Subgroups — C₂×C₅₃×#1567 #27225303

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#423	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1567	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#335	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	—	2	2	no	—
6	C₂³×C₈×C₅₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#423	—	2	4	no	—
9	C₅₃×#423	C₂²	1	8	yes	—
10	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#335	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	—	2	8	no	—
22	C₂×M₁₆×C₅₃	—	4	8	no	—
23	C₂²×C₈×C₅₃	—	2	8	no	—
24	C₂²×C₈×C₅₃	—	2	4	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	8	no	—
37	C₄⋊C₈×C₅₃	—	2	4	no	—
38	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂³	1	4	yes	—
43	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	D₈	1	4	yes	—
53	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
60	M₁₆×C₅₃	—	4	16	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	16	no	—
63	C₂×C₈×C₅₃	—	2	8	no	—
64	C₂×C₈×C₅₃	—	4	8	no	—
65	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
66	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂×D₄	1	2	yes	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
76	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	#107	1	4	yes	—
81	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	#107	1	8	yes	—
90	C₂×C₄×C₅₃	#107	1	4	yes	—
91	C₂×C₄×C₅₃	#108	1	4	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
99	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
100	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
102	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	—	4	8	no	—
108	C₈×C₅₃	C₂×M₁₆	1	8	yes	complemented
109	C₂×#1567	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₅₃	—	2	8	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	#423	1	2	yes	—
114	C₂²×C₅₃	C₂×#107	1	4	yes	—
115	C₂²×C₅₃	C₂×#107	1	2	yes	—
116	C₂²×C₅₃	C₂×#108	1	2	yes	—
117	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	#423	1	2	yes	—
122	C₄×C₅₃	C₂×#107	1	4	yes	—
123	C₄×C₅₃	C₂×#107	1	2	yes	—
124	C₄×C₅₃	C₂×#108	1	4	yes	—
125	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
126	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
127	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
128	C₂×#423	C₂×C₅₃	1	2	yes	—
129	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
130	#1567	C₂×C₅₃	1	8	yes	direct product factor
131	C₂×#335	C₂×C₅₃	1	4	yes	complemented
132	C₂×C₅₃	—	2	4	no	—
133	C₂×C₅₃	#1567	1	4	yes	direct product factor
134	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
135	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
136	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
137	C₂²×M₁₆	—	2	2	no	—
138	C₂³×C₈	C₄×C₅₃	1	2	yes	—
139	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
140	#423	—	2	4	no	—
141	#423	C₂²×C₅₃	1	8	yes	—
142	C₂×C₄⋊C₈	—	2	2	no	—
143	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
144	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
146	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
149	#335	C₂²×C₅₃	1	16	yes	complemented
150	C₅₃	C₂×#1567	1	1	yes	characteristic, direct product factor, Sylow
151	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
152	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
153	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
154	C₂×M₁₆	—	2	8	no	—
155	C₂×M₁₆	—	4	8	no	—
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	2	4	no	—
158	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
160	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
162	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
163	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄²	D₄×C₅₃	1	2	yes	—
165	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
166	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
167	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
169	C₄⋊C₈	—	2	8	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
172	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
173	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
174	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
175	C₄×C₈	C₂³×C₅₃	1	4	yes	—
176	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
180	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
182	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
183	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	D₈×C₅₃	1	4	yes	—
186	C₂²×C₄	Q₁₆×C₅₃	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
193	M₁₆	—	4	16	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	16	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	4	8	no	—
198	C₂×C₈	M₁₆×C₅₃	1	8	yes	complemented
199	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
200	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
201	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
202	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
203	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
204	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
205	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
206	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₅₃	1	2	yes	—
208	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
209	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
210	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#107×C₅₃	1	4	yes	—
214	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
215	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	#107×C₅₃	1	8	yes	—
223	C₂×C₄	#107×C₅₃	1	4	yes	—
224	C₂×C₄	#108×C₅₃	1	4	yes	—
225	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
227	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
231	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
232	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
234	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
235	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
236	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	4	8	no	—
241	C₈	C₂×M₁₆×C₅₃	1	8	yes	complemented
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	C₅₃×#423	1	2	yes	—
246	C₂²	C₂×#107×C₅₃	1	4	yes	—
247	C₂²	C₂×#107×C₅₃	1	2	yes	—
248	C₂²	C₂×#108×C₅₃	1	2	yes	—
249	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	C₅₃×#423	1	2	yes	—
254	C₄	C₂×#107×C₅₃	1	2	yes	—
255	C₄	C₂×#107×C₅₃	1	4	yes	—
256	C₄	C₂×#108×C₅₃	1	4	yes	—
257	C₄	C₂²×M₁₆×C₅₃	1	2	yes	—
258	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
259	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
260	C₂	—	2	4	no	—
261	C₂	C₅₃×#1567	1	4	yes	direct product factor
262	C₂	C₂×C₅₃×#423	1	1	yes	characteristic
263	C₂	C₂²×#107×C₅₃	1	1	yes	characteristic
264	C₂	C₂²×#108×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1567	C₂	direct (N × H)
4. C₂×C₅₃×#335	C₂	semidirect
17. C₅₃×#335	C₂²	semidirect
65. C₂×C₈×C₅₃	M₁₆	semidirect
108. C₈×C₅₃	C₂×M₁₆	semidirect
109. C₂×#1567	C₅₃	direct (N × H)
130. #1567	C₂×C₅₃	direct (N × H)
131. C₂×#335	C₂×C₅₃	semidirect
133. C₂×C₅₃	#1567	direct (N × H)
149. #335	C₂²×C₅₃	semidirect
150. C₅₃	C₂×#1567	direct (N × H)
198. C₂×C₈	M₁₆×C₅₃	semidirect
241. C₈	C₂×M₁₆×C₅₃	semidirect
261. C₂	C₅₃×#1567	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×#423	2
C₂²×C₄×C₈×C₅₃	2
C₅₃×#1567	2
C₂×C₅₃×#335	2
C₂×#1567	53

Subgroups — C2×C53×#1567 #27225303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1567 #27225303