Subgroups — C₅₃×#10723 #27218235

Order 13568.

1422 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#407	C₂	1	2	yes	maximal
3	C₂×C₅₃×#341	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₅₃	C₂²	1	4	yes	—
6	C₂³×C₈×C₅₃	—	2	4	no	—
7	C₂³×C₈×C₅₃	—	2	8	no	—
8	C₂³×C₈×C₅₃	C₄	1	4	yes	—
9	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#407	—	2	4	no	—
11	C₅₃×#407	C₄	1	8	yes	—
12	C₂×C₂²⋊C₈×C₅₃	—	2	4	no	—
13	C₅₃×#341	—	2	8	no	—
14	C₂⁵×C₅₃	Q₈	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	—	2	4	no	—
16	C₂³×C₄×C₅₃	—	2	4	no	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₅₃	D₄	1	4	yes	—
20	C₂³×C₄×C₅₃	D₄	1	2	yes	—
21	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	—	2	4	no	—
24	C₂×M₁₆×C₅₃	—	2	8	no	—
25	C₂×M₁₆×C₅₃	—	2	8	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₅₃	—	2	32	no	—
28	C₂²×C₈×C₅₃	—	2	16	no	—
29	C₂²×C₈×C₅₃	—	2	8	no	—
30	C₂²×C₈×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	—	2	4	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₅₃	D₄	1	8	yes	—
35	C₂²⋊C₈×C₅₃	—	2	16	no	—
36	C₂⁴×C₅₃	—	2	8	no	—
37	C₂⁴×C₅₃	—	2	4	no	—
38	C₂⁴×C₅₃	C₄⋊C₄	1	4	yes	—
39	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	16	no	—
45	C₂²×C₄×C₅₃	—	2	8	no	—
46	C₂²×C₄×C₅₃	—	2	8	no	—
47	C₂²×C₄×C₅₃	—	2	8	no	—
48	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	4	yes	—
59	M₁₆×C₅₃	—	2	16	no	—
60	M₁₆×C₅₃	—	4	4	no	—
61	M₁₆×C₅₃	—	4	4	no	—
62	C₂×C₈×C₅₃	—	2	32	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	16	no	—
66	C₂×C₈×C₅₃	—	2	8	no	—
67	C₂×C₈×C₅₃	—	4	8	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	8	yes	—
69	C₂³×C₅₃	—	2	32	no	—
70	C₂³×C₅₃	—	2	16	no	—
71	C₂³×C₅₃	—	2	4	no	—
72	C₂³×C₅₃	—	2	8	no	—
73	C₂³×C₅₃	—	2	4	no	—
74	C₂³×C₅₃	—	2	4	no	—
75	C₂³×C₅₃	#95	1	4	yes	—
76	C₂³×C₅₃	#108	1	8	yes	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₅₃	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	8	no	—
82	C₂×C₄×C₅₃	—	2	8	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	16	no	—
86	C₂×C₄×C₅₃	#95	1	4	yes	—
87	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	#117	1	2	yes	—
92	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
93	C₂×C₄×C₅₃	#120	1	4	yes	—
94	C₂×C₄×C₅₃	#121	1	4	yes	—
95	C₂×C₄×C₅₃	#123	1	4	yes	—
96	C₈×C₅₃	—	2	8	no	—
97	C₈×C₅₃	—	4	8	no	—
98	#10723	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	2	32	no	—
100	C₂²×C₅₃	—	2	16	no	—
101	C₂²×C₅₃	—	2	8	no	—
102	C₂²×C₅₃	—	2	4	no	—
103	C₂²×C₅₃	—	2	4	no	—
104	C₂²×C₅₃	—	2	4	no	—
105	C₂²×C₅₃	#341	1	8	yes	complemented
106	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
107	C₂²×C₅₃	#380	1	2	yes	—
108	C₂²×C₅₃	#385	1	4	yes	—
109	C₂²×C₅₃	C₂×#108	1	4	yes	—
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
112	C₄×C₅₃	#379	1	1	yes	characteristic
113	C₄×C₅₃	#381	1	1	yes	characteristic
114	C₄×C₅₃	#386	1	4	yes	—
115	C₂⁴×C₈	C₂×C₅₃	1	1	yes	characteristic
116	C₂×#407	C₂×C₅₃	1	2	yes	—
117	C₂×#341	C₂×C₅₃	1	4	yes	complemented
118	C₂×C₅₃	—	2	8	no	—
119	C₂×C₅₃	—	2	4	no	—
120	C₂×C₅₃	C₂²⋊#95	1	1	yes	characteristic
121	C₂×C₅₃	C₂×#341	1	2	yes	—
122	C₂×C₅₃	#1810	1	4	yes	—
123	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
124	C₂²×M₁₆	C₂²×C₅₃	1	4	yes	—
125	C₂³×C₈	—	2	4	no	—
126	C₂³×C₈	—	2	8	no	—
127	C₂³×C₈	C₄×C₅₃	1	4	yes	—
128	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
129	#407	—	2	4	no	—
130	#407	C₄×C₅₃	1	8	yes	—
131	C₂×C₂²⋊C₈	—	2	4	no	—
132	#341	—	2	8	no	—
133	C₅₃	#10723	1	1	yes	characteristic, direct product factor, Sylow
134	C₂⁵	Q₈×C₅₃	1	1	yes	characteristic
135	C₂³×C₄	—	2	4	no	—
136	C₂³×C₄	—	2	4	no	—
137	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
138	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
139	C₂³×C₄	D₄×C₅₃	1	4	yes	—
140	C₂³×C₄	D₄×C₅₃	1	2	yes	—
141	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	2	8	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
147	C₂²×C₈	—	2	32	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	2	8	no	—
151	C₂²×C₈	—	2	16	no	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
154	C₂²×C₈	D₄×C₅₃	1	8	yes	—
155	C₂²⋊C₈	—	2	16	no	—
156	C₂⁴	—	2	8	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	C₄⋊C₄×C₅₃	1	4	yes	—
159	C₂⁴	C₄⋊C₄×C₅₃	1	2	yes	—
160	C₂⁴	C₂×Q₈×C₅₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	16	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
169	C₂²×C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
170	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
171	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
172	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
173	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
175	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
178	C₂²×C₄	#41×C₅₃	1	4	yes	—
179	M₁₆	—	2	16	no	—
180	M₁₆	—	4	4	no	—
181	M₁₆	—	4	4	no	—
182	C₂×C₈	—	2	32	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	16	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	4	8	no	—
188	C₂×C₈	C₂²⋊C₄×C₅₃	1	8	yes	—
189	C₂³	—	2	32	no	—
190	C₂³	—	2	16	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	8	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	4	no	—
195	C₂³	#95×C₅₃	1	4	yes	—
196	C₂³	#108×C₅₃	1	8	yes	—
197	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
198	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
199	C₂³	C₂²⋊Q₈×C₅₃	1	4	yes	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	#95×C₅₃	1	4	yes	—
207	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
211	C₂×C₄	#117×C₅₃	1	2	yes	—
212	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
213	C₂×C₄	#120×C₅₃	1	4	yes	—
214	C₂×C₄	#121×C₅₃	1	4	yes	—
215	C₂×C₄	#123×C₅₃	1	4	yes	—
216	C₈	—	2	8	no	—
217	C₈	—	4	8	no	—
218	C₂²	—	2	32	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	16	no	—
223	C₂²	—	2	8	no	—
224	C₂²	C₅₃×#341	1	8	yes	complemented
225	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
226	C₂²	C₅₃×#380	1	2	yes	—
227	C₂²	C₅₃×#385	1	4	yes	—
228	C₂²	C₂×#108×C₅₃	1	4	yes	—
229	C₄	—	2	4	no	—
230	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
231	C₄	C₅₃×#379	1	1	yes	characteristic
232	C₄	C₅₃×#381	1	1	yes	characteristic
233	C₄	C₅₃×#386	1	4	yes	—
234	C₂	—	2	8	no	—
235	C₂	—	2	4	no	—
236	C₂	C₅₃×C₂²⋊#95	1	1	yes	characteristic
237	C₂	C₂×C₅₃×#341	1	2	yes	—
238	C₂	C₅₃×#1810	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁵×C₅₃	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂⁴×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#341	C₂	semidirect
98. #10723	C₅₃	direct (N × H)
105. C₂²×C₅₃	#341	semidirect
117. C₂×#341	C₂×C₅₃	semidirect
133. C₅₃	#10723	direct (N × H)
224. C₂²	C₅₃×#341	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₅₃	2
C₂×C₅₃×#407	2
C₂×C₅₃×#341	2
#10723	53

Subgroups — C53×#10723 #27218235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10723 #27218235