Subgroups — C₅₃×#33905 #27241417

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3338	C₂	1	2	yes	maximal
2	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2236	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2227	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2216	C₂	1	2	yes	maximal, complemented
6	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#366	C₂	1	2	yes	maximal, complemented
9	C₅₃×#554	—	2	2	no	—
10	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#500	C₂²	1	2	yes	—
13	C₅₃×#484	—	2	4	no	—
14	C₅₃×#477	—	2	2	no	—
15	C₅₃×#467	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
17	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#107×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#366	C₂²	1	1	yes	characteristic
20	C₅₃×#366	C₂²	1	2	yes	—
21	C₅₃×#366	C₂²	1	4	yes	—
22	C₅₃×#366	C₂²	1	2	yes	—
23	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₅₃×#358	C₂²	1	2	yes	—
25	C₅₃×#358	C₂²	1	4	yes	—
26	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	#125×C₅₃	—	4	2	no	—
32	C₂²⋊Q₈×C₅₃	—	2	4	no	—
33	C₂²⋊Q₈×C₅₃	—	4	4	no	—
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×D₄×C₅₃	—	4	2	no	—
36	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
38	Q₃₂×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	—	2	4	no	—
40	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
43	#107×C₅₃	—	2	2	no	—
44	#107×C₅₃	—	2	4	no	—
45	#107×C₅₃	C₂³	1	4	yes	—
46	#107×C₅₃	C₂³	1	2	yes	—
47	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
48	#103×C₅₃	—	2	2	no	—
49	#103×C₅₃	—	2	4	no	—
50	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	—	2	2	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	4	4	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	Q₁₆×C₅₃	—	2	2	no	—
58	Q₁₆×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	4	4	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	4	no	—
70	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
71	C₂²⋊C₄×C₅₃	—	4	4	no	—
72	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
73	C₄²×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₅₃	—	2	2	no	—
76	C₁₆×C₅₃	—	2	1	no	—
77	C₁₆×C₅₃	#41	1	2	yes	—
78	C₂³×C₅₃	C₂×D₈	1	2	yes	—
79	Q₈×C₅₃	—	4	2	no	—
80	Q₈×C₅₃	—	4	2	no	—
81	Q₈×C₅₃	—	16	2	no	—
82	D₄×C₅₃	Q₃₂	1	4	yes	complemented
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	16	4	no	—
89	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
96	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
97	#33905	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	C₂×Q₃₂	1	4	yes	—
101	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	8	2	no	—
105	C₄×C₅₃	—	8	2	no	—
106	C₄×C₅₃	—	8	4	no	—
107	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
108	C₄×C₅₃	#547	1	1	yes	characteristic
109	C₄×C₅₃	#578	1	1	yes	characteristic
110	#3338	C₂×C₅₃	1	2	yes	—
111	#2252	C₂×C₅₃	1	1	yes	characteristic, complemented
112	#2236	C₂×C₅₃	1	4	yes	complemented
113	#2227	C₂×C₅₃	1	2	yes	complemented
114	#2216	C₂×C₅₃	1	2	yes	complemented
115	C₄×Q₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
116	D₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
117	C₂×#366	C₂×C₅₃	1	2	yes	complemented
118	C₂×C₅₃	—	2	4	no	—
119	C₂×C₅₃	#3334	1	1	yes	characteristic
120	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
121	C₂×C₅₃	#3415	1	1	yes	characteristic
122	#554	—	2	2	no	—
123	C₂×Q₃₂	C₂²×C₅₃	1	1	yes	characteristic
124	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
125	#500	C₂²×C₅₃	1	2	yes	—
126	#484	—	2	4	no	—
127	#477	—	2	2	no	—
128	#467	C₂²×C₅₃	1	4	yes	—
129	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
130	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
131	C₂×#107	C₂²×C₅₃	1	4	yes	—
132	#366	C₂²×C₅₃	1	1	yes	characteristic
133	#366	C₂²×C₅₃	1	2	yes	—
134	#366	C₂²×C₅₃	1	4	yes	—
135	#366	C₂²×C₅₃	1	2	yes	—
136	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
137	#358	C₂²×C₅₃	1	2	yes	—
138	#358	C₂²×C₅₃	1	4	yes	—
139	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
140	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
141	C₅₃	#33905	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
143	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
144	C₄⋊Q₈	—	2	2	no	—
145	#125	—	4	2	no	—
146	C₂²⋊Q₈	—	2	4	no	—
147	C₂²⋊Q₈	—	4	4	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×D₄	—	4	2	no	—
150	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
151	C₂×C₄⋊C₄	—	2	4	no	—
152	Q₃₂	—	2	2	no	—
153	C₂×C₁₆	—	2	4	no	—
154	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
155	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
156	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
157	#107	—	2	4	no	—
158	#107	—	2	2	no	—
159	#107	C₂³×C₅₃	1	2	yes	—
160	#107	C₂³×C₅₃	1	4	yes	—
161	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
162	#103	—	2	2	no	—
163	#103	—	2	4	no	—
164	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
165	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
171	Q₁₆	—	2	2	no	—
172	Q₁₆	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
177	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
185	C₂²⋊C₄	—	4	4	no	—
186	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
187	C₄²	—	4	2	no	—
188	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
189	C₁₆	—	2	2	no	—
190	C₁₆	—	2	1	no	—
191	C₁₆	#41×C₅₃	1	2	yes	—
192	C₂³	C₂×D₈×C₅₃	1	2	yes	—
193	Q₈	—	4	2	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	16	2	no	—
196	D₄	Q₃₂×C₅₃	1	4	yes	complemented
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	16	4	no	—
203	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
205	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	1	no	—
209	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
210	C₈	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	C₂×Q₃₂×C₅₃	1	4	yes	—
214	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	4	no	—
220	C₄	C₂×Q₃₂×C₅₃	1	2	yes	—
221	C₄	C₅₃×#547	1	1	yes	characteristic
222	C₄	C₅₃×#578	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	C₅₃×#3334	1	1	yes	characteristic
225	C₂	C₂²×Q₃₂×C₅₃	1	1	yes	characteristic
226	C₂	C₅₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂×D₄×C₅₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×D₄×C₅₃ ◁ C₈×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2252	C₂	semidirect
3. C₅₃×#2236	C₂	semidirect
4. C₅₃×#2227	C₂	semidirect
5. C₅₃×#2216	C₂	semidirect
6. C₄×Q₃₂×C₅₃	C₂	semidirect
8. C₂×C₅₃×#366	C₂	semidirect
82. D₄×C₅₃	Q₃₂	semidirect
97. #33905	C₅₃	direct (N × H)
111. #2252	C₂×C₅₃	semidirect
112. #2236	C₂×C₅₃	semidirect
113. #2227	C₂×C₅₃	semidirect
114. #2216	C₂×C₅₃	semidirect
115. C₄×Q₃₂	C₂×C₅₃	semidirect
117. C₂×#366	C₂×C₅₃	semidirect
141. C₅₃	#33905	direct (N × H)
196. D₄	Q₃₂×C₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#3338	2
C₅₃×#2252	2
C₅₃×#2236	2
C₅₃×#2227	2
C₅₃×#2216	2
C₄×Q₃₂×C₅₃	2
D₄×C₁₆×C₅₃	2
C₂×C₅₃×#366	2
#33905	53

Subgroups — C53×#33905 #27241417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33905 #27241417