Subgroups — C₂×C₅₃×#1376 #27220060

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 264 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×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1376	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×C₄×C₈×C₅₃	—	2	4	no	—
13	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₅₃	—	2	1	no	—
16	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	C₈	1	2	yes	—
20	C₂²×C₈×C₅₃	C₈	1	2	yes	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	D₄	1	4	yes	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	—	2	1	no	—
32	C₂×C₄²×C₅₃	C₈	1	4	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	M₃₂×C₅₃	—	2	8	no	—
35	M₃₂×C₅₃	—	2	4	no	—
36	M₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
37	C₂×C₁₆×C₅₃	—	2	8	no	—
38	C₂×C₁₆×C₅₃	—	2	4	no	—
39	C₄×C₈×C₅₃	—	2	4	no	—
40	C₄×C₈×C₅₃	—	2	4	no	—
41	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	8	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	C₄²	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
63	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	C₂×C₈	1	4	yes	—
75	C₁₆×C₅₃	—	2	8	no	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	#97	1	2	yes	—
78	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
79	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
80	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
81	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	—	2	16	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	#95	1	2	yes	—
86	C₂×C₄×C₅₃	#95	1	2	yes	—
87	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
89	C₂×C₄×C₅₃	#97	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₈×C₅₃	—	2	4	no	—
104	C₈×C₅₃	#95	1	2	yes	—
105	C₈×C₅₃	#95	1	2	yes	—
106	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
107	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×#1376	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₅₃	—	2	4	no	—
110	C₂²×C₅₃	—	2	4	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	#336	1	2	yes	—
114	C₂²×C₅₃	#336	1	2	yes	—
115	C₂²×C₅₃	#364	1	4	yes	—
116	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₄×C₅₃	—	2	8	no	—
120	C₄×C₅₃	#336	1	2	yes	—
121	C₄×C₅₃	#336	1	2	yes	—
122	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
123	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
124	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₂²×M₃₂	C₂×C₅₃	1	2	yes	—
127	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
128	#1376	C₂×C₅₃	1	4	yes	direct product factor
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	#1376	1	4	yes	direct product factor
133	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#364	1	2	yes	—
135	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic
136	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
137	C₂²×C₄²	C₄×C₅₃	1	1	yes	characteristic
138	C₂×M₃₂	—	2	2	no	—
139	C₂×M₃₂	—	2	4	no	—
140	C₂×M₃₂	C₄×C₅₃	1	8	yes	complemented
141	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
142	C₂²×C₁₆	—	2	2	no	—
143	C₂×C₄×C₈	—	2	4	no	—
144	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
146	C₅₃	C₂×#1376	1	1	yes	characteristic, direct product factor, Sylow
147	C₂³×C₄	—	2	1	no	—
148	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	C₈×C₅₃	1	2	yes	—
152	C₂²×C₈	C₈×C₅₃	1	2	yes	—
153	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
154	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
155	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
156	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₅₃	1	4	yes	—
159	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
160	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₈×C₅₃	1	4	yes	—
165	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
166	M₃₂	—	2	4	no	—
167	M₃₂	—	2	8	no	—
168	M₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
169	C₂×C₁₆	—	2	4	no	—
170	C₂×C₁₆	—	2	8	no	—
171	C₄×C₈	—	2	4	no	—
172	C₄×C₈	—	2	4	no	—
173	C₂⁴	M₁₆×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
184	C₂²×C₄	M₁₆×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₄²×C₅₃	1	2	yes	—
192	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
193	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
194	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
195	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
196	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
197	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
198	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
199	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
200	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
203	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	C₂×C₈×C₅₃	1	4	yes	—
207	C₁₆	—	2	8	no	—
208	C₂³	—	2	4	no	—
209	C₂³	#97×C₅₃	1	2	yes	—
210	C₂³	C₂²⋊C₈×C₅₃	1	2	yes	—
211	C₂³	C₄⋊C₈×C₅₃	1	1	yes	characteristic
212	C₂³	C₄⋊C₈×C₅₃	1	1	yes	characteristic
213	C₂³	C₂×M₁₆×C₅₃	1	1	yes	characteristic
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	#95×C₅₃	1	2	yes	—
218	C₂×C₄	#95×C₅₃	1	2	yes	—
219	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
220	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
221	C₂×C₄	#97×C₅₃	1	2	yes	—
222	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
223	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
224	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
225	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
226	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
228	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
235	C₈	—	2	4	no	—
236	C₈	#95×C₅₃	1	2	yes	—
237	C₈	#95×C₅₃	1	2	yes	—
238	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
239	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	C₅₃×#336	1	2	yes	—
245	C₂²	C₅₃×#336	1	2	yes	—
246	C₂²	C₅₃×#364	1	4	yes	—
247	C₂²	C₂×#97×C₅₃	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
249	C₂²	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	C₅₃×#336	1	2	yes	—
252	C₄	C₅₃×#336	1	2	yes	—
253	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
254	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
255	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
256	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	2	1	no	—
260	C₂	C₅₃×#1376	1	4	yes	direct product factor
261	C₂	C₂×C₅₃×#336	1	1	yes	characteristic
262	C₂	C₂×C₅₃×#364	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₅₃	D₄
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂³×C₅₃	C₄⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1376	C₂	direct (N × H)
9. C₂×M₃₂×C₅₃	C₄	semidirect
36. M₃₂×C₅₃	C₂×C₄	semidirect
108. C₂×#1376	C₅₃	direct (N × H)
128. #1376	C₂×C₅₃	direct (N × H)
132. C₂×C₅₃	#1376	direct (N × H)
140. C₂×M₃₂	C₄×C₅₃	semidirect
146. C₅₃	C₂×#1376	direct (N × H)
168. M₃₂	C₂×C₄×C₅₃	semidirect
260. C₂	C₅₃×#1376	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×C₄×C₈×C₅₃	2
C₅₃×#1376	2
C₂×#1376	53

Subgroups — C2×C53×#1376 #27220060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1376 #27220060