Subgroups — C₅₃×#33441 #27240953

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#504	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2955	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2171	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2170	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#2117	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#575	C₄	1	1	yes	characteristic
9	C₂×#131×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#504	—	2	2	no	—
11	C₅₃×#504	—	2	2	no	—
12	C₅₃×#504	C₂²	1	4	yes	complemented
13	C₅₃×#504	C₂²	1	4	yes	complemented
14	C₂×M₃₂×C₅₃	—	2	2	no	—
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₃	—	2	2	no	—
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#444	C₄	1	4	yes	—
20	C₅₃×#443	C₂²	1	4	yes	—
21	C₅₃×#430	C₄	1	1	yes	characteristic
22	C₅₃×#421	C₄	1	2	yes	—
23	C₅₃×#405	C₂²	1	1	yes	characteristic
24	C₅₃×#364	C₂²	1	4	yes	complemented
25	C₅₃×#350	C₂²	1	8	yes	complemented
26	C₅₃×#346	C₂²	1	2	yes	complemented
27	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
28	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
29	#137×C₅₃	C₈	1	4	yes	—
30	AGL(1,ℤ/8)×C₅₃	C₈	1	4	yes	—
31	#135×C₅₃	C₂×C₄	1	4	yes	—
32	#131×C₅₃	—	2	2	no	—
33	#131×C₅₃	—	2	2	no	—
34	#131×C₅₃	C₂×C₄	1	4	yes	—
35	#131×C₅₃	C₂³	1	4	yes	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	2	no	—
43	M₃₂×C₅₃	—	2	2	no	—
44	M₃₂×C₅₃	—	2	4	no	—
45	M₃₂×C₅₃	—	2	4	no	—
46	M₃₂×C₅₃	C₂³	1	4	yes	—
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	—	2	4	no	—
49	C₂×C₁₆×C₅₃	—	2	4	no	—
50	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
51	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
52	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
53	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
55	#108×C₅₃	C₂×C₄	1	4	yes	—
56	#104×C₅₃	C₂×C₄	1	8	yes	—
57	#97×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₃	C₂³	1	2	yes	—
59	#41×C₅₃	—	2	2	no	—
60	#41×C₅₃	—	2	2	no	—
61	#41×C₅₃	C₂×C₈	1	4	yes	—
62	#41×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
69	SD₁₆×C₅₃	C₂×C₈	1	8	yes	—
70	D₈×C₅₃	C₂×C₈	1	4	yes	—
71	M₁₆×C₅₃	—	2	2	no	—
72	M₁₆×C₅₃	—	2	4	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
75	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₅₃	#41	1	2	yes	—
83	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₂²⋊C₄×C₅₃	—	2	1	no	—
86	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
87	C₁₆×C₅₃	—	2	4	no	—
88	C₁₆×C₅₃	—	2	2	no	—
89	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
90	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	#131	1	1	yes	characteristic
94	Q₈×C₅₃	—	2	2	no	—
95	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
96	D₄×C₅₃	—	2	4	no	—
97	D₄×C₅₃	—	2	4	no	—
98	D₄×C₅₃	—	2	2	no	—
99	D₄×C₅₃	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₅₃	—	2	4	no	—
109	C₈×C₅₃	C₄×D₄	1	2	yes	—
110	C₈×C₅₃	C₂²×C₈	1	4	yes	—
111	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
113	#33441	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	4	1	no	—
116	C₂²×C₅₃	—	4	4	no	—
117	C₂²×C₅₃	C₈×D₄	1	2	yes	—
118	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	C₈×D₄	1	2	yes	—
122	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
124	C₂×#504	C₂×C₅₃	1	2	yes	complemented
125	#2955	C₂×C₅₃	1	1	yes	characteristic
126	#2171	C₂×C₅₃	1	4	yes	complemented
127	#2170	C₂×C₅₃	1	4	yes	complemented
128	#2153	C₂×C₅₃	1	1	yes	characteristic, complemented
129	#2117	C₂×C₅₃	1	2	yes	complemented
130	#2108	C₂×C₅₃	1	1	yes	characteristic, complemented
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	4	4	no	—
134	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
135	#575	C₄×C₅₃	1	1	yes	characteristic
136	C₂×#131	C₂²×C₅₃	1	2	yes	—
137	#504	—	2	2	no	—
138	#504	—	2	2	no	—
139	#504	C₂²×C₅₃	1	4	yes	complemented
140	#504	C₂²×C₅₃	1	4	yes	complemented
141	C₂×M₃₂	—	2	2	no	—
142	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
143	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
144	C₂²×C₁₆	—	2	2	no	—
145	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
146	#444	C₄×C₅₃	1	4	yes	—
147	#443	C₂²×C₅₃	1	4	yes	—
148	#430	C₄×C₅₃	1	1	yes	characteristic
149	#421	C₄×C₅₃	1	2	yes	—
150	#405	C₂²×C₅₃	1	1	yes	characteristic
151	#364	C₂²×C₅₃	1	4	yes	complemented
152	#350	C₂²×C₅₃	1	8	yes	complemented
153	#346	C₂²×C₅₃	1	2	yes	complemented
154	C₄×C₁₆	C₂²×C₅₃	1	2	yes	complemented
155	C₅₃	#33441	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
157	#137	C₈×C₅₃	1	4	yes	—
158	AGL(1,ℤ/8)	C₈×C₅₃	1	4	yes	—
159	#135	C₂×C₄×C₅₃	1	4	yes	—
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	C₂×C₄×C₅₃	1	4	yes	—
163	#131	C₂³×C₅₃	1	4	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
166	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
169	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	2	no	—
172	M₃₂	—	2	4	no	—
173	M₃₂	—	2	4	no	—
174	M₃₂	C₂³×C₅₃	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	—	2	4	no	—
178	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
179	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
180	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
181	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
182	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
183	#108	C₂×C₄×C₅₃	1	4	yes	—
184	#104	C₂×C₄×C₅₃	1	8	yes	—
185	#97	C₂×C₄×C₅₃	1	2	yes	—
186	C₄×C₈	C₂³×C₅₃	1	2	yes	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×C₈×C₅₃	1	4	yes	—
190	#41	C₂²×C₄×C₅₃	1	4	yes	—
191	C₂×Q₈	C₂×C₈×C₅₃	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×C₈×C₅₃	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
196	Q₁₆	C₂×C₈×C₅₃	1	4	yes	—
197	SD₁₆	C₂×C₈×C₅₃	1	8	yes	—
198	D₈	C₂×C₈×C₅₃	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	C₂×C₈×C₅₃	1	4	yes	—
203	M₁₆	C₂²×C₄×C₅₃	1	4	yes	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
210	C₂×C₈	#41×C₅₃	1	2	yes	—
211	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
215	C₁₆	—	2	4	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₁₆	C₂×D₄×C₅₃	1	2	yes	—
219	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	#131×C₅₃	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	C₂²×C₈×C₅₃	1	4	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂²×C₈×C₅₃	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
233	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
234	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	C₄×D₄×C₅₃	1	2	yes	—
238	C₈	C₂²×C₈×C₅₃	1	4	yes	—
239	C₈	C₂²×D₄×C₅₃	1	1	yes	characteristic
240	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	C₈×D₄×C₅₃	1	2	yes	—
245	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	C₈×D₄×C₅₃	1	2	yes	—
249	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
250	C₄	C₂³×C₈×C₅₃	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×C₈×D₄×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₅₃	C₂×D₄
commutator	C₄	C₂³×C₈×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	C₅₃×#575	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#504	C₂	semidirect
3. C₅₃×#2171	C₂	semidirect
4. C₅₃×#2170	C₂	semidirect
5. C₅₃×#2153	C₂	semidirect
6. C₅₃×#2117	C₂	semidirect
7. C₅₃×#2108	C₂	semidirect
12. C₅₃×#504	C₂²	semidirect
13. C₅₃×#504	C₂²	semidirect
24. C₅₃×#364	C₂²	semidirect
25. C₅₃×#350	C₂²	semidirect
26. C₅₃×#346	C₂²	semidirect
27. C₄×C₁₆×C₅₃	C₂²	semidirect
113. #33441	C₅₃	direct (N × H)
124. C₂×#504	C₂×C₅₃	semidirect
126. #2171	C₂×C₅₃	semidirect
127. #2170	C₂×C₅₃	semidirect
128. #2153	C₂×C₅₃	semidirect
129. #2117	C₂×C₅₃	semidirect
130. #2108	C₂×C₅₃	semidirect
139. #504	C₂²×C₅₃	semidirect
140. #504	C₂²×C₅₃	semidirect
151. #364	C₂²×C₅₃	semidirect
152. #350	C₂²×C₅₃	semidirect
153. #346	C₂²×C₅₃	semidirect
154. C₄×C₁₆	C₂²×C₅₃	semidirect
155. C₅₃	#33441	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×#504	2
C₅₃×#2955	2
C₅₃×#2171	2
C₅₃×#2170	2
C₅₃×#2153	2
C₅₃×#2117	2
C₅₃×#2108	2
#33441	53

Subgroups — C53×#33441 #27240953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33441 #27240953