Subgroups — C₅₃×#9598 #27217110

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1939	C₂	1	2	yes	maximal
2	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1279	C₂	1	2	yes	maximal
6	C₂×#107×C₅₃	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#340	C₄	1	1	yes	characteristic
14	C₅₃×#340	C₄	1	1	yes	characteristic
15	C₅₃×#336	—	2	2	no	—
16	C₅₃×#335	—	2	2	no	—
17	C₅₃×#322	—	2	2	no	—
18	C₂²×C₈×C₅₃	—	2	2	no	—
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	—	2	2	no	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#107×C₅₃	—	2	4	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₄⋊C₈×C₅₃	—	4	4	no	—
32	C₄×C₈×C₅₃	—	2	1	no	—
33	C₄×C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	#95×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	D₈	1	2	yes	—
58	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
63	C₄²×C₅₃	—	4	2	no	—
64	C₄²×C₅₃	—	4	2	no	—
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₅₃	#41	1	1	yes	characteristic
69	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₅₃	#131	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	—	2	2	no	—
93	C₈×C₅₃	—	4	2	no	—
94	C₈×C₅₃	—	8	2	no	—
95	#9598	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₅₃	#389	1	1	yes	characteristic
97	C₂²×C₅₃	#435	1	2	yes	—
98	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
99	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
100	C₂²×C₅₃	#444	1	2	yes	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	4	4	no	—
104	C₄×C₅₃	—	8	2	no	—
105	C₄×C₅₃	#423	1	1	yes	characteristic
106	C₄×C₅₃	#424	1	1	yes	characteristic
107	C₄×C₅₃	#446	1	2	yes	—
108	C₄×C₅₃	#466	1	1	yes	characteristic
109	C₄×C₅₃	#467	1	1	yes	characteristic
110	C₄×C₅₃	#499	1	1	yes	characteristic
111	C₄×C₅₃	#500	1	1	yes	characteristic
112	#1939	C₂×C₅₃	1	2	yes	—
113	C₄×#107	C₂×C₅₃	1	1	yes	characteristic
114	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
115	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
116	#1279	C₂×C₅₃	1	2	yes	—
117	C₂×C₅₃	#1581	1	2	yes	—
118	C₂×C₅₃	#1584	1	2	yes	—
119	C₂×C₅₃	#1940	1	1	yes	characteristic
120	C₂×C₅₃	#1942	1	1	yes	characteristic
121	C₂×C₅₃	#1946	1	1	yes	characteristic
122	C₂×#107	C₄×C₅₃	1	2	yes	—
123	C₂×C₄⋊C₈	—	2	2	no	—
124	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
128	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
129	#340	C₄×C₅₃	1	1	yes	characteristic
130	#340	C₄×C₅₃	1	1	yes	characteristic
131	#336	—	2	2	no	—
132	#335	—	2	2	no	—
133	#322	—	2	2	no	—
134	C₅₃	#9598	1	1	yes	characteristic, direct product factor, Sylow
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
140	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
145	#107	—	2	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	4	4	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	D₄×C₅₃	1	2	yes	—
153	C₄×C₈	Q₈×C₅₃	1	2	yes	—
154	C₄×C₈	Q₈×C₅₃	1	2	yes	—
155	#95	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
160	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
173	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
174	C₂×C₈	D₈×C₅₃	1	2	yes	—
175	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
180	C₄²	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
183	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
184	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
185	C₄²	#41×C₅₃	1	1	yes	characteristic
186	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
195	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
196	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
198	C₂×C₄	#125×C₅₃	1	1	yes	characteristic
199	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
200	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
201	C₂×C₄	#131×C₅₃	1	2	yes	—
202	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
203	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
204	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
205	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	8	2	no	—
212	C₂²	C₅₃×#389	1	1	yes	characteristic
213	C₂²	C₅₃×#435	1	2	yes	—
214	C₂²	C₄×D₈×C₅₃	1	1	yes	characteristic
215	C₂²	C₄×Q₁₆×C₅₃	1	1	yes	characteristic
216	C₂²	C₅₃×#444	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	C₅₃×#423	1	1	yes	characteristic
222	C₄	C₅₃×#424	1	1	yes	characteristic
223	C₄	C₅₃×#446	1	2	yes	—
224	C₄	C₅₃×#466	1	1	yes	characteristic
225	C₄	C₅₃×#467	1	1	yes	characteristic
226	C₄	C₅₃×#499	1	1	yes	characteristic
227	C₄	C₅₃×#500	1	1	yes	characteristic
228	C₂	C₅₃×#1581	1	2	yes	—
229	C₂	C₅₃×#1584	1	2	yes	—
230	C₂	C₅₃×#1940	1	1	yes	characteristic
231	C₂	C₅₃×#1942	1	1	yes	characteristic
232	C₂	C₅₃×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
95. #9598	C₅₃	direct (N × H)
134. C₅₃	#9598	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×#1939	2
C₄×#107×C₅₃	2
C₂×C₅₃×#335	2
C₂×C₅₃×#322	2
C₅₃×#1279	2
#9598	53

Subgroups — C53×#9598 #27217110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9598 #27217110