Subgroups — C₅₃×#7855 #27215367

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2920	C₂	1	2	yes	maximal
2	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1274	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₅₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#432	—	2	2	no	—
7	C₅₃×#432	—	2	2	no	—
8	C₅₃×#431	C₄	1	4	yes	complemented
9	C₅₃×#431	C₂²	1	4	yes	—
10	C₅₃×#423	—	2	2	no	—
11	C₅₃×#407	—	2	2	no	—
12	C₄×M₁₆×C₅₃	C₄	1	4	yes	complemented
13	C₅₃×#376	—	2	2	no	—
14	C₅₃×#376	C₄	1	2	yes	—
15	C₅₃×#376	C₂²	1	2	yes	—
16	C₂×#95×C₅₃	—	2	2	no	—
17	C₂³×C₄×C₅₃	—	2	2	no	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	D₄	1	2	yes	—
20	C₂×M₁₆×C₅₃	—	2	4	no	—
21	C₂×C₄²×C₅₃	—	2	8	no	—
22	C₂×C₄²×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	4	no	—
32	C₄⋊C₈×C₅₃	—	2	4	no	—
33	C₂²⋊C₈×C₅₃	—	2	4	no	—
34	C₂²⋊C₈×C₅₃	—	2	4	no	—
35	#97×C₅₃	—	2	4	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
37	#95×C₅₃	—	2	4	no	—
38	#95×C₅₃	—	2	4	no	—
39	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	2	8	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	4	no	—
44	C₂²×C₄×C₅₃	—	4	8	no	—
45	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	M₁₆×C₅₃	—	4	8	no	—
56	C₂×C₈×C₅₃	—	2	8	no	—
57	C₄²×C₅₃	—	2	8	no	—
58	C₄²×C₅₃	—	2	4	no	—
59	C₄²×C₅₃	C₄²	1	1	yes	characteristic
60	C₄²×C₅₃	C₄²	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₃	M₁₆	1	8	yes	complemented
67	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₅₃	—	2	4	no	—
71	C₂³×C₅₃	—	2	2	no	—
72	C₂³×C₅₃	#95	1	2	yes	—
73	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	8	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	16	no	—
85	C₂×C₄×C₅₃	—	4	8	no	—
86	C₂×C₄×C₅₃	#95	1	2	yes	—
87	C₂×C₄×C₅₃	#95	1	2	yes	—
88	C₂×C₄×C₅₃	#95	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
94	C₈×C₅₃	—	4	8	no	—
95	#7855	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₅₃	—	2	8	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	H(3,ℤ/4)	1	4	yes	—
101	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
102	C₂²×C₅₃	#407	1	2	yes	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	4	8	no	—
109	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
110	C₄×C₅₃	#423	1	2	yes	—
111	#2920	C₂×C₅₃	1	2	yes	—
112	C₂×#376	C₂×C₅₃	1	1	yes	characteristic
113	#1274	C₂×C₅₃	1	4	yes	complemented
114	C₂×C₅₃	—	2	2	no	—
115	C₂×C₅₃	—	2	4	no	—
116	C₂×C₅₃	#1728	1	1	yes	characteristic
117	C₂×C₅₃	C₂×H(3,ℤ/4)	1	2	yes	—
118	C₂²×C₄²	C₄×C₅₃	1	2	yes	—
119	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
120	#432	—	2	2	no	—
121	#432	—	2	2	no	—
122	#431	C₄×C₅₃	1	4	yes	complemented
123	#431	C₂²×C₅₃	1	4	yes	—
124	#423	—	2	2	no	—
125	#407	—	2	2	no	—
126	C₄×M₁₆	C₄×C₅₃	1	4	yes	complemented
127	#376	—	2	2	no	—
128	#376	C₄×C₅₃	1	2	yes	—
129	#376	C₂²×C₅₃	1	2	yes	—
130	C₂×#95	—	2	2	no	—
131	C₅₃	#7855	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
134	C₂³×C₄	D₄×C₅₃	1	2	yes	—
135	C₂×M₁₆	—	2	4	no	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	—	2	8	no	—
138	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
139	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
141	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
142	C₂×C₄²	D₄×C₅₃	1	2	yes	—
143	C₂×C₄²	D₄×C₅₃	1	2	yes	—
144	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
145	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	2	4	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	C₂²⋊C₈	—	2	4	no	—
150	#97	—	2	4	no	—
151	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	complemented
152	#95	—	2	4	no	—
153	#95	—	2	4	no	—
154	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
163	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
165	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
166	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
167	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
168	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
170	M₁₆	—	4	8	no	—
171	C₂×C₈	—	2	8	no	—
172	C₄²	—	2	8	no	—
173	C₄²	—	2	4	no	—
174	C₄²	C₄²×C₅₃	1	1	yes	characteristic
175	C₄²	C₄²×C₅₃	1	1	yes	characteristic
176	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
177	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
178	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
179	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
180	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
181	C₄²	M₁₆×C₅₃	1	8	yes	complemented
182	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
183	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
184	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
185	C₂³	—	2	4	no	—
186	C₂³	—	2	2	no	—
187	C₂³	#95×C₅₃	1	2	yes	—
188	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	16	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	#95×C₅₃	1	2	yes	—
202	C₂×C₄	#95×C₅₃	1	2	yes	—
203	C₂×C₄	#95×C₅₃	1	2	yes	—
204	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
209	C₈	—	4	8	no	—
210	C₂²	—	2	8	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	C₅₃×H(3,ℤ/4)	1	4	yes	—
215	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
216	C₂²	C₅₃×#407	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	8	no	—
223	C₄	C₄×M₁₆×C₅₃	1	2	yes	—
224	C₄	C₅₃×#423	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	—	2	4	no	—
227	C₂	C₅₃×#1728	1	1	yes	characteristic
228	C₂	C₂×C₅₃×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1274	C₂	semidirect
8. C₅₃×#431	C₄	semidirect
12. C₄×M₁₆×C₅₃	C₄	semidirect
36. C₄×C₈×C₅₃	C₂×C₄	semidirect
66. C₄²×C₅₃	M₁₆	semidirect
95. #7855	C₅₃	direct (N × H)
113. #1274	C₂×C₅₃	semidirect
122. #431	C₄×C₅₃	semidirect
126. C₄×M₁₆	C₄×C₅₃	semidirect
131. C₅₃	#7855	direct (N × H)
151. C₄×C₈	C₂×C₄×C₅₃	semidirect
181. C₄²	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2920	2
C₂×C₅₃×#376	2
C₅₃×#1274	2
#7855	53

Subgroups — C53×#7855 #27215367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7855 #27215367