Subgroups — C₅₃×#23825 #27231337

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2100	C₂	1	2	yes	maximal
4	C₅₃×#2074	C₂	1	2	yes	maximal
5	C₅₃×#2043	C₂	1	2	yes	maximal
6	C₅₃×#2026	C₂	1	2	yes	maximal
7	C₅₃×#1763	C₂	1	4	yes	maximal
8	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#125×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#103×C₅₃	C₂²	1	4	yes	—
13	C₂×#103×C₅₃	C₂²	1	4	yes	—
14	C₂×#97×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#391	C₂²	1	2	yes	—
16	C₅₃×#389	—	2	2	no	—
17	C₅₃×#389	C₂²	1	2	yes	—
18	C₅₃×#387	—	2	2	no	—
19	C₅₃×#384	C₂²	1	1	yes	characteristic
20	C₅₃×#382	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₄³×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#340	C₂²	1	4	yes	—
26	C₅₃×#340	C₂²	1	4	yes	—
27	C₅₃×#336	C₂²	1	2	yes	—
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₅₃	—	2	4	no	—
32	#125×C₅₃	—	2	4	no	—
33	C₄×Q₈×C₅₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	D₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#103×C₅₃	—	2	8	no	—
50	#103×C₅₃	—	2	8	no	—
51	#97×C₅₃	—	2	4	no	—
52	#95×C₅₃	—	2	2	no	—
53	#95×C₅₃	—	2	2	no	—
54	#95×C₅₃	—	2	4	no	—
55	#95×C₅₃	—	2	2	no	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	—	2	4	no	—
58	C₂×Q₈×C₅₃	—	2	4	no	—
59	C₂×Q₈×C₅₃	#41	1	4	yes	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	—	2	8	no	—
73	C₄⋊C₄×C₅₃	—	2	8	no	—
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	2	4	no	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	4	4	no	—
78	C₄⋊C₄×C₅₃	—	4	4	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₄⋊C₄×C₅₃	#41	1	4	yes	—
81	C₄²×C₅₃	—	2	4	no	—
82	C₄²×C₅₃	—	2	2	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	4	2	no	—
85	C₄²×C₅₃	—	4	4	no	—
86	C₄²×C₅₃	—	4	4	no	—
87	C₄²×C₅₃	#41	1	4	yes	—
88	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₅₃	—	2	8	no	—
90	Q₈×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	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	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	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	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	#124	1	2	yes	—
110	C₂×C₄×C₅₃	#124	1	2	yes	—
111	C₂×C₄×C₅₃	#137	1	4	yes	—
112	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
115	C₈×C₅₃	—	4	4	no	—
116	#23825	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₅₃	#525	1	1	yes	characteristic
118	C₂²×C₅₃	#525	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#137	1	2	yes	—
121	C₂²×C₅₃	#575	1	2	yes	—
122	C₄×C₅₃	—	2	4	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	4	2	no	—
126	C₄×C₅₃	—	4	4	no	—
127	C₄×C₅₃	—	4	4	no	—
128	C₄×C₅₃	—	8	2	no	—
129	C₄×C₅₃	—	8	2	no	—
130	C₄×C₅₃	#529	1	4	yes	—
131	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
132	C₄×#125	C₂×C₅₃	1	1	yes	characteristic
133	#2100	C₂×C₅₃	1	2	yes	—
134	#2074	C₂×C₅₃	1	2	yes	—
135	#2043	C₂×C₅₃	1	2	yes	—
136	#2026	C₂×C₅₃	1	2	yes	—
137	#1763	C₂×C₅₃	1	4	yes	—
138	#1745	C₂×C₅₃	1	1	yes	characteristic
139	C₂×C₅₃	#2398	1	1	yes	characteristic
140	C₂×C₅₃	#3116	1	2	yes	—
141	C₂×C₅₃	#3117	1	2	yes	—
142	C₂×C₅₃	#3139	1	1	yes	characteristic
143	C₂×C₅₃	#3142	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
145	C₂×#125	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
147	C₂×#103	C₂²×C₅₃	1	4	yes	—
148	C₂×#103	C₂²×C₅₃	1	4	yes	—
149	C₂×#97	C₂²×C₅₃	1	2	yes	—
150	#391	C₂²×C₅₃	1	2	yes	—
151	#389	—	2	2	no	—
152	#389	C₂²×C₅₃	1	2	yes	—
153	#387	—	2	2	no	—
154	#384	C₂²×C₅₃	1	1	yes	characteristic
155	#382	C₂²×C₅₃	1	1	yes	characteristic
156	C₄×C₄⋊C₄	—	2	2	no	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
159	C₄³	C₂²×C₅₃	1	1	yes	characteristic
160	#340	C₂²×C₅₃	1	4	yes	—
161	#340	C₂²×C₅₃	1	4	yes	—
162	#336	C₂²×C₅₃	1	2	yes	—
163	#336	C₂²×C₅₃	1	2	yes	—
164	C₅₃	#23825	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
166	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
167	C₄⋊Q₈	—	2	4	no	—
168	#125	—	2	4	no	—
169	C₄×Q₈	—	2	8	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
177	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	D₄×C₅₃	1	2	yes	—
181	C₂×C₄²	D₄×C₅₃	1	2	yes	—
182	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
185	#103	—	2	8	no	—
186	#103	—	2	8	no	—
187	#97	—	2	4	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	#95	—	2	4	no	—
191	#95	—	2	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	#41×C₅₃	1	4	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
204	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	#41×C₅₃	1	4	yes	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	#41×C₅₃	1	4	yes	—
224	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	4	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	#124×C₅₃	1	2	yes	—
246	C₂×C₄	#124×C₅₃	1	2	yes	—
247	C₂×C₄	#137×C₅₃	1	4	yes	—
248	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
249	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
250	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
251	C₈	—	4	4	no	—
252	C₂²	C₅₃×#525	1	1	yes	characteristic
253	C₂²	C₅₃×#525	1	1	yes	characteristic
254	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
255	C₂²	C₂×#137×C₅₃	1	2	yes	—
256	C₂²	C₅₃×#575	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₄	—	8	2	no	—
265	C₄	C₅₃×#529	1	4	yes	—
266	C₂	C₅₃×#2398	1	1	yes	characteristic
267	C₂	C₅₃×#3116	1	2	yes	—
268	C₂	C₅₃×#3117	1	2	yes	—
269	C₂	C₅₃×#3139	1	1	yes	characteristic
270	C₂	C₅₃×#3142	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂²×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
116. #23825	C₅₃	direct (N × H)
164. C₅₃	#23825	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₅₃	2
C₄×#125×C₅₃	2
C₅₃×#2100	2
C₅₃×#2074	2
C₅₃×#2043	2
C₅₃×#2026	2
C₅₃×#1763	2
C₅₃×#1745	2
#23825	53

Subgroups — C53×#23825 #27231337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#23825 #27231337