Subgroups — C₅₃×#19662 #27227174

Order 13568.

838 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	C₅₃×#3226	C₂	1	4	yes	maximal, complemented
2	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1722	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1713	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1709	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	—	2	2	no	—
9	C₂²×Q₁₆×C₅₃	—	2	2	no	—
10	C₅₃×#532	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#500	C₂²	1	8	yes	—
13	C₅₃×#495	—	2	2	no	—
14	C₅₃×#494	—	2	2	no	—
15	C₅₃×#494	C₂²	1	2	yes	—
16	C₅₃×#494	C₂²	1	1	yes	characteristic
17	C₅₃×#494	C₂²	1	2	yes	—
18	C₅₃×#493	C₂²	1	1	yes	characteristic
19	C₅₃×#492	C₂²	1	2	yes	—
20	C₅₃×#486	C₂²	1	8	yes	—
21	C₅₃×#480	—	2	8	no	—
22	C₅₃×#474	—	2	8	no	—
23	C₅₃×#431	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#322	C₂²	1	2	yes	—
26	C₈²×C₅₃	C₂²	1	2	yes	—
27	C₂×#41×C₅₃	—	2	2	no	—
28	C₂²×Q₈×C₅₃	—	2	2	no	—
29	#135×C₅₃	—	4	8	no	—
30	C₂×Q₁₆×C₅₃	—	2	4	no	—
31	C₂×Q₁₆×C₅₃	—	2	4	no	—
32	C₂×Q₁₆×C₅₃	—	2	2	no	—
33	C₂×Q₁₆×C₅₃	—	4	8	no	—
34	C₂×SD₁₆×C₅₃	—	2	4	no	—
35	C₂×D₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₄⋊Q₈×C₅₃	—	2	2	no	—
38	C₄⋊Q₈×C₅₃	—	2	4	no	—
39	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
41	#127×C₅₃	C₂³	1	2	yes	—
42	#124×C₅₃	—	2	2	no	—
43	#121×C₅₃	—	2	4	no	—
44	C₄×D₄×C₅₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#107×C₅₃	—	2	16	no	—
48	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
49	#102×C₅₃	—	2	16	no	—
50	C₂²⋊C₈×C₅₃	D₄	1	4	yes	—
51	C₄×C₈×C₅₃	D₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	4	yes	—
53	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	#41×C₅₃	—	4	8	no	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	—	2	4	no	—
58	C₂×Q₈×C₅₃	—	4	2	no	—
59	C₂×Q₈×C₅₃	—	4	8	no	—
60	C₂×D₄×C₅₃	—	2	4	no	—
61	C₂×D₄×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₅₃	—	4	8	no	—
68	Q₁₆×C₅₃	—	4	8	no	—
69	Q₁₆×C₅₃	—	4	4	no	—
70	SD₁₆×C₅₃	—	4	8	no	—
71	D₈×C₅₃	—	4	4	no	—
72	C₂×C₈×C₅₃	D₈	1	8	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	2	8	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	—	4	4	no	—
79	C₂²⋊C₄×C₅₃	—	4	4	no	—
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	2	yes	—
82	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₂³×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	#127	1	1	yes	characteristic
85	Q₈×C₅₃	—	4	8	no	—
86	Q₈×C₅₃	—	4	4	no	—
87	Q₈×C₅₃	—	8	4	no	—
88	D₄×C₅₃	—	4	4	no	—
89	D₄×C₅₃	—	4	8	no	—
90	D₄×C₅₃	—	8	4	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	8	4	no	—
95	C₂×C₄×C₅₃	—	8	4	no	—
96	C₂×C₄×C₅₃	#127	1	2	yes	—
97	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
99	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
101	C₈×C₅₃	—	2	8	no	—
102	C₈×C₅₃	C₂×D₈	1	4	yes	—
103	C₈×C₅₃	C₂×D₈	1	4	yes	—
104	#19662	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	8	2	no	—
107	C₂²×C₅₃	—	8	4	no	—
108	C₂²×C₅₃	#493	1	2	yes	—
109	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	8	2	no	—
112	C₄×C₅₃	—	8	4	no	—
113	C₄×C₅₃	#493	1	2	yes	—
114	C₄×C₅₃	C₂²×D₈	1	2	yes	—
115	C₄×C₅₃	#578	1	4	yes	—
116	#3226	C₂×C₅₃	1	4	yes	complemented
117	#3147	C₂×C₅₃	1	1	yes	characteristic
118	C₂×#494	C₂×C₅₃	1	1	yes	characteristic, complemented
119	#1722	C₂×C₅₃	1	4	yes	complemented
120	#1713	C₂×C₅₃	1	2	yes	complemented
121	#1709	C₂×C₅₃	1	2	yes	complemented
122	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	—	16	2	no	—
125	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
126	C₂×C₅₃	#3156	1	2	yes	—
127	C₂×#135	—	2	2	no	—
128	C₂²×Q₁₆	—	2	2	no	—
129	#532	C₂²×C₅₃	1	2	yes	—
130	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
131	#500	C₂²×C₅₃	1	8	yes	—
132	#495	—	2	2	no	—
133	#494	—	2	2	no	—
134	#494	C₂²×C₅₃	1	2	yes	—
135	#494	C₂²×C₅₃	1	2	yes	—
136	#494	C₂²×C₅₃	1	1	yes	characteristic
137	#493	C₂²×C₅₃	1	1	yes	characteristic
138	#492	C₂²×C₅₃	1	2	yes	—
139	#486	C₂²×C₅₃	1	8	yes	—
140	#480	—	2	8	no	—
141	#474	—	2	8	no	—
142	#431	C₂²×C₅₃	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
144	#322	C₂²×C₅₃	1	2	yes	—
145	C₈²	C₂²×C₅₃	1	2	yes	—
146	C₅₃	#19662	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#41	—	2	2	no	—
148	C₂²×Q₈	—	2	2	no	—
149	#135	—	4	8	no	—
150	C₂×Q₁₆	—	2	4	no	—
151	C₂×Q₁₆	—	2	4	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	4	8	no	—
154	C₂×SD₁₆	—	2	4	no	—
155	C₂×D₈	—	2	2	no	—
156	C₂²×C₈	D₄×C₅₃	1	2	yes	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
160	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
161	#127	C₂³×C₅₃	1	2	yes	—
162	#124	—	2	2	no	—
163	#121	—	2	4	no	—
164	C₄×D₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
167	#107	—	2	16	no	—
168	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
169	#102	—	2	16	no	—
170	C₂²⋊C₈	D₄×C₅₃	1	4	yes	—
171	C₄×C₈	D₄×C₅₃	1	2	yes	—
172	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₅₃	1	4	yes	—
174	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
175	#41	—	4	8	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	8	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
187	Q₁₆	—	4	8	no	—
188	Q₁₆	—	4	8	no	—
189	Q₁₆	—	4	4	no	—
190	SD₁₆	—	4	8	no	—
191	D₈	—	4	4	no	—
192	C₂×C₈	D₈×C₅₃	1	8	yes	—
193	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
194	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
196	C₄⋊C₄	—	2	8	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	—	4	4	no	—
200	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₅₃	1	2	yes	—
202	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
203	C₂³	—	4	2	no	—
204	C₂³	#127×C₅₃	1	1	yes	characteristic
205	Q₈	—	4	4	no	—
206	Q₈	—	4	8	no	—
207	Q₈	—	8	4	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	8	no	—
210	D₄	—	8	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	#127×C₅₃	1	2	yes	—
217	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
219	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
221	C₈	—	2	8	no	—
222	C₈	C₂×D₈×C₅₃	1	4	yes	—
223	C₈	C₂×D₈×C₅₃	1	4	yes	—
224	C₂²	—	2	2	no	—
225	C₂²	—	8	2	no	—
226	C₂²	—	8	4	no	—
227	C₂²	C₅₃×#493	1	2	yes	—
228	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₅₃×#493	1	2	yes	—
233	C₄	C₂²×D₈×C₅₃	1	2	yes	—
234	C₄	C₅₃×#578	1	4	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	16	2	no	—
237	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
238	C₂	C₅₃×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3226	C₂	semidirect
3. C₂×C₅₃×#494	C₂	semidirect
4. C₅₃×#1722	C₂	semidirect
5. C₅₃×#1713	C₂	semidirect
6. C₅₃×#1709	C₂	semidirect
7. C₅₃×#1450	C₂	semidirect
104. #19662	C₅₃	direct (N × H)
116. #3226	C₂×C₅₃	semidirect
118. C₂×#494	C₂×C₅₃	semidirect
119. #1722	C₂×C₅₃	semidirect
120. #1713	C₂×C₅₃	semidirect
121. #1709	C₂×C₅₃	semidirect
122. #1450	C₂×C₅₃	semidirect
146. C₅₃	#19662	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×#3226	2
C₅₃×#3147	2
C₂×C₅₃×#494	2
C₅₃×#1722	2
C₅₃×#1713	2
C₅₃×#1709	2
C₅₃×#1450	2
#19662	53

Subgroups — C53×#19662 #27227174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19662 #27227174