Subgroups — C₅₃×#3084 #20272021

Order 6784.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#520	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×#476	C₂	1	2	yes	maximal, complemented
6	C₅₃×#475	C₂	1	2	yes	maximal, complemented
7	C₅₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#417	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#416	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₅₃	—	2	1	no	—
16	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
17	#125×C₅₃	—	2	1	no	—
18	#125×C₅₃	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₅₃	—	2	2	no	—
20	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
21	C₄×D₄×C₅₃	—	2	2	no	—
22	C₄×D₄×C₅₃	C₂²	1	2	yes	—
23	#117×C₅₃	—	2	1	no	—
24	#117×C₅₃	C₂²	1	1	yes	characteristic
25	#117×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	#107×C₅₃	C₂²	1	2	yes	—
30	#106×C₅₃	C₂²	1	2	yes	—
31	#106×C₅₃	C₂²	1	2	yes	—
32	#106×C₅₃	C₂²	1	2	yes	—
33	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
34	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
35	#103×C₅₃	C₂²	1	1	yes	characteristic
36	#103×C₅₃	C₂²	1	1	yes	characteristic
37	#103×C₅₃	C₂²	1	2	yes	—
38	#102×C₅₃	C₂²	1	1	yes	characteristic
39	#102×C₅₃	C₂²	1	1	yes	characteristic
40	#102×C₅₃	C₂²	1	2	yes	—
41	#41×C₅₃	—	2	2	no	—
42	#41×C₅₃	Q₈	1	4	yes	complemented
43	C₂×Q₈×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₅₃	—	2	1	no	—
46	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	M₁₆×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
69	C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₂²⋊C₄×C₅₃	—	2	2	no	—
71	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	C₂³	1	2	yes	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₅₃	—	2	1	no	—
78	Q₈×C₅₃	—	8	2	no	—
79	Q₈×C₅₃	C₂×Q₈	1	2	yes	—
80	D₄×C₅₃	—	2	4	no	—
81	D₄×C₅₃	—	2	1	no	—
82	D₄×C₅₃	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	8	2	no	—
90	C₂×C₄×C₅₃	—	8	1	no	—
91	C₂×C₄×C₅₃	—	8	1	no	—
92	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₅₃	#41	1	2	yes	—
97	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	4	1	no	—
105	C₂²×C₅₃	—	4	1	no	—
106	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	4	1	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	—	4	1	no	—
115	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
118	#3084	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₅₃	#577	1	1	yes	characteristic
123	C₂×C₅₃	#577	1	1	yes	characteristic
124	#544	C₂×C₅₃	1	1	yes	characteristic, complemented
125	#520	C₂×C₅₃	1	1	yes	characteristic
126	#479	C₂×C₅₃	1	2	yes	complemented
127	#478	C₂×C₅₃	1	2	yes	complemented
128	#476	C₂×C₅₃	1	2	yes	complemented
129	#475	C₂×C₅₃	1	2	yes	complemented
130	#428	C₂×C₅₃	1	1	yes	characteristic, complemented
131	C₂×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
132	#424	C₂×C₅₃	1	1	yes	characteristic, complemented
133	#417	C₂×C₅₃	1	1	yes	characteristic
134	#416	C₂×C₅₃	1	1	yes	characteristic
135	C₅₃	#3084	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
137	C₂×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
141	#125	—	2	1	no	—
142	#125	C₂²×C₅₃	1	2	yes	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	C₂²×C₅₃	1	2	yes	—
147	#117	—	2	1	no	—
148	#117	C₂²×C₅₃	1	1	yes	characteristic
149	#117	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
153	#107	C₂²×C₅₃	1	2	yes	—
154	#106	C₂²×C₅₃	1	2	yes	—
155	#106	C₂²×C₅₃	1	2	yes	—
156	#106	C₂²×C₅₃	1	2	yes	—
157	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
158	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
159	#103	C₂²×C₅₃	1	1	yes	characteristic
160	#103	C₂²×C₅₃	1	1	yes	characteristic
161	#103	C₂²×C₅₃	1	2	yes	—
162	#102	C₂²×C₅₃	1	1	yes	characteristic
163	#102	C₂²×C₅₃	1	1	yes	characteristic
164	#102	C₂²×C₅₃	1	2	yes	—
165	#41	—	2	2	no	—
166	#41	Q₈×C₅₃	1	4	yes	complemented
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
176	M₁₆	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
180	C₂×C₈	C₂³×C₅₃	1	2	yes	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
188	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
190	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
192	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂³×C₅₃	1	2	yes	—
199	C₂³	—	2	1	no	—
200	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	8	2	no	—
203	Q₈	C₂×Q₈×C₅₃	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Q₈×C₅₃	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
220	C₂×C₄	#41×C₅₃	1	2	yes	—
221	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₂²⋊Q₈×C₅₃	1	2	yes	—
231	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
240	C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
241	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
245	C₂	C₅₃×#577	1	1	yes	characteristic
246	C₂	C₅₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
prime radical	C₂×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ #117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#544	C₂	semidirect
3. C₅₃×#479	C₂	semidirect
4. C₅₃×#478	C₂	semidirect
5. C₅₃×#476	C₂	semidirect
6. C₅₃×#475	C₂	semidirect
7. C₅₃×#428	C₂	semidirect
8. C₂×#106×C₅₃	C₂	semidirect
9. C₅₃×#424	C₂	semidirect
42. #41×C₅₃	Q₈	semidirect
118. #3084	C₅₃	direct (N × H)
124. #544	C₂×C₅₃	semidirect
126. #479	C₂×C₅₃	semidirect
127. #478	C₂×C₅₃	semidirect
128. #476	C₂×C₅₃	semidirect
129. #475	C₂×C₅₃	semidirect
130. #428	C₂×C₅₃	semidirect
131. C₂×#106	C₂×C₅₃	semidirect
132. #424	C₂×C₅₃	semidirect
135. C₅₃	#3084	direct (N × H)
166. #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×#544	2
C₅₃×#520	2
C₅₃×#479	2
C₅₃×#478	2
C₅₃×#476	2
C₅₃×#475	2
C₅₃×#428	2
C₂×#106×C₅₃	2
C₅₃×#424	2
C₅₃×#417	2
C₅₃×#416	2
#3084	53

Subgroups — C53×#3084 #20272021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3084 #20272021