subgroups — C53×#12867 #27220379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12867 #27220379