Subgroups — C₅₃×#7868 #27215380

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 250 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	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1276	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₅₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#432	—	2	2	no	—
7	C₅₃×#432	C₄	1	4	yes	complemented
8	C₅₃×#431	—	2	2	no	—
9	C₅₃×#431	C₂²	1	4	yes	—
10	C₅₃×#423	C₄	1	4	yes	complemented
11	C₅₃×#407	—	2	2	no	—
12	C₄×M₁₆×C₅₃	—	2	2	no	—
13	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
15	C₅₃×#376	C₄	1	2	yes	—
16	C₄³×C₅₃	C₂²	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	D₄	1	2	yes	—
19	C₂×M₁₆×C₅₃	—	2	4	no	—
20	#117×C₅₃	—	2	8	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	—	2	4	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	4	no	—
35	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
36	C₂²⋊C₈×C₅₃	—	2	4	no	—
37	C₂²⋊C₈×C₅₃	—	2	4	no	—
38	#97×C₅₃	—	2	4	no	—
39	C₄×C₈×C₅₃	—	2	4	no	—
40	#95×C₅₃	—	2	4	no	—
41	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	4	4	no	—
45	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	M₁₆×C₅₃	—	4	8	no	—
58	C₂×C₈×C₅₃	—	2	8	no	—
59	C₄⋊C₄×C₅₃	—	2	8	no	—
60	C₂²⋊C₄×C₅₃	—	2	8	no	—
61	C₄²×C₅₃	—	2	4	no	—
62	C₄²×C₅₃	—	4	8	no	—
63	C₄²×C₅₃	C₄²	1	2	yes	—
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	M₁₆	1	8	yes	complemented
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	—	4	2	no	—
76	C₂³×C₅₃	#95	1	1	yes	characteristic
77	C₂³×C₅₃	#95	1	1	yes	characteristic
78	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	8	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	8	no	—
92	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#95	1	2	yes	—
95	C₂×C₄×C₅₃	#95	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
101	C₈×C₅₃	—	4	8	no	—
102	#7868	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	4	4	no	—
109	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
110	C₂²×C₅₃	#407	1	2	yes	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	4	8	no	—
117	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
118	C₄×C₅₃	#421	1	4	yes	—
119	C₄×C₅₃	#423	1	2	yes	—
120	#2920	C₂×C₅₃	1	2	yes	—
121	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
122	#1276	C₂×C₅₃	1	4	yes	complemented
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	—	4	1	no	—
125	C₂×C₅₃	—	4	1	no	—
126	C₂×C₅₃	#1728	1	1	yes	characteristic
127	C₂×C₅₃	#1733	1	2	yes	—
128	C₂×#117	C₄×C₅₃	1	2	yes	—
129	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
130	#432	—	2	2	no	—
131	#432	C₄×C₅₃	1	4	yes	complemented
132	#431	—	2	2	no	—
133	#431	C₂²×C₅₃	1	4	yes	—
134	#423	C₄×C₅₃	1	4	yes	complemented
135	#407	—	2	2	no	—
136	C₄×M₁₆	—	2	2	no	—
137	C₄×C₄⋊C₄	—	2	2	no	—
138	C₄×C₂²⋊C₄	—	2	2	no	—
139	#376	C₄×C₅₃	1	2	yes	—
140	C₄³	C₂²×C₅₃	1	2	yes	—
141	C₅₃	#7868	1	1	yes	characteristic, direct product factor, Sylow
142	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
143	C₂³×C₄	D₄×C₅₃	1	2	yes	—
144	C₂×M₁₆	—	2	4	no	—
145	#117	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	2	2	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
150	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
151	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
152	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
153	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
155	C₂×C₄²	D₄×C₅₃	1	2	yes	—
156	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
157	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
158	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
161	C₂²⋊C₈	—	2	4	no	—
162	C₂²⋊C₈	—	2	4	no	—
163	#97	—	2	4	no	—
164	C₄×C₈	—	2	4	no	—
165	#95	—	2	4	no	—
166	C₂⁴	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
174	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
175	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
182	M₁₆	—	4	8	no	—
183	C₂×C₈	—	2	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	8	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	8	no	—
188	C₄²	C₄²×C₅₃	1	2	yes	—
189	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
190	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
191	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
192	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
193	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
194	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
195	C₄²	M₁₆×C₅₃	1	8	yes	complemented
196	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
197	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
198	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	—	4	2	no	—
201	C₂³	#95×C₅₃	1	1	yes	characteristic
202	C₂³	#95×C₅₃	1	1	yes	characteristic
203	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	#95×C₅₃	1	2	yes	—
218	C₂×C₄	#95×C₅₃	1	2	yes	—
219	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
220	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
226	C₈	—	4	8	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
234	C₂²	C₅₃×#407	1	2	yes	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	C₄×M₁₆×C₅₃	1	2	yes	—
242	C₄	C₅₃×#421	1	4	yes	—
243	C₄	C₅₃×#423	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	C₅₃×#1728	1	1	yes	characteristic
248	C₂	C₅₃×#1733	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1276	C₂	semidirect
7. C₅₃×#432	C₄	semidirect
10. C₅₃×#423	C₄	semidirect
70. C₄²×C₅₃	M₁₆	semidirect
102. #7868	C₅₃	direct (N × H)
122. #1276	C₂×C₅₃	semidirect
131. #432	C₄×C₅₃	semidirect
134. #423	C₄×C₅₃	semidirect
141. C₅₃	#7868	direct (N × H)
195. 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
C₄×#117×C₅₃	2
C₅₃×#1276	2
#7868	53

Subgroups — C53×#7868 #27215380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7868 #27215380