Subgroups — C₄×C₅₃×#382 #27221285

Order 13568.

1742 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#382	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
3	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×C₄⋊#95	C₂	1	2	yes	maximal
5	C₅₃×#1434	C₂	1	4	yes	maximal
6	C₄×#95×C₅₃	C₂	1	4	yes	maximal
7	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#382	C₄	1	16	yes	direct product factor
15	C₄×C₄⋊C₄×C₅₃	—	2	8	no	—
16	C₂×#95×C₅₃	C₂²	1	8	yes	—
17	C₂×#95×C₅₃	C₂²	1	4	yes	—
18	C₂×#95×C₅₃	C₂²	1	8	yes	—
19	C₄³×C₅₃	C₄	1	8	yes	complemented
20	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
28	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
29	C₂×C₄²×C₅₃	—	2	8	no	—
30	C₂×C₄²×C₅₃	—	2	8	no	—
31	C₂×C₄²×C₅₃	—	2	4	no	—
32	C₂×C₄²×C₅₃	—	2	8	no	—
33	C₂×C₄²×C₅₃	—	2	16	no	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₅₃	D₄	1	4	yes	—
41	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
42	#95×C₅₃	—	2	8	no	—
43	#95×C₅₃	—	2	16	no	—
44	#95×C₅₃	C₂×C₄	1	32	yes	—
45	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	—	2	4	no	—
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	8	no	—
52	C₂²×C₄×C₅₃	—	2	16	no	—
53	C₂²×C₄×C₅₃	—	2	16	no	—
54	C₂²×C₄×C₅₃	—	2	8	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅₃	#41	1	8	yes	—
70	C₂²×C₄×C₅₃	#41	1	4	yes	—
71	C₂²×C₄×C₅₃	#41	1	8	yes	—
72	C₄⋊C₄×C₅₃	—	2	32	no	—
73	C₄²×C₅₃	—	2	16	no	—
74	C₄²×C₅₃	—	2	8	no	—
75	C₄²×C₅₃	—	2	16	no	—
76	C₄²×C₅₃	—	4	16	no	—
77	C₄²×C₅₃	C₄²	1	16	yes	complemented
78	C₄²×C₅₃	C₄⋊C₄	1	16	yes	—
79	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
80	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
81	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂³×C₅₃	C₂×#41	1	2	yes	—
84	C₂³×C₅₃	C₂×#41	1	4	yes	—
85	C₂³×C₅₃	C₂×#41	1	4	yes	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	16	no	—
92	C₂×C₄×C₅₃	—	2	16	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	16	no	—
96	C₂×C₄×C₅₃	—	2	32	no	—
97	C₂×C₄×C₅₃	—	4	8	no	—
98	C₂×C₄×C₅₃	—	4	16	no	—
99	C₂×C₄×C₅₃	—	4	8	no	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₅₃	#117	1	8	yes	—
106	C₂×C₄×C₅₃	#117	1	4	yes	—
107	C₂×C₄×C₅₃	#117	1	8	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
109	C₂×C₄×C₅₃	C₄×Q₈	1	8	yes	—
110	C₂×C₄×C₅₃	#124	1	8	yes	—
111	C₂×C₄×C₅₃	#125	1	8	yes	—
112	C₄×#382	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
114	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#117	1	2	yes	—
117	C₂²×C₅₃	C₂×#117	1	2	yes	—
118	C₂²×C₅₃	C₂×C₄×D₄	1	2	yes	—
119	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
120	C₂²×C₅₃	C₄×#41	1	8	yes	—
121	C₂²×C₅₃	C₂×#124	1	2	yes	—
122	C₂²×C₅₃	C₂×#125	1	2	yes	—
123	C₂²×C₅₃	#529	1	8	yes	—
124	C₂²×C₅₃	#532	1	2	yes	—
125	C₂²×C₅₃	#533	1	2	yes	—
126	C₄×C₅₃	—	2	8	no	—
127	C₄×C₅₃	—	2	8	no	—
128	C₄×C₅₃	—	2	4	no	—
129	C₄×C₅₃	—	4	16	no	—
130	C₄×C₅₃	C₄×C₄⋊C₄	1	8	yes	—
131	C₄×C₅₃	#382	1	8	yes	direct product factor
132	C₂×#382	C₂×C₅₃	1	2	yes	—
133	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
134	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic
135	C₄⋊#95	C₂×C₅₃	1	2	yes	—
136	#1434	C₂×C₅₃	1	4	yes	—
137	C₄×#95	C₂×C₅₃	1	4	yes	—
138	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₅₃	C₄×#117	1	2	yes	—
140	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
141	C₂×C₅₃	#2283	1	2	yes	—
142	C₂×C₅₃	#2285	1	1	yes	characteristic
143	C₂×C₅₃	C₄×#124	1	4	yes	—
144	C₂×C₅₃	C₄×#125	1	4	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
146	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
147	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
148	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
149	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
150	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
151	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
152	#382	C₄×C₅₃	1	16	yes	direct product factor
153	C₄×C₄⋊C₄	—	2	8	no	—
154	C₂×#95	C₂²×C₅₃	1	8	yes	—
155	C₂×#95	C₂²×C₅₃	1	4	yes	—
156	C₂×#95	C₂²×C₅₃	1	8	yes	—
157	C₄³	C₄×C₅₃	1	8	yes	complemented
158	C₅₃	C₄×#382	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
160	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
161	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
162	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
163	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
164	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
165	C₂×C₄⋊C₄	—	2	8	no	—
166	C₂×C₄⋊C₄	—	2	8	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	16	yes	—
168	C₂×C₄²	—	2	8	no	—
169	C₂×C₄²	—	2	4	no	—
170	C₂×C₄²	—	2	8	no	—
171	C₂×C₄²	—	2	8	no	—
172	C₂×C₄²	—	2	16	no	—
173	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
174	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
175	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
178	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
179	C₂×C₄²	D₄×C₅₃	1	4	yes	—
180	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
181	#95	—	2	8	no	—
182	#95	—	2	16	no	—
183	#95	C₂×C₄×C₅₃	1	32	yes	—
184	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	16	no	—
192	C₂²×C₄	—	2	16	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
195	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
196	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
200	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
201	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
205	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
206	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
207	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
208	C₂²×C₄	#41×C₅₃	1	8	yes	—
209	C₂²×C₄	#41×C₅₃	1	4	yes	—
210	C₂²×C₄	#41×C₅₃	1	8	yes	—
211	C₄⋊C₄	—	2	32	no	—
212	C₄²	—	2	16	no	—
213	C₄²	—	2	8	no	—
214	C₄²	—	2	16	no	—
215	C₄²	—	4	16	no	—
216	C₄²	C₄²×C₅₃	1	16	yes	complemented
217	C₄²	C₄⋊C₄×C₅₃	1	16	yes	—
218	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
219	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
220	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
221	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
222	C₂³	C₂×#41×C₅₃	1	2	yes	—
223	C₂³	C₂×#41×C₅₃	1	4	yes	—
224	C₂³	C₂×#41×C₅₃	1	4	yes	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	—	2	32	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	16	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₂×C₄²×C₅₃	1	8	yes	—
240	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
242	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
244	C₂×C₄	#117×C₅₃	1	8	yes	—
245	C₂×C₄	#117×C₅₃	1	4	yes	—
246	C₂×C₄	#117×C₅₃	1	8	yes	—
247	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
248	C₂×C₄	C₄×Q₈×C₅₃	1	8	yes	—
249	C₂×C₄	#124×C₅₃	1	8	yes	—
250	C₂×C₄	#125×C₅₃	1	8	yes	—
251	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
252	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
253	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
254	C₂²	C₂×#117×C₅₃	1	2	yes	—
255	C₂²	C₂×#117×C₅₃	1	2	yes	—
256	C₂²	C₂×C₄×D₄×C₅₃	1	2	yes	—
257	C₂²	C₂×C₄×Q₈×C₅₃	1	2	yes	—
258	C₂²	C₄×#41×C₅₃	1	8	yes	—
259	C₂²	C₂×#124×C₅₃	1	2	yes	—
260	C₂²	C₂×#125×C₅₃	1	2	yes	—
261	C₂²	C₅₃×#529	1	8	yes	—
262	C₂²	C₅₃×#532	1	2	yes	—
263	C₂²	C₅₃×#533	1	2	yes	—
264	C₄	—	2	8	no	—
265	C₄	—	2	8	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	16	no	—
268	C₄	C₄×C₄⋊C₄×C₅₃	1	8	yes	—
269	C₄	C₅₃×#382	1	8	yes	direct product factor
270	C₂	C₂×C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
271	C₂	C₄×#117×C₅₃	1	2	yes	—
272	C₂	C₂×C₅₃×#382	1	1	yes	characteristic
273	C₂	C₅₃×#2283	1	2	yes	—
274	C₂	C₅₃×#2285	1	1	yes	characteristic
275	C₂	C₄×#124×C₅₃	1	4	yes	—
276	C₂	C₄×#125×C₅₃	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅₃	C₂³
commutator	C₂²	C₂²×C₄²×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₅₃×#382	C₄	direct (N × H)
19. C₄³×C₅₃	C₄	semidirect
77. C₄²×C₅₃	C₄²	semidirect
112. C₄×#382	C₅₃	direct (N × H)
131. C₄×C₅₃	#382	direct (N × H)
152. #382	C₄×C₅₃	direct (N × H)
157. C₄³	C₄×C₅₃	semidirect
158. C₅₃	C₄×#382	direct (N × H)
216. C₄²	C₄²×C₅₃	semidirect
269. C₄	C₅₃×#382	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#382	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₄³×C₅₃	2
C₅₃×C₄⋊#95	2
C₅₃×#1434	2
C₄×#95×C₅₃	2
C₄×#382	53

Subgroups — C4×C53×#382 #27221285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#382 #27221285