Subgroups — C₂×C₅₃×#1377 #27220062

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1377	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₅₃	—	2	4	no	—
8	C₂×M₃₂×C₅₃	—	2	2	no	—
9	C₂×M₃₂×C₅₃	C₄	1	8	yes	complemented
10	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₅₃	—	2	2	no	—
12	C₂×#97×C₅₃	—	2	2	no	—
13	C₂×#97×C₅₃	—	2	2	no	—
14	C₂×#97×C₅₃	C₄	1	2	yes	—
15	C₂×#97×C₅₃	C₂²	1	2	yes	—
16	C₂³×C₄×C₅₃	C₈	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	—	2	1	no	—
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	C₈	1	4	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	C₂²×C₈×C₅₃	D₄	1	2	yes	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	—	2	1	no	—
31	C₂×C₄²×C₅₃	—	2	1	no	—
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	M₃₂×C₅₃	—	2	8	no	—
37	M₃₂×C₅₃	—	2	4	no	—
38	M₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
39	C₂×C₁₆×C₅₃	—	2	8	no	—
40	C₂×C₁₆×C₅₃	—	2	4	no	—
41	#97×C₅₃	—	2	4	no	—
42	#97×C₅₃	—	2	4	no	—
43	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	4	4	no	—
47	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
55	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₄²	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	—	2	2	no	—
78	C₁₆×C₅₃	—	2	8	no	—
79	C₂³×C₅₃	—	2	4	no	—
80	C₂³×C₅₃	C₄×C₈	1	2	yes	—
81	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
82	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
84	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	#95	1	2	yes	—
90	C₂×C₄×C₅₃	#95	1	2	yes	—
91	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₅₃	#97	1	2	yes	—
93	C₂×C₄×C₅₃	#97	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	C₈×C₅₃	—	2	4	no	—
108	C₈×C₅₃	#95	1	4	yes	—
109	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
110	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×#1377	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	2	4	no	—
116	C₂²×C₅₃	#336	1	2	yes	—
117	C₂²×C₅₃	#336	1	2	yes	—
118	C₂²×C₅₃	#349	1	4	yes	—
119	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₄×C₅₃	—	4	4	no	—
123	C₄×C₅₃	#336	1	2	yes	—
124	C₄×C₅₃	#336	1	2	yes	—
125	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
127	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂²×M₃₂	C₂×C₅₃	1	2	yes	—
130	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
131	#1377	C₂×C₅₃	1	4	yes	direct product factor
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	#1377	1	4	yes	direct product factor
136	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#349	1	2	yes	—
138	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₂²×C₄²	C₄×C₅₃	1	1	yes	characteristic
141	C₂×M₃₂	—	2	2	no	—
142	C₂×M₃₂	—	2	4	no	—
143	C₂×M₃₂	C₄×C₅₃	1	8	yes	complemented
144	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
145	C₂²×C₁₆	—	2	2	no	—
146	C₂×#97	—	2	2	no	—
147	C₂×#97	—	2	2	no	—
148	C₂×#97	C₄×C₅₃	1	2	yes	—
149	C₂×#97	C₂²×C₅₃	1	2	yes	—
150	C₅₃	C₂×#1377	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³×C₄	C₈×C₅₃	1	2	yes	—
152	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	C₈×C₅₃	1	4	yes	—
157	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
159	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₅₃	1	2	yes	—
161	C₂²×C₈	D₄×C₅₃	1	2	yes	—
162	C₂²×C₈	D₄×C₅₃	1	2	yes	—
163	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
164	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₈×C₅₃	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
171	M₃₂	—	2	4	no	—
172	M₃₂	—	2	8	no	—
173	M₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
174	C₂×C₁₆	—	2	4	no	—
175	C₂×C₁₆	—	2	8	no	—
176	#97	—	2	4	no	—
177	#97	—	2	4	no	—
178	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
187	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
189	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
190	C₂²×C₄	M₁₆×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	M₁₆×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	C₄²×C₅₃	1	2	yes	—
198	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
199	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
200	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
201	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
202	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
203	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
204	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
205	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
208	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₁₆	—	2	8	no	—
214	C₂³	—	2	4	no	—
215	C₂³	C₄×C₈×C₅₃	1	2	yes	—
216	C₂³	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
217	C₂³	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
218	C₂³	C₄⋊C₈×C₅₃	1	2	yes	—
219	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	#95×C₅₃	1	2	yes	—
225	C₂×C₄	#95×C₅₃	1	2	yes	—
226	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
227	C₂×C₄	#97×C₅₃	1	2	yes	—
228	C₂×C₄	#97×C₅₃	1	2	yes	—
229	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
232	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
233	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
234	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
235	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
236	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
240	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	#95×C₅₃	1	4	yes	—
244	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
245	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	C₅₃×#336	1	2	yes	—
251	C₂²	C₅₃×#336	1	2	yes	—
252	C₂²	C₅₃×#349	1	4	yes	—
253	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
254	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
255	C₂²	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
256	C₄	—	4	4	no	—
257	C₄	C₅₃×#336	1	2	yes	—
258	C₄	C₅₃×#336	1	2	yes	—
259	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
260	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
261	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
262	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	C₅₃×#1377	1	4	yes	direct product factor
267	C₂	C₂×C₅₃×#336	1	1	yes	characteristic
268	C₂	C₂×C₅₃×#349	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1377	C₂	direct (N × H)
9. C₂×M₃₂×C₅₃	C₄	semidirect
38. M₃₂×C₅₃	C₂×C₄	semidirect
111. C₂×#1377	C₅₃	direct (N × H)
131. #1377	C₂×C₅₃	direct (N × H)
135. C₂×C₅₃	#1377	direct (N × H)
143. C₂×M₃₂	C₄×C₅₃	semidirect
150. C₅₃	C₂×#1377	direct (N × H)
173. M₃₂	C₂×C₄×C₅₃	semidirect
266. C₂	C₅₃×#1377	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×M₃₂×C₅₃	2
C₂²×#97×C₅₃	2
C₅₃×#1377	2
C₂×#1377	53

Subgroups — C2×C53×#1377 #27220062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1377 #27220062