Subgroups — C₂×C₅₃×#2219 #27241114

Order 13568.

682 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	C₂²×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#467	C₂	1	2	yes	maximal
4	C₅₃×#2219	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#367	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	C₂×M₃₂×C₅₃	—	2	4	no	—
13	C₂×M₃₂×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×#367	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	M₃₂×C₅₃	—	2	8	no	—
41	C₂×C₁₆×C₅₃	—	2	2	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	4	no	—
81	C₂³×C₅₃	C₂×D₈	1	2	yes	—
82	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₅₃	—	4	4	no	—
84	Q₈×C₅₃	—	4	4	no	—
85	Q₈×C₅₃	—	8	4	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	8	4	no	—
93	C₂×C₄×C₅₃	—	8	4	no	—
94	C₂×C₄×C₅₃	#121	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#135	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	#121	1	2	yes	—
101	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
103	C₂×#2219	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₅₃	—	2	4	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	4	4	no	—
107	C₂²×C₅₃	#466	1	2	yes	—
108	C₂²×C₅₃	#510	1	4	yes	—
109	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	8	4	no	—
112	C₄×C₅₃	—	8	4	no	—
113	C₄×C₅₃	#466	1	2	yes	—
114	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
116	C₂²×Q₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
117	C₂²×M₃₂	C₂×C₅₃	1	1	yes	characteristic
118	C₂×#467	C₂×C₅₃	1	2	yes	—
119	#2219	C₂×C₅₃	1	8	yes	direct product factor
120	C₂×#367	C₂×C₅₃	1	1	yes	characteristic, complemented
121	C₂×#358	C₂×C₅₃	1	2	yes	complemented
122	C₂×C₅₃	—	4	2	no	—
123	C₂×C₅₃	#2219	1	4	yes	direct product factor
124	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#510	1	2	yes	—
126	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
127	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
128	C₂×C₂²⋊Q₈	—	2	2	no	—
129	C₂×Q₃₂	—	2	4	no	—
130	C₂×Q₃₂	C₂²×C₅₃	1	4	yes	complemented
131	C₂×M₃₂	—	2	4	no	—
132	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
133	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
134	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
135	#467	—	2	4	no	—
136	#467	C₂²×C₅₃	1	8	yes	—
137	C₂×#107	C₂²×C₅₃	1	2	yes	—
138	C₂×#103	—	2	2	no	—
139	#367	C₂²×C₅₃	1	4	yes	complemented
140	#358	C₂²×C₅₃	1	8	yes	complemented
141	C₅₃	C₂×#2219	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₈	—	2	2	no	—
143	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
144	C₂×Q₁₆	—	2	4	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	4	no	—
147	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₅₃	1	2	yes	—
152	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
153	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
154	C₂²⋊Q₈	—	2	8	no	—
155	C₂²⋊Q₈	—	4	4	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	4	2	no	—
158	C₂×C₂²⋊C₄	—	4	2	no	—
159	Q₃₂	—	2	8	no	—
160	M₃₂	—	2	8	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
163	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
164	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
165	#107	—	2	2	no	—
166	#107	C₂³×C₅₃	1	4	yes	—
167	#103	—	2	8	no	—
168	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	D₈×C₅₃	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
181	Q₁₆	—	2	4	no	—
182	Q₁₆	—	2	4	no	—
183	Q₁₆	—	4	4	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂×D₄×C₅₃	1	2	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	2	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	C₁₆	—	2	2	no	—
198	C₁₆	C₂×D₄×C₅₃	1	4	yes	—
199	C₂³	—	2	4	no	—
200	C₂³	—	2	2	no	—
201	C₂³	C₂×D₈×C₅₃	1	2	yes	—
202	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
203	Q₈	—	4	4	no	—
204	Q₈	—	4	4	no	—
205	Q₈	—	8	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	8	4	no	—
214	C₂×C₄	#121×C₅₃	1	2	yes	—
215	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
216	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
217	C₂×C₄	#135×C₅₃	1	2	yes	—
218	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	#121×C₅₃	1	2	yes	—
221	C₈	C₂²×D₄×C₅₃	1	1	yes	characteristic
222	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	4	4	no	—
226	C₂²	C₅₃×#466	1	2	yes	—
227	C₂²	C₅₃×#510	1	4	yes	—
228	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	8	4	no	—
231	C₄	—	8	4	no	—
232	C₄	C₅₃×#466	1	2	yes	—
233	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
234	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
235	C₂	—	4	2	no	—
236	C₂	C₅₃×#2219	1	4	yes	direct product factor
237	C₂	C₂×C₅₃×#466	1	1	yes	characteristic
238	C₂	C₂×C₅₃×#510	1	2	yes	—

	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×#2219	C₂	direct (N × H)
5. C₂×C₅₃×#367	C₂	semidirect
6. C₂×C₅₃×#358	C₂	semidirect
11. C₂×Q₃₂×C₅₃	C₂²	semidirect
20. C₅₃×#367	C₂²	semidirect
21. C₅₃×#358	C₂²	semidirect
103. C₂×#2219	C₅₃	direct (N × H)
116. C₂²×Q₃₂	C₂×C₅₃	semidirect
119. #2219	C₂×C₅₃	direct (N × H)
120. C₂×#367	C₂×C₅₃	semidirect
121. C₂×#358	C₂×C₅₃	semidirect
123. C₂×C₅₃	#2219	direct (N × H)
130. C₂×Q₃₂	C₂²×C₅₃	semidirect
139. #367	C₂²×C₅₃	semidirect
140. #358	C₂²×C₅₃	semidirect
141. C₅₃	C₂×#2219	direct (N × H)
236. C₂	C₅₃×#2219	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
C₂²×M₃₂×C₅₃	2
C₂×C₅₃×#467	2
C₅₃×#2219	2
C₂×C₅₃×#367	2
C₂×C₅₃×#358	2
C₂×#2219	53

Subgroups — C2×C53×#2219 #27241114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2219 #27241114