Subgroups — C₄×C₅₃×#508 #27240959

Order 13568.

690 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	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₄×#135×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₄×Q₃₂×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₄×SD₃₂×C₅₃	C₂	1	4	yes	maximal, complemented
5	C₄×D₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2138	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₅₃	C₂²	1	2	yes	—
10	C₄×#41×C₅₃	—	2	2	no	—
11	C₅₃×#508	C₄	1	8	yes	direct product factor
12	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	—
14	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
18	C₄×SD₁₆×C₅₃	—	2	4	no	—
19	C₄×D₈×C₅₃	C₂²	1	4	yes	—
20	C₅₃×#427	C₂²	1	2	yes	—
21	C₅₃×#416	—	2	2	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#367	C₂²	1	2	yes	—
24	C₅₃×#366	C₂²	1	2	yes	—
25	C₅₃×#358	C₂²	1	4	yes	—
26	C₅₃×#357	C₂²	1	4	yes	—
27	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
28	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
29	C₂×#41×C₅₃	—	2	2	no	—
30	#135×C₅₃	—	2	4	no	—
31	#135×C₅₃	C₂×C₄	1	8	yes	complemented
32	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₅₃	—	2	4	no	—
38	C₄×D₄×C₅₃	—	2	4	no	—
39	C₄×D₄×C₅₃	—	4	4	no	—
40	#117×C₅₃	—	2	2	no	—
41	#117×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	—	4	2	no	—
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	Q₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
45	SD₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
46	D₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
52	#107×C₅₃	C₂³	1	4	yes	—
53	#106×C₅₃	—	2	2	no	—
54	#103×C₅₃	—	2	4	no	—
55	#102×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	D₄	1	2	yes	—
57	C₄×C₈×C₅₃	C₂³	1	2	yes	—
58	#41×C₅₃	—	2	8	no	—
59	#41×C₅₃	—	4	4	no	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	—	4	4	no	—
65	C₂²×C₄×C₅₃	D₈	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₅₃	—	2	2	no	—
68	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₅₃	—	2	8	no	—
70	D₈×C₅₃	—	2	2	no	—
71	D₈×C₅₃	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₂²⋊C₄×C₅₃	—	4	2	no	—
81	C₂²⋊C₄×C₅₃	—	4	4	no	—
82	C₄²×C₅₃	—	4	2	no	—
83	C₄²×C₅₃	—	4	4	no	—
84	C₄²×C₅₃	D₈	1	1	yes	characteristic
85	C₄²×C₅₃	D₈	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	2	yes	—
87	C₁₆×C₅₃	—	2	2	no	—
88	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂³×C₅₃	—	4	2	no	—
91	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₅₃	—	2	4	no	—
93	Q₈×C₅₃	—	4	2	no	—
94	D₄×C₅₃	—	2	4	no	—
95	D₄×C₅₃	—	4	8	no	—
96	D₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	8	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	8	8	no	—
102	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₅₃	#135	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	C₄×D₄	1	2	yes	—
110	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
112	C₄×#508	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	4	4	no	—
115	C₂²×C₅₃	—	8	4	no	—
116	C₂²×C₅₃	C₄×D₈	1	2	yes	—
117	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	4	4	no	—
120	C₄×C₅₃	—	8	4	no	—
121	C₄×C₅₃	C₄×D₈	1	2	yes	—
122	C₄×C₅₃	#508	1	4	yes	direct product factor
123	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
124	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
125	C₂×#508	C₂×C₅₃	1	1	yes	characteristic
126	C₄×#135	C₂×C₅₃	1	2	yes	complemented
127	C₄×Q₃₂	C₂×C₅₃	1	2	yes	complemented
128	C₄×SD₃₂	C₂×C₅₃	1	4	yes	complemented
129	C₄×D₁₆	C₂×C₅₃	1	2	yes	complemented
130	#2158	C₂×C₅₃	1	1	yes	characteristic, complemented
131	#2138	C₂×C₅₃	1	2	yes	complemented
132	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	—	8	4	no	—
135	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
136	C₂×C₅₃	C₂×#508	1	2	yes	—
137	C₂×#135	C₂²×C₅₃	1	2	yes	—
138	C₄×#41	—	2	2	no	—
139	#508	C₄×C₅₃	1	8	yes	direct product factor
140	C₂×Q₃₂	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×SD₃₂	C₂²×C₅₃	1	2	yes	—
142	C₂×D₁₆	C₂²×C₅₃	1	1	yes	characteristic
143	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
144	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
145	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	—
146	C₄×SD₁₆	—	2	4	no	—
147	C₄×D₈	C₂²×C₅₃	1	4	yes	—
148	#427	C₂²×C₅₃	1	2	yes	—
149	#416	—	2	2	no	—
150	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
151	#367	C₂²×C₅₃	1	2	yes	—
152	#366	C₂²×C₅₃	1	2	yes	—
153	#358	C₂²×C₅₃	1	4	yes	—
154	#357	C₂²×C₅₃	1	4	yes	—
155	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
156	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
157	C₅₃	C₄×#508	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×#41	—	2	2	no	—
159	#135	—	2	4	no	—
160	#135	C₂×C₄×C₅₃	1	8	yes	complemented
161	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	C₂³×C₅₃	1	2	yes	—
164	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
166	C₄×Q₈	—	2	4	no	—
167	C₄×D₄	—	2	4	no	—
168	C₄×D₄	—	4	4	no	—
169	#117	—	2	2	no	—
170	#117	—	4	2	no	—
171	C₂×C₄²	—	4	2	no	—
172	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
173	Q₃₂	C₂×C₄×C₅₃	1	4	yes	complemented
174	SD₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
175	D₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
178	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
179	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
180	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
181	#107	C₂³×C₅₃	1	4	yes	—
182	#106	—	2	2	no	—
183	#103	—	2	4	no	—
184	#102	—	2	4	no	—
185	C₄×C₈	D₄×C₅₃	1	2	yes	—
186	C₄×C₈	C₂³×C₅₃	1	2	yes	—
187	#41	—	2	8	no	—
188	#41	—	4	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	D₈×C₅₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
196	Q₁₆	—	2	2	no	—
197	Q₁₆	C₂²×C₄×C₅₃	1	4	yes	—
198	SD₁₆	—	2	8	no	—
199	D₈	—	2	2	no	—
200	D₈	C₂²×C₄×C₅₃	1	4	yes	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
204	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
205	C₂×C₈	#41×C₅₃	1	2	yes	—
206	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	4	no	—
213	C₄²	D₈×C₅₃	1	1	yes	characteristic
214	C₄²	D₈×C₅₃	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₅₃	1	2	yes	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
218	C₁₆	C₂²×C₄×C₅₃	1	2	yes	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	4	8	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
232	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
234	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
235	C₂×C₄	#135×C₅₃	1	2	yes	—
236	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	C₄×D₄×C₅₃	1	2	yes	—
239	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
240	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	8	4	no	—
244	C₂²	C₄×D₈×C₅₃	1	2	yes	—
245	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	4	no	—
249	C₄	C₄×D₈×C₅₃	1	2	yes	—
250	C₄	C₅₃×#508	1	4	yes	direct product factor
251	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
252	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	8	4	no	—
255	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
256	C₂	C₂×C₅₃×#508	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×C₅₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#135×C₅₃	C₂	semidirect
3. C₄×Q₃₂×C₅₃	C₂	semidirect
4. C₄×SD₃₂×C₅₃	C₂	semidirect
5. C₄×D₁₆×C₅₃	C₂	semidirect
6. C₅₃×#2158	C₂	semidirect
7. C₅₃×#2138	C₂	semidirect
8. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
11. C₅₃×#508	C₄	direct (N × H)
31. #135×C₅₃	C₂×C₄	semidirect
44. Q₃₂×C₅₃	C₂×C₄	semidirect
45. SD₃₂×C₅₃	C₂×C₄	semidirect
46. D₁₆×C₅₃	C₂×C₄	semidirect
48. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
112. C₄×#508	C₅₃	direct (N × H)
122. C₄×C₅₃	#508	direct (N × H)
126. C₄×#135	C₂×C₅₃	semidirect
127. C₄×Q₃₂	C₂×C₅₃	semidirect
128. C₄×SD₃₂	C₂×C₅₃	semidirect
129. C₄×D₁₆	C₂×C₅₃	semidirect
130. #2158	C₂×C₅₃	semidirect
131. #2138	C₂×C₅₃	semidirect
132. C₂×C₄×C₁₆	C₂×C₅₃	semidirect
139. #508	C₄×C₅₃	direct (N × H)
157. C₅₃	C₄×#508	direct (N × H)
160. #135	C₂×C₄×C₅₃	semidirect
173. Q₃₂	C₂×C₄×C₅₃	semidirect
174. SD₃₂	C₂×C₄×C₅₃	semidirect
175. D₁₆	C₂×C₄×C₅₃	semidirect
177. C₂×C₁₆	C₂×C₄×C₅₃	semidirect
250. C₄	C₅₃×#508	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×#508	2
C₄×#135×C₅₃	2
C₄×Q₃₂×C₅₃	2
C₄×SD₃₂×C₅₃	2
C₄×D₁₆×C₅₃	2
C₅₃×#2158	2
C₅₃×#2138	2
C₂×C₄×C₁₆×C₅₃	2
C₄×#508	53

Subgroups — C4×C53×#508 #27240959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#508 #27240959