Subgroups — C₅₃×#10766 #27218278

Order 13568.

1180 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#410	C₂	1	2	yes	maximal
3	C₅₃×#1733	C₂	1	4	yes	maximal, complemented
4	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
5	C₅₃×#568	—	2	8	no	—
6	C₂×#131×C₅₃	—	2	4	no	—
7	C₂²×M₁₆×C₅₃	C₄	1	4	yes	complemented
8	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂×#117×C₅₃	C₂²	1	4	yes	—
10	C₅₃×#410	—	2	4	no	—
11	C₅₃×#410	C₄	1	8	yes	complemented
12	C₂×#99×C₅₃	—	2	4	no	—
13	C₅₃×#339	—	2	8	no	—
14	C₂×#41×C₅₃	—	2	4	no	—
15	C₂×#41×C₅₃	—	2	4	no	—
16	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
17	C₂×#41×C₅₃	D₄	1	4	yes	—
18	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
19	C₂²×D₄×C₅₃	D₄	1	2	yes	—
20	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	#131×C₅₃	—	2	32	no	—
24	C₂×M₁₆×C₅₃	—	2	16	no	—
25	C₂×M₁₆×C₅₃	—	2	4	no	—
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂×M₁₆×C₅₃	—	2	4	no	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₅₃	D₄	1	8	yes	—
30	C₂²×C₈×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	#117×C₅₃	—	2	4	no	—
33	#117×C₅₃	—	2	4	no	—
34	#117×C₅₃	—	2	4	no	—
35	#117×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
38	C₂×C₄²×C₅₃	—	2	4	no	—
39	#99×C₅₃	—	2	16	no	—
40	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
42	#41×C₅₃	—	2	8	no	—
43	#41×C₅₃	—	2	8	no	—
44	#41×C₅₃	—	2	8	no	—
45	#41×C₅₃	—	2	8	no	—
46	C₂×Q₈×C₅₃	—	2	4	no	—
47	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
48	C₂×D₄×C₅₃	—	2	8	no	—
49	C₂×D₄×C₅₃	—	2	4	no	—
50	C₂×D₄×C₅₃	C₂²⋊C₄	1	8	yes	—
51	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	4	4	no	—
55	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	4	yes	—
64	M₁₆×C₅₃	—	2	16	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	2	8	no	—
67	M₁₆×C₅₃	C₂²⋊C₄	1	8	yes	—
68	C₂×C₈×C₅₃	—	2	16	no	—
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	8	no	—
71	C₄⋊C₄×C₅₃	—	2	8	no	—
72	C₄⋊C₄×C₅₃	—	4	4	no	—
73	C₂²⋊C₄×C₅₃	—	2	8	no	—
74	C₂²⋊C₄×C₅₃	—	4	4	no	—
75	C₄²×C₅₃	—	2	8	no	—
76	C₄²×C₅₃	—	4	4	no	—
77	C₂³×C₅₃	—	2	4	no	—
78	C₂³×C₅₃	—	8	1	no	—
79	C₂³×C₅₃	—	8	2	no	—
80	C₂³×C₅₃	#95	1	4	yes	—
81	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₅₃	#121	1	4	yes	—
84	Q₈×C₅₃	—	2	8	no	—
85	D₄×C₅₃	—	2	16	no	—
86	D₄×C₅₃	—	2	8	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	8	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	4	8	no	—
92	C₂×C₄×C₅₃	—	8	8	no	—
93	C₂×C₄×C₅₃	#95	1	4	yes	—
94	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	#117	1	2	yes	—
99	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₅₃	#120	1	4	yes	—
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₅₃	#123	1	4	yes	—
104	C₈×C₅₃	—	2	8	no	—
105	#10766	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₅₃	—	2	4	no	—
107	C₂²×C₅₃	—	4	4	no	—
108	C₂²×C₅₃	—	4	1	no	—
109	C₂²×C₅₃	—	4	2	no	—
110	C₂²×C₅₃	—	4	4	no	—
111	C₂²×C₅₃	—	8	4	no	—
112	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
113	C₂²×C₅₃	#380	1	2	yes	—
114	C₂²×C₅₃	#386	1	4	yes	—
115	C₄×C₅₃	—	2	4	no	—
116	C₄×C₅₃	—	8	8	no	—
117	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
118	C₄×C₅₃	#379	1	1	yes	characteristic
119	C₄×C₅₃	#381	1	1	yes	characteristic
120	C₄×C₅₃	#385	1	4	yes	—
121	C₂×#568	C₂×C₅₃	1	1	yes	characteristic
122	C₂×#410	C₂×C₅₃	1	2	yes	—
123	#1733	C₂×C₅₃	1	4	yes	complemented
124	C₂×C₅₃	—	2	4	no	—
125	C₂×C₅₃	—	2	1	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	—	4	4	no	—
128	C₂×C₅₃	C₂²⋊#95	1	1	yes	characteristic
129	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
130	#568	—	2	8	no	—
131	C₂×#131	—	2	4	no	—
132	C₂²×M₁₆	C₄×C₅₃	1	4	yes	complemented
133	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
134	C₂×#117	C₂²×C₅₃	1	4	yes	—
135	#410	—	2	4	no	—
136	#410	C₄×C₅₃	1	8	yes	complemented
137	C₂×#99	—	2	4	no	—
138	#339	—	2	8	no	—
139	C₅₃	#10766	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×#41	—	2	4	no	—
141	C₂×#41	—	2	4	no	—
142	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
143	C₂×#41	D₄×C₅₃	1	4	yes	—
144	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
145	C₂²×D₄	D₄×C₅₃	1	2	yes	—
146	C₂²×D₄	Q₈×C₅₃	1	1	yes	characteristic
147	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
148	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
149	#131	—	2	32	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	16	no	—
154	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
155	C₂×M₁₆	D₄×C₅₃	1	8	yes	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	4	no	—
158	#117	—	2	4	no	—
159	#117	—	2	4	no	—
160	#117	—	2	4	no	—
161	#117	C₂×C₄×C₅₃	1	8	yes	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₄²	—	2	4	no	—
165	#99	—	2	16	no	—
166	C₂⁴	C₄⋊C₄×C₅₃	1	2	yes	—
167	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
168	#41	—	2	8	no	—
169	#41	—	2	8	no	—
170	#41	—	2	8	no	—
171	#41	—	2	8	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	C₄⋊C₄×C₅₃	1	4	yes	—
174	C₂×D₄	—	2	8	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	C₂²⋊C₄×C₅₃	1	8	yes	—
177	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
182	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
183	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
188	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	#41×C₅₃	1	4	yes	—
190	M₁₆	—	2	16	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	2	8	no	—
193	M₁₆	C₂²⋊C₄×C₅₃	1	8	yes	—
194	C₂×C₈	—	2	16	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	8	no	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	—	2	8	no	—
200	C₂²⋊C₄	—	4	4	no	—
201	C₄²	—	2	8	no	—
202	C₄²	—	4	4	no	—
203	C₂³	—	2	4	no	—
204	C₂³	—	8	1	no	—
205	C₂³	—	8	2	no	—
206	C₂³	#95×C₅₃	1	4	yes	—
207	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
208	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
209	C₂³	#121×C₅₃	1	4	yes	—
210	Q₈	—	2	8	no	—
211	D₄	—	2	16	no	—
212	D₄	—	2	8	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	—	8	8	no	—
219	C₂×C₄	#95×C₅₃	1	4	yes	—
220	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
224	C₂×C₄	#117×C₅₃	1	2	yes	—
225	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
226	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
227	C₂×C₄	#120×C₅₃	1	4	yes	—
228	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
229	C₂×C₄	#123×C₅₃	1	4	yes	—
230	C₈	—	2	8	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	8	4	no	—
237	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
238	C₂²	C₅₃×#380	1	2	yes	—
239	C₂²	C₅₃×#386	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	8	8	no	—
242	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
243	C₄	C₅₃×#379	1	1	yes	characteristic
244	C₄	C₅₃×#381	1	1	yes	characteristic
245	C₄	C₅₃×#385	1	4	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	4	no	—
250	C₂	C₅₃×C₂²⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#379
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂×C₅₃	C₂²⋊#95
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1733	C₂	semidirect
7. C₂²×M₁₆×C₅₃	C₄	semidirect
11. C₅₃×#410	C₄	semidirect
105. #10766	C₅₃	direct (N × H)
123. #1733	C₂×C₅₃	semidirect
132. C₂²×M₁₆	C₄×C₅₃	semidirect
136. #410	C₄×C₅₃	semidirect
139. C₅₃	#10766	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#568	2
C₂×C₅₃×#410	2
C₅₃×#1733	2
#10766	53

Subgroups — C53×#10766 #27218278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10766 #27218278