Subgroups — C₅₃×#22248 #27229760

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1990	C₂	1	2	yes	maximal
4	C₅₃×#1921	C₂	1	8	yes	maximal
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
9	C₅₃×#500	C₄	1	8	yes	complemented
10	C₂×#107×C₅₃	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#391	—	2	2	no	—
14	C₅₃×#389	—	2	2	no	—
15	C₅₃×#389	C₂²	1	4	yes	—
16	C₅₃×#384	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#340	C₂²	1	8	yes	—
21	C₅₃×#340	C₂²	1	4	yes	—
22	C₅₃×#336	C₂²	1	4	yes	—
23	C₂²×Q₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
26	C₄⋊Q₈×C₅₃	—	2	4	no	—
27	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
28	C₄×Q₈×C₅₃	—	4	8	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
35	C₂×C₄²×C₅₃	—	2	4	no	—
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#107×C₅₃	C₂×C₄	1	16	yes	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
44	#95×C₅₃	—	2	4	no	—
45	#95×C₅₃	—	2	4	no	—
46	C₂×Q₈×C₅₃	—	2	8	no	—
47	C₂×Q₈×C₅₃	—	4	4	no	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
60	C₄⋊C₄×C₅₃	—	2	8	no	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	2	8	no	—
63	C₄⋊C₄×C₅₃	—	4	8	no	—
64	C₄⋊C₄×C₅₃	—	4	4	no	—
65	C₄⋊C₄×C₅₃	—	4	4	no	—
66	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	4	no	—
70	C₄²×C₅₃	—	4	8	no	—
71	C₄²×C₅₃	Q₁₆	1	4	yes	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	#41	1	2	yes	—
74	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	16	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	8	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	—	4	4	no	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
90	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	#135	1	2	yes	—
92	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
93	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
95	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
96	C₈×C₅₃	—	2	4	no	—
97	C₈×C₅₃	—	4	2	no	—
98	#22248	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂²×C₅₃	#536	1	2	yes	—
101	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	—	4	8	no	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	—	8	4	no	—
108	C₄×C₅₃	C₄×Q₁₆	1	4	yes	—
109	C₄×C₅₃	#442	1	4	yes	—
110	C₄×C₅₃	#519	1	2	yes	—
111	C₄×C₅₃	#541	1	2	yes	—
112	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
113	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	—
114	#1990	C₂×C₅₃	1	2	yes	—
115	#1921	C₂×C₅₃	1	8	yes	—
116	#1834	C₂×C₅₃	1	1	yes	characteristic
117	#1744	C₂×C₅₃	1	1	yes	characteristic
118	C₂×C₅₃	#2327	1	1	yes	characteristic
119	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
121	C₂×C₅₃	#3228	1	2	yes	—
122	C₂×C₅₃	#3230	1	2	yes	—
123	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
124	C₂×C₄×Q₈	—	2	2	no	—
125	#500	C₄×C₅₃	1	8	yes	complemented
126	C₂×#107	C₂²×C₅₃	1	4	yes	—
127	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
129	#391	—	2	2	no	—
130	#389	—	2	2	no	—
131	#389	C₂²×C₅₃	1	4	yes	—
132	#384	C₂²×C₅₃	1	2	yes	—
133	C₄×C₄⋊C₄	—	2	2	no	—
134	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
135	C₄³	C₂²×C₅₃	1	1	yes	characteristic
136	#340	C₂²×C₅₃	1	4	yes	—
137	#340	C₂²×C₅₃	1	8	yes	—
138	#336	C₂²×C₅₃	1	4	yes	—
139	C₅₃	#22248	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²×Q₈	—	2	2	no	—
141	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
142	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
145	C₄×Q₈	—	4	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
151	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
152	C₂×C₄²	—	2	4	no	—
153	C₂×C₄²	D₄×C₅₃	1	2	yes	—
154	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
155	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
157	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
158	#107	C₂×C₄×C₅₃	1	16	yes	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
161	#95	—	2	4	no	—
162	#95	—	2	4	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
177	C₄⋊C₄	—	2	8	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	8	no	—
188	C₄²	Q₁₆×C₅₃	1	4	yes	—
189	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
190	C₄²	#41×C₅₃	1	2	yes	—
191	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
192	Q₈	—	4	16	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
205	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
206	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
207	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
208	C₂×C₄	#135×C₅₃	1	2	yes	—
209	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
210	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
212	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	4	2	no	—
215	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
216	C₂²	C₅₃×#536	1	2	yes	—
217	C₂²	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
218	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
219	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	4	8	no	—
222	C₄	—	4	2	no	—
223	C₄	—	8	4	no	—
224	C₄	C₄×Q₁₆×C₅₃	1	4	yes	—
225	C₄	C₅₃×#442	1	4	yes	—
226	C₄	C₅₃×#519	1	2	yes	—
227	C₄	C₅₃×#541	1	2	yes	—
228	C₂	C₅₃×#2327	1	1	yes	characteristic
229	C₂	C₂×C₄×Q₁₆×C₅₃	1	1	yes	characteristic
230	C₂	C₂×C₅₃×#442	1	1	yes	characteristic
231	C₂	C₅₃×#3228	1	2	yes	—
232	C₂	C₅₃×#3230	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅₃×#500	C₄	semidirect
98. #22248	C₅₃	direct (N × H)
125. #500	C₄×C₅₃	semidirect
139. C₅₃	#22248	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×#500	2
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1990	2
C₅₃×#1921	2
C₅₃×#1834	2
C₅₃×#1744	2
#22248	53

Subgroups — C53×#22248 #27229760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22248 #27229760