Subgroups — C₅₃×#10694 #27218206

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#336	C₂	1	4	yes	maximal
5	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
6	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#97×C₅₃	—	2	2	no	—
10	C₂×#97×C₅₃	—	2	2	no	—
11	C₂×#97×C₅₃	C₄	1	4	yes	complemented
12	C₂×C₄×C₈×C₅₃	—	2	2	no	—
13	C₂×C₄×C₈×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
15	C₅₃×#376	—	2	2	no	—
16	C₅₃×#376	—	2	2	no	—
17	C₂×#95×C₅₃	C₄	1	4	yes	—
18	C₅₃×#336	C₄	1	16	yes	complemented
19	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	4	no	—
27	C₂²×C₈×C₅₃	—	2	4	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	—	2	4	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
42	#97×C₅₃	—	2	8	no	—
43	C₄×C₈×C₅₃	—	2	8	no	—
44	#95×C₅₃	—	2	8	no	—
45	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	8	no	—
47	C₂²×C₄×C₅₃	—	2	4	no	—
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	—	2	16	no	—
50	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
51	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	4	yes	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	8	no	—
67	C₂×C₈×C₅₃	—	2	8	no	—
68	C₂×C₈×C₅₃	—	2	8	no	—
69	C₂×C₈×C₅₃	—	2	8	no	—
70	C₂×C₈×C₅₃	C₄²	1	8	yes	—
71	C₂×C₈×C₅₃	C₄²	1	8	yes	—
72	C₄²×C₅₃	—	2	8	no	—
73	C₄²×C₅₃	—	2	8	no	—
74	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
75	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
77	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
78	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
80	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₅₃	#117	1	2	yes	—
83	C₂³×C₅₃	#131	1	8	yes	—
84	C₂×C₄×C₅₃	—	2	8	no	—
85	C₂×C₄×C₅₃	—	2	8	no	—
86	C₂×C₄×C₅₃	—	2	8	no	—
87	C₂×C₄×C₅₃	—	2	8	no	—
88	C₂×C₄×C₅₃	—	2	16	no	—
89	C₂×C₄×C₅₃	—	4	8	no	—
90	C₂×C₄×C₅₃	—	4	8	no	—
91	C₂×C₄×C₅₃	#95	1	8	yes	—
92	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#117	1	2	yes	—
99	C₂×C₄×C₅₃	#117	1	2	yes	—
100	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
104	C₈×C₅₃	—	2	8	no	—
105	C₈×C₅₃	—	2	8	no	—
106	#10694	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₅₃	C₄³	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
109	C₂²×C₅₃	#376	1	1	yes	characteristic
110	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
111	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
112	C₂²×C₅₃	#405	1	4	yes	—
113	C₂²×C₅₃	#405	1	8	yes	—
114	C₂²×C₅₃	#408	1	4	yes	—
115	C₂²×C₅₃	#424	1	4	yes	—
116	C₂²×C₅₃	#433	1	8	yes	—
117	C₄×C₅₃	—	2	8	no	—
118	C₄×C₅₃	—	4	8	no	—
119	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
120	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
121	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
122	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
123	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
124	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
125	C₂×#376	C₂×C₅₃	1	1	yes	characteristic
126	C₂×#336	C₂×C₅₃	1	4	yes	—
127	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
128	C₂×C₅₃	#1726	1	2	yes	—
129	C₂×C₅₃	#1729	1	2	yes	—
130	C₂×C₅₃	#1746	1	2	yes	—
131	C₂×C₅₃	#1750	1	4	yes	—
132	C₂×C₅₃	#1768	1	4	yes	—
133	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
134	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
136	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
137	C₂×#97	—	2	2	no	—
138	C₂×#97	—	2	2	no	—
139	C₂×#97	C₄×C₅₃	1	4	yes	complemented
140	C₂×C₄×C₈	—	2	2	no	—
141	C₂×C₄×C₈	—	2	2	no	—
142	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	complemented
143	#376	—	2	2	no	—
144	#376	—	2	2	no	—
145	C₂×#95	C₄×C₅₃	1	4	yes	—
146	#336	C₄×C₅₃	1	16	yes	complemented
147	C₅₃	#10694	1	1	yes	characteristic, direct product factor, Sylow
148	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
149	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
150	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
158	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
159	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
160	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	4	no	—
164	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
165	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
166	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
167	C₂×C₄²	D₄×C₅₃	1	2	yes	—
168	C₂×C₄²	D₄×C₅₃	1	2	yes	—
169	C₂×C₄²	D₄×C₅₃	1	2	yes	—
170	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
171	#97	—	2	8	no	—
172	C₄×C₈	—	2	8	no	—
173	#95	—	2	8	no	—
174	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	16	no	—
179	C₂²×C₄	C₄²×C₅₃	1	8	yes	—
180	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
181	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
182	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
183	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	#41×C₅₃	1	4	yes	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	8	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	C₄²×C₅₃	1	8	yes	—
200	C₂×C₈	C₄²×C₅₃	1	8	yes	—
201	C₄²	—	2	8	no	—
202	C₄²	—	2	8	no	—
203	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
204	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
205	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
206	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
207	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
208	C₂³	C₂×C₄²×C₅₃	1	2	yes	—
209	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
211	C₂³	#117×C₅₃	1	2	yes	—
212	C₂³	#131×C₅₃	1	8	yes	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	16	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	#95×C₅₃	1	8	yes	—
221	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
224	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₄	#117×C₅₃	1	2	yes	—
228	C₂×C₄	#117×C₅₃	1	2	yes	—
229	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
230	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
231	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
232	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
233	C₈	—	2	8	no	—
234	C₈	—	2	8	no	—
235	C₂²	C₄³×C₅₃	1	1	yes	characteristic
236	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
237	C₂²	C₅₃×#376	1	1	yes	characteristic
238	C₂²	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
239	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
240	C₂²	C₅₃×#405	1	4	yes	—
241	C₂²	C₅₃×#405	1	8	yes	—
242	C₂²	C₅₃×#408	1	4	yes	—
243	C₂²	C₅₃×#424	1	4	yes	—
244	C₂²	C₅₃×#433	1	8	yes	—
245	C₄	—	2	8	no	—
246	C₄	—	4	8	no	—
247	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
248	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
249	C₄	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
250	C₄	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
251	C₂	C₄×#95×C₅₃	1	1	yes	characteristic
252	C₂	C₅₃×#1726	1	2	yes	—
253	C₂	C₅₃×#1729	1	2	yes	—
254	C₂	C₅₃×#1746	1	2	yes	—
255	C₂	C₅₃×#1750	1	4	yes	—
256	C₂	C₅₃×#1768	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅₃	C₂³
commutator	C₂²	C₄³×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×#97×C₅₃	C₄	semidirect
14. C₂×C₄×C₈×C₅₃	C₄	semidirect
18. C₅₃×#336	C₄	semidirect
106. #10694	C₅₃	direct (N × H)
139. C₂×#97	C₄×C₅₃	semidirect
142. C₂×C₄×C₈	C₄×C₅₃	semidirect
146. #336	C₄×C₅₃	semidirect
147. C₅₃	#10694	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×#97×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#376	2
C₂×C₅₃×#336	2
#10694	53

Subgroups — C53×#10694 #27218206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10694 #27218206