Subgroups — C₅₃×#11298 #27218810

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#336	C₂	1	3	yes	maximal
2	C₂×C₅₃×#336	C₂	1	3	yes	maximal
3	C₄×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₅₃	C₂²	1	3	yes	—
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
7	C₂×#95×C₅₃	C₄	1	3	yes	—
8	C₂×#95×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×#336	—	2	6	no	—
10	C₅₃×#336	—	2	6	no	—
11	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
12	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
13	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	—	2	3	no	—
15	C₂²×C₈×C₅₃	—	2	3	no	—
16	C₂²×C₈×C₅₃	—	2	3	no	—
17	C₂²×C₈×C₅₃	—	2	3	no	—
18	C₂²×C₈×C₅₃	—	2	3	no	—
19	C₂²×C₈×C₅₃	—	2	3	no	—
20	C₂²×C₈×C₅₃	—	2	3	no	—
21	C₂²×C₈×C₅₃	—	2	3	no	—
22	C₂²×C₈×C₅₃	D₄	1	6	yes	—
23	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
24	C₂×C₄²×C₅₃	—	2	3	no	—
25	C₂×C₄²×C₅₃	—	2	3	no	—
26	C₂×C₄²×C₅₃	—	2	3	no	—
27	C₂×C₄²×C₅₃	—	2	3	no	—
28	C₂×C₄²×C₅₃	—	2	3	no	—
29	C₂×C₄²×C₅₃	—	2	3	no	—
30	#95×C₅₃	—	2	6	no	—
31	#95×C₅₃	C₈	1	4	yes	complemented
32	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	2	3	no	—
34	C₂²×C₄×C₅₃	—	2	3	no	—
35	C₂²×C₄×C₅₃	—	2	3	no	—
36	C₂²×C₄×C₅₃	—	2	3	no	—
37	C₂²×C₄×C₅₃	—	2	3	no	—
38	C₂²×C₄×C₅₃	—	2	3	no	—
39	C₂²×C₄×C₅₃	—	2	3	no	—
40	C₂²×C₄×C₅₃	—	2	3	no	—
41	C₂²×C₄×C₅₃	—	2	3	no	—
42	C₂²×C₄×C₅₃	—	2	3	no	—
43	C₂²×C₄×C₅₃	—	2	3	no	—
44	C₂²×C₄×C₅₃	—	2	3	no	—
45	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
46	C₂²×C₄×C₅₃	M₁₆	1	6	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	3	yes	—
50	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	3	yes	—
52	C₂²×C₄×C₅₃	#41	1	3	yes	—
53	C₂×C₈×C₅₃	—	2	6	no	—
54	C₂×C₈×C₅₃	—	2	6	no	—
55	C₂×C₈×C₅₃	—	2	6	no	—
56	C₂×C₈×C₅₃	—	2	6	no	—
57	C₂×C₈×C₅₃	—	4	3	no	—
58	C₂×C₈×C₅₃	—	4	3	no	—
59	C₂×C₈×C₅₃	—	4	3	no	—
60	C₂×C₈×C₅₃	—	4	1	no	—
61	C₂×C₈×C₅₃	—	4	3	no	—
62	C₂×C₈×C₅₃	—	4	3	no	—
63	C₄²×C₅₃	—	4	3	no	—
64	C₄²×C₅₃	—	4	3	no	—
65	C₄²×C₅₃	—	4	3	no	—
66	C₄²×C₅₃	—	4	3	no	—
67	C₂³×C₅₃	#117	1	3	yes	—
68	C₂³×C₅₃	C₄×D₄	1	3	yes	—
69	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
71	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
72	C₂³×C₅₃	#131	1	3	yes	—
73	C₂³×C₅₃	#131	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	6	no	—
75	C₂×C₄×C₅₃	—	2	6	no	—
76	C₂×C₄×C₅₃	—	2	6	no	—
77	C₂×C₄×C₅₃	—	2	6	no	—
78	C₂×C₄×C₅₃	—	2	6	no	—
79	C₂×C₄×C₅₃	—	2	6	no	—
80	C₂×C₄×C₅₃	—	4	3	no	—
81	C₂×C₄×C₅₃	—	4	3	no	—
82	C₂×C₄×C₅₃	—	4	3	no	—
83	C₂×C₄×C₅₃	—	4	3	no	—
84	C₂×C₄×C₅₃	—	4	3	no	—
85	C₂×C₄×C₅₃	—	4	3	no	—
86	C₂×C₄×C₅₃	—	4	3	no	—
87	C₂×C₄×C₅₃	—	4	3	no	—
88	C₂×C₄×C₅₃	#121	1	3	yes	—
89	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
90	C₂×C₄×C₅₃	#123	1	3	yes	—
91	C₂×C₄×C₅₃	#123	1	3	yes	—
92	C₂×C₄×C₅₃	#124	1	3	yes	—
93	C₂×C₄×C₅₃	#125	1	3	yes	—
94	C₂×C₄×C₅₃	#126	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#126	1	3	yes	—
96	C₂×C₄×C₅₃	#126	1	3	yes	—
97	C₂×C₄×C₅₃	#126	1	3	yes	—
98	C₈×C₅₃	—	4	2	no	—
99	C₈×C₅₃	—	4	6	no	—
100	#11298	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₅₃	#383	1	1	yes	characteristic
102	C₂²×C₅₃	#387	1	3	yes	—
103	C₂²×C₅₃	#388	1	3	yes	—
104	C₂²×C₅₃	#431	1	3	yes	—
105	C₂²×C₅₃	#432	1	3	yes	—
106	C₂²×C₅₃	#433	1	3	yes	—
107	C₂²×C₅₃	#433	1	3	yes	—
108	C₂²×C₅₃	C₈×D₄	1	3	yes	—
109	C₂²×C₅₃	#435	1	3	yes	—
110	C₂²×C₅₃	#435	1	3	yes	—
111	C₂²×C₅₃	#436	1	3	yes	—
112	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
113	C₂²×C₅₃	#446	1	3	yes	—
114	C₄×C₅₃	—	4	6	no	—
115	C₄×C₅₃	—	4	6	no	—
116	C₄×C₅₃	#396	1	1	yes	characteristic
117	C₄×C₅₃	#397	1	3	yes	—
118	C₄×C₅₃	#400	1	3	yes	—
119	C₄×C₅₃	#401	1	1	yes	characteristic
120	C₂×#336	C₂×C₅₃	1	3	yes	—
121	C₂×#336	C₂×C₅₃	1	3	yes	—
122	C₄×#95	C₂×C₅₃	1	1	yes	characteristic
123	C₂×C₅₃	#1443	1	1	yes	characteristic
124	C₂×C₅₃	#1839	1	1	yes	characteristic
125	C₂×C₅₃	#1840	1	1	yes	characteristic
126	C₂×C₅₃	#1916	1	3	yes	—
127	C₂×C₅₃	#1917	1	3	yes	—
128	C₂×C₅₃	#1923	1	3	yes	—
129	C₂×C₅₃	#1924	1	3	yes	—
130	C₂³×C₈	C₂²×C₅₃	1	3	yes	—
131	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
132	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
133	C₂×#95	C₄×C₅₃	1	3	yes	—
134	C₂×#95	C₄×C₅₃	1	1	yes	characteristic
135	#336	—	2	6	no	—
136	#336	—	2	6	no	—
137	C₅₃	#11298	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
139	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
140	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	3	no	—
142	C₂²×C₈	—	2	3	no	—
143	C₂²×C₈	—	2	3	no	—
144	C₂²×C₈	—	2	3	no	—
145	C₂²×C₈	—	2	3	no	—
146	C₂²×C₈	—	2	3	no	—
147	C₂²×C₈	—	2	3	no	—
148	C₂²×C₈	—	2	3	no	—
149	C₂²×C₈	D₄×C₅₃	1	6	yes	—
150	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
151	C₂×C₄²	—	2	3	no	—
152	C₂×C₄²	—	2	3	no	—
153	C₂×C₄²	—	2	3	no	—
154	C₂×C₄²	—	2	3	no	—
155	C₂×C₄²	—	2	3	no	—
156	C₂×C₄²	—	2	3	no	—
157	#95	—	2	6	no	—
158	#95	C₈×C₅₃	1	4	yes	complemented
159	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	3	no	—
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	—	2	3	no	—
163	C₂²×C₄	—	2	3	no	—
164	C₂²×C₄	—	2	3	no	—
165	C₂²×C₄	—	2	3	no	—
166	C₂²×C₄	—	2	3	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	—	2	3	no	—
169	C₂²×C₄	—	2	3	no	—
170	C₂²×C₄	—	2	3	no	—
171	C₂²×C₄	—	2	3	no	—
172	C₂²×C₄	C₂×C₈×C₅₃	1	6	yes	—
173	C₂²×C₄	M₁₆×C₅₃	1	6	yes	—
174	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
175	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	#41×C₅₃	1	3	yes	—
177	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	#41×C₅₃	1	3	yes	—
179	C₂²×C₄	#41×C₅₃	1	3	yes	—
180	C₂×C₈	—	2	6	no	—
181	C₂×C₈	—	2	6	no	—
182	C₂×C₈	—	2	6	no	—
183	C₂×C₈	—	2	6	no	—
184	C₂×C₈	—	4	3	no	—
185	C₂×C₈	—	4	3	no	—
186	C₂×C₈	—	4	3	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	3	no	—
189	C₂×C₈	—	4	3	no	—
190	C₄²	—	4	3	no	—
191	C₄²	—	4	3	no	—
192	C₄²	—	4	3	no	—
193	C₄²	—	4	3	no	—
194	C₂³	#117×C₅₃	1	3	yes	—
195	C₂³	C₄×D₄×C₅₃	1	3	yes	—
196	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
197	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
198	C₂³	C₂×M₁₆×C₅₃	1	3	yes	—
199	C₂³	#131×C₅₃	1	3	yes	—
200	C₂³	#131×C₅₃	1	1	yes	characteristic
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	2	6	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	4	3	no	—
208	C₂×C₄	—	4	3	no	—
209	C₂×C₄	—	4	3	no	—
210	C₂×C₄	—	4	3	no	—
211	C₂×C₄	—	4	3	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	—	4	3	no	—
214	C₂×C₄	—	4	3	no	—
215	C₂×C₄	#121×C₅₃	1	3	yes	—
216	C₂×C₄	C₂²⋊Q₈×C₅₃	1	3	yes	—
217	C₂×C₄	#123×C₅₃	1	3	yes	—
218	C₂×C₄	#123×C₅₃	1	3	yes	—
219	C₂×C₄	#124×C₅₃	1	3	yes	—
220	C₂×C₄	#125×C₅₃	1	3	yes	—
221	C₂×C₄	#126×C₅₃	1	1	yes	characteristic
222	C₂×C₄	#126×C₅₃	1	3	yes	—
223	C₂×C₄	#126×C₅₃	1	3	yes	—
224	C₂×C₄	#126×C₅₃	1	3	yes	—
225	C₈	—	4	2	no	—
226	C₈	—	4	6	no	—
227	C₂²	C₅₃×#383	1	1	yes	characteristic
228	C₂²	C₅₃×#387	1	3	yes	—
229	C₂²	C₅₃×#388	1	3	yes	—
230	C₂²	C₅₃×#431	1	3	yes	—
231	C₂²	C₅₃×#432	1	3	yes	—
232	C₂²	C₅₃×#433	1	3	yes	—
233	C₂²	C₅₃×#433	1	3	yes	—
234	C₂²	C₈×D₄×C₅₃	1	3	yes	—
235	C₂²	C₅₃×#435	1	3	yes	—
236	C₂²	C₅₃×#435	1	3	yes	—
237	C₂²	C₅₃×#436	1	3	yes	—
238	C₂²	C₈×Q₈×C₅₃	1	1	yes	characteristic
239	C₂²	C₅₃×#446	1	3	yes	—
240	C₄	—	4	6	no	—
241	C₄	—	4	6	no	—
242	C₄	C₅₃×#396	1	1	yes	characteristic
243	C₄	C₅₃×#397	1	3	yes	—
244	C₄	C₅₃×#400	1	3	yes	—
245	C₄	C₅₃×#401	1	1	yes	characteristic
246	C₂	C₅₃×#1443	1	1	yes	characteristic
247	C₂	C₅₃×#1839	1	1	yes	characteristic
248	C₂	C₅₃×#1840	1	1	yes	characteristic
249	C₂	C₅₃×#1916	1	3	yes	—
250	C₂	C₅₃×#1917	1	3	yes	—
251	C₂	C₅₃×#1923	1	3	yes	—
252	C₂	C₅₃×#1924	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
31. #95×C₅₃	C₈	semidirect
100. #11298	C₅₃	direct (N × H)
137. C₅₃	#11298	direct (N × H)
158. #95	C₈×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×#336	2
C₄×#95×C₅₃	2
#11298	53

Subgroups — C53×#11298 #27218810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11298 #27218810