Subgroups — C₅₃×#8158 #27215670

Order 13568.

400 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#433	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1458	C₂	1	2	yes	maximal
3	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1281	C₂	1	2	yes	maximal, complemented
6	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#433	—	2	1	no	—
8	C₅₃×#433	—	2	1	no	—
9	C₅₃×#433	—	2	2	no	—
10	C₅₃×#431	C₂²	1	2	yes	—
11	C₅₃×#431	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#324	C₄	1	4	yes	—
17	C₅₃×#323	C₄	1	4	yes	—
18	C₅₃×#322	—	2	1	no	—
19	C₅₃×#322	—	2	2	no	—
20	C₈²×C₅₃	—	2	1	no	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	#117×C₅₃	—	2	1	no	—
25	#117×C₅₃	—	2	1	no	—
26	#117×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₄⋊C₈×C₅₃	—	2	2	no	—
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₂²⋊C₈×C₅₃	—	2	2	no	—
36	C₂²⋊C₈×C₅₃	—	2	2	no	—
37	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
38	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
39	#97×C₅₃	—	2	1	no	—
40	#97×C₅₃	—	2	1	no	—
41	C₄×C₈×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊C₄×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	C₄²×C₅₃	—	2	1	no	—
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	#41	1	2	yes	—
70	C₂³×C₅₃	—	2	1	no	—
71	C₂³×C₅₃	—	2	2	no	—
72	C₂³×C₅₃	—	4	2	no	—
73	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	#131	1	4	yes	—
86	C₈×C₅₃	—	2	2	no	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	4	4	no	—
89	C₈×C₅₃	—	4	4	no	—
90	C₈×C₅₃	—	4	2	no	—
91	#8158	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	—	4	1	no	—
94	C₂²×C₅₃	—	4	2	no	—
95	C₂²×C₅₃	—	8	2	no	—
96	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₅₃	#405	1	2	yes	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	—	4	1	no	—
101	C₄×C₅₃	—	4	2	no	—
102	C₄×C₅₃	#408	1	2	yes	—
103	C₄×C₅₃	#433	1	2	yes	—
104	C₂×#433	C₂×C₅₃	1	1	yes	characteristic
105	#1458	C₂×C₅₃	1	2	yes	—
106	#1452	C₂×C₅₃	1	1	yes	characteristic, complemented
107	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
108	#1281	C₂×C₅₃	1	2	yes	complemented
109	C₂×C₅₃	—	2	2	no	—
110	C₂×C₅₃	—	8	1	no	—
111	C₂×C₅₃	#1750	1	1	yes	characteristic
112	C₂×C₅₃	#1757	1	2	yes	—
113	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
114	#433	—	2	1	no	—
115	#433	—	2	1	no	—
116	#433	—	2	2	no	—
117	#431	C₂²×C₅₃	1	2	yes	—
118	#431	C₂²×C₅₃	1	2	yes	—
119	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
120	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	—
121	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
123	#324	C₄×C₅₃	1	4	yes	—
124	#323	C₄×C₅₃	1	4	yes	—
125	#322	—	2	1	no	—
126	#322	—	2	2	no	—
127	C₈²	—	2	1	no	—
128	C₅₃	#8158	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
131	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
132	#117	—	2	1	no	—
133	#117	—	2	1	no	—
134	#117	—	2	2	no	—
135	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
138	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₂²⋊C₈	—	2	2	no	—
144	C₂²⋊C₈	—	2	2	no	—
145	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
146	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
147	#97	—	2	1	no	—
148	#97	—	2	1	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
152	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
153	C₄×C₈	D₄×C₅₃	1	2	yes	—
154	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
161	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
168	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
177	C₄²	#41×C₅₃	1	2	yes	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
191	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
192	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
193	C₂×C₄	#131×C₅₃	1	4	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	4	4	no	—
197	C₈	—	4	4	no	—
198	C₈	—	4	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	8	2	no	—
203	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
204	C₂²	C₅₃×#405	1	2	yes	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	2	no	—
209	C₄	C₅₃×#408	1	2	yes	—
210	C₄	C₅₃×#433	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	—	8	1	no	—
213	C₂	C₅₃×#1750	1	1	yes	characteristic
214	C₂	C₅₃×#1757	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1452	C₂	semidirect
4. C₅₃×#1450	C₂	semidirect
5. C₅₃×#1281	C₂	semidirect
91. #8158	C₅₃	direct (N × H)
106. #1452	C₂×C₅₃	semidirect
107. #1450	C₂×C₅₃	semidirect
108. #1281	C₂×C₅₃	semidirect
128. C₅₃	#8158	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×#433	2
C₅₃×#1458	2
C₅₃×#1452	2
C₅₃×#1450	2
C₅₃×#1281	2
#8158	53

Subgroups — C53×#8158 #27215670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8158 #27215670