Subgroups — C₅₃×#8224 #27215736

Order 13568.

528 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1745	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#329	C₂	1	2	yes	maximal
4	C₅₃×#1295	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
8	C₂×#97×C₅₃	C₂²	1	4	yes	—
9	C₅₃×#391	C₄	1	1	yes	characteristic
10	C₅₃×#390	C₄	1	1	yes	characteristic
11	C₅₃×#388	—	2	2	no	—
12	C₅₃×#382	C₂²	1	1	yes	characteristic
13	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#336	—	2	2	no	—
16	C₅₃×#336	—	2	2	no	—
17	C₅₃×#329	C₄	1	8	yes	complemented
18	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	—	2	1	no	—
21	C₂²×C₈×C₅₃	—	2	4	no	—
22	#124×C₅₃	—	2	2	no	—
23	#124×C₅₃	C₂×C₄	1	4	yes	—
24	C₄×Q₈×C₅₃	—	4	2	no	—
25	C₄×D₄×C₅₃	—	4	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#97×C₅₃	—	2	4	no	—
39	#97×C₅₃	C₂×C₄	1	8	yes	complemented
40	#95×C₅₃	—	2	2	no	—
41	#95×C₅₃	—	2	2	no	—
42	C₂⁴×C₅₃	—	2	1	no	—
43	C₂×Q₈×C₅₃	—	2	1	no	—
44	C₂×Q₈×C₅₃	—	4	1	no	—
45	C₂×Q₈×C₅₃	—	4	1	no	—
46	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
47	C₂×D₄×C₅₃	—	2	1	no	—
48	C₂×D₄×C₅₃	—	4	1	no	—
49	C₂×D₄×C₅₃	—	4	1	no	—
50	C₂×D₄×C₅₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	4	4	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	—	4	4	no	—
63	C₂×C₈×C₅₃	—	4	4	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	1	no	—
66	C₄⋊C₄×C₅₃	—	4	1	no	—
67	C₂²⋊C₄×C₅₃	—	2	8	no	—
68	C₂²⋊C₄×C₅₃	—	4	4	no	—
69	C₂²⋊C₄×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	4	1	no	—
72	C₄²×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	—	4	2	no	—
75	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₅₃	#41	1	2	yes	—
77	C₂³×C₅₃	—	2	2	no	—
78	C₂³×C₅₃	—	4	1	no	—
79	C₂³×C₅₃	—	4	1	no	—
80	C₂³×C₅₃	—	4	2	no	—
81	C₂³×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	4	2	no	—
84	Q₈×C₅₃	—	4	2	no	—
85	D₄×C₅₃	—	4	2	no	—
86	D₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	1	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	—	4	1	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	8	4	no	—
105	C₂×C₄×C₅₃	#100	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
109	C₈×C₅₃	—	4	8	no	—
110	#8224	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₅₃	—	4	2	no	—
112	C₂²×C₅₃	—	4	4	no	—
113	C₂²×C₅₃	—	8	1	no	—
114	C₂²×C₅₃	—	8	1	no	—
115	C₂²×C₅₃	—	8	2	no	—
116	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
118	C₂²×C₅₃	#413	1	1	yes	characteristic
119	C₂²×C₅₃	#421	1	2	yes	—
120	C₂²×C₅₃	#421	1	2	yes	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	4	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	4	1	no	—
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	—	8	1	no	—
129	C₄×C₅₃	—	8	1	no	—
130	C₄×#124	C₂×C₅₃	1	1	yes	characteristic
131	#1745	C₂×C₅₃	1	2	yes	complemented
132	C₂×#329	C₂×C₅₃	1	2	yes	—
133	#1295	C₂×C₅₃	1	2	yes	complemented
134	C₂×C₅₃	—	8	2	no	—
135	C₂×C₅₃	#1533	1	1	yes	characteristic
136	C₂×C₅₃	#1535	1	2	yes	—
137	C₂×C₅₃	#1536	1	1	yes	characteristic
138	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
139	C₂×C₅₃	#1759	1	2	yes	—
140	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic
142	C₂×C₄×D₄	C₄×C₅₃	1	1	yes	characteristic
143	C₂×#97	C₂²×C₅₃	1	4	yes	—
144	#391	C₄×C₅₃	1	1	yes	characteristic
145	#390	C₄×C₅₃	1	1	yes	characteristic
146	#388	—	2	2	no	—
147	#382	C₂²×C₅₃	1	1	yes	characteristic
148	C₄×C₂²⋊C₄	—	2	2	no	—
149	C₄³	C₂²×C₅₃	1	1	yes	characteristic
150	#336	—	2	2	no	—
151	#336	—	2	2	no	—
152	#329	C₄×C₅₃	1	8	yes	complemented
153	C₅₃	#8224	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
155	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
156	C₂³×C₄	—	2	1	no	—
157	C₂²×C₈	—	2	4	no	—
158	#124	—	2	2	no	—
159	#124	C₂×C₄×C₅₃	1	4	yes	—
160	C₄×Q₈	—	4	2	no	—
161	C₄×D₄	—	4	2	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	—	2	2	no	—
166	C₂×C₂²⋊C₄	—	2	1	no	—
167	C₂×C₂²⋊C₄	—	2	2	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₅₃	1	2	yes	—
172	C₂×C₄²	D₄×C₅₃	1	2	yes	—
173	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
174	#97	—	2	4	no	—
175	#97	C₂×C₄×C₅₃	1	8	yes	complemented
176	#95	—	2	2	no	—
177	#95	—	2	2	no	—
178	C₂⁴	—	2	1	no	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	C₄²×C₅₃	1	2	yes	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	C₄²×C₅₃	1	2	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	2	8	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₂²⋊C₄	—	4	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	1	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	4	2	no	—
211	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
212	C₄²	#41×C₅₃	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	1	no	—
215	C₂³	—	4	1	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	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	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	#100×C₅₃	1	2	yes	—
242	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
243	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
245	C₈	—	4	8	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	8	1	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	1	no	—
251	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
252	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
253	C₂²	C₅₃×#413	1	1	yes	characteristic
254	C₂²	C₅₃×#421	1	2	yes	—
255	C₂²	C₅₃×#421	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	8	1	no	—
264	C₄	—	8	1	no	—
265	C₂	—	8	2	no	—
266	C₂	C₅₃×#1533	1	1	yes	characteristic
267	C₂	C₅₃×#1535	1	2	yes	—
268	C₂	C₅₃×#1536	1	1	yes	characteristic
269	C₂	C₄×#100×C₅₃	1	1	yes	characteristic
270	C₂	C₅₃×#1759	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂×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×#1745	C₂	semidirect
4. C₅₃×#1295	C₂	semidirect
17. C₅₃×#329	C₄	semidirect
39. #97×C₅₃	C₂×C₄	semidirect
110. #8224	C₅₃	direct (N × H)
131. #1745	C₂×C₅₃	semidirect
133. #1295	C₂×C₅₃	semidirect
152. #329	C₄×C₅₃	semidirect
153. C₅₃	#8224	direct (N × H)
175. #97	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#124×C₅₃	2
C₅₃×#1745	2
C₂×C₅₃×#329	2
C₅₃×#1295	2
#8224	53

Subgroups — C53×#8224 #27215736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8224 #27215736