Subgroups — C₅₃×#3368 #20272305

Order 6784.

382 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#550	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#490	C₂	1	2	yes	maximal, complemented
4	C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#478	C₂	1	2	yes	maximal, complemented
6	C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#460	C₂	1	2	yes	maximal, complemented
8	C₅₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#440	C₂	1	2	yes	maximal, complemented
11	C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₅₃	—	2	1	no	—
13	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₅₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₅₃	—	2	1	no	—
16	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	#127×C₅₃	C₂²	1	1	yes	characteristic
18	#127×C₅₃	C₂²	1	2	yes	—
19	#125×C₅₃	—	2	2	no	—
20	#125×C₅₃	C₂²	1	2	yes	—
21	#121×C₅₃	—	2	1	no	—
22	#121×C₅₃	—	2	2	no	—
23	C₄×Q₈×C₅₃	—	2	1	no	—
24	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
26	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₅₃	C₂²	1	2	yes	—
29	#107×C₅₃	C₂²	1	2	yes	—
30	#106×C₅₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
33	#103×C₅₃	C₂²	1	1	yes	characteristic
34	#102×C₅₃	C₂²	1	2	yes	—
35	#102×C₅₃	C₂²	1	2	yes	—
36	#102×C₅₃	C₂²	1	1	yes	characteristic
37	#102×C₅₃	C₂²	1	2	yes	—
38	#102×C₅₃	C₂²	1	2	yes	—
39	#97×C₅₃	C₂²	1	2	yes	—
40	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	#41×C₅₃	—	4	4	no	—
42	C₂×Q₈×C₅₃	—	2	1	no	—
43	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₅₃	—	2	2	no	—
46	C₂×D₄×C₅₃	—	2	1	no	—
47	C₂×D₄×C₅₃	—	2	2	no	—
48	C₂×D₄×C₅₃	—	2	1	no	—
49	C₂×D₄×C₅₃	C₂³	1	2	yes	—
50	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	SD₁₆×C₅₃	—	2	2	no	—
54	SD₁₆×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂³	1	2	yes	—
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	1	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
65	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₅₃	—	2	2	no	—
69	C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₄²×C₅₃	C₂³	1	2	yes	—
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	Q₈×C₅₃	—	2	2	no	—
77	Q₈×C₅₃	—	2	1	no	—
78	Q₈×C₅₃	—	4	2	no	—
79	Q₈×C₅₃	#41	1	2	yes	complemented
80	D₄×C₅₃	—	2	2	no	—
81	D₄×C₅₃	—	2	1	no	—
82	D₄×C₅₃	—	2	2	no	—
83	D₄×C₅₃	—	2	1	no	—
84	D₄×C₅₃	—	4	4	no	—
85	D₄×C₅₃	—	4	2	no	—
86	D₄×C₅₃	—	8	2	no	—
87	D₄×C₅₃	—	8	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	—	4	1	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	—	4	1	no	—
106	C₂²×C₅₃	—	4	2	no	—
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	4	1	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	—	4	1	no	—
116	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
117	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
118	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
119	#3368	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₅₃	—	8	2	no	—
121	C₂×C₅₃	—	8	1	no	—
122	C₂×C₅₃	#548	1	1	yes	characteristic
123	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₅₃	#577	1	1	yes	characteristic
125	#557	C₂×C₅₃	1	1	yes	characteristic, complemented
126	#550	C₂×C₅₃	1	1	yes	characteristic
127	#490	C₂×C₅₃	1	2	yes	complemented
128	#486	C₂×C₅₃	1	1	yes	characteristic, complemented
129	#478	C₂×C₅₃	1	2	yes	complemented
130	#474	C₂×C₅₃	1	1	yes	characteristic, complemented
131	#460	C₂×C₅₃	1	2	yes	complemented
132	#460	C₂×C₅₃	1	1	yes	characteristic, complemented
133	#446	C₂×C₅₃	1	1	yes	characteristic, complemented
134	#440	C₂×C₅₃	1	2	yes	complemented
135	C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
136	C₅₃	#3368	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#41	—	2	1	no	—
138	C₂×SD₁₆	C₂²×C₅₃	1	1	yes	characteristic
139	C₂×SD₁₆	C₂²×C₅₃	1	2	yes	—
140	C₄⋊Q₈	—	2	1	no	—
141	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
142	#127	C₂²×C₅₃	1	1	yes	characteristic
143	#127	C₂²×C₅₃	1	2	yes	—
144	#125	—	2	2	no	—
145	#125	C₂²×C₅₃	1	2	yes	—
146	#121	—	2	1	no	—
147	#121	—	2	2	no	—
148	C₄×Q₈	—	2	1	no	—
149	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
150	C₄×Q₈	C₂²×C₅₃	1	2	yes	—
151	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
152	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
153	C₄×D₄	C₂²×C₅₃	1	2	yes	—
154	#107	C₂²×C₅₃	1	2	yes	—
155	#106	C₂²×C₅₃	1	1	yes	characteristic
156	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
157	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
158	#103	C₂²×C₅₃	1	1	yes	characteristic
159	#102	C₂²×C₅₃	1	1	yes	characteristic
160	#102	C₂²×C₅₃	1	2	yes	—
161	#102	C₂²×C₅₃	1	2	yes	—
162	#102	C₂²×C₅₃	1	2	yes	—
163	#102	C₂²×C₅₃	1	2	yes	—
164	#97	C₂²×C₅₃	1	2	yes	—
165	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
166	#41	—	4	4	no	—
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	D₄×C₅₃	1	1	yes	characteristic
169	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
175	C₂×D₄	C₂³×C₅₃	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	SD₁₆	—	2	2	no	—
179	SD₁₆	—	2	4	no	—
180	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
181	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₅₃	1	2	yes	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
189	C₄⋊C₄	D₄×C₅₃	1	2	yes	—
190	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
192	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	1	no	—
197	C₄²	C₂³×C₅₃	1	1	yes	characteristic
198	C₄²	C₂³×C₅₃	1	2	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	4	2	no	—
204	Q₈	#41×C₅₃	1	2	yes	complemented
205	D₄	—	2	1	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	2	no	—
211	D₄	—	8	2	no	—
212	D₄	—	8	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
220	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	1	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
242	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
243	C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
244	C₂	—	8	2	no	—
245	C₂	—	8	1	no	—
246	C₂	C₅₃×#548	1	1	yes	characteristic
247	C₂	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
248	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	C₅₃×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#557	C₂	semidirect
3. C₅₃×#490	C₂	semidirect
4. C₅₃×#486	C₂	semidirect
5. C₅₃×#478	C₂	semidirect
6. C₅₃×#474	C₂	semidirect
7. C₅₃×#460	C₂	semidirect
8. C₅₃×#460	C₂	semidirect
9. C₅₃×#446	C₂	semidirect
10. C₅₃×#440	C₂	semidirect
11. C₄×SD₁₆×C₅₃	C₂	semidirect
79. Q₈×C₅₃	#41	semidirect
119. #3368	C₅₃	direct (N × H)
125. #557	C₂×C₅₃	semidirect
127. #490	C₂×C₅₃	semidirect
128. #486	C₂×C₅₃	semidirect
129. #478	C₂×C₅₃	semidirect
130. #474	C₂×C₅₃	semidirect
131. #460	C₂×C₅₃	semidirect
132. #460	C₂×C₅₃	semidirect
133. #446	C₂×C₅₃	semidirect
134. #440	C₂×C₅₃	semidirect
135. C₄×SD₁₆	C₂×C₅₃	semidirect
136. C₅₃	#3368	direct (N × H)
204. Q₈	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#557	2
C₅₃×#550	2
C₅₃×#490	2
C₅₃×#486	2
C₅₃×#478	2
C₅₃×#474	2
C₅₃×#460	2
C₅₃×#446	2
C₅₃×#440	2
C₄×SD₁₆×C₅₃	2
#3368	53

Subgroups — C53×#3368 #20272305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3368 #20272305