Subgroups — C₅₃×#10538 #27218050

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1988	C₂	1	2	yes	maximal
3	C₅₃×#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#332	C₂	1	2	yes	maximal
5	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₃	—	2	1	no	—
7	C₂²×Q₁₆×C₅₃	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#494	—	2	4	no	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#340	—	2	2	no	—
16	C₅₃×#340	—	2	2	no	—
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₅₃	—	2	2	no	—
20	C₂×Q₁₆×C₅₃	—	2	4	no	—
21	C₂×Q₁₆×C₅₃	—	2	4	no	—
22	C₂×Q₁₆×C₅₃	—	4	2	no	—
23	C₂×Q₁₆×C₅₃	—	4	2	no	—
24	C₂×Q₁₆×C₅₃	—	4	2	no	—
25	C₂×Q₁₆×C₅₃	—	4	2	no	—
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	D₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	4	2	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	#95×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	4	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	Q₁₆×C₅₃	—	4	8	no	—
59	Q₁₆×C₅₃	—	4	8	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	4	1	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	4	1	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	4	1	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	D₈	1	2	yes	—
71	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
72	C₄⋊C₄×C₅₃	—	2	1	no	—
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	—	2	4	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	4	2	no	—
77	C₄⋊C₄×C₅₃	—	4	2	no	—
78	C₄⋊C₄×C₅₃	Q₁₆	1	2	yes	complemented
79	C₄²×C₅₃	—	4	1	no	—
80	C₄²×C₅₃	—	4	1	no	—
81	C₄²×C₅₃	C₂×D₄	1	2	yes	—
82	C₄²×C₅₃	#41	1	2	yes	—
83	C₂³×C₅₃	#121	1	1	yes	characteristic
84	Q₈×C₅₃	—	4	4	no	—
85	Q₈×C₅₃	—	4	4	no	—
86	Q₈×C₅₃	—	8	2	no	—
87	Q₈×C₅₃	—	8	2	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	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	1	no	—
96	C₂×C₄×C₅₃	—	4	1	no	—
97	C₂×C₄×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	—	8	2	no	—
99	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#121	1	2	yes	—
101	C₂×C₄×C₅₃	#123	1	2	yes	—
102	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
107	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#137	1	2	yes	—
110	C₈×C₅₃	—	2	2	no	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	—	8	1	no	—
113	C₈×C₅₃	—	8	2	no	—
114	#10538	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₅₃	#394	1	1	yes	characteristic
116	C₂²×C₅₃	#459	1	1	yes	characteristic
117	C₂²×C₅₃	#460	1	1	yes	characteristic
118	C₂²×C₅₃	#467	1	1	yes	characteristic
119	C₂²×C₅₃	#470	1	1	yes	characteristic
120	C₂²×C₅₃	#473	1	1	yes	characteristic
121	C₂²×C₅₃	#473	1	1	yes	characteristic
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	4	2	no	—
124	C₄×C₅₃	—	8	1	no	—
125	C₄×C₅₃	—	8	4	no	—
126	C₄×C₅₃	#447	1	1	yes	characteristic
127	C₄×C₅₃	#450	1	1	yes	characteristic
128	C₄×C₅₃	#462	1	2	yes	—
129	C₄×C₅₃	#464	1	2	yes	—
130	C₄×C₅₃	#486	1	1	yes	characteristic
131	C₄×C₅₃	#490	1	1	yes	characteristic
132	C₂×#494	C₂×C₅₃	1	1	yes	characteristic
133	#1988	C₂×C₅₃	1	2	yes	—
134	#1939	C₂×C₅₃	1	1	yes	characteristic
135	C₂×#332	C₂×C₅₃	1	2	yes	—
136	#1277	C₂×C₅₃	1	1	yes	characteristic
137	C₂×C₅₃	#1669	1	1	yes	characteristic
138	C₂×C₅₃	#1670	1	1	yes	characteristic
139	C₂×C₅₃	#1689	1	1	yes	characteristic
140	C₂×C₅₃	#1690	1	1	yes	characteristic
141	C₂×C₅₃	#2041	1	1	yes	characteristic
142	C₂×C₅₃	#2047	1	1	yes	characteristic
143	C₂×C₅₃	#2056	1	1	yes	characteristic
144	C₂²×Q₁₆	—	2	1	no	—
145	C₂²×Q₁₆	—	2	1	no	—
146	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
147	#494	—	2	4	no	—
148	C₂×C₄⋊C₈	—	2	1	no	—
149	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
152	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
153	#340	—	2	2	no	—
154	#340	—	2	2	no	—
155	#336	—	2	1	no	—
156	#332	—	2	4	no	—
157	C₅₃	#10538	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×Q₈	—	2	2	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	4	2	no	—
162	C₂×Q₁₆	—	4	2	no	—
163	C₂×Q₁₆	—	4	2	no	—
164	C₂×Q₁₆	—	4	2	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	D₄×C₅₃	1	4	yes	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	4	2	no	—
183	C₄×C₈	—	2	2	no	—
184	#95	—	2	1	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
197	Q₁₆	—	4	8	no	—
198	Q₁₆	—	4	8	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	D₈×C₅₃	1	2	yes	—
210	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	Q₁₆×C₅₃	1	2	yes	complemented
218	C₄²	—	4	1	no	—
219	C₄²	—	4	1	no	—
220	C₄²	C₂×D₄×C₅₃	1	2	yes	—
221	C₄²	#41×C₅₃	1	2	yes	—
222	C₂³	#121×C₅₃	1	1	yes	characteristic
223	Q₈	—	4	4	no	—
224	Q₈	—	4	4	no	—
225	Q₈	—	8	2	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
239	C₂×C₄	#121×C₅₃	1	2	yes	—
240	C₂×C₄	#123×C₅₃	1	2	yes	—
241	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
245	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
246	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
247	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
248	C₂×C₄	#137×C₅₃	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	8	1	no	—
252	C₈	—	8	2	no	—
253	C₂²	C₅₃×#394	1	1	yes	characteristic
254	C₂²	C₅₃×#459	1	1	yes	characteristic
255	C₂²	C₅₃×#460	1	1	yes	characteristic
256	C₂²	C₅₃×#467	1	1	yes	characteristic
257	C₂²	C₅₃×#470	1	1	yes	characteristic
258	C₂²	C₅₃×#473	1	1	yes	characteristic
259	C₂²	C₅₃×#473	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	8	1	no	—
263	C₄	—	8	4	no	—
264	C₄	C₅₃×#447	1	1	yes	characteristic
265	C₄	C₅₃×#450	1	1	yes	characteristic
266	C₄	C₅₃×#462	1	2	yes	—
267	C₄	C₅₃×#464	1	2	yes	—
268	C₄	C₅₃×#486	1	1	yes	characteristic
269	C₄	C₅₃×#490	1	1	yes	characteristic
270	C₂	C₅₃×#1669	1	1	yes	characteristic
271	C₂	C₅₃×#1670	1	1	yes	characteristic
272	C₂	C₅₃×#1689	1	1	yes	characteristic
273	C₂	C₅₃×#1690	1	1	yes	characteristic
274	C₂	C₅₃×#2041	1	1	yes	characteristic
275	C₂	C₅₃×#2047	1	1	yes	characteristic
276	C₂	C₅₃×#2056	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
78. C₄⋊C₄×C₅₃	Q₁₆	semidirect
114. #10538	C₅₃	direct (N × H)
157. C₅₃	#10538	direct (N × H)
217. C₄⋊C₄	Q₁₆×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×#494	2
C₅₃×#1988	2
C₅₃×#1939	2
C₂×C₅₃×#332	2
C₅₃×#1277	2
#10538	53

Subgroups — C53×#10538 #27218050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10538 #27218050