Subgroups — C₅₃×#1441 #20270378

Order 6784.

564 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×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#95×C₅₃	C₂	1	2	yes	maximal
4	C₂×#95×C₅₃	C₂	1	2	yes	maximal
5	C₂×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
9	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
12	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
15	C₂×C₄⋊C₄×C₅₃	C₄	1	4	yes	complemented
16	#95×C₅₃	—	2	4	no	—
17	#95×C₅₃	—	2	2	no	—
18	#95×C₅₃	—	2	4	no	—
19	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	—	2	2	no	—
26	C₂²×C₄×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	—	2	2	no	—
28	C₂²×C₄×C₅₃	—	2	2	no	—
29	C₂²×C₄×C₅₃	—	2	2	no	—
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	—	2	1	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	D₄	1	4	yes	—
46	C₂²×C₄×C₅₃	D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
50	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
51	C₄⋊C₄×C₅₃	—	2	8	no	—
52	C₄⋊C₄×C₅₃	—	2	8	no	—
53	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂³×C₅₃	C₂×D₄	1	2	yes	—
58	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂³×C₅₃	#41	1	1	yes	characteristic
61	C₂³×C₅₃	#41	1	1	yes	characteristic
62	C₂³×C₅₃	#41	1	2	yes	—
63	C₂³×C₅₃	#41	1	2	yes	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	4	1	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
94	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	complemented
95	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₅₃	#117	1	1	yes	characteristic
98	C₂²×C₅₃	C₄×D₄	1	2	yes	—
99	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
100	C₂²×C₅₃	#120	1	1	yes	characteristic
101	C₂²×C₅₃	#121	1	1	yes	characteristic
102	C₂²×C₅₃	#121	1	1	yes	characteristic
103	C₂²×C₅₃	#121	1	2	yes	—
104	C₂²×C₅₃	#121	1	2	yes	—
105	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
108	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
110	C₂²×C₅₃	#123	1	1	yes	characteristic
111	C₂²×C₅₃	#123	1	2	yes	—
112	C₂²×C₅₃	#124	1	1	yes	characteristic
113	C₂²×C₅₃	#125	1	2	yes	—
114	C₂²×C₅₃	#125	1	2	yes	—
115	C₂²×C₅₃	#126	1	2	yes	—
116	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
117	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
118	C₄×C₅₃	—	2	4	no	—
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	—	4	4	no	—
122	C₄×C₅₃	—	4	4	no	—
123	#1441	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₅₃	#380	1	1	yes	characteristic
125	C₂×C₅₃	#386	1	1	yes	characteristic
126	C₂×C₅₃	#387	1	2	yes	—
127	C₂×C₅₃	#389	1	2	yes	—
128	C₂×C₅₃	#391	1	1	yes	characteristic
129	C₂×C₅₃	#394	1	1	yes	characteristic
130	C₂×C₅₃	#395	1	1	yes	characteristic
131	C₂×C₅₃	#396	1	2	yes	—
132	C₂×C₅₃	#398	1	1	yes	characteristic
133	C₂×C₅₃	#398	1	2	yes	—
134	C₂×C₅₃	#400	1	1	yes	characteristic
135	C₂²×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
136	C₂²×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
137	C₂×#95	C₂×C₅₃	1	2	yes	—
138	C₂×#95	C₂×C₅₃	1	2	yes	—
139	C₂×#95	C₂×C₅₃	1	1	yes	characteristic
140	C₅₃	#1441	1	1	yes	characteristic, direct product factor, Sylow
141	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
143	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
144	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
145	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₄×C₅₃	1	4	yes	complemented
151	#95	—	2	4	no	—
152	#95	—	2	2	no	—
153	#95	—	2	4	no	—
154	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
178	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
179	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
180	C₂²×C₄	D₄×C₅₃	1	4	yes	—
181	C₂²×C₄	D₄×C₅₃	1	2	yes	—
182	C₂²×C₄	D₄×C₅₃	1	2	yes	—
183	C₂²×C₄	D₄×C₅₃	1	2	yes	—
184	C₂²×C₄	Q₈×C₅₃	1	4	yes	—
185	C₂²×C₄	Q₈×C₅₃	1	2	yes	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
189	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
190	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
191	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
192	C₂³	C₂×D₄×C₅₃	1	2	yes	—
193	C₂³	C₂×Q₈×C₅₃	1	1	yes	characteristic
194	C₂³	C₂×Q₈×C₅₃	1	2	yes	—
195	C₂³	#41×C₅₃	1	1	yes	characteristic
196	C₂³	#41×C₅₃	1	1	yes	characteristic
197	C₂³	#41×C₅₃	1	2	yes	—
198	C₂³	#41×C₅₃	1	2	yes	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
229	C₂×C₄	C₄⋊C₄×C₅₃	1	4	yes	complemented
230	C₂²	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
231	C₂²	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
232	C₂²	#117×C₅₃	1	1	yes	characteristic
233	C₂²	C₄×D₄×C₅₃	1	2	yes	—
234	C₂²	C₄×Q₈×C₅₃	1	2	yes	—
235	C₂²	#120×C₅₃	1	1	yes	characteristic
236	C₂²	#121×C₅₃	1	1	yes	characteristic
237	C₂²	#121×C₅₃	1	1	yes	characteristic
238	C₂²	#121×C₅₃	1	2	yes	—
239	C₂²	#121×C₅₃	1	2	yes	—
240	C₂²	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
241	C₂²	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
242	C₂²	C₂²⋊Q₈×C₅₃	1	2	yes	—
243	C₂²	C₂²⋊Q₈×C₅₃	1	2	yes	—
244	C₂²	C₂²⋊Q₈×C₅₃	1	2	yes	—
245	C₂²	#123×C₅₃	1	1	yes	characteristic
246	C₂²	#123×C₅₃	1	2	yes	—
247	C₂²	#124×C₅₃	1	1	yes	characteristic
248	C₂²	#125×C₅₃	1	2	yes	—
249	C₂²	#125×C₅₃	1	2	yes	—
250	C₂²	#126×C₅₃	1	2	yes	—
251	C₂²	C₄⋊Q₈×C₅₃	1	2	yes	—
252	C₂²	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	4	4	no	—
258	C₂	C₅₃×#380	1	1	yes	characteristic
259	C₂	C₅₃×#386	1	1	yes	characteristic
260	C₂	C₅₃×#387	1	2	yes	—
261	C₂	C₅₃×#389	1	2	yes	—
262	C₂	C₅₃×#391	1	1	yes	characteristic
263	C₂	C₅₃×#394	1	1	yes	characteristic
264	C₂	C₅₃×#395	1	1	yes	characteristic
265	C₂	C₅₃×#396	1	2	yes	—
266	C₂	C₅₃×#398	1	1	yes	characteristic
267	C₂	C₅₃×#398	1	2	yes	—
268	C₂	C₅₃×#400	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
15. C₂×C₄⋊C₄×C₅₃	C₄	semidirect
94. C₂×C₄×C₅₃	C₄⋊C₄	semidirect
123. #1441	C₅₃	direct (N × H)
140. C₅₃	#1441	direct (N × H)
150. C₂×C₄⋊C₄	C₄×C₅₃	semidirect
229. C₂×C₄	C₄⋊C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₄×C₅₃	2
C₂×#95×C₅₃	2
#1441	53

Subgroups — C53×#1441 #20270378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1441 #20270378