Subgroups — C₂×C₅₃×#2242 #27241208

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#495	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2242	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#358	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#357	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×SD₁₆×C₅₃	—	2	1	no	—
9	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#125×C₅₃	—	2	1	no	—
11	C₂×#124×C₅₃	—	2	1	no	—
12	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#499	—	2	2	no	—
14	C₅₃×#499	C₂²	1	4	yes	complemented
15	C₅₃×#495	—	2	2	no	—
16	C₅₃×#495	C₂²	1	4	yes	—
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×#106×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#358	C₂²	1	8	yes	complemented
21	C₅₃×#357	C₂²	1	8	yes	—
22	C₄×C₁₆×C₅₃	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₅₃	—	2	1	no	—
24	C₂²×D₄×C₅₃	—	2	1	no	—
25	C₂×Q₁₆×C₅₃	—	2	1	no	—
26	C₂×Q₁₆×C₅₃	—	2	4	no	—
27	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
28	C₂×SD₁₆×C₅₃	—	2	4	no	—
29	C₂×SD₁₆×C₅₃	—	4	4	no	—
30	C₂×D₈×C₅₃	—	2	1	no	—
31	C₂×D₈×C₅₃	—	2	4	no	—
32	C₂×D₈×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	#125×C₅₃	—	2	4	no	—
36	#125×C₅₃	—	4	2	no	—
37	#124×C₅₃	—	2	4	no	—
38	#124×C₅₃	—	4	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
41	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	—	2	4	no	—
44	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
45	#107×C₅₃	—	2	2	no	—
46	#107×C₅₃	C₂³	1	4	yes	—
47	#106×C₅₃	—	2	4	no	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂⁴×C₅₃	—	4	1	no	—
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	1	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	D₈	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₅₃	—	2	4	no	—
64	Q₁₆×C₅₃	—	4	2	no	—
65	SD₁₆×C₅₃	—	4	8	no	—
66	D₈×C₅₃	—	2	4	no	—
67	D₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	SD₁₆	1	4	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	4	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	C₄²×C₅₃	D₈	1	2	yes	—
78	C₄²×C₅₃	C₂×D₄	1	2	yes	—
79	C₁₆×C₅₃	—	2	4	no	—
80	C₂³×C₅₃	—	4	2	no	—
81	C₂³×C₅₃	—	8	2	no	—
82	C₂³×C₅₃	—	8	4	no	—
83	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₅₃	—	4	2	no	—
85	Q₈×C₅₃	—	4	2	no	—
86	Q₈×C₅₃	—	8	2	no	—
87	D₄×C₅₃	—	4	2	no	—
88	D₄×C₅₃	—	4	2	no	—
89	D₄×C₅₃	—	8	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	—	8	4	no	—
95	C₂×C₄×C₅₃	#124	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
97	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₅₃	#124	1	2	yes	—
101	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
102	C₈×C₅₃	C₂×#41	1	2	yes	—
103	C₂×#2242	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₅₃	—	8	2	no	—
105	C₂²×C₅₃	—	8	4	no	—
106	C₂²×C₅₃	—	16	4	no	—
107	C₂²×C₅₃	#486	1	2	yes	—
108	C₂²×C₅₃	#508	1	4	yes	—
109	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	—	8	2	no	—
112	C₄×C₅₃	—	8	4	no	—
113	C₄×C₅₃	#486	1	2	yes	—
114	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
115	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
116	C₂×#499	C₂×C₅₃	1	1	yes	characteristic, complemented
117	C₂×#495	C₂×C₅₃	1	1	yes	characteristic
118	#2242	C₂×C₅₃	1	8	yes	direct product factor
119	C₂×#358	C₂×C₅₃	1	2	yes	complemented
120	C₂×#357	C₂×C₅₃	1	2	yes	—
121	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
122	C₂×C₅₃	—	16	2	no	—
123	C₂×C₅₃	#2242	1	4	yes	direct product factor
124	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#508	1	2	yes	—
126	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
127	C₂²×SD₁₆	—	2	1	no	—
128	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂×#125	—	2	1	no	—
130	C₂×#124	—	2	1	no	—
131	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
132	#499	—	2	2	no	—
133	#499	C₂²×C₅₃	1	4	yes	complemented
134	#495	—	2	2	no	—
135	#495	C₂²×C₅₃	1	4	yes	—
136	C₂×#107	C₂²×C₅₃	1	2	yes	—
137	C₂×#106	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
139	#358	C₂²×C₅₃	1	8	yes	complemented
140	#357	C₂²×C₅₃	1	8	yes	—
141	C₄×C₁₆	C₂²×C₅₃	1	4	yes	complemented
142	C₅₃	C₂×#2242	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×Q₈	—	2	1	no	—
144	C₂²×D₄	—	2	1	no	—
145	C₂×Q₁₆	—	2	1	no	—
146	C₂×Q₁₆	—	2	4	no	—
147	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
148	C₂×SD₁₆	—	2	4	no	—
149	C₂×SD₁₆	—	4	4	no	—
150	C₂×D₈	—	2	1	no	—
151	C₂×D₈	—	2	4	no	—
152	C₂×D₈	C₂³×C₅₃	1	2	yes	—
153	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
155	#125	—	2	4	no	—
156	#125	—	4	2	no	—
157	#124	—	2	4	no	—
158	#124	—	4	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	4	2	no	—
161	C₂×C₂²⋊C₄	—	4	2	no	—
162	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
163	C₂×C₁₆	—	2	4	no	—
164	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
165	#107	—	2	2	no	—
166	#107	C₂³×C₅₃	1	4	yes	—
167	#106	—	2	4	no	—
168	C₄×C₈	D₄×C₅₃	1	2	yes	—
169	C₄×C₈	C₂³×C₅₃	1	2	yes	—
170	C₂⁴	—	4	1	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	D₈×C₅₃	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
183	Q₁₆	—	2	4	no	—
184	Q₁₆	—	4	2	no	—
185	SD₁₆	—	4	8	no	—
186	D₈	—	2	4	no	—
187	D₈	—	4	2	no	—
188	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
189	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
190	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
191	C₂×C₈	#41×C₅₃	1	4	yes	—
192	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	8	no	—
196	C₂²⋊C₄	—	4	8	no	—
197	C₄²	D₈×C₅₃	1	2	yes	—
198	C₄²	C₂×D₄×C₅₃	1	2	yes	—
199	C₁₆	—	2	4	no	—
200	C₂³	—	4	2	no	—
201	C₂³	—	8	2	no	—
202	C₂³	—	8	4	no	—
203	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
204	Q₈	—	4	2	no	—
205	Q₈	—	4	2	no	—
206	Q₈	—	8	2	no	—
207	D₄	—	4	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	#124×C₅₃	1	2	yes	—
216	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
217	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
219	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
220	C₈	#124×C₅₃	1	2	yes	—
221	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
222	C₈	C₂×#41×C₅₃	1	2	yes	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	4	no	—
225	C₂²	—	16	4	no	—
226	C₂²	C₅₃×#486	1	2	yes	—
227	C₂²	C₅₃×#508	1	4	yes	—
228	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₅₃×#486	1	2	yes	—
233	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
234	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
235	C₂	—	16	2	no	—
236	C₂	C₅₃×#2242	1	4	yes	direct product factor
237	C₂	C₂×C₅₃×#486	1	1	yes	characteristic
238	C₂	C₂×C₅₃×#508	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
1. C₂×C₅₃×#499	C₂	semidirect
3. C₅₃×#2242	C₂	direct (N × H)
4. C₂×C₅₃×#358	C₂	semidirect
6. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
14. C₅₃×#499	C₂²	semidirect
20. C₅₃×#358	C₂²	semidirect
22. C₄×C₁₆×C₅₃	C₂²	semidirect
103. C₂×#2242	C₅₃	direct (N × H)
116. C₂×#499	C₂×C₅₃	semidirect
118. #2242	C₂×C₅₃	direct (N × H)
119. C₂×#358	C₂×C₅₃	semidirect
121. C₂×C₄×C₁₆	C₂×C₅₃	semidirect
123. C₂×C₅₃	#2242	direct (N × H)
133. #499	C₂²×C₅₃	semidirect
139. #358	C₂²×C₅₃	semidirect
141. C₄×C₁₆	C₂²×C₅₃	semidirect
142. C₅₃	C₂×#2242	direct (N × H)
236. C₂	C₅₃×#2242	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×#499	2
C₂×C₅₃×#495	2
C₅₃×#2242	2
C₂×C₅₃×#358	2
C₂×C₅₃×#357	2
C₂×C₄×C₁₆×C₅₃	2
C₂×#2242	53

Subgroups — C2×C53×#2242 #27241208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2242 #27241208