Subgroups — C₅₃×#3076 #20272013

Order 6784.

388 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	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#479	C₂	1	2	yes	maximal, complemented
4	C₅₃×#478	C₂	1	2	yes	maximal, complemented
5	C₅₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#427	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#416	C₂	1	2	yes	maximal
12	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	#125×C₅₃	C₂²	1	2	yes	—
17	C₂²⋊Q₈×C₅₃	—	2	1	no	—
18	C₂²⋊Q₈×C₅₃	—	2	1	no	—
19	C₄×Q₈×C₅₃	—	2	1	no	—
20	C₄×Q₈×C₅₃	—	2	1	no	—
21	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₃	—	2	1	no	—
24	C₄×D₄×C₅₃	—	2	1	no	—
25	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
27	#117×C₅₃	—	2	1	no	—
28	#117×C₅₃	C₂²	1	1	yes	characteristic
29	#117×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₅₃	—	2	1	no	—
31	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	#107×C₅₃	C₂²	1	2	yes	—
33	#107×C₅₃	C₂²	1	2	yes	—
34	#106×C₅₃	C₂²	1	2	yes	—
35	#106×C₅₃	C₂²	1	2	yes	—
36	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
39	#103×C₅₃	C₂²	1	2	yes	—
40	#103×C₅₃	C₂²	1	2	yes	—
41	#102×C₅₃	C₂²	1	2	yes	—
42	#102×C₅₃	C₂²	1	2	yes	—
43	#41×C₅₃	—	2	2	no	—
44	#41×C₅₃	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	1	no	—
47	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₅₃	—	2	1	no	—
49	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	C₂³	1	2	yes	—
58	C₂×C₈×C₅₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	1	no	—
64	C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₂²⋊C₄×C₅₃	—	2	1	no	—
72	C₂²⋊C₄×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	D₄	1	1	yes	characteristic
78	C₄²×C₅₃	D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	2	1	no	—
84	Q₈×C₅₃	—	4	2	no	—
85	Q₈×C₅₃	—	4	2	no	—
86	Q₈×C₅₃	C₂×Q₈	1	2	yes	—
87	D₄×C₅₃	—	2	4	no	—
88	D₄×C₅₃	—	2	1	no	—
89	D₄×C₅₃	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	4	1	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₅₃	#41	1	2	yes	—
107	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	4	1	no	—
114	C₂²×C₅₃	—	4	1	no	—
115	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	4	1	no	—
122	C₄×C₅₃	—	4	2	no	—
123	C₄×C₅₃	—	4	1	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
126	C₄×C₅₃	#135	1	2	yes	—
127	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
128	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	#3076	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	—	4	2	no	—
132	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
134	C₂×C₅₃	#575	1	1	yes	characteristic
135	#533	C₂×C₅₃	1	1	yes	characteristic, complemented
136	C₄×#41	C₂×C₅₃	1	1	yes	characteristic
137	#479	C₂×C₅₃	1	2	yes	complemented
138	#478	C₂×C₅₃	1	2	yes	complemented
139	#477	C₂×C₅₃	1	1	yes	characteristic, complemented
140	#476	C₂×C₅₃	1	1	yes	characteristic, complemented
141	#475	C₂×C₅₃	1	1	yes	characteristic, complemented
142	#474	C₂×C₅₃	1	1	yes	characteristic, complemented
143	#427	C₂×C₅₃	1	2	yes	complemented
144	C₂×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
145	#416	C₂×C₅₃	1	2	yes	—
146	C₅₃	#3076	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
149	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
150	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
151	#125	C₂²×C₅₃	1	2	yes	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
157	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
161	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
162	#117	—	2	1	no	—
163	#117	C₂²×C₅₃	1	1	yes	characteristic
164	#117	C₂²×C₅₃	1	2	yes	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
167	#107	C₂²×C₅₃	1	2	yes	—
168	#107	C₂²×C₅₃	1	2	yes	—
169	#106	C₂²×C₅₃	1	2	yes	—
170	#106	C₂²×C₅₃	1	2	yes	—
171	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
172	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
173	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
174	#103	C₂²×C₅₃	1	2	yes	—
175	#103	C₂²×C₅₃	1	2	yes	—
176	#102	C₂²×C₅₃	1	2	yes	—
177	#102	C₂²×C₅₃	1	2	yes	—
178	#41	—	2	2	no	—
179	#41	Q₈×C₅₃	1	4	yes	complemented
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	C₂³×C₅₃	1	2	yes	—
193	C₂×C₈	C₂³×C₅₃	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
201	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
202	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	D₄×C₅₃	1	1	yes	characteristic
213	C₄²	D₄×C₅₃	1	1	yes	characteristic
214	C₄²	C₂³×C₅₃	1	1	yes	characteristic
215	C₄²	C₂³×C₅₃	1	1	yes	characteristic
216	C₂³	—	2	1	no	—
217	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
218	Q₈	—	2	1	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	Q₈	C₂×Q₈×C₅₃	1	2	yes	—
222	D₄	—	2	4	no	—
223	D₄	—	2	1	no	—
224	D₄	C₂×Q₈×C₅₃	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
240	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
241	C₂×C₄	#41×C₅₃	1	2	yes	—
242	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	C₂²⋊Q₈×C₅₃	1	2	yes	—
251	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
261	C₄	#135×C₅₃	1	2	yes	—
262	C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
263	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
267	C₂	C₂×#135×C₅₃	1	1	yes	characteristic
268	C₂	C₅₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×D₄
prime radical	C₂×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#533	C₂	semidirect
3. C₅₃×#479	C₂	semidirect
4. C₅₃×#478	C₂	semidirect
5. C₅₃×#477	C₂	semidirect
6. C₅₃×#476	C₂	semidirect
7. C₅₃×#475	C₂	semidirect
8. C₅₃×#474	C₂	semidirect
9. C₅₃×#427	C₂	semidirect
10. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
44. #41×C₅₃	Q₈	semidirect
129. #3076	C₅₃	direct (N × H)
135. #533	C₂×C₅₃	semidirect
137. #479	C₂×C₅₃	semidirect
138. #478	C₂×C₅₃	semidirect
139. #477	C₂×C₅₃	semidirect
140. #476	C₂×C₅₃	semidirect
141. #475	C₂×C₅₃	semidirect
142. #474	C₂×C₅₃	semidirect
143. #427	C₂×C₅₃	semidirect
144. C₂×C₄⋊C₈	C₂×C₅₃	semidirect
146. C₅₃	#3076	direct (N × H)
179. #41	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#533	2
C₄×#41×C₅₃	2
C₅₃×#479	2
C₅₃×#478	2
C₅₃×#477	2
C₅₃×#476	2
C₅₃×#475	2
C₅₃×#474	2
C₅₃×#427	2
C₂×C₄⋊C₈×C₅₃	2
C₅₃×#416	2
#3076	53

Subgroups — C53×#3076 #20272013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3076 #20272013