Subgroups — C₅₃×#9849 #27217361

Order 13568.

530 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×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1962	C₂	1	2	yes	maximal, complemented
3	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1492	C₂	1	2	yes	maximal, complemented
6	C₅₃×#532	C₂²	1	2	yes	—
7	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
8	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
9	C₅₃×#486	C₄	1	2	yes	complemented
10	C₅₃×#466	—	2	2	no	—
11	C₄×D₈×C₅₃	—	2	2	no	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₂×#104×C₅₃	—	2	4	no	—
14	C₂×#97×C₅₃	—	2	2	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#334	—	2	2	no	—
19	C₅₃×D₄⋊C₈	—	2	4	no	—
20	C₂×#41×C₅₃	—	2	2	no	—
21	C₂×D₈×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	D₄	1	2	yes	—
25	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
26	#127×C₅₃	C₂×C₄	1	2	yes	—
27	#124×C₅₃	—	2	2	no	—
28	#121×C₅₃	—	2	2	no	—
29	#121×C₅₃	—	2	2	no	—
30	C₄×D₄×C₅₃	—	2	2	no	—
31	C₄×D₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#107×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	#104×C₅₃	—	4	8	no	—
40	#102×C₅₃	—	2	4	no	—
41	#97×C₅₃	—	2	8	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	#41×C₅₃	—	4	4	no	—
48	#41×C₅₃	—	4	4	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
59	D₈×C₅₃	—	4	4	no	—
60	M₁₆×C₅₃	—	4	4	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₂²⋊C₄×C₅₃	—	4	2	no	—
70	C₂²⋊C₄×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	#41	1	1	yes	characteristic
78	C₂³×C₅₃	—	4	2	no	—
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	Q₈×C₅₃	—	4	4	no	—
81	D₄×C₅₃	—	4	4	no	—
82	D₄×C₅₃	—	4	4	no	—
83	D₄×C₅₃	—	4	4	no	—
84	D₄×C₅₃	—	8	4	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	4	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	—	8	2	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
100	C₈×C₅₃	—	2	4	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	8	2	no	—
103	C₈×C₅₃	#104	1	4	yes	complemented
104	#9849	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	8	2	no	—
108	C₂²×C₅₃	—	8	2	no	—
109	C₂²×C₅₃	—	8	2	no	—
110	C₂²×C₅₃	#390	1	1	yes	characteristic
111	C₂²×C₅₃	#469	1	2	yes	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	4	4	no	—
115	C₄×C₅₃	—	8	2	no	—
116	C₄×C₅₃	C₂×#104	1	2	yes	—
117	C₄×C₅₃	#442	1	2	yes	—
118	C₄×C₅₃	#444	1	2	yes	—
119	C₄×C₅₃	#496	1	2	yes	—
120	#3169	C₂×C₅₃	1	1	yes	characteristic
121	#1962	C₂×C₅₃	1	2	yes	complemented
122	C₄×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
123	#1566	C₂×C₅₃	1	1	yes	characteristic, complemented
124	#1492	C₂×C₅₃	1	2	yes	complemented
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	—	16	2	no	—
127	C₂×C₅₃	#1962	1	1	yes	characteristic
128	C₂×C₅₃	#1972	1	1	yes	characteristic
129	C₂×C₅₃	#1975	1	1	yes	characteristic
130	#532	C₂²×C₅₃	1	2	yes	—
131	#500	C₄×C₅₃	1	1	yes	characteristic, complemented
132	#493	C₄×C₅₃	1	1	yes	characteristic, complemented
133	#486	C₄×C₅₃	1	2	yes	complemented
134	#466	—	2	2	no	—
135	C₄×D₈	—	2	2	no	—
136	#423	C₂²×C₅₃	1	2	yes	—
137	C₂×#104	—	2	4	no	—
138	C₂×#97	—	2	2	no	—
139	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
141	C₄³	C₂²×C₅₃	1	1	yes	characteristic
142	#334	—	2	2	no	—
143	D₄⋊C₈	—	2	4	no	—
144	C₅₃	#9849	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×#41	—	2	2	no	—
146	C₂×D₈	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	D₄×C₅₃	1	2	yes	—
150	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
151	#127	C₂×C₄×C₅₃	1	2	yes	—
152	#124	—	2	2	no	—
153	#121	—	2	2	no	—
154	#121	—	2	2	no	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	—	2	2	no	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	D₄×C₅₃	1	2	yes	—
160	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
161	#107	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	#104	—	4	8	no	—
165	#102	—	2	4	no	—
166	#97	—	2	8	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
170	C₄×C₈	D₄×C₅₃	1	2	yes	—
171	C₄×C₈	D₄×C₅₃	1	2	yes	—
172	#41	—	4	4	no	—
173	#41	—	4	4	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
183	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
184	D₈	—	4	4	no	—
185	M₁₆	—	4	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
191	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	4	2	no	—
199	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₄²	#41×C₅₃	1	1	yes	characteristic
203	C₂³	—	4	2	no	—
204	C₂³	#121×C₅₃	1	1	yes	characteristic
205	Q₈	—	4	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	4	4	no	—
208	D₄	—	4	4	no	—
209	D₄	—	8	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
220	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
222	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
223	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
224	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	2	2	no	—
227	C₈	—	8	2	no	—
228	C₈	#104×C₅₃	1	4	yes	complemented
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	—	8	2	no	—
234	C₂²	C₅₃×#390	1	1	yes	characteristic
235	C₂²	C₅₃×#469	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	4	no	—
239	C₄	—	8	2	no	—
240	C₄	C₂×#104×C₅₃	1	2	yes	—
241	C₄	C₅₃×#442	1	2	yes	—
242	C₄	C₅₃×#444	1	2	yes	—
243	C₄	C₅₃×#496	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	16	2	no	—
246	C₂	C₅₃×#1962	1	1	yes	characteristic
247	C₂	C₅₃×#1972	1	1	yes	characteristic
248	C₂	C₅₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#390
prime radical	C₅₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1962	C₂	semidirect
3. C₄×#97×C₅₃	C₂	semidirect
4. C₅₃×#1566	C₂	semidirect
5. C₅₃×#1492	C₂	semidirect
7. C₅₃×#500	C₄	semidirect
8. C₅₃×#493	C₄	semidirect
9. C₅₃×#486	C₄	semidirect
103. C₈×C₅₃	#104	semidirect
104. #9849	C₅₃	direct (N × H)
121. #1962	C₂×C₅₃	semidirect
122. C₄×#97	C₂×C₅₃	semidirect
123. #1566	C₂×C₅₃	semidirect
124. #1492	C₂×C₅₃	semidirect
131. #500	C₄×C₅₃	semidirect
132. #493	C₄×C₅₃	semidirect
133. #486	C₄×C₅₃	semidirect
144. C₅₃	#9849	direct (N × H)
228. C₈	#104×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3169	2
C₅₃×#1962	2
C₄×#97×C₅₃	2
C₅₃×#1566	2
C₅₃×#1492	2
#9849	53

Subgroups — C53×#9849 #27217361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9849 #27217361