Subgroups — C₅₃×#10867 #27218379

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1742	C₂	1	2	yes	maximal
4	C₂×C₅₃×#338	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₅₃	C₂²	1	4	yes	—
6	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#108×C₅₃	—	2	4	no	—
10	C₂×#108×C₅₃	—	2	4	no	—
11	C₂×#108×C₅₃	C₄	1	8	yes	complemented
12	C₅₃×#405	—	2	4	no	—
13	C₂×#97×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
15	C₅₃×#343	—	2	4	no	—
16	C₅₃×#338	—	2	4	no	—
17	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	—	2	2	no	—
19	C₂×M₁₆×C₅₃	—	2	4	no	—
20	C₂×M₁₆×C₅₃	—	2	4	no	—
21	C₂×M₁₆×C₅₃	—	2	4	no	—
22	C₂×M₁₆×C₅₃	—	2	4	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	4	no	—
25	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
26	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
27	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
28	C₂²×C₈×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	D₄	1	4	yes	—
34	C₂²×C₈×C₅₃	D₄	1	2	yes	—
35	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
36	#117×C₅₃	—	2	2	no	—
37	#117×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	#108×C₅₃	—	2	16	no	—
42	#97×C₅₃	—	2	4	no	—
43	C₄×C₈×C₅₃	—	2	4	no	—
44	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	#41	1	2	yes	—
55	C₂²×C₄×C₅₃	#41	1	2	yes	—
56	M₁₆×C₅₃	—	2	4	no	—
57	M₁₆×C₅₃	—	2	4	no	—
58	M₁₆×C₅₃	—	2	8	no	—
59	M₁₆×C₅₃	—	2	8	no	—
60	M₁₆×C₅₃	—	4	4	no	—
61	M₁₆×C₅₃	—	4	4	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	8	no	—
65	C₂×C₈×C₅₃	—	4	4	no	—
66	C₂×C₈×C₅₃	—	4	4	no	—
67	C₂×C₈×C₅₃	C₄²	1	2	yes	—
68	C₂×C₈×C₅₃	C₄²	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	8	yes	—
72	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
73	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₂²⋊C₄×C₅₃	—	2	4	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	—	4	2	no	—
81	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
83	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
84	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₅₃	#121	1	2	yes	—
100	C₂×C₄×C₅₃	#121	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₅₃	#124	1	2	yes	—
103	C₂×C₄×C₅₃	#125	1	2	yes	—
104	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	—	4	8	no	—
108	C₈×C₅₃	#95	1	4	yes	—
109	C₈×C₅₃	#95	1	4	yes	—
110	#10867	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	—	4	1	no	—
115	C₂²×C₅₃	—	4	1	no	—
116	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₅₃	#380	1	1	yes	characteristic
118	C₂²×C₅₃	#384	1	1	yes	characteristic
119	C₂²×C₅₃	#389	1	2	yes	—
120	C₂²×C₅₃	#391	1	2	yes	—
121	C₄×C₅₃	—	4	4	no	—
122	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
123	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₅₃	#380	1	1	yes	characteristic
125	C₄×C₅₃	#382	1	1	yes	characteristic
126	C₄×C₅₃	#389	1	2	yes	—
127	C₄×C₅₃	#390	1	2	yes	—
128	C₂²×#108	C₂×C₅₃	1	2	yes	—
129	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
130	#1742	C₂×C₅₃	1	2	yes	—
131	C₂×#338	C₂×C₅₃	1	2	yes	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	2	no	—
135	C₂×C₅₃	—	2	2	no	—
136	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
137	C₂×C₅₃	#1813	1	2	yes	—
138	C₂²×M₁₆	C₂²×C₅₃	1	4	yes	—
139	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
140	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×#108	—	2	4	no	—
143	C₂×#108	—	2	4	no	—
144	C₂×#108	C₄×C₅₃	1	8	yes	complemented
145	#405	—	2	4	no	—
146	C₂×#97	C₄×C₅₃	1	2	yes	—
147	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
148	#343	—	2	4	no	—
149	#338	—	2	4	no	—
150	C₅₃	#10867	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
160	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
161	C₂×M₁₆	Q₈×C₅₃	1	2	yes	—
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
164	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
166	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
167	C₂²×C₈	D₄×C₅₃	1	4	yes	—
168	C₂²×C₈	D₄×C₅₃	1	2	yes	—
169	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
170	#117	—	2	2	no	—
171	#117	—	2	2	no	—
172	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
173	C₂×C₂²⋊C₄	Q₈×C₅₃	1	2	yes	—
174	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
175	#108	—	2	16	no	—
176	#97	—	2	4	no	—
177	C₄×C₈	—	2	4	no	—
178	C₂⁴	C₂×Q₈×C₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
180	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	#41×C₅₃	1	2	yes	—
189	C₂²×C₄	#41×C₅₃	1	2	yes	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	2	8	no	—
193	M₁₆	—	2	8	no	—
194	M₁₆	—	4	4	no	—
195	M₁₆	—	4	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	4	4	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	C₄²×C₅₃	1	2	yes	—
202	C₂×C₈	C₄²×C₅₃	1	2	yes	—
203	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
204	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
205	C₂×C₈	C₂²⋊C₄×C₅₃	1	8	yes	—
206	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
207	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
208	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
209	C₂×C₈	C₄⋊C₄×C₅₃	1	8	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
216	C₂³	C₄×Q₈×C₅₃	1	2	yes	—
217	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
218	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
219	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
225	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
227	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
229	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
230	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
231	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
232	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
233	C₂×C₄	#121×C₅₃	1	2	yes	—
234	C₂×C₄	#121×C₅₃	1	2	yes	—
235	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
236	C₂×C₄	#124×C₅₃	1	2	yes	—
237	C₂×C₄	#125×C₅₃	1	2	yes	—
238	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
239	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	4	8	no	—
242	C₈	#95×C₅₃	1	4	yes	—
243	C₈	#95×C₅₃	1	4	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
250	C₂²	C₅₃×#380	1	1	yes	characteristic
251	C₂²	C₅₃×#384	1	1	yes	characteristic
252	C₂²	C₅₃×#389	1	2	yes	—
253	C₂²	C₅₃×#391	1	2	yes	—
254	C₄	—	4	4	no	—
255	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
256	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
257	C₄	C₅₃×#380	1	1	yes	characteristic
258	C₄	C₅₃×#382	1	1	yes	characteristic
259	C₄	C₅₃×#389	1	2	yes	—
260	C₄	C₅₃×#390	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	C₅₃×C₄⋊#95	1	1	yes	characteristic
266	C₂	C₅₃×#1813	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×#108×C₅₃	C₄	semidirect
110. #10867	C₅₃	direct (N × H)
144. C₂×#108	C₄×C₅₃	semidirect
150. C₅₃	#10867	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#108×C₅₃	2
C₂×C₅₃×#405	2
C₅₃×#1742	2
C₂×C₅₃×#338	2
#10867	53

Subgroups — C53×#10867 #27218379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10867 #27218379