Subgroups — C₅₃×#22618 #27230130

Order 13568.

810 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	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2345	C₂	1	2	yes	maximal
3	C₅₃×#1943	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1918	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1824	C₂	1	2	yes	maximal
6	C₅₃×#1807	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1776	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₂×#107×C₅₃	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#106×C₅₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#389	C₂²	1	4	yes	—
19	C₅₃×#387	—	2	4	no	—
20	C₅₃×#380	—	2	2	no	—
21	C₅₃×#380	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
24	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#340	C₂²	1	2	yes	—
26	C₅₃×#340	C₂²	1	4	yes	—
27	C₅₃×#340	C₂²	1	4	yes	—
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₂³×C₄×C₅₃	—	2	2	no	—
30	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	—	2	4	no	—
46	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
47	#107×C₅₃	C₂×C₄	1	8	yes	—
48	#107×C₅₃	C₂×C₄	1	4	yes	—
49	#107×C₅₃	C₂×C₄	1	4	yes	—
50	#106×C₅₃	—	2	4	no	—
51	C₂²⋊C₈×C₅₃	—	2	2	no	—
52	#97×C₅₃	—	2	4	no	—
53	#95×C₅₃	—	2	4	no	—
54	#95×C₅₃	—	2	4	no	—
55	#95×C₅₃	—	2	2	no	—
56	#95×C₅₃	D₄	1	2	yes	—
57	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	8	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	—	8	4	no	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	8	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	#41	1	4	yes	—
78	C₄⋊C₄×C₅₃	—	2	8	no	—
79	C₄⋊C₄×C₅₃	—	2	2	no	—
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	8	4	no	—
84	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
85	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂²⋊C₄×C₅₃	—	2	4	no	—
87	C₂²⋊C₄×C₅₃	—	4	4	no	—
88	C₄²×C₅₃	—	2	2	no	—
89	C₄²×C₅₃	—	2	2	no	—
90	C₄²×C₅₃	—	8	4	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	C₄×D₄	1	2	yes	—
94	C₂³×C₅₃	#137	1	4	yes	—
95	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	8	8	no	—
111	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	—	4	2	no	—
118	C₈×C₅₃	#117	1	4	yes	—
119	#22618	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₅₃	—	2	8	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	#441	1	4	yes	—
124	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₅₃	#547	1	2	yes	—
126	C₂²×C₅₃	C₂×#137	1	2	yes	—
127	C₂²×C₅₃	#576	1	2	yes	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	4	4	no	—
130	C₄×C₅₃	—	4	4	no	—
131	C₄×C₅₃	—	4	4	no	—
132	C₄×C₅₃	—	8	4	no	—
133	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
134	C₄×C₅₃	#519	1	1	yes	characteristic
135	C₄×C₅₃	#552	1	2	yes	—
136	C₂²×#107	C₂×C₅₃	1	1	yes	characteristic
137	#2345	C₂×C₅₃	1	2	yes	—
138	#1943	C₂×C₅₃	1	2	yes	complemented
139	#1918	C₂×C₅₃	1	4	yes	complemented
140	#1824	C₂×C₅₃	1	2	yes	—
141	#1807	C₂×C₅₃	1	1	yes	characteristic
142	#1776	C₂×C₅₃	1	2	yes	complemented
143	#1752	C₂×C₅₃	1	1	yes	characteristic
144	C₂×C₅₃	—	2	4	no	—
145	C₂×C₅₃	#2344	1	1	yes	characteristic
146	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
147	C₂×C₅₃	#2948	1	1	yes	characteristic
148	C₂×C₅₃	#3346	1	4	yes	—
149	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
150	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
151	C₂×#107	C₄×C₅₃	1	8	yes	complemented
152	C₂×#107	C₂²×C₅₃	1	2	yes	—
153	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
155	C₂×#106	C₂²×C₅₃	1	2	yes	—
156	C₂×C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×#97	C₂²×C₅₃	1	2	yes	—
158	#389	C₂²×C₅₃	1	4	yes	—
159	#387	—	2	4	no	—
160	#380	—	2	2	no	—
161	#380	C₂²×C₅₃	1	2	yes	—
162	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
163	C₄×C₂²⋊C₄	—	2	2	no	—
164	C₄×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
165	#340	C₂²×C₅₃	1	2	yes	—
166	#340	C₂²×C₅₃	1	4	yes	—
167	#340	C₂²×C₅₃	1	4	yes	—
168	#336	C₂²×C₅₃	1	2	yes	—
169	C₅₃	#22618	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
174	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
175	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
181	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
188	#107	C₂×C₄×C₅₃	1	8	yes	—
189	#107	C₂×C₄×C₅₃	1	4	yes	—
190	#107	C₂×C₄×C₅₃	1	4	yes	—
191	#106	—	2	4	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	#97	—	2	4	no	—
194	#95	—	2	4	no	—
195	#95	—	2	4	no	—
196	#95	—	2	2	no	—
197	#95	D₄×C₅₃	1	2	yes	—
198	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	8	4	no	—
206	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
208	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	#41×C₅₃	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
218	C₂×C₈	#41×C₅₃	1	4	yes	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	8	4	no	—
225	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
226	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	4	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	8	4	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₄×D₄×C₅₃	1	2	yes	—
235	C₂³	#137×C₅₃	1	4	yes	—
236	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
253	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
255	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
256	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
257	C₈	—	2	2	no	—
258	C₈	—	4	2	no	—
259	C₈	#117×C₅₃	1	4	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	C₅₃×#441	1	4	yes	—
264	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
265	C₂²	C₅₃×#547	1	2	yes	—
266	C₂²	C₂×#137×C₅₃	1	2	yes	—
267	C₂²	C₅₃×#576	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	—	4	4	no	—
272	C₄	—	8	4	no	—
273	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
274	C₄	C₅₃×#519	1	1	yes	characteristic
275	C₄	C₅₃×#552	1	2	yes	—
276	C₂	—	2	4	no	—
277	C₂	C₅₃×#2344	1	1	yes	characteristic
278	C₂	C₂×C₅₃×#441	1	1	yes	characteristic
279	C₂	C₅₃×#2948	1	1	yes	characteristic
280	C₂	C₅₃×#3346	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1943	C₂	semidirect
4. C₅₃×#1918	C₂	semidirect
7. C₅₃×#1776	C₂	semidirect
11. C₂×#107×C₅₃	C₄	semidirect
119. #22618	C₅₃	direct (N × H)
138. #1943	C₂×C₅₃	semidirect
139. #1918	C₂×C₅₃	semidirect
142. #1776	C₂×C₅₃	semidirect
151. C₂×#107	C₄×C₅₃	semidirect
169. C₅₃	#22618	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#107×C₅₃	2
C₅₃×#2345	2
C₅₃×#1943	2
C₅₃×#1918	2
C₅₃×#1824	2
C₅₃×#1807	2
C₅₃×#1776	2
C₅₃×#1752	2
#22618	53

Subgroups — C53×#22618 #27230130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22618 #27230130