Subgroups — C₅₃×#33908 #27241420

Order 13568.

492 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3341	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2234	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2226	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2215	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2158	C₂	1	2	yes	maximal, complemented
9	C₅₃×#555	—	2	2	no	—
10	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#500	C₂²	1	2	yes	—
13	C₅₃×#482	—	2	4	no	—
14	C₅₃×#476	—	2	2	no	—
15	C₅₃×#466	C₂²	1	4	yes	—
16	C₄×D₈×C₅₃	C₂²	1	2	yes	—
17	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#427	C₂²	1	4	yes	—
19	C₅₃×#367	C₂²	1	4	yes	—
20	C₅₃×#366	C₂²	1	1	yes	characteristic
21	C₅₃×#366	C₂²	1	2	yes	—
22	C₅₃×#366	C₂²	1	2	yes	—
23	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₅₃×#357	C₂²	1	2	yes	—
25	C₅₃×#357	C₂²	1	4	yes	—
26	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×D₈×C₅₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	#124×C₅₃	—	4	4	no	—
32	#121×C₅₃	—	2	4	no	—
33	#121×C₅₃	—	4	2	no	—
34	C₄×D₄×C₅₃	—	2	2	no	—
35	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
36	#117×C₅₃	—	2	4	no	—
37	#117×C₅₃	—	4	2	no	—
38	D₁₆×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	C₂³	1	4	yes	—
45	#107×C₅₃	C₂³	1	2	yes	—
46	#106×C₅₃	—	2	4	no	—
47	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
48	#102×C₅₃	—	2	2	no	—
49	#102×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	—	4	2	no	—
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	4	4	no	—
55	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
58	D₈×C₅₃	—	2	2	no	—
59	D₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
69	C₂²⋊C₄×C₅₃	—	4	4	no	—
70	C₂²⋊C₄×C₅₃	—	4	4	no	—
71	C₂²⋊C₄×C₅₃	—	4	2	no	—
72	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
73	C₄²×C₅₃	—	4	4	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	—	4	2	no	—
79	C₂³×C₅₃	C₂×D₈	1	2	yes	—
80	Q₈×C₅₃	—	16	2	no	—
81	D₄×C₅₃	—	2	2	no	—
82	D₄×C₅₃	—	4	2	no	—
83	D₄×C₅₃	—	4	2	no	—
84	D₄×C₅₃	—	16	4	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	8	4	no	—
90	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	2	1	no	—
96	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
97	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
98	#33908	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	8	2	no	—
103	C₂²×C₅₃	—	8	2	no	—
104	C₂²×C₅₃	—	8	2	no	—
105	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	8	2	no	—
109	C₄×C₅₃	—	8	4	no	—
110	C₄×C₅₃	#508	1	2	yes	—
111	C₄×C₅₃	#547	1	1	yes	characteristic
112	C₄×C₅₃	#578	1	1	yes	characteristic
113	#3341	C₂×C₅₃	1	2	yes	complemented
114	#2252	C₂×C₅₃	1	1	yes	characteristic, complemented
115	#2234	C₂×C₅₃	1	4	yes	complemented
116	#2226	C₂×C₅₃	1	2	yes	complemented
117	#2215	C₂×C₅₃	1	2	yes	complemented
118	C₄×D₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
119	D₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
120	#2158	C₂×C₅₃	1	2	yes	complemented
121	C₂×C₅₃	—	4	2	no	—
122	C₂×C₅₃	—	16	2	no	—
123	C₂×C₅₃	#3334	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
125	C₂×C₅₃	#3415	1	1	yes	characteristic
126	#555	—	2	2	no	—
127	C₂×D₁₆	C₂²×C₅₃	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
129	#500	C₂²×C₅₃	1	2	yes	—
130	#482	—	2	4	no	—
131	#476	—	2	2	no	—
132	#466	C₂²×C₅₃	1	4	yes	—
133	C₄×D₈	C₂²×C₅₃	1	2	yes	—
134	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
135	#427	C₂²×C₅₃	1	4	yes	—
136	#367	C₂²×C₅₃	1	4	yes	—
137	#366	C₂²×C₅₃	1	1	yes	characteristic
138	#366	C₂²×C₅₃	1	2	yes	—
139	#366	C₂²×C₅₃	1	2	yes	—
140	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
141	#357	C₂²×C₅₃	1	2	yes	—
142	#357	C₂²×C₅₃	1	4	yes	—
143	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
144	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
145	C₅₃	#33908	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×D₈	C₂³×C₅₃	1	2	yes	—
147	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
148	C₄⋊Q₈	—	2	2	no	—
149	#124	—	4	4	no	—
150	#121	—	2	4	no	—
151	#121	—	4	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
154	#117	—	2	4	no	—
155	#117	—	4	2	no	—
156	D₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	4	no	—
158	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
159	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
160	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
161	#107	—	2	2	no	—
162	#107	C₂³×C₅₃	1	4	yes	—
163	#107	C₂³×C₅₃	1	2	yes	—
164	#106	—	2	4	no	—
165	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
166	#102	—	2	2	no	—
167	#102	—	2	4	no	—
168	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
169	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
170	C₂×Q₈	—	4	2	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
176	D₈	—	2	2	no	—
177	D₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
182	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
187	C₂²⋊C₄	—	4	4	no	—
188	C₂²⋊C₄	—	4	4	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
191	C₄²	—	4	4	no	—
192	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
193	C₁₆	—	2	2	no	—
194	C₁₆	—	2	1	no	—
195	C₁₆	#41×C₅₃	1	2	yes	—
196	C₂³	—	4	2	no	—
197	C₂³	C₂×D₈×C₅₃	1	2	yes	—
198	Q₈	—	16	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	16	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
215	C₈	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	2	no	—
222	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	2	no	—
226	C₄	—	8	4	no	—
227	C₄	C₅₃×#508	1	2	yes	—
228	C₄	C₅₃×#547	1	1	yes	characteristic
229	C₄	C₅₃×#578	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	—	16	2	no	—
232	C₂	C₅₃×#3334	1	1	yes	characteristic
233	C₂	C₂×C₅₃×#508	1	1	yes	characteristic
234	C₂	C₅₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
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
1. C₅₃×#3341	C₂	semidirect
2. C₅₃×#2252	C₂	semidirect
3. C₅₃×#2234	C₂	semidirect
4. C₅₃×#2226	C₂	semidirect
5. C₅₃×#2215	C₂	semidirect
6. C₄×D₁₆×C₅₃	C₂	semidirect
7. D₄×C₁₆×C₅₃	C₂	semidirect
8. C₅₃×#2158	C₂	semidirect
98. #33908	C₅₃	direct (N × H)
113. #3341	C₂×C₅₃	semidirect
114. #2252	C₂×C₅₃	semidirect
115. #2234	C₂×C₅₃	semidirect
116. #2226	C₂×C₅₃	semidirect
117. #2215	C₂×C₅₃	semidirect
118. C₄×D₁₆	C₂×C₅₃	semidirect
119. D₄×C₁₆	C₂×C₅₃	semidirect
120. #2158	C₂×C₅₃	semidirect
145. C₅₃	#33908	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3341	2
C₅₃×#2252	2
C₅₃×#2234	2
C₅₃×#2226	2
C₅₃×#2215	2
C₄×D₁₆×C₅₃	2
D₄×C₁₆×C₅₃	2
C₅₃×#2158	2
#33908	53

Subgroups — C53×#33908 #27241420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33908 #27241420