Subgroups — C₅₃×#9845 #27217357

Order 13568.

478 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×#3170	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1994	C₂	1	2	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1493	C₂	1	2	yes	maximal
6	C₅₃×#533	C₂²	1	2	yes	—
7	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
8	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
9	C₅₃×#487	C₄	1	2	yes	complemented
10	C₅₃×#467	—	2	2	no	—
11	C₄×Q₁₆×C₅₃	—	2	2	no	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#339	—	2	4	no	—
18	C₅₃×#335	—	2	2	no	—
19	C₅₃×Q₈⋊C₈	—	2	4	no	—
20	C₂×Q₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	1	no	—
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
26	#125×C₅₃	—	2	2	no	—
27	C₂²⋊Q₈×C₅₃	—	2	2	no	—
28	C₂²⋊Q₈×C₅₃	—	2	2	no	—
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	#117×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#107×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	#103×C₅₃	—	2	4	no	—
40	C₄×C₈×C₅₃	—	2	8	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
52	Q₁₆×C₅₃	—	4	4	no	—
53	M₁₆×C₅₃	—	4	4	no	—
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	—	4	2	no	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₂²⋊C₄×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	—	2	4	no	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	Q₁₆	1	4	yes	complemented
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₂³×C₅₃	#121	1	1	yes	characteristic
77	Q₈×C₅₃	—	4	4	no	—
78	Q₈×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₅₃	#135	1	2	yes	—
94	C₈×C₅₃	—	2	4	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	—	8	2	no	—
97	C₈×C₅₃	#104	1	4	yes	—
98	#9845	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	#390	1	1	yes	characteristic
102	C₂²×C₅₃	#467	1	2	yes	—
103	C₄×C₅₃	—	2	8	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	8	2	no	—
107	C₄×C₅₃	—	8	2	no	—
108	C₄×C₅₃	C₂×#104	1	2	yes	—
109	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
110	C₄×C₅₃	#443	1	2	yes	—
111	C₄×C₅₃	#494	1	1	yes	characteristic
112	C₄×C₅₃	#495	1	1	yes	characteristic
113	#3170	C₂×C₅₃	1	1	yes	characteristic
114	#1994	C₂×C₅₃	1	2	yes	—
115	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
116	#1567	C₂×C₅₃	1	1	yes	characteristic
117	#1493	C₂×C₅₃	1	2	yes	—
118	C₂×C₅₃	—	2	2	no	—
119	C₂×C₅₃	#1962	1	1	yes	characteristic
120	C₂×C₅₃	#1969	1	1	yes	characteristic
121	C₂×C₅₃	#1974	1	1	yes	characteristic
122	#533	C₂²×C₅₃	1	2	yes	—
123	#500	C₄×C₅₃	1	1	yes	characteristic, complemented
124	#494	C₄×C₅₃	1	1	yes	characteristic, complemented
125	#487	C₄×C₅₃	1	2	yes	complemented
126	#467	—	2	2	no	—
127	C₄×Q₁₆	—	2	2	no	—
128	#423	C₂²×C₅₃	1	2	yes	—
129	C₂×C₄×C₈	—	2	2	no	—
130	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
132	C₄³	C₂²×C₅₃	1	1	yes	characteristic
133	#339	—	2	4	no	—
134	#335	—	2	2	no	—
135	Q₈⋊C₈	—	2	4	no	—
136	C₅₃	#9845	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₂²×C₈	D₄×C₅₃	1	2	yes	—
141	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
142	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
143	#125	—	2	2	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₂²⋊Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	#117	—	2	2	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	D₄×C₅₃	1	2	yes	—
152	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
153	#107	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	#103	—	2	4	no	—
157	C₄×C₈	—	2	8	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
161	C₄×C₈	D₄×C₅₃	1	2	yes	—
162	C₄×C₈	D₄×C₅₃	1	2	yes	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
168	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
169	Q₁₆	—	4	4	no	—
170	M₁₆	—	4	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
176	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	Q₁₆×C₅₃	1	4	yes	complemented
189	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
190	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
192	C₄²	#41×C₅₃	1	1	yes	characteristic
193	C₂³	#121×C₅₃	1	1	yes	characteristic
194	Q₈	—	4	4	no	—
195	Q₈	—	4	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
204	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
205	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
206	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
207	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
208	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
210	C₂×C₄	#135×C₅₃	1	2	yes	—
211	C₈	—	2	4	no	—
212	C₈	—	2	2	no	—
213	C₈	—	8	2	no	—
214	C₈	#104×C₅₃	1	4	yes	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	C₅₃×#390	1	1	yes	characteristic
218	C₂²	C₅₃×#467	1	2	yes	—
219	C₄	—	2	8	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	8	2	no	—
223	C₄	—	8	2	no	—
224	C₄	C₂×#104×C₅₃	1	2	yes	—
225	C₄	C₄×Q₁₆×C₅₃	1	2	yes	—
226	C₄	C₅₃×#443	1	2	yes	—
227	C₄	C₅₃×#494	1	1	yes	characteristic
228	C₄	C₅₃×#495	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₅₃×#1962	1	1	yes	characteristic
231	C₂	C₅₃×#1969	1	1	yes	characteristic
232	C₂	C₅₃×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₅₃×#500	C₄	semidirect
8. C₅₃×#494	C₄	semidirect
9. C₅₃×#487	C₄	semidirect
71. C₄²×C₅₃	Q₁₆	semidirect
98. #9845	C₅₃	direct (N × H)
123. #500	C₄×C₅₃	semidirect
124. #494	C₄×C₅₃	semidirect
125. #487	C₄×C₅₃	semidirect
136. C₅₃	#9845	direct (N × H)
188. C₄²	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3170	2
C₅₃×#1994	2
C₄²×C₈×C₅₃	2
C₅₃×#1567	2
C₅₃×#1493	2
#9845	53

Subgroups — C53×#9845 #27217357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9845 #27217357