Subgroups — C₅₃×#22270 #27229782

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1956	C₂	1	2	yes	maximal
4	C₅₃×#1878	C₂	1	8	yes	maximal
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₅₃	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
10	C₅₃×#494	C₄	1	8	yes	complemented
11	C₂×#103×C₅₃	C₂²	1	8	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#391	C₂²	1	4	yes	—
15	C₅₃×#389	—	2	2	no	—
16	C₅₃×#389	—	2	2	no	—
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₄³×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#340	C₂²	1	4	yes	—
22	C₅₃×#336	C₂²	1	4	yes	—
23	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
24	C₂×Q₁₆×C₅₃	—	4	8	no	—
25	C₂×Q₁₆×C₅₃	C₂×C₄	1	16	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₄⋊Q₈×C₅₃	—	2	4	no	—
29	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
30	C₄×Q₈×C₅₃	—	4	8	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₅₃	—	2	4	no	—
38	C₂×C₄²×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#103×C₅₃	—	4	8	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
46	#95×C₅₃	—	2	4	no	—
47	#95×C₅₃	—	2	4	no	—
48	C₂×Q₈×C₅₃	—	2	4	no	—
49	C₂×Q₈×C₅₃	—	2	8	no	—
50	C₂×Q₈×C₅₃	—	2	8	no	—
51	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₅₃	—	4	16	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₄⋊C₄×C₅₃	—	2	8	no	—
65	C₄⋊C₄×C₅₃	—	2	8	no	—
66	C₄⋊C₄×C₅₃	—	2	8	no	—
67	C₄⋊C₄×C₅₃	—	4	8	no	—
68	C₄⋊C₄×C₅₃	—	4	4	no	—
69	C₄⋊C₄×C₅₃	—	4	4	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	—	4	8	no	—
74	C₄²×C₅₃	Q₁₆	1	4	yes	complemented
75	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₅₃	#41	1	2	yes	—
77	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₅₃	—	2	16	no	—
79	Q₈×C₅₃	—	4	8	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	8	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₅₃	#135	1	2	yes	—
96	C₂×C₄×C₅₃	#137	1	4	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
100	C₈×C₅₃	—	2	4	no	—
101	C₈×C₅₃	—	4	2	no	—
102	#22270	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂²×C₅₃	#535	1	2	yes	—
105	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
106	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×#137	1	2	yes	—
108	C₄×C₅₃	—	2	4	no	—
109	C₄×C₅₃	—	4	8	no	—
110	C₄×C₅₃	—	4	2	no	—
111	C₄×C₅₃	—	8	4	no	—
112	C₄×C₅₃	C₄×Q₁₆	1	4	yes	—
113	C₄×C₅₃	#441	1	4	yes	—
114	C₄×C₅₃	#518	1	2	yes	—
115	C₄×C₅₃	#540	1	2	yes	—
116	C₂×#494	C₂×C₅₃	1	1	yes	characteristic
117	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	—
118	#1956	C₂×C₅₃	1	2	yes	—
119	#1878	C₂×C₅₃	1	8	yes	—
120	#1834	C₂×C₅₃	1	1	yes	characteristic
121	#1744	C₂×C₅₃	1	1	yes	characteristic
122	C₂×C₅₃	#2328	1	1	yes	characteristic
123	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
125	C₂×C₅₃	#3209	1	2	yes	—
126	C₂×C₅₃	#3210	1	2	yes	—
127	C₂²×Q₁₆	C₂²×C₅₃	1	4	yes	—
128	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
129	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	—
130	#494	C₄×C₅₃	1	8	yes	complemented
131	C₂×#103	C₂²×C₅₃	1	8	yes	—
132	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
134	#391	C₂²×C₅₃	1	4	yes	—
135	#389	—	2	2	no	—
136	#389	—	2	2	no	—
137	#384	C₂²×C₅₃	1	2	yes	—
138	C₄×C₄⋊C₄	—	2	2	no	—
139	C₄×C₄⋊C₄	—	2	2	no	—
140	C₄³	C₂²×C₅₃	1	1	yes	characteristic
141	#340	C₂²×C₅₃	1	4	yes	—
142	#336	C₂²×C₅₃	1	4	yes	—
143	C₅₃	#22270	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
145	C₂×Q₁₆	—	4	8	no	—
146	C₂×Q₁₆	C₂×C₄×C₅₃	1	16	yes	—
147	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
148	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
151	C₄×Q₈	—	4	8	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
158	C₂×C₄²	—	2	4	no	—
159	C₂×C₄²	D₄×C₅₃	1	2	yes	—
160	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
161	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
163	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
164	#103	—	4	8	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
167	#95	—	2	4	no	—
168	#95	—	2	4	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	2	8	no	—
171	C₂×Q₈	—	2	8	no	—
172	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
180	Q₁₆	—	4	16	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	4	8	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	4	8	no	—
195	C₄²	Q₁₆×C₅₃	1	4	yes	complemented
196	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
197	C₄²	#41×C₅₃	1	2	yes	—
198	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
199	Q₈	—	2	16	no	—
200	Q₈	—	4	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
213	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
214	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
215	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
216	C₂×C₄	#135×C₅₃	1	2	yes	—
217	C₂×C₄	#137×C₅₃	1	4	yes	—
218	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
220	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
224	C₂²	C₅₃×#535	1	2	yes	—
225	C₂²	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
226	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
227	C₂²	C₂×#137×C₅₃	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	8	no	—
231	C₄	—	8	4	no	—
232	C₄	C₄×Q₁₆×C₅₃	1	4	yes	—
233	C₄	C₅₃×#441	1	4	yes	—
234	C₄	C₅₃×#518	1	2	yes	—
235	C₄	C₅₃×#540	1	2	yes	—
236	C₂	C₅₃×#2328	1	1	yes	characteristic
237	C₂	C₂×C₄×Q₁₆×C₅₃	1	1	yes	characteristic
238	C₂	C₂×C₅₃×#441	1	1	yes	characteristic
239	C₂	C₅₃×#3209	1	2	yes	—
240	C₂	C₅₃×#3210	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
10. C₅₃×#494	C₄	semidirect
74. C₄²×C₅₃	Q₁₆	semidirect
102. #22270	C₅₃	direct (N × H)
130. #494	C₄×C₅₃	semidirect
143. C₅₃	#22270	direct (N × H)
195. C₄²	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#494	2
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1956	2
C₅₃×#1878	2
C₅₃×#1834	2
C₅₃×#1744	2
#22270	53

Subgroups — C53×#22270 #27229782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22270 #27229782