Subgroups — C₅₃×#21727 #27229239

Order 13568.

854 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	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1990	C₂	1	2	yes	maximal
4	C₅₃×#1944	C₂	1	4	yes	maximal
5	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1776	C₂	1	4	yes	maximal
7	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
10	C₅₃×#498	C₄	1	8	yes	complemented
11	C₂×#107×C₅₃	C₂²	1	4	yes	—
12	C₂×#106×C₅₃	C₂²	1	4	yes	—
13	C₂×#97×C₅₃	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#391	—	2	2	no	—
18	C₅₃×#389	—	2	2	no	—
19	C₅₃×#389	C₂²	1	4	yes	—
20	C₅₃×#384	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#340	C₂²	1	8	yes	—
25	C₂²×Q₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₄⋊Q₈×C₅₃	—	2	4	no	—
29	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
30	C₄×Q₈×C₅₃	—	4	8	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₅₃	—	2	4	no	—
38	C₂×C₄²×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#107×C₅₃	—	2	8	no	—
44	#106×C₅₃	C₂×C₄	1	16	yes	—
45	#97×C₅₃	—	2	4	no	—
46	#97×C₅₃	Q₈	1	8	yes	complemented
47	C₄×C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
49	#95×C₅₃	—	2	4	no	—
50	#95×C₅₃	—	2	4	no	—
51	C₂×Q₈×C₅₃	—	2	8	no	—
52	C₂×Q₈×C₅₃	—	4	4	no	—
53	C₂×Q₈×C₅₃	—	4	4	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₅₃	#41	1	4	yes	—
67	C₄⋊C₄×C₅₃	—	2	8	no	—
68	C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₄⋊C₄×C₅₃	—	2	8	no	—
70	C₄⋊C₄×C₅₃	—	4	8	no	—
71	C₄⋊C₄×C₅₃	—	4	4	no	—
72	C₄⋊C₄×C₅₃	—	4	4	no	—
73	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
74	C₄²×C₅₃	—	2	4	no	—
75	C₄²×C₅₃	—	2	4	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	—	4	8	no	—
78	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
80	C₄²×C₅₃	#41	1	2	yes	—
81	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₅₃	—	4	16	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	8	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×C₄×C₅₃	#137	1	4	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
103	C₈×C₅₃	—	2	4	no	—
104	C₈×C₅₃	C₄×Q₈	1	8	yes	complemented
105	#21727	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₅₃	#532	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
110	C₂²×C₅₃	C₂×#137	1	2	yes	—
111	C₄×C₅₃	—	2	4	no	—
112	C₄×C₅₃	—	4	1	no	—
113	C₄×C₅₃	—	4	1	no	—
114	C₄×C₅₃	—	4	8	no	—
115	C₄×C₅₃	—	8	4	no	—
116	C₄×C₅₃	#440	1	8	yes	—
117	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
118	C₄×C₅₃	#533	1	2	yes	—
119	C₂×#498	C₂×C₅₃	1	1	yes	characteristic
120	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	—
121	#1990	C₂×C₅₃	1	2	yes	—
122	#1944	C₂×C₅₃	1	4	yes	—
123	#1845	C₂×C₅₃	1	1	yes	characteristic
124	#1776	C₂×C₅₃	1	4	yes	—
125	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
126	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#440	1	2	yes	—
128	C₂×C₅₃	#3165	1	2	yes	—
129	C₂×C₅₃	#3184	1	1	yes	characteristic
130	C₂×C₅₃	#3185	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
132	C₂×C₄×Q₈	—	2	2	no	—
133	#498	C₄×C₅₃	1	8	yes	complemented
134	C₂×#107	C₂²×C₅₃	1	4	yes	—
135	C₂×#106	C₂²×C₅₃	1	4	yes	—
136	C₂×#97	C₂²×C₅₃	1	2	yes	—
137	C₂×#97	C₂²×C₅₃	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
140	#391	—	2	2	no	—
141	#389	—	2	2	no	—
142	#389	C₂²×C₅₃	1	4	yes	—
143	#384	C₂²×C₅₃	1	2	yes	—
144	C₄×C₄⋊C₄	—	2	2	no	—
145	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
146	C₄³	C₂²×C₅₃	1	1	yes	characteristic
147	#340	C₂²×C₅₃	1	8	yes	—
148	C₅₃	#21727	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
151	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
154	C₄×Q₈	—	4	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	D₄×C₅₃	1	2	yes	—
163	C₂×C₄²	D₄×C₅₃	1	2	yes	—
164	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
167	#107	—	2	8	no	—
168	#106	C₂×C₄×C₅₃	1	16	yes	—
169	#97	—	2	4	no	—
170	#97	Q₈×C₅₃	1	8	yes	complemented
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
173	#95	—	2	4	no	—
174	#95	—	2	4	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
184	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
188	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
189	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
190	C₂×C₈	#41×C₅₃	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	8	no	—
202	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
203	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
204	C₄²	#41×C₅₃	1	2	yes	—
205	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
206	Q₈	—	4	16	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
220	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
221	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
222	C₂×C₄	#137×C₅₃	1	4	yes	—
223	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
225	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
226	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
227	C₈	—	2	4	no	—
228	C₈	C₄×Q₈×C₅₃	1	8	yes	complemented
229	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
230	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
231	C₂²	C₅₃×#532	1	1	yes	characteristic
232	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
233	C₂²	C₂×#137×C₅₃	1	2	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	4	8	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	1	no	—
238	C₄	—	8	4	no	—
239	C₄	C₅₃×#440	1	8	yes	—
240	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
241	C₄	C₅₃×#533	1	2	yes	—
242	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
243	C₂	C₂×C₅₃×#440	1	2	yes	—
244	C₂	C₅₃×#3165	1	2	yes	—
245	C₂	C₅₃×#3184	1	1	yes	characteristic
246	C₂	C₅₃×#3185	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅₃×#498	C₄	semidirect
46. #97×C₅₃	Q₈	semidirect
104. C₈×C₅₃	C₄×Q₈	semidirect
105. #21727	C₅₃	direct (N × H)
133. #498	C₄×C₅₃	semidirect
148. C₅₃	#21727	direct (N × H)
170. #97	Q₈×C₅₃	semidirect
228. C₈	C₄×Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#498	2
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1990	2
C₅₃×#1944	2
C₅₃×#1845	2
C₅₃×#1776	2
C₄×#97×C₅₃	2
#21727	53

Subgroups — C53×#21727 #27229239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21727 #27229239