Subgroups — C₅₃×#11966 #27219478

Order 13568.

400 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1362	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₅₃	—	2	1	no	—
7	C₂×#135×C₅₃	C₄	1	2	yes	—
8	C₅₃×#532	—	2	1	no	—
9	C₅₃×#495	—	2	1	no	—
10	C₅₃×#495	C₄	1	2	yes	—
11	C₅₃×#494	C₂²	1	1	yes	characteristic
12	C₅₃×#493	C₂²	1	1	yes	characteristic
13	C₅₃×#492	—	2	1	no	—
14	C₅₃×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₅₃×#346	—	2	1	no	—
19	C₅₃×#335	—	2	1	no	—
20	C₅₃×#335	C₂²	1	2	yes	—
21	C₂×#41×C₅₃	—	2	2	no	—
22	#135×C₅₃	—	4	4	no	—
23	#135×C₅₃	—	4	4	no	—
24	C₂×Q₁₆×C₅₃	—	2	1	no	—
25	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×SD₁₆×C₅₃	—	2	2	no	—
27	C₂×SD₁₆×C₅₃	—	2	2	no	—
28	C₂×D₈×C₅₃	—	2	1	no	—
29	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₅₃	—	4	1	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₅₃	—	2	1	no	—
34	#127×C₅₃	—	2	1	no	—
35	#124×C₅₃	—	2	2	no	—
36	#121×C₅₃	—	4	1	no	—
37	#121×C₅₃	—	4	1	no	—
38	C₄×D₄×C₅₃	—	4	1	no	—
39	C₄×D₄×C₅₃	—	4	1	no	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	4	1	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	#41×C₅₃	—	4	4	no	—
49	#41×C₅₃	—	4	4	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	1	no	—
54	C₂×D₄×C₅₃	—	4	1	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
59	Q₁₆×C₅₃	—	4	2	no	—
60	Q₁₆×C₅₃	—	4	2	no	—
61	SD₁₆×C₅₃	—	4	4	no	—
62	SD₁₆×C₅₃	—	4	4	no	—
63	D₈×C₅₃	—	4	2	no	—
64	D₈×C₅₃	—	4	2	no	—
65	M₁₆×C₅₃	—	8	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	—	4	1	no	—
74	C₄⋊C₄×C₅₃	—	4	1	no	—
75	C₂²⋊C₄×C₅₃	—	4	2	no	—
76	C₂²⋊C₄×C₅₃	—	4	2	no	—
77	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅₃	D₈	1	1	yes	characteristic
79	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	#102	1	1	yes	characteristic
84	Q₈×C₅₃	—	4	2	no	—
85	Q₈×C₅₃	—	4	2	no	—
86	D₄×C₅₃	—	4	2	no	—
87	D₄×C₅₃	—	4	2	no	—
88	D₄×C₅₃	—	8	2	no	—
89	D₄×C₅₃	—	8	4	no	—
90	D₄×C₅₃	—	8	1	no	—
91	D₄×C₅₃	—	8	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	4	1	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	8	4	no	—
99	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	8	2	no	—
108	C₈×C₅₃	#100	1	2	yes	—
109	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
110	#11966	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	8	2	no	—
115	C₂²×C₅₃	—	8	2	no	—
116	C₂²×C₅₃	—	8	2	no	—
117	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	8	2	no	—
123	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
124	C₄×C₅₃	#418	1	1	yes	characteristic
125	C₄×C₅₃	#509	1	1	yes	characteristic
126	C₄×C₅₃	#510	1	1	yes	characteristic
127	#3147	C₂×C₅₃	1	1	yes	characteristic
128	#2163	C₂×C₅₃	1	1	yes	characteristic, complemented
129	#1567	C₂×C₅₃	1	1	yes	characteristic, complemented
130	#1362	C₂×C₅₃	1	2	yes	complemented
131	#1361	C₂×C₅₃	1	2	yes	complemented
132	C₂×C₅₃	—	4	1	no	—
133	C₂×C₅₃	—	16	2	no	—
134	C₂×C₅₃	#1545	1	1	yes	characteristic
135	C₂×C₅₃	#2139	1	1	yes	characteristic
136	C₂×C₅₃	#2148	1	1	yes	characteristic
137	C₂×#135	—	2	1	no	—
138	C₂×#135	C₄×C₅₃	1	2	yes	—
139	#532	—	2	1	no	—
140	#495	—	2	1	no	—
141	#495	C₄×C₅₃	1	2	yes	—
142	#494	C₂²×C₅₃	1	1	yes	characteristic
143	#493	C₂²×C₅₃	1	1	yes	characteristic
144	#492	—	2	1	no	—
145	#423	—	2	1	no	—
146	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
147	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
148	C₂²⋊C₁₆	—	2	1	no	—
149	#346	—	2	1	no	—
150	#335	—	2	1	no	—
151	#335	C₂²×C₅₃	1	2	yes	—
152	C₅₃	#11966	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#41	—	2	2	no	—
154	#135	—	4	4	no	—
155	#135	—	4	4	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
162	C₂×M₁₆	—	4	1	no	—
163	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
164	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	1	no	—
166	#127	—	2	1	no	—
167	#124	—	2	2	no	—
168	#121	—	4	1	no	—
169	#121	—	4	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₄×D₄	—	4	1	no	—
172	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
173	C₂×C₁₆	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	4	1	no	—
176	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
177	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
178	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
180	#41	—	4	4	no	—
181	#41	—	4	4	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
191	Q₁₆	—	4	2	no	—
192	Q₁₆	—	4	2	no	—
193	SD₁₆	—	4	4	no	—
194	SD₁₆	—	4	4	no	—
195	D₈	—	4	2	no	—
196	D₈	—	4	2	no	—
197	M₁₆	—	8	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₄²	D₈×C₅₃	1	1	yes	characteristic
211	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
212	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
213	C₁₆	—	4	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	#102×C₅₃	1	1	yes	characteristic
216	Q₈	—	4	2	no	—
217	Q₈	—	4	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	8	2	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	1	no	—
223	D₄	—	8	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
232	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
233	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	8	2	no	—
240	C₈	#100×C₅₃	1	2	yes	—
241	C₈	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
255	C₄	C₅₃×#418	1	1	yes	characteristic
256	C₄	C₅₃×#509	1	1	yes	characteristic
257	C₄	C₅₃×#510	1	1	yes	characteristic
258	C₂	—	4	1	no	—
259	C₂	—	16	2	no	—
260	C₂	C₅₃×#1545	1	1	yes	characteristic
261	C₂	C₅₃×#2139	1	1	yes	characteristic
262	C₂	C₅₃×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₅₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2163	C₂	semidirect
3. C₅₃×#1567	C₂	semidirect
4. C₅₃×#1362	C₂	semidirect
5. C₅₃×#1361	C₂	semidirect
110. #11966	C₅₃	direct (N × H)
128. #2163	C₂×C₅₃	semidirect
129. #1567	C₂×C₅₃	semidirect
130. #1362	C₂×C₅₃	semidirect
131. #1361	C₂×C₅₃	semidirect
152. C₅₃	#11966	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×#3147	2
C₅₃×#2163	2
C₅₃×#1567	2
C₅₃×#1362	2
C₅₃×#1361	2
#11966	53

Subgroups — C53×#11966 #27219478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11966 #27219478