Subgroups — C₅₃×#8356 #27215868

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3502	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1801	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1550	C₂	1	2	yes	maximal
5	C₅₃×#1493	C₂	1	2	yes	maximal
6	Q₈²×C₅₃	—	2	2	no	—
7	C₅₃×#557	—	2	2	no	—
8	C₅₃×#557	—	2	4	no	—
9	C₅₃×#544	—	2	1	no	—
10	C₅₃×#544	C₄	1	4	yes	—
11	C₅₃×#533	—	2	2	no	—
12	C₅₃×#533	C₂²	1	2	yes	—
13	C₅₃×#519	—	2	1	no	—
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#423	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#101×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#343	—	2	1	no	—
20	C₅₃×#336	—	2	2	no	—
21	C₅₃×#333	—	2	2	no	—
22	C₅₃×#332	—	2	2	no	—
23	C₅₃×Q₈⋊C₈	—	2	4	no	—
24	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₄⋊Q₈×C₅₃	—	2	2	no	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	—	2	4	no	—
31	C₄⋊Q₈×C₅₃	—	2	8	no	—
32	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
33	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
34	#125×C₅₃	—	2	4	no	—
35	#125×C₅₃	—	2	4	no	—
36	#125×C₅₃	—	2	8	no	—
37	#125×C₅₃	—	2	2	no	—
38	C₂²⋊Q₈×C₅₃	—	2	4	no	—
39	C₂²⋊Q₈×C₅₃	—	2	2	no	—
40	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₅₃	—	2	4	no	—
42	C₄×Q₈×C₅₃	—	2	4	no	—
43	C₄×Q₈×C₅₃	—	2	4	no	—
44	C₄×Q₈×C₅₃	—	2	4	no	—
45	C₄×Q₈×C₅₃	—	2	2	no	—
46	C₄×Q₈×C₅₃	—	2	2	no	—
47	C₄×Q₈×C₅₃	—	2	2	no	—
48	C₄×Q₈×C₅₃	D₄	1	4	yes	—
49	#117×C₅₃	—	2	4	no	—
50	#117×C₅₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	2	no	—
56	C₄⋊C₈×C₅₃	—	2	2	no	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	#101×C₅₃	—	4	2	no	—
59	C₄×C₈×C₅₃	—	2	2	no	—
60	C₂×Q₈×C₅₃	—	2	4	no	—
61	C₂×Q₈×C₅₃	—	2	2	no	—
62	C₂×Q₈×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	—	2	2	no	—
64	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
65	C₂×Q₈×C₅₃	SD₁₆	1	2	yes	—
66	C₂×Q₈×C₅₃	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₅₃	—	4	2	no	—
73	M₁₆×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	2	8	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	8	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	4	no	—
97	C₄²×C₅₃	C₂×D₄	1	2	yes	—
98	C₄²×C₅₃	C₂×D₄	1	2	yes	—
99	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	Q₈×C₅₃	—	2	4	no	—
101	Q₈×C₅₃	—	2	4	no	—
102	Q₈×C₅₃	—	4	4	no	—
103	Q₈×C₅₃	—	4	2	no	—
104	Q₈×C₅₃	—	4	2	no	—
105	Q₈×C₅₃	—	8	4	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	#103	1	4	yes	—
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#120	1	2	yes	—
119	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#137	1	2	yes	—
123	C₈×C₅₃	—	4	2	no	—
124	C₈×C₅₃	—	8	2	no	—
125	C₈×C₅₃	—	8	1	no	—
126	#8356	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	#379	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
131	C₂²×C₅₃	#419	1	1	yes	characteristic
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	4	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	4	4	no	—
136	C₄×C₅₃	—	4	2	no	—
137	C₄×C₅₃	—	4	4	no	—
138	C₄×C₅₃	#448	1	2	yes	—
139	C₄×C₅₃	#451	1	2	yes	—
140	C₄×C₅₃	#454	1	1	yes	characteristic
141	C₄×C₅₃	#456	1	1	yes	characteristic
142	#3502	C₂×C₅₃	1	1	yes	characteristic
143	#1984	C₂×C₅₃	1	1	yes	characteristic
144	#1801	C₂×C₅₃	1	1	yes	characteristic
145	#1550	C₂×C₅₃	1	2	yes	—
146	#1493	C₂×C₅₃	1	2	yes	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	#1787	1	1	yes	characteristic
150	C₂×C₅₃	#1791	1	1	yes	characteristic
151	C₂×C₅₃	#1793	1	1	yes	characteristic
152	Q₈²	—	2	2	no	—
153	#557	—	2	2	no	—
154	#557	—	2	4	no	—
155	#544	—	2	1	no	—
156	#544	C₄×C₅₃	1	4	yes	—
157	#533	—	2	2	no	—
158	#533	C₂²×C₅₃	1	2	yes	—
159	#519	—	2	1	no	—
160	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
161	#423	C₂²×C₅₃	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×#101	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
165	#343	—	2	1	no	—
166	#336	—	2	2	no	—
167	#333	—	2	2	no	—
168	#332	—	2	2	no	—
169	Q₈⋊C₈	—	2	4	no	—
170	C₅₃	#8356	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	2	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	4	no	—
178	C₄⋊Q₈	—	2	8	no	—
179	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
180	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
181	#125	—	2	4	no	—
182	#125	—	2	4	no	—
183	#125	—	2	8	no	—
184	#125	—	2	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	C₂²⋊Q₈	—	2	4	no	—
187	C₂²⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
188	C₄×Q₈	—	2	4	no	—
189	C₄×Q₈	—	2	4	no	—
190	C₄×Q₈	—	2	4	no	—
191	C₄×Q₈	—	2	4	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×Q₈	D₄×C₅₃	1	4	yes	—
196	#117	—	2	2	no	—
197	#117	—	2	4	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
199	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₅₃	1	2	yes	—
201	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#101	—	4	2	no	—
206	C₄×C₈	—	2	2	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	C₂²⋊C₄×C₅₃	1	4	yes	—
212	C₂×Q₈	SD₁₆×C₅₃	1	2	yes	—
213	C₂×Q₈	Q₁₆×C₅₃	1	2	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
215	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₄²	—	2	8	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	4	no	—
244	C₄²	C₂×D₄×C₅₃	1	2	yes	—
245	C₄²	C₂×D₄×C₅₃	1	2	yes	—
246	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	2	4	no	—
249	Q₈	—	4	4	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	8	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	#103×C₅₃	1	4	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
264	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
265	C₂×C₄	#120×C₅₃	1	2	yes	—
266	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
269	C₂×C₄	#137×C₅₃	1	2	yes	—
270	C₈	—	4	2	no	—
271	C₈	—	8	2	no	—
272	C₈	—	8	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	C₅₃×#379	1	1	yes	characteristic
276	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
277	C₂²	C₅₃×#419	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	C₅₃×#448	1	2	yes	—
285	C₄	C₅₃×#451	1	2	yes	—
286	C₄	C₅₃×#454	1	1	yes	characteristic
287	C₄	C₅₃×#456	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	C₅₃×#1787	1	1	yes	characteristic
291	C₂	C₅₃×#1791	1	1	yes	characteristic
292	C₂	C₅₃×#1793	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
126. #8356	C₅₃	direct (N × H)
170. C₅₃	#8356	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3502	2
C₅₃×#1984	2
C₅₃×#1801	2
C₅₃×#1550	2
C₅₃×#1493	2
#8356	53

Subgroups — C53×#8356 #27215868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8356 #27215868