Subgroups — C₅₃×#12947 #27220459

Order 13568.

380 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2957	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2143	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#359	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#568	—	2	1	no	—
10	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#444	—	2	2	no	—
13	C₅₃×#443	—	2	2	no	—
14	C₅₃×#430	—	2	1	no	—
15	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#421	—	2	1	no	—
19	C₅₃×#405	C₂²	1	1	yes	characteristic
20	C₅₃×#368	—	2	2	no	—
21	C₅₃×#359	—	2	2	no	—
22	C₅₃×#359	C₄	1	4	yes	complemented
23	C₅₃×#344	—	2	1	no	—
24	C₂×#41×C₅₃	—	2	1	no	—
25	#135×C₅₃	—	2	1	no	—
26	#135×C₅₃	—	2	1	no	—
27	#135×C₅₃	C₂×C₄	1	2	yes	—
28	#135×C₅₃	D₄	1	2	yes	—
29	C₂×Q₁₆×C₅₃	Q₈	1	1	yes	characteristic
30	C₂×SD₁₆×C₅₃	—	2	1	no	—
31	C₂×D₈×C₅₃	Q₈	1	1	yes	characteristic
32	#131×C₅₃	—	2	4	no	—
33	#131×C₅₃	—	4	2	no	—
34	C₂×M₁₆×C₅₃	—	2	1	no	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	—	2	1	no	—
37	C₂×M₁₆×C₅₃	—	4	2	no	—
38	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	#117×C₅₃	D₄	1	1	yes	characteristic
41	M₃₂×C₅₃	—	2	1	no	—
42	M₃₂×C₅₃	—	2	1	no	—
43	C₂×C₁₆×C₅₃	—	2	1	no	—
44	C₂×C₁₆×C₅₃	—	2	1	no	—
45	C₂×C₁₆×C₅₃	—	2	1	no	—
46	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	#108×C₅₃	—	2	1	no	—
48	#108×C₅₃	—	2	1	no	—
49	#108×C₅₃	—	2	1	no	—
50	#108×C₅₃	—	2	1	no	—
51	#108×C₅₃	—	4	2	no	—
52	#108×C₅₃	C₂×C₄	1	2	yes	—
53	#108×C₅₃	D₄	1	2	yes	—
54	#104×C₅₃	—	2	4	no	—
55	#97×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	#41×C₅₃	—	2	2	no	—
58	#41×C₅₃	—	2	2	no	—
59	#41×C₅₃	—	4	1	no	—
60	#41×C₅₃	—	4	1	no	—
61	C₂×Q₈×C₅₃	—	2	1	no	—
62	C₂×D₄×C₅₃	—	2	1	no	—
63	C₂×D₄×C₅₃	—	4	1	no	—
64	C₂²×C₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₅₃	—	2	1	no	—
67	Q₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
68	SD₁₆×C₅₃	—	2	2	no	—
69	SD₁₆×C₅₃	—	2	2	no	—
70	D₈×C₅₃	—	2	1	no	—
71	D₈×C₅₃	C₄⋊C₄	1	2	yes	—
72	M₁₆×C₅₃	—	2	1	no	—
73	M₁₆×C₅₃	—	2	2	no	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	—	4	4	no	—
76	M₁₆×C₅₃	—	4	1	no	—
77	M₁₆×C₅₃	—	4	1	no	—
78	M₁₆×C₅₃	—	4	1	no	—
79	M₁₆×C₅₃	—	4	1	no	—
80	M₁₆×C₅₃	D₈	1	2	yes	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	4	4	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	—	4	1	no	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₂²⋊C₄×C₅₃	—	2	1	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₁₆×C₅₃	—	2	2	no	—
96	C₁₆×C₅₃	—	4	1	no	—
97	C₂³×C₅₃	—	4	1	no	—
98	C₂³×C₅₃	#137	1	1	yes	characteristic
99	Q₈×C₅₃	—	2	2	no	—
100	Q₈×C₅₃	—	4	1	no	—
101	D₄×C₅₃	—	2	2	no	—
102	D₄×C₅₃	—	4	2	no	—
103	D₄×C₅₃	—	4	2	no	—
104	D₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	1	no	—
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
114	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	—	4	2	no	—
117	C₈×C₅₃	—	4	2	no	—
118	C₈×C₅₃	—	8	1	no	—
119	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₈×C₅₃	#123	1	1	yes	characteristic
123	#12947	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	—	16	1	no	—
127	C₂²×C₅₃	#440	1	1	yes	characteristic
128	C₂²×C₅₃	#452	1	1	yes	characteristic
129	C₂²×C₅₃	#474	1	1	yes	characteristic
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	—	8	1	no	—
132	C₄×C₅₃	#385	1	1	yes	characteristic
133	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
134	C₄×C₅₃	#447	1	1	yes	characteristic
135	C₄×C₅₃	#478	1	1	yes	characteristic
136	#2957	C₂×C₅₃	1	1	yes	characteristic
137	C₂×#368	C₂×C₅₃	1	1	yes	characteristic, complemented
138	#2143	C₂×C₅₃	1	1	yes	characteristic
139	C₂×#359	C₂×C₅₃	1	1	yes	characteristic
140	#1953	C₂×C₅₃	1	1	yes	characteristic, complemented
141	#1388	C₂×C₅₃	1	1	yes	characteristic, complemented
142	#1379	C₂×C₅₃	1	1	yes	characteristic, complemented
143	C₂×C₅₃	—	2	1	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	8	2	no	—
147	C₂×C₅₃	#1862	1	1	yes	characteristic
148	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
149	#568	—	2	1	no	—
150	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
151	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
152	#444	—	2	2	no	—
153	#443	—	2	2	no	—
154	#430	—	2	1	no	—
155	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
156	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
158	#421	—	2	1	no	—
159	#405	C₂²×C₅₃	1	1	yes	characteristic
160	#368	—	2	2	no	—
161	#359	—	2	2	no	—
162	#359	C₄×C₅₃	1	4	yes	complemented
163	#344	—	2	1	no	—
164	C₅₃	#12947	1	1	yes	characteristic, direct product factor, Sylow
165	C₂×#41	—	2	1	no	—
166	#135	—	2	1	no	—
167	#135	—	2	1	no	—
168	#135	C₂×C₄×C₅₃	1	2	yes	—
169	#135	D₄×C₅₃	1	2	yes	—
170	C₂×Q₁₆	Q₈×C₅₃	1	1	yes	characteristic
171	C₂×SD₁₆	—	2	1	no	—
172	C₂×D₈	Q₈×C₅₃	1	1	yes	characteristic
173	#131	—	2	4	no	—
174	#131	—	4	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	4	2	no	—
179	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
181	#117	D₄×C₅₃	1	1	yes	characteristic
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
188	#108	—	2	1	no	—
189	#108	—	2	1	no	—
190	#108	—	2	1	no	—
191	#108	—	2	1	no	—
192	#108	—	4	2	no	—
193	#108	C₂×C₄×C₅₃	1	2	yes	—
194	#108	D₄×C₅₃	1	2	yes	—
195	#104	—	2	4	no	—
196	#97	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	4	1	no	—
201	#41	—	4	1	no	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
207	Q₁₆	—	2	1	no	—
208	Q₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
209	SD₁₆	—	2	2	no	—
210	SD₁₆	—	2	2	no	—
211	D₈	—	2	1	no	—
212	D₈	C₄⋊C₄×C₅₃	1	2	yes	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	4	4	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	D₈×C₅₃	1	2	yes	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
231	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
232	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₂²⋊C₄	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	4	1	no	—
238	C₂³	—	4	1	no	—
239	C₂³	#137×C₅₃	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	4	1	no	—
242	D₄	—	2	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
251	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
254	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
255	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	8	1	no	—
260	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
261	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
262	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
263	C₈	#123×C₅₃	1	1	yes	characteristic
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	16	1	no	—
267	C₂²	C₅₃×#440	1	1	yes	characteristic
268	C₂²	C₅₃×#452	1	1	yes	characteristic
269	C₂²	C₅₃×#474	1	1	yes	characteristic
270	C₄	—	4	2	no	—
271	C₄	—	8	1	no	—
272	C₄	C₅₃×#385	1	1	yes	characteristic
273	C₄	C₄×D₈×C₅₃	1	1	yes	characteristic
274	C₄	C₅₃×#447	1	1	yes	characteristic
275	C₄	C₅₃×#478	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₅₃×#1862	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#447
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1862
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#368	C₂	semidirect
5. C₅₃×#1953	C₂	semidirect
6. C₅₃×#1388	C₂	semidirect
7. C₅₃×#1379	C₂	semidirect
22. C₅₃×#359	C₄	semidirect
123. #12947	C₅₃	direct (N × H)
137. C₂×#368	C₂×C₅₃	semidirect
140. #1953	C₂×C₅₃	semidirect
141. #1388	C₂×C₅₃	semidirect
142. #1379	C₂×C₅₃	semidirect
162. #359	C₄×C₅₃	semidirect
164. C₅₃	#12947	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×#2957	2
C₂×C₅₃×#368	2
C₅₃×#2143	2
C₂×C₅₃×#359	2
C₅₃×#1953	2
C₅₃×#1388	2
C₅₃×#1379	2
#12947	53

Subgroups — C53×#12947 #27220459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12947 #27220459