Subgroups — C₅₃×#12826 #27220338

Order 13568.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2948	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#361	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2140	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#518	—	2	1	no	—
10	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#442	C₄	1	4	yes	complemented
13	C₄×D₈×C₅₃	—	2	2	no	—
14	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#427	C₂²	1	1	yes	characteristic
16	C₅₃×#427	C₂²	1	1	yes	characteristic
17	C₅₃×#418	—	2	1	no	—
18	C₅₃×#405	C₂²	1	1	yes	characteristic
19	C₅₃×#364	—	2	2	no	—
20	C₅₃×#361	—	2	2	no	—
21	C₅₃×#357	—	2	2	no	—
22	C₅₃×#344	—	2	1	no	—
23	C₅₃×H(3,ℤ/4)	—	2	1	no	—
24	C₂²×D₄×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	C₂×C₄	1	2	yes	—
29	C₂×D₈×C₅₃	D₄	1	2	yes	—
30	C₂×D₈×C₅₃	Q₈	1	2	yes	—
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	C₄×D₄×C₅₃	—	2	4	no	—
35	C₄×D₄×C₅₃	—	4	2	no	—
36	#117×C₅₃	—	2	1	no	—
37	#117×C₅₃	—	2	1	no	—
38	#117×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
40	M₃₂×C₅₃	—	2	1	no	—
41	M₃₂×C₅₃	—	2	1	no	—
42	M₃₂×C₅₃	—	2	1	no	—
43	M₃₂×C₅₃	—	2	1	no	—
44	C₂×C₁₆×C₅₃	—	2	1	no	—
45	C₂×C₁₆×C₅₃	—	2	1	no	—
46	#108×C₅₃	—	2	1	no	—
47	#108×C₅₃	—	2	1	no	—
48	#107×C₅₃	—	2	1	no	—
49	#107×C₅₃	—	2	1	no	—
50	#106×C₅₃	—	2	1	no	—
51	#106×C₅₃	C₂×C₄	1	2	yes	—
52	#102×C₅₃	—	2	4	no	—
53	#97×C₅₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₅₃	D₄	1	2	yes	—
55	C₂⁴×C₅₃	—	4	1	no	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	2	2	no	—
58	C₂×D₄×C₅₃	—	2	2	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	4	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	—	2	2	no	—
67	D₈×C₅₃	C₄⋊C₄	1	4	yes	—
68	M₁₆×C₅₃	—	2	1	no	—
69	M₁₆×C₅₃	—	2	1	no	—
70	M₁₆×C₅₃	—	4	1	no	—
71	M₁₆×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	4	1	no	—
83	C₂²⋊C₄×C₅₃	—	2	1	no	—
84	C₂²⋊C₄×C₅₃	—	4	4	no	—
85	C₂²⋊C₄×C₅₃	—	4	1	no	—
86	C₂²⋊C₄×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	—	4	1	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
90	C₁₆×C₅₃	—	4	1	no	—
91	C₁₆×C₅₃	—	4	1	no	—
92	C₂³×C₅₃	—	4	2	no	—
93	C₂³×C₅₃	—	4	2	no	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	—	16	2	no	—
96	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	D₄×C₅₃	—	2	4	no	—
98	D₄×C₅₃	—	4	2	no	—
99	D₄×C₅₃	—	4	2	no	—
100	D₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	8	1	no	—
104	C₂×C₄×C₅₃	—	16	4	no	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	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	#135	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
110	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	—	8	1	no	—
113	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₈×C₅₃	#117	1	1	yes	characteristic
115	C₈×C₅₃	#121	1	1	yes	characteristic
116	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	#12826	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₅₃	—	4	1	no	—
119	C₂²×C₅₃	—	4	4	no	—
120	C₂²×C₅₃	—	8	4	no	—
121	C₂²×C₅₃	—	16	2	no	—
122	C₂²×C₅₃	—	16	2	no	—
123	C₂²×C₅₃	#418	1	1	yes	characteristic
124	C₂²×C₅₃	#463	1	1	yes	characteristic
125	C₂²×C₅₃	#478	1	1	yes	characteristic
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	—	8	2	no	—
128	C₄×C₅₃	—	16	1	no	—
129	C₄×C₅₃	#380	1	1	yes	characteristic
130	C₄×C₅₃	#416	1	1	yes	characteristic
131	C₄×C₅₃	#463	1	1	yes	characteristic
132	C₄×C₅₃	#478	1	1	yes	characteristic
133	#2948	C₂×C₅₃	1	1	yes	characteristic
134	#2153	C₂×C₅₃	1	1	yes	characteristic, complemented
135	C₂×#361	C₂×C₅₃	1	1	yes	characteristic
136	#2140	C₂×C₅₃	1	1	yes	characteristic
137	#1833	C₂×C₅₃	1	1	yes	characteristic, complemented
138	#1389	C₂×C₅₃	1	1	yes	characteristic, complemented
139	#1384	C₂×C₅₃	1	1	yes	characteristic, complemented
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	8	4	no	—
144	C₂×C₅₃	#1804	1	1	yes	characteristic
145	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
146	#518	—	2	1	no	—
147	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
149	#442	C₄×C₅₃	1	4	yes	complemented
150	C₄×D₈	—	2	2	no	—
151	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
152	#427	C₂²×C₅₃	1	1	yes	characteristic
153	#427	C₂²×C₅₃	1	1	yes	characteristic
154	#418	—	2	1	no	—
155	#405	C₂²×C₅₃	1	1	yes	characteristic
156	#364	—	2	2	no	—
157	#361	—	2	2	no	—
158	#357	—	2	2	no	—
159	#344	—	2	1	no	—
160	H(3,ℤ/4)	—	2	1	no	—
161	C₅₃	#12826	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×D₄	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
167	C₂×D₈	D₄×C₅₃	1	2	yes	—
168	C₂×D₈	Q₈×C₅₃	1	2	yes	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
172	C₄×D₄	—	2	4	no	—
173	C₄×D₄	—	4	2	no	—
174	#117	—	2	1	no	—
175	#117	—	2	1	no	—
176	#117	D₄×C₅₃	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	4	2	no	—
178	M₃₂	—	2	1	no	—
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#107	—	2	1	no	—
187	#107	—	2	1	no	—
188	#106	—	2	1	no	—
189	#106	C₂×C₄×C₅₃	1	2	yes	—
190	#102	—	2	4	no	—
191	#97	C₂×C₄×C₅₃	1	2	yes	—
192	C₄×C₈	D₄×C₅₃	1	2	yes	—
193	C₂⁴	—	4	1	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
202	D₈	—	2	2	no	—
203	D₈	—	2	2	no	—
204	D₈	—	2	2	no	—
205	D₈	C₄⋊C₄×C₅₃	1	4	yes	—
206	M₁₆	—	2	1	no	—
207	M₁₆	—	2	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
215	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
216	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
217	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₂²⋊C₄	—	4	1	no	—
224	C₂²⋊C₄	—	4	1	no	—
225	C₄²	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
228	C₁₆	—	4	1	no	—
229	C₁₆	—	4	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	16	2	no	—
234	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
235	D₄	—	2	4	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	16	4	no	—
243	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
244	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
246	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
247	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
248	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	8	1	no	—
251	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
252	C₈	#117×C₅₃	1	1	yes	characteristic
253	C₈	#121×C₅₃	1	1	yes	characteristic
254	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
255	C₂²	—	4	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	16	2	no	—
259	C₂²	—	16	2	no	—
260	C₂²	C₅₃×#418	1	1	yes	characteristic
261	C₂²	C₅₃×#463	1	1	yes	characteristic
262	C₂²	C₅₃×#478	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	16	1	no	—
266	C₄	C₅₃×#380	1	1	yes	characteristic
267	C₄	C₅₃×#416	1	1	yes	characteristic
268	C₄	C₅₃×#463	1	1	yes	characteristic
269	C₄	C₅₃×#478	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	8	4	no	—
274	C₂	C₅₃×#1804	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1804
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2153	C₂	semidirect
5. C₅₃×#1833	C₂	semidirect
6. C₅₃×#1389	C₂	semidirect
7. C₅₃×#1384	C₂	semidirect
12. C₅₃×#442	C₄	semidirect
117. #12826	C₅₃	direct (N × H)
134. #2153	C₂×C₅₃	semidirect
137. #1833	C₂×C₅₃	semidirect
138. #1389	C₂×C₅₃	semidirect
139. #1384	C₂×C₅₃	semidirect
149. #442	C₄×C₅₃	semidirect
161. C₅₃	#12826	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×#2948	2
C₅₃×#2153	2
C₂×C₅₃×#361	2
C₅₃×#2140	2
C₅₃×#1833	2
C₅₃×#1389	2
C₅₃×#1384	2
#12826	53

Subgroups — C53×#12826 #27220338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12826 #27220338