Subgroups — C₅₃×#9696 #27217208

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2858	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#332	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#331	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#124×C₅₃	—	2	1	no	—
9	C₂×#124×C₅₃	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	—	2	1	no	—
13	C₅₃×#390	—	2	2	no	—
14	C₅₃×#390	C₄	1	4	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	1	no	—
17	C₅₃×#336	—	2	2	no	—
18	C₅₃×#332	—	2	4	no	—
19	C₅₃×#331	—	2	4	no	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×D₄×C₅₃	—	2	1	no	—
22	C₂²×D₄×C₅₃	—	2	1	no	—
23	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	#127×C₅₃	—	2	2	no	—
29	#127×C₅₃	—	2	2	no	—
30	#124×C₅₃	—	4	4	no	—
31	#124×C₅₃	—	4	2	no	—
32	#124×C₅₃	—	4	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
46	C₄×C₈×C₅₃	—	4	2	no	—
47	C₂⁴×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂×Q₈×C₅₃	—	4	1	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	2	4	no	—
53	C₂×D₄×C₅₃	—	4	4	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	1	no	—
56	C₂×D₄×C₅₃	—	4	1	no	—
57	C₂×D₄×C₅₃	—	4	1	no	—
58	C₂×D₄×C₅₃	—	4	1	no	—
59	C₂×D₄×C₅₃	—	4	4	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	4	4	no	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	—	4	1	no	—
74	C₄⋊C₄×C₅₃	—	4	1	no	—
75	C₂²⋊C₄×C₅₃	—	4	8	no	—
76	C₂²⋊C₄×C₅₃	—	4	8	no	—
77	C₄²×C₅₃	—	2	4	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	D₈	1	2	yes	—
81	C₄²×C₅₃	SD₁₆	1	2	yes	—
82	C₄²×C₅₃	C₂×D₄	1	2	yes	—
83	C₄²×C₅₃	#41	1	2	yes	—
84	C₂³×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	—	4	4	no	—
86	C₂³×C₅₃	—	4	2	no	—
87	C₂³×C₅₃	—	4	4	no	—
88	C₂³×C₅₃	—	4	2	no	—
89	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₅₃	—	8	2	no	—
91	Q₈×C₅₃	—	8	2	no	—
92	D₄×C₅₃	—	8	4	no	—
93	D₄×C₅₃	—	8	2	no	—
94	D₄×C₅₃	—	8	2	no	—
95	D₄×C₅₃	—	8	2	no	—
96	D₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	#102	1	4	yes	—
107	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₃	#121	1	2	yes	—
109	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#135	1	4	yes	—
114	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₈×C₅₃	—	4	4	no	—
116	C₈×C₅₃	—	8	2	no	—
117	#9696	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₅₃	—	8	4	no	—
119	C₂²×C₅₃	—	8	4	no	—
120	C₂²×C₅₃	—	8	2	no	—
121	C₂²×C₅₃	—	8	2	no	—
122	C₂²×C₅₃	—	8	2	no	—
123	C₂²×C₅₃	#390	1	1	yes	characteristic
124	C₂²×C₅₃	#413	1	2	yes	—
125	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
126	C₂²×C₅₃	#416	1	2	yes	—
127	C₂²×C₅₃	#418	1	1	yes	characteristic
128	C₄×C₅₃	—	2	4	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	4	2	no	—
131	C₄×C₅₃	—	8	2	no	—
132	C₄×C₅₃	#459	1	2	yes	—
133	C₄×C₅₃	#460	1	2	yes	—
134	C₄×C₅₃	#488	1	2	yes	—
135	C₄×C₅₃	#495	1	2	yes	—
136	#2858	C₂×C₅₃	1	1	yes	characteristic
137	#1831	C₂×C₅₃	1	1	yes	characteristic, complemented
138	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
139	C₂×#332	C₂×C₅₃	1	2	yes	complemented
140	C₂×#331	C₂×C₅₃	1	2	yes	—
141	C₂×C₅₃	—	16	2	no	—
142	C₂×C₅₃	#1540	1	2	yes	—
143	C₂×C₅₃	#1553	1	2	yes	—
144	C₂×C₅₃	#1958	1	1	yes	characteristic
145	C₂×C₅₃	#1967	1	1	yes	characteristic
146	C₂×C₅₃	#1982	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
149	C₂×#124	—	2	1	no	—
150	C₂×#124	—	2	1	no	—
151	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	#390	—	2	2	no	—
155	#390	C₄×C₅₃	1	4	yes	—
156	C₄³	C₂²×C₅₃	1	1	yes	characteristic
157	#336	—	2	2	no	—
158	#336	—	2	1	no	—
159	#332	—	2	4	no	—
160	#331	—	2	4	no	—
161	C₅₃	#9696	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	—	2	1	no	—
165	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	#127	—	2	2	no	—
171	#127	—	2	2	no	—
172	#124	—	4	4	no	—
173	#124	—	4	2	no	—
174	#124	—	4	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	4	no	—
178	C₂×C₂²⋊C₄	—	2	4	no	—
179	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
180	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₅₃	1	2	yes	—
182	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
188	C₄×C₈	—	4	2	no	—
189	C₂⁴	—	2	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	4	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
205	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	8	no	—
218	C₂²⋊C₄	—	4	8	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	D₈×C₅₃	1	2	yes	—
223	C₄²	SD₁₆×C₅₃	1	2	yes	—
224	C₄²	C₂×D₄×C₅₃	1	2	yes	—
225	C₄²	#41×C₅₃	1	2	yes	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
232	Q₈	—	8	2	no	—
233	Q₈	—	8	2	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	#102×C₅₃	1	4	yes	—
249	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
250	C₂×C₄	#121×C₅₃	1	2	yes	—
251	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
252	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
255	C₂×C₄	#135×C₅₃	1	4	yes	—
256	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
257	C₈	—	4	4	no	—
258	C₈	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	4	no	—
263	C₂²	—	8	4	no	—
264	C₂²	C₅₃×#390	1	1	yes	characteristic
265	C₂²	C₅₃×#413	1	2	yes	—
266	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
267	C₂²	C₅₃×#416	1	2	yes	—
268	C₂²	C₅₃×#418	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	2	no	—
272	C₄	—	8	2	no	—
273	C₄	C₅₃×#459	1	2	yes	—
274	C₄	C₅₃×#460	1	2	yes	—
275	C₄	C₅₃×#488	1	2	yes	—
276	C₄	C₅₃×#495	1	2	yes	—
277	C₂	—	16	2	no	—
278	C₂	C₅₃×#1540	1	2	yes	—
279	C₂	C₅₃×#1553	1	2	yes	—
280	C₂	C₅₃×#1958	1	1	yes	characteristic
281	C₂	C₅₃×#1967	1	1	yes	characteristic
282	C₂	C₅₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1831	C₂	semidirect
3. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
4. C₂×C₅₃×#332	C₂	semidirect
117. #9696	C₅₃	direct (N × H)
137. #1831	C₂×C₅₃	semidirect
138. C₄×C₄⋊C₈	C₂×C₅₃	semidirect
139. C₂×#332	C₂×C₅₃	semidirect
161. C₅₃	#9696	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×#2858	2
C₅₃×#1831	2
C₄×C₄⋊C₈×C₅₃	2
C₂×C₅₃×#332	2
C₂×C₅₃×#331	2
#9696	53

Subgroups — C53×#9696 #27217208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9696 #27217208