Subgroups — C₅₃×#11485 #27218997

Order 13568.

372 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2115	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1327	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1321	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
6	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#532	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₅₃	—	2	2	no	—
9	C₅₃×#436	—	2	2	no	—
10	C₈×D₄×C₅₃	—	2	2	no	—
11	C₅₃×#423	C₂²	1	1	yes	characteristic
12	C₅₃×#408	C₄	1	1	yes	characteristic
13	C₅₃×#408	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₅₃×#349	—	2	2	no	—
16	C₅₃×#349	C₂²	1	4	yes	—
17	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#131×C₅₃	—	4	1	no	—
20	#131×C₅₃	—	4	1	no	—
21	#131×C₅₃	—	4	1	no	—
22	#131×C₅₃	—	4	1	no	—
23	C₂×M₁₆×C₅₃	—	2	1	no	—
24	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
26	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
32	#127×C₅₃	C₈	1	1	yes	characteristic
33	#124×C₅₃	C₈	1	1	yes	characteristic
34	#124×C₅₃	C₈	1	1	yes	characteristic
35	#121×C₅₃	—	2	2	no	—
36	C₄×D₄×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₅₃	—	2	4	no	—
39	M₃₂×C₅₃	—	4	2	no	—
40	C₂×C₁₆×C₅₃	—	4	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	C₂²⋊C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	—	2	1	no	—
45	C₄×C₈×C₅₃	—	2	1	no	—
46	#41×C₅₃	—	4	1	no	—
47	#41×C₅₃	—	4	1	no	—
48	#41×C₅₃	—	4	1	no	—
49	#41×C₅₃	—	4	1	no	—
50	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	—	2	1	no	—
54	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂×D₄×C₅₃	M₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	—	2	1	no	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₅₃	—	4	2	no	—
62	M₁₆×C₅₃	—	4	1	no	—
63	M₁₆×C₅₃	—	4	2	no	—
64	M₁₆×C₅₃	—	4	1	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₂²⋊C₄×C₅₃	—	2	2	no	—
79	C₂²⋊C₄×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₁₆×C₅₃	—	4	4	no	—
84	C₂³×C₅₃	—	2	1	no	—
85	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
86	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
87	Q₈×C₅₃	—	4	1	no	—
88	Q₈×C₅₃	—	4	1	no	—
89	D₄×C₅₃	—	4	1	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	4	1	no	—
92	D₄×C₅₃	—	4	4	no	—
93	D₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	4	1	no	—
110	C₈×C₅₃	—	4	1	no	—
111	C₈×C₅₃	#99	1	1	yes	characteristic
112	C₈×C₅₃	#99	1	1	yes	characteristic
113	#11485	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	4	1	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	—	4	2	no	—
119	C₂²×C₅₃	—	4	1	no	—
120	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	4	1	no	—
125	C₄×C₅₃	—	4	1	no	—
126	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
127	C₄×C₅₃	#413	1	1	yes	characteristic
128	#2931	C₂×C₅₃	1	1	yes	characteristic
129	#2115	C₂×C₅₃	1	2	yes	complemented
130	#1327	C₂×C₅₃	1	2	yes	complemented
131	#1321	C₂×C₅₃	1	2	yes	complemented
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	4	1	no	—
134	C₂×C₅₃	—	4	2	no	—
135	C₂×C₅₃	—	8	1	no	—
136	C₂×C₅₃	#1459	1	1	yes	characteristic
137	C₂×#131	C₄×C₅₃	1	1	yes	characteristic
138	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
139	#532	C₄×C₅₃	1	1	yes	characteristic
140	C₂×M₃₂	—	2	2	no	—
141	#436	—	2	2	no	—
142	C₈×D₄	—	2	2	no	—
143	#423	C₂²×C₅₃	1	1	yes	characteristic
144	#408	C₄×C₅₃	1	1	yes	characteristic
145	#408	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
147	#349	—	2	2	no	—
148	#349	C₂²×C₅₃	1	4	yes	—
149	C₅₃	#11485	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
151	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
152	#131	—	4	1	no	—
153	#131	—	4	1	no	—
154	#131	—	4	1	no	—
155	#131	—	4	1	no	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
158	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
159	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₅₃	1	2	yes	—
164	C₄⋊Q₈	C₈×C₅₃	1	1	yes	characteristic
165	#127	C₈×C₅₃	1	1	yes	characteristic
166	#124	C₈×C₅₃	1	1	yes	characteristic
167	#124	C₈×C₅₃	1	1	yes	characteristic
168	#121	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
171	M₃₂	—	2	4	no	—
172	M₃₂	—	4	2	no	—
173	C₂×C₁₆	—	4	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	#41	—	4	1	no	—
180	#41	—	4	1	no	—
181	#41	—	4	1	no	—
182	#41	—	4	1	no	—
183	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
184	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	C₂×C₈×C₅₃	1	1	yes	characteristic
188	C₂×D₄	C₂×C₈×C₅₃	1	1	yes	characteristic
189	C₂×D₄	M₁₆×C₅₃	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	1	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
210	C₄⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	—	2	1	no	—
214	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
215	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
216	C₁₆	—	4	4	no	—
217	C₂³	—	2	1	no	—
218	C₂³	C₂²⋊C₈×C₅₃	1	2	yes	—
219	C₂³	C₂×M₁₆×C₅₃	1	1	yes	characteristic
220	Q₈	—	4	1	no	—
221	Q₈	—	4	1	no	—
222	D₄	—	4	1	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	1	no	—
243	C₈	—	4	1	no	—
244	C₈	#99×C₅₃	1	1	yes	characteristic
245	C₈	#99×C₅₃	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	C₂×#99×C₅₃	1	1	yes	characteristic
259	C₄	C₅₃×#413	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	4	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₅₃×#1459	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	#99
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×M₁₆	C₂³×C₅₃
socle	C₂×C₅₃	#1459
prime radical	C₅₃×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2115	C₂	semidirect
3. C₅₃×#1327	C₂	semidirect
4. C₅₃×#1321	C₂	semidirect
113. #11485	C₅₃	direct (N × H)
129. #2115	C₂×C₅₃	semidirect
130. #1327	C₂×C₅₃	semidirect
131. #1321	C₂×C₅₃	semidirect
149. C₅₃	#11485	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×#2931	2
C₅₃×#2115	2
C₅₃×#1327	2
C₅₃×#1321	2
#11485	53

Subgroups — C53×#11485 #27218997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11485 #27218997