Subgroups — C₅₃×#12663 #27220175

Order 13568.

374 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	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	C₄	1	4	yes	complemented
18	C₅₃×#346	—	2	2	no	—
19	C₅₃×#346	C₄	1	4	yes	complemented
20	C₄×C₁₆×C₅₃	—	2	2	no	—
21	C₅₃×#336	C₄	1	1	yes	characteristic
22	C₅₃×#336	C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	—	2	1	no	—
32	C₂×C₁₆×C₅₃	—	2	1	no	—
33	C₂×C₁₆×C₅₃	—	2	1	no	—
34	C₂×C₁₆×C₅₃	—	2	1	no	—
35	C₂×C₁₆×C₅₃	—	2	1	no	—
36	C₂×C₁₆×C₅₃	—	2	1	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
42	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
43	C₄⋊C₈×C₅₃	C₈	1	4	yes	—
44	C₄×C₈×C₅₃	—	2	1	no	—
45	C₄×C₈×C₅₃	—	2	1	no	—
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	#95×C₅₃	C₈	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	C₄²	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	C₄²	1	2	yes	—
77	C₄²×C₅₃	C₂×C₈	1	2	yes	—
78	C₁₆×C₅₃	—	2	2	no	—
79	C₁₆×C₅₃	—	2	2	no	—
80	C₁₆×C₅₃	—	2	2	no	—
81	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
82	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
93	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₈×C₅₃	#117	1	1	yes	characteristic
104	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
106	#12663	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
109	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
111	C₂²×C₅₃	#504	1	2	yes	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	4	no	—
115	C₄×C₅₃	—	4	2	no	—
116	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₅₃	#405	1	1	yes	characteristic
118	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
119	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
120	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
121	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
122	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
123	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
124	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
125	#1374	C₂×C₅₃	1	1	yes	characteristic
126	#1374	C₂×C₅₃	1	1	yes	characteristic
127	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
129	C₂×C₅₃	#2108	1	1	yes	characteristic
130	C₂×C₅₃	#2168	1	2	yes	—
131	C₂×C₅₃	#2183	1	2	yes	—
132	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
133	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
134	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
135	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₈	C₄×C₅₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₄×C₄⋊C₄	C₄×C₅₃	1	1	yes	characteristic
141	C₄⋊C₁₆	C₄×C₅₃	1	4	yes	complemented
142	#346	—	2	2	no	—
143	#346	C₄×C₅₃	1	4	yes	complemented
144	C₄×C₁₆	—	2	2	no	—
145	#336	C₄×C₅₃	1	1	yes	characteristic
146	#336	C₄×C₅₃	1	1	yes	characteristic
147	C₅₃	#12663	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
149	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₈×C₅₃	1	2	yes	—
153	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
155	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	2	1	no	—
160	C₂×C₁₆	—	2	1	no	—
161	C₂×C₁₆	—	2	1	no	—
162	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
163	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
164	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
165	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
166	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
167	C₂×C₁₆	Q₈×C₅₃	1	2	yes	—
168	C₄⋊C₈	C₈×C₅₃	1	4	yes	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
174	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
175	#95	C₈×C₅₃	1	2	yes	—
176	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
177	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	C₄²×C₅₃	1	2	yes	—
188	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
189	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
190	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
191	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
193	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
194	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
195	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	4	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	C₄²×C₅₃	1	2	yes	—
202	C₄²	C₂×C₈×C₅₃	1	2	yes	—
203	C₁₆	—	2	2	no	—
204	C₁₆	—	2	2	no	—
205	C₁₆	—	2	2	no	—
206	C₁₆	C₄⋊C₄×C₅₃	1	4	yes	complemented
207	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
216	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
217	C₂×C₄	M₃₂×C₅₃	1	4	yes	—
218	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
219	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
220	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
222	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
223	C₂×C₄	#131×C₅₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
228	C₈	#117×C₅₃	1	1	yes	characteristic
229	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
230	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
231	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
232	C₂²	C₈×D₄×C₅₃	1	1	yes	characteristic
233	C₂²	C₈×Q₈×C₅₃	1	1	yes	characteristic
234	C₂²	C₂×M₃₂×C₅₃	1	2	yes	—
235	C₂²	C₅₃×#504	1	2	yes	—
236	C₄	—	2	4	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	2	no	—
240	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
241	C₄	C₅₃×#405	1	1	yes	characteristic
242	C₄	C₈×D₄×C₅₃	1	1	yes	characteristic
243	C₄	C₈×Q₈×C₅₃	1	1	yes	characteristic
244	C₂	C₈×C₄⋊C₄×C₅₃	1	1	yes	characteristic
245	C₂	C₄×M₃₂×C₅₃	1	1	yes	characteristic
246	C₂	C₅₃×#2108	1	1	yes	characteristic
247	C₂	C₅₃×#2168	1	2	yes	—
248	C₂	C₅₃×#2183	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
17. C₅₃×C₄⋊C₁₆	C₄	semidirect
19. C₅₃×#346	C₄	semidirect
81. C₁₆×C₅₃	C₄⋊C₄	semidirect
106. #12663	C₅₃	direct (N × H)
141. C₄⋊C₁₆	C₄×C₅₃	semidirect
143. #346	C₄×C₅₃	semidirect
147. C₅₃	#12663	direct (N × H)
206. C₁₆	C₄⋊C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×C₄⋊C₁₆	2
C₂×C₅₃×#346	2
C₂×C₄×C₁₆×C₅₃	2
C₈×C₄⋊C₄×C₅₃	2
C₅₃×#1374	2
#12663	53

Subgroups — C53×#12663 #27220175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12663 #27220175