Subgroups — C₅₃×#13470 #27220982

Order 13568.

326 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#361	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#357	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1978	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#428	—	2	1	no	—
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#418	—	2	1	no	—
16	C₂×#100×C₅₃	—	2	1	no	—
17	C₅₃×#405	C₂²	1	1	yes	characteristic
18	C₅₃×#368	—	2	2	no	—
19	C₅₃×C₄⋊C₁₆	—	2	2	no	—
20	C₅₃×#361	—	2	2	no	—
21	C₅₃×#357	—	2	2	no	—
22	C₅₃×#343	—	2	1	no	—
23	C₂²×D₄×C₅₃	—	2	1	no	—
24	C₂×D₈×C₅₃	—	2	1	no	—
25	C₂×D₈×C₅₃	—	2	1	no	—
26	C₂×D₈×C₅₃	—	2	1	no	—
27	C₂×D₈×C₅₃	—	2	1	no	—
28	C₂×D₈×C₅₃	—	2	1	no	—
29	C₂×D₈×C₅₃	Q₈	1	2	yes	—
30	C₂×M₁₆×C₅₃	—	2	1	no	—
31	C₂×M₁₆×C₅₃	—	2	1	no	—
32	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	#117×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	M₃₂×C₅₃	—	2	1	no	—
37	M₃₂×C₅₃	—	2	1	no	—
38	C₂×C₁₆×C₅₃	—	2	1	no	—
39	C₂×C₁₆×C₅₃	—	2	1	no	—
40	C₂×C₁₆×C₅₃	—	2	1	no	—
41	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
42	#108×C₅₃	—	2	1	no	—
43	#108×C₅₃	—	2	1	no	—
44	#108×C₅₃	—	2	1	no	—
45	#108×C₅₃	D₄	1	2	yes	—
46	#107×C₅₃	—	2	1	no	—
47	#107×C₅₃	—	2	1	no	—
48	#107×C₅₃	—	4	1	no	—
49	#106×C₅₃	—	4	1	no	—
50	#102×C₅₃	—	4	1	no	—
51	#102×C₅₃	—	4	1	no	—
52	#100×C₅₃	—	4	2	no	—
53	#97×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₂⁴×C₅₃	—	4	1	no	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	4	1	no	—
58	C₂×D₄×C₅₃	—	4	1	no	—
59	C₂×D₄×C₅₃	—	4	1	no	—
60	C₂×D₄×C₅₃	—	4	1	no	—
61	C₂×D₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	D₈×C₅₃	—	2	2	no	—
65	D₈×C₅₃	—	2	2	no	—
66	D₈×C₅₃	—	4	1	no	—
67	D₈×C₅₃	—	4	1	no	—
68	M₁₆×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	2	1	no	—
70	M₁₆×C₅₃	—	2	1	no	—
71	M₁₆×C₅₃	—	4	1	no	—
72	M₁₆×C₅₃	—	4	1	no	—
73	M₁₆×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₅₃	—	4	1	no	—
84	C₄⋊C₄×C₅₃	—	4	1	no	—
85	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
87	C₂²⋊C₄×C₅₃	—	2	1	no	—
88	C₄²×C₅₃	—	2	1	no	—
89	C₁₆×C₅₃	—	2	2	no	—
90	C₁₆×C₅₃	—	4	1	no	—
91	C₂³×C₅₃	—	4	2	no	—
92	C₂³×C₅₃	—	8	1	no	—
93	C₂³×C₅₃	—	8	1	no	—
94	C₂³×C₅₃	—	16	1	no	—
95	C₂³×C₅₃	—	16	1	no	—
96	C₂³×C₅₃	#135	1	1	yes	characteristic
97	D₄×C₅₃	—	4	2	no	—
98	D₄×C₅₃	—	4	2	no	—
99	D₄×C₅₃	—	8	1	no	—
100	D₄×C₅₃	—	8	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	8	1	no	—
104	C₂×C₄×C₅₃	—	16	1	no	—
105	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
111	C₈×C₅₃	—	4	2	no	—
112	C₈×C₅₃	—	4	1	no	—
113	C₈×C₅₃	—	8	1	no	—
114	C₈×C₅₃	#121	1	1	yes	characteristic
115	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₈×C₅₃	#126	1	1	yes	characteristic
118	#13470	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	8	2	no	—
122	C₂²×C₅₃	—	16	2	no	—
123	C₂²×C₅₃	—	16	2	no	—
124	C₂²×C₅₃	—	16	1	no	—
125	C₂²×C₅₃	—	16	1	no	—
126	C₂²×C₅₃	#464	1	1	yes	characteristic
127	C₂²×C₅₃	#466	1	1	yes	characteristic
128	C₂²×C₅₃	#478	1	1	yes	characteristic
129	C₄×C₅₃	—	4	1	no	—
130	C₄×C₅₃	—	4	1	no	—
131	C₄×C₅₃	—	16	1	no	—
132	C₄×C₅₃	#396	1	1	yes	characteristic
133	C₄×C₅₃	#459	1	1	yes	characteristic
134	C₄×C₅₃	#468	1	1	yes	characteristic
135	C₄×C₅₃	#474	1	1	yes	characteristic
136	C₂×#368	C₂×C₅₃	1	1	yes	characteristic, complemented
137	#2151	C₂×C₅₃	1	1	yes	characteristic, complemented
138	C₂×#361	C₂×C₅₃	1	1	yes	characteristic
139	C₂×#357	C₂×C₅₃	1	1	yes	characteristic
140	#1978	C₂×C₅₃	1	1	yes	characteristic
141	#1952	C₂×C₅₃	1	1	yes	characteristic, complemented
142	#1387	C₂×C₅₃	1	1	yes	characteristic, complemented
143	C₂×C₅₃	—	2	1	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	16	2	no	—
146	C₂×C₅₃	#2067	1	1	yes	characteristic
147	C₂×C₅₃	#2221	1	1	yes	characteristic
148	C₂×C₅₃	#2223	1	1	yes	characteristic
149	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
151	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
153	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
154	#428	—	2	1	no	—
155	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
156	#418	—	2	1	no	—
157	C₂×#100	—	2	1	no	—
158	#405	C₂²×C₅₃	1	1	yes	characteristic
159	#368	—	2	2	no	—
160	C₄⋊C₁₆	—	2	2	no	—
161	#361	—	2	2	no	—
162	#357	—	2	2	no	—
163	#343	—	2	1	no	—
164	C₅₃	#13470	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×D₄	—	2	1	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	Q₈×C₅₃	1	2	yes	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
176	#117	D₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#108	—	2	1	no	—
187	#108	D₄×C₅₃	1	2	yes	—
188	#107	—	2	1	no	—
189	#107	—	2	1	no	—
190	#107	—	4	1	no	—
191	#106	—	4	1	no	—
192	#102	—	4	1	no	—
193	#102	—	4	1	no	—
194	#100	—	4	2	no	—
195	#97	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₂⁴	—	4	1	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
206	D₈	—	2	2	no	—
207	D₈	—	2	2	no	—
208	D₈	—	4	1	no	—
209	D₈	—	4	1	no	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	—	4	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
220	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
221	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
222	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
223	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
224	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
228	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
229	C₂²⋊C₄	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₁₆	—	2	2	no	—
232	C₁₆	—	4	1	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	8	1	no	—
235	C₂³	—	8	1	no	—
236	C₂³	—	16	1	no	—
237	C₂³	—	16	1	no	—
238	C₂³	#135×C₅₃	1	1	yes	characteristic
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	8	1	no	—
242	D₄	—	8	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	8	1	no	—
246	C₂×C₄	—	16	1	no	—
247	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
248	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
251	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
252	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
253	C₈	—	4	2	no	—
254	C₈	—	4	1	no	—
255	C₈	—	8	1	no	—
256	C₈	#121×C₅₃	1	1	yes	characteristic
257	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
258	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
259	C₈	#126×C₅₃	1	1	yes	characteristic
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	16	2	no	—
264	C₂²	—	16	2	no	—
265	C₂²	—	16	1	no	—
266	C₂²	—	16	1	no	—
267	C₂²	C₅₃×#464	1	1	yes	characteristic
268	C₂²	C₅₃×#466	1	1	yes	characteristic
269	C₂²	C₅₃×#478	1	1	yes	characteristic
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	16	1	no	—
273	C₄	C₅₃×#396	1	1	yes	characteristic
274	C₄	C₅₃×#459	1	1	yes	characteristic
275	C₄	C₅₃×#468	1	1	yes	characteristic
276	C₄	C₅₃×#474	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	16	2	no	—
280	C₂	C₅₃×#2067	1	1	yes	characteristic
281	C₂	C₅₃×#2221	1	1	yes	characteristic
282	C₂	C₅₃×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#368	C₂	semidirect
2. C₅₃×#2151	C₂	semidirect
6. C₅₃×#1952	C₂	semidirect
7. C₅₃×#1387	C₂	semidirect
118. #13470	C₅₃	direct (N × H)
136. C₂×#368	C₂×C₅₃	semidirect
137. #2151	C₂×C₅₃	semidirect
141. #1952	C₂×C₅₃	semidirect
142. #1387	C₂×C₅₃	semidirect
164. C₅₃	#13470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#368	2
C₅₃×#2151	2
C₂×C₅₃×#361	2
C₂×C₅₃×#357	2
C₅₃×#1978	2
C₅₃×#1952	2
C₅₃×#1387	2
#13470	53

Subgroups — C53×#13470 #27220982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13470 #27220982