Subgroups — C₅₃×#7862 #27215374

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 248 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×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1275	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×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	—
9	C₅₃×#431	C₂²	1	4	yes	—
10	C₅₃×#423	—	2	2	no	—
11	C₅₃×#407	—	2	2	no	—
12	C₄×M₁₆×C₅₃	—	2	2	no	—
13	C₄×M₁₆×C₅₃	—	2	2	no	—
14	C₄×M₁₆×C₅₃	C₄	1	4	yes	—
15	C₂×#97×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	8	no	—
21	C₂×M₁₆×C₅₃	—	2	4	no	—
22	C₂×M₁₆×C₅₃	—	2	4	no	—
23	C₂²×C₈×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	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	—	2	4	no	—
33	C₄⋊C₈×C₅₃	—	2	4	no	—
34	C₂²⋊C₈×C₅₃	—	2	4	no	—
35	C₂²⋊C₈×C₅₃	—	2	4	no	—
36	#97×C₅₃	—	2	4	no	—
37	#97×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
41	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	M₁₆×C₅₃	—	2	16	no	—
58	M₁₆×C₅₃	—	4	8	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
63	C₄²×C₅₃	C₄²	1	1	yes	characteristic
64	C₄²×C₅₃	C₄²	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₃	C₂×D₄	1	2	yes	—
73	C₂³×C₅₃	—	2	2	no	—
74	C₂³×C₅₃	—	4	1	no	—
75	C₂³×C₅₃	—	4	1	no	—
76	C₂³×C₅₃	#95	1	2	yes	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	#95	1	2	yes	—
90	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
99	C₈×C₅₃	—	2	8	no	—
100	C₈×C₅₃	—	4	8	no	—
101	#7862	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	4	4	no	—
108	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
109	C₂²×C₅₃	#423	1	2	yes	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
116	C₄×C₅₃	#338	1	2	yes	—
117	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
118	C₄×C₅₃	#407	1	2	yes	—
119	#2920	C₂×C₅₃	1	2	yes	—
120	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
121	#1275	C₂×C₅₃	1	4	yes	complemented
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	—	4	2	no	—
124	C₂×C₅₃	#1728	1	1	yes	characteristic
125	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#338	1	1	yes	characteristic
127	C₂²×M₁₆	C₄×C₅₃	1	2	yes	—
128	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
129	#432	—	2	2	no	—
130	#432	—	2	2	no	—
131	#431	C₄×C₅₃	1	4	yes	—
132	#431	C₂²×C₅₃	1	4	yes	—
133	#423	—	2	2	no	—
134	#407	—	2	2	no	—
135	C₄×M₁₆	—	2	2	no	—
136	C₄×M₁₆	—	2	2	no	—
137	C₄×M₁₆	C₄×C₅₃	1	4	yes	—
138	C₂×#97	C₂²×C₅₃	1	2	yes	—
139	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
140	C₅₃	#7862	1	1	yes	characteristic, direct product factor, Sylow
141	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
142	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
143	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	—	2	8	no	—
147	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
148	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
149	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
150	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄²	D₄×C₅₃	1	2	yes	—
152	C₂×C₄²	D₄×C₅₃	1	2	yes	—
153	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
156	C₄⋊C₈	—	2	4	no	—
157	C₄⋊C₈	—	2	4	no	—
158	C₂²⋊C₈	—	2	4	no	—
159	C₂²⋊C₈	—	2	4	no	—
160	#97	—	2	4	no	—
161	#97	—	2	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
165	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
170	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
171	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
172	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
173	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
177	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
181	M₁₆	—	2	16	no	—
182	M₁₆	—	4	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	M₁₆×C₅₃	1	8	yes	complemented
187	C₄²	C₄²×C₅₃	1	1	yes	characteristic
188	C₄²	C₄²×C₅₃	1	1	yes	characteristic
189	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
190	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
191	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
192	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
193	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
194	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
195	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
196	C₄²	C₂×D₄×C₅₃	1	2	yes	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	#95×C₅₃	1	2	yes	—
201	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
214	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
215	C₂×C₄	#95×C₅₃	1	2	yes	—
216	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
217	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
223	C₈	—	2	8	no	—
224	C₈	—	4	8	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
232	C₂²	C₅₃×#423	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	C₅₃×H(3,ℤ/4)	1	2	yes	—
239	C₄	C₅₃×#338	1	2	yes	—
240	C₄	C₄×M₁₆×C₅₃	1	2	yes	—
241	C₄	C₅₃×#407	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₅₃×#1728	1	1	yes	characteristic
245	C₂	C₂×C₅₃×H(3,ℤ/4)	1	1	yes	characteristic
246	C₂	C₂×C₅₃×#338	1	1	yes	characteristic

	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	C₄⋊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×#1275	C₂	semidirect
62. C₂×C₈×C₅₃	M₁₆	semidirect
101. #7862	C₅₃	direct (N × H)
121. #1275	C₂×C₅₃	semidirect
140. C₅₃	#7862	direct (N × H)
186. C₂×C₈	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×C₄×M₁₆×C₅₃	2
C₅₃×#1275	2
#7862	53

Subgroups — C53×#7862 #27215374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7862 #27215374