Subgroups — C₅₃×#9710 #27217222

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2858	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#332	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#331	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#124×C₅₃	—	2	2	no	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×#97×C₅₃	—	2	1	no	—
12	C₅₃×#390	—	2	1	no	—
13	C₅₃×#390	—	2	1	no	—
14	C₅₃×#390	C₄	1	4	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	1	no	—
17	C₅₃×#336	—	2	2	no	—
18	C₅₃×#332	—	2	4	no	—
19	C₅₃×#331	—	2	4	no	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×D₄×C₅₃	—	2	1	no	—
22	C₂²×D₄×C₅₃	—	2	1	no	—
23	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	#127×C₅₃	—	2	2	no	—
29	#127×C₅₃	—	2	2	no	—
30	#124×C₅₃	—	4	8	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
44	#97×C₅₃	—	4	2	no	—
45	C₂⁴×C₅₃	—	2	2	no	—
46	C₂×Q₈×C₅₃	—	4	1	no	—
47	C₂×Q₈×C₅₃	—	4	1	no	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂×D₄×C₅₃	—	2	4	no	—
50	C₂×D₄×C₅₃	—	4	4	no	—
51	C₂×D₄×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	4	1	no	—
53	C₂×D₄×C₅₃	—	4	1	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	4	no	—
56	C₂×D₄×C₅₃	—	4	1	no	—
57	C₂×D₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	4	no	—
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₂²⋊C₄×C₅₃	—	4	8	no	—
72	C₂²⋊C₄×C₅₃	—	4	8	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	4	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	—	4	2	no	—
78	C₄²×C₅₃	C₂×D₄	1	2	yes	—
79	C₄²×C₅₃	#41	1	2	yes	—
80	C₂³×C₅₃	—	4	2	no	—
81	C₂³×C₅₃	—	4	4	no	—
82	C₂³×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	—	4	4	no	—
85	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₅₃	—	8	2	no	—
87	Q₈×C₅₃	—	8	2	no	—
88	D₄×C₅₃	—	8	4	no	—
89	D₄×C₅₃	—	8	2	no	—
90	D₄×C₅₃	—	8	2	no	—
91	D₄×C₅₃	—	8	2	no	—
92	D₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	#121	1	2	yes	—
104	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#135	1	2	yes	—
107	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
109	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
110	C₈×C₅₃	—	4	4	no	—
111	C₈×C₅₃	—	8	2	no	—
112	#9710	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	8	4	no	—
114	C₂²×C₅₃	—	8	4	no	—
115	C₂²×C₅₃	—	8	2	no	—
116	C₂²×C₅₃	—	8	2	no	—
117	C₂²×C₅₃	—	8	2	no	—
118	C₂²×C₅₃	#390	1	1	yes	characteristic
119	C₂²×C₅₃	#413	1	2	yes	—
120	C₂²×C₅₃	#416	1	1	yes	characteristic
121	C₂²×C₅₃	#417	1	1	yes	characteristic
122	C₂²×C₅₃	#418	1	2	yes	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	—	4	2	no	—
126	C₄×C₅₃	—	8	2	no	—
127	C₄×C₅₃	#463	1	2	yes	—
128	C₄×C₅₃	#464	1	2	yes	—
129	C₄×C₅₃	#490	1	2	yes	—
130	C₄×C₅₃	#496	1	2	yes	—
131	#2858	C₂×C₅₃	1	1	yes	characteristic
132	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
133	#1767	C₂×C₅₃	1	1	yes	characteristic, complemented
134	C₂×#332	C₂×C₅₃	1	2	yes	complemented
135	C₂×#331	C₂×C₅₃	1	2	yes	—
136	C₂×C₅₃	—	16	2	no	—
137	C₂×C₅₃	#1553	1	2	yes	—
138	C₂×C₅₃	#1557	1	2	yes	—
139	C₂×C₅₃	#1961	1	1	yes	characteristic
140	C₂×C₅₃	#1967	1	1	yes	characteristic
141	C₂×C₅₃	#1983	1	1	yes	characteristic
142	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
143	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
144	C₂×#124	—	2	2	no	—
145	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
147	C₂×#97	—	2	1	no	—
148	#390	—	2	1	no	—
149	#390	—	2	1	no	—
150	#390	C₄×C₅₃	1	4	yes	—
151	C₄³	C₂²×C₅₃	1	1	yes	characteristic
152	#336	—	2	2	no	—
153	#336	—	2	1	no	—
154	#332	—	2	4	no	—
155	#331	—	2	4	no	—
156	C₅₃	#9710	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×D₄	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	#127	—	2	2	no	—
166	#127	—	2	2	no	—
167	#124	—	4	8	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	—	2	4	no	—
172	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
173	C₂×C₄²	D₄×C₅₃	1	2	yes	—
174	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
181	#97	—	4	2	no	—
182	C₂⁴	—	2	2	no	—
183	C₂×Q₈	—	4	1	no	—
184	C₂×Q₈	—	4	1	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	8	no	—
209	C₂²⋊C₄	—	4	8	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	2	no	—
215	C₄²	C₂×D₄×C₅₃	1	2	yes	—
216	C₄²	#41×C₅₃	1	2	yes	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	4	no	—
222	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
223	Q₈	—	8	2	no	—
224	Q₈	—	8	2	no	—
225	D₄	—	8	4	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	2	no	—
228	D₄	—	8	2	no	—
229	D₄	—	8	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
240	C₂×C₄	#121×C₅₃	1	2	yes	—
241	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
242	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
243	C₂×C₄	#135×C₅₃	1	2	yes	—
244	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
245	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
246	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
247	C₈	—	4	4	no	—
248	C₈	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	C₅₃×#390	1	1	yes	characteristic
255	C₂²	C₅₃×#413	1	2	yes	—
256	C₂²	C₅₃×#416	1	1	yes	characteristic
257	C₂²	C₅₃×#417	1	1	yes	characteristic
258	C₂²	C₅₃×#418	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	2	no	—
263	C₄	C₅₃×#463	1	2	yes	—
264	C₄	C₅₃×#464	1	2	yes	—
265	C₄	C₅₃×#490	1	2	yes	—
266	C₄	C₅₃×#496	1	2	yes	—
267	C₂	—	16	2	no	—
268	C₂	C₅₃×#1553	1	2	yes	—
269	C₂	C₅₃×#1557	1	2	yes	—
270	C₂	C₅₃×#1961	1	1	yes	characteristic
271	C₂	C₅₃×#1967	1	1	yes	characteristic
272	C₂	C₅₃×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1831	C₂	semidirect
3. C₅₃×#1767	C₂	semidirect
4. C₂×C₅₃×#332	C₂	semidirect
112. #9710	C₅₃	direct (N × H)
132. #1831	C₂×C₅₃	semidirect
133. #1767	C₂×C₅₃	semidirect
134. C₂×#332	C₂×C₅₃	semidirect
156. C₅₃	#9710	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2858	2
C₅₃×#1831	2
C₅₃×#1767	2
C₂×C₅₃×#332	2
C₂×C₅₃×#331	2
#9710	53

Subgroups — C53×#9710 #27217222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9710 #27217222