Subgroups — C₂×C₅₃×#2216 #27241110

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#467	C₂	1	2	yes	maximal
4	C₅₃×#2216	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#358	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
10	C₂×Q₃₂×C₅₃	—	2	4	no	—
11	C₂×Q₃₂×C₅₃	C₂²	1	4	yes	complemented
12	C₂²×C₁₆×C₅₃	—	2	4	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	4	yes	—
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#467	—	2	4	no	—
17	C₅₃×#467	C₂²	1	8	yes	—
18	C₂×#107×C₅₃	C₂²	1	2	yes	—
19	C₂×#103×C₅₃	—	2	2	no	—
20	C₅₃×#366	C₂²	1	4	yes	complemented
21	C₅₃×#358	C₂²	1	8	yes	complemented
22	C₂²×Q₈×C₅₃	—	2	2	no	—
23	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂×Q₁₆×C₅₃	—	2	2	no	—
25	C₂×Q₁₆×C₅₃	—	2	4	no	—
26	C₂×Q₁₆×C₅₃	—	2	4	no	—
27	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²⋊Q₈×C₅₃	—	2	8	no	—
35	C₂²⋊Q₈×C₅₃	—	4	4	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
39	Q₃₂×C₅₃	—	2	8	no	—
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	C₂×C₁₆×C₅₃	—	2	8	no	—
42	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
43	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
44	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
45	#107×C₅₃	—	2	2	no	—
46	#107×C₅₃	C₂³	1	4	yes	—
47	#103×C₅₃	—	2	8	no	—
48	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
49	C₂×Q₈×C₅₃	—	2	4	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	—	4	4	no	—
52	C₂×Q₈×C₅₃	—	4	4	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	D₈	1	2	yes	—
58	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₅₃	—	2	4	no	—
62	Q₁₆×C₅₃	—	2	4	no	—
63	Q₁₆×C₅₃	—	4	4	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₃	#41	1	2	yes	—
72	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	—	4	2	no	—
75	C₄⋊C₄×C₅₃	—	4	8	no	—
76	C₂²⋊C₄×C₅₃	—	4	8	no	—
77	C₁₆×C₅₃	—	2	2	no	—
78	C₁₆×C₅₃	C₂×D₄	1	4	yes	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	Q₃₂	1	4	yes	complemented
82	C₂³×C₅₃	C₂×D₈	1	2	yes	—
83	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₅₃	—	4	4	no	—
85	Q₈×C₅₃	—	4	4	no	—
86	Q₈×C₅₃	—	8	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	8	4	no	—
94	C₂×C₄×C₅₃	—	8	4	no	—
95	C₂×C₄×C₅₃	#121	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#135	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	#121	1	2	yes	—
102	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
103	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
104	C₂×#2216	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	8	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	#466	1	2	yes	—
109	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
110	C₂²×C₅₃	C₂×Q₃₂	1	4	yes	complemented
111	C₂²×C₅₃	#508	1	2	yes	—
112	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	8	4	no	—
115	C₄×C₅₃	—	8	4	no	—
116	C₄×C₅₃	#466	1	2	yes	—
117	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
118	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
119	C₂²×Q₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
120	C₂³×C₁₆	C₂×C₅₃	1	1	yes	characteristic
121	C₂×#467	C₂×C₅₃	1	2	yes	—
122	#2216	C₂×C₅₃	1	8	yes	direct product factor
123	C₂×#366	C₂×C₅₃	1	1	yes	characteristic, complemented
124	C₂×#358	C₂×C₅₃	1	2	yes	complemented
125	C₂×C₅₃	—	2	4	no	—
126	C₂×C₅₃	#2216	1	4	yes	direct product factor
127	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
128	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
130	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
131	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
132	C₂×C₂²⋊Q₈	—	2	2	no	—
133	C₂×Q₃₂	—	2	4	no	—
134	C₂×Q₃₂	C₂²×C₅₃	1	4	yes	complemented
135	C₂²×C₁₆	—	2	4	no	—
136	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
137	C₂²×C₁₆	C₂²×C₅₃	1	4	yes	—
138	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
139	#467	—	2	4	no	—
140	#467	C₂²×C₅₃	1	8	yes	—
141	C₂×#107	C₂²×C₅₃	1	2	yes	—
142	C₂×#103	—	2	2	no	—
143	#366	C₂²×C₅₃	1	4	yes	complemented
144	#358	C₂²×C₅₃	1	8	yes	complemented
145	C₅₃	C₂×#2216	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×Q₈	—	2	2	no	—
147	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
148	C₂×Q₁₆	—	2	4	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	—	2	4	no	—
151	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
155	C₂²×C₈	D₄×C₅₃	1	2	yes	—
156	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
157	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
158	C₂²⋊Q₈	—	2	8	no	—
159	C₂²⋊Q₈	—	4	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	4	2	no	—
162	C₂×C₂²⋊C₄	—	4	2	no	—
163	Q₃₂	—	2	8	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	8	no	—
166	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
167	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
168	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
169	#107	—	2	2	no	—
170	#107	C₂³×C₅₃	1	4	yes	—
171	#103	—	2	8	no	—
172	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	D₈×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	2	4	no	—
187	Q₁₆	—	4	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
195	C₂×C₈	#41×C₅₃	1	2	yes	—
196	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₁₆	—	2	2	no	—
202	C₁₆	C₂×D₄×C₅₃	1	4	yes	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	Q₃₂×C₅₃	1	4	yes	complemented
206	C₂³	C₂×D₈×C₅₃	1	2	yes	—
207	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
208	Q₈	—	4	4	no	—
209	Q₈	—	4	4	no	—
210	Q₈	—	8	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	#121×C₅₃	1	2	yes	—
220	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
221	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
222	C₂×C₄	#135×C₅₃	1	2	yes	—
223	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	#121×C₅₃	1	2	yes	—
226	C₈	C₂²×D₄×C₅₃	1	1	yes	characteristic
227	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
228	C₂²	—	2	8	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	C₅₃×#466	1	2	yes	—
232	C₂²	C₂×Q₃₂×C₅₃	1	2	yes	—
233	C₂²	C₂×Q₃₂×C₅₃	1	4	yes	complemented
234	C₂²	C₅₃×#508	1	2	yes	—
235	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	4	no	—
239	C₄	C₅₃×#466	1	2	yes	—
240	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
241	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	C₅₃×#2216	1	4	yes	direct product factor
244	C₂	C₂×C₅₃×#466	1	1	yes	characteristic
245	C₂	C₂²×Q₃₂×C₅₃	1	1	yes	characteristic
246	C₂	C₂×C₅₃×#508	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₃₂×C₅₃	C₂	semidirect
4. C₅₃×#2216	C₂	direct (N × H)
5. C₂×C₅₃×#366	C₂	semidirect
6. C₂×C₅₃×#358	C₂	semidirect
11. C₂×Q₃₂×C₅₃	C₂²	semidirect
20. C₅₃×#366	C₂²	semidirect
21. C₅₃×#358	C₂²	semidirect
81. C₂³×C₅₃	Q₃₂	semidirect
104. C₂×#2216	C₅₃	direct (N × H)
110. C₂²×C₅₃	C₂×Q₃₂	semidirect
119. C₂²×Q₃₂	C₂×C₅₃	semidirect
122. #2216	C₂×C₅₃	direct (N × H)
123. C₂×#366	C₂×C₅₃	semidirect
124. C₂×#358	C₂×C₅₃	semidirect
126. C₂×C₅₃	#2216	direct (N × H)
134. C₂×Q₃₂	C₂²×C₅₃	semidirect
143. #366	C₂²×C₅₃	semidirect
144. #358	C₂²×C₅₃	semidirect
145. C₅₃	C₂×#2216	direct (N × H)
205. C₂³	Q₃₂×C₅₃	semidirect
233. C₂²	C₂×Q₃₂×C₅₃	semidirect
243. C₂	C₅₃×#2216	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×Q₃₂×C₅₃	2
C₂³×C₁₆×C₅₃	2
C₂×C₅₃×#467	2
C₅₃×#2216	2
C₂×C₅₃×#366	2
C₂×C₅₃×#358	2
C₂×#2216	53

Subgroups — C2×C53×#2216 #27241110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2216 #27241110