Subgroups — C₅₃×#17977 #27225489

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#2960	C₂	1	2	yes	maximal
3	C₅₃×#1584	C₂	1	8	yes	maximal, complemented
4	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1493	C₂	1	2	yes	maximal
6	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂²	1	2	yes	—
8	C₅₃×#500	C₄	1	1	yes	characteristic
9	C₅₃×#494	C₄	1	1	yes	characteristic
10	C₅₃×#487	C₄	1	2	yes	—
11	C₅₃×#467	C₄	1	4	yes	—
12	C₅₃×#446	—	2	2	no	—
13	C₅₃×#446	C₂²	1	4	yes	—
14	C₈×Q₈×C₅₃	—	2	2	no	—
15	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#424	—	2	2	no	—
18	C₅₃×#423	C₂²	1	2	yes	—
19	C₅₃×#405	—	2	2	no	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#335	C₂²	1	4	yes	—
24	C₅₃×Q₈⋊C₈	C₂²	1	8	yes	—
25	C₅₃×#322	C₂²	1	4	yes	—
26	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
34	#125×C₅₃	—	2	2	no	—
35	C₂²⋊Q₈×C₅₃	—	2	2	no	—
36	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₅₃	—	2	2	no	—
38	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
39	#117×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#107×C₅₃	C₂×C₄	1	4	yes	—
42	C₄⋊C₈×C₅₃	—	2	4	no	—
43	C₄⋊C₈×C₅₃	—	2	4	no	—
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
46	#103×C₅₃	C₂×C₄	1	8	yes	—
47	#97×C₅₃	—	2	4	no	—
48	C₄×C₈×C₅₃	—	2	4	no	—
49	C₄×C₈×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	C₄×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	Q₁₆×C₅₃	—	4	4	no	—
62	M₁₆×C₅₃	—	4	4	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂²⋊C₄×C₅₃	—	2	4	no	—
80	C₄²×C₅₃	—	2	4	no	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	#41	1	1	yes	characteristic
84	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₅₃	—	2	8	no	—
87	Q₈×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₅₃	#135	1	2	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	—	4	2	no	—
104	C₈×C₅₃	—	4	4	no	—
105	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
106	#17977	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
110	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	4	4	no	—
114	C₄×C₅₃	—	4	4	no	—
115	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
116	C₄×C₅₃	#444	1	2	yes	—
117	C₄×C₅₃	#568	1	2	yes	—
118	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
119	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
120	#3170	C₂×C₅₃	1	1	yes	characteristic
121	#2960	C₂×C₅₃	1	2	yes	—
122	#1584	C₂×C₅₃	1	8	yes	complemented
123	#1567	C₂×C₅₃	1	1	yes	characteristic
124	#1493	C₂×C₅₃	1	2	yes	—
125	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	#2932	1	1	yes	characteristic
128	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
130	#533	C₂²×C₅₃	1	2	yes	—
131	#500	C₄×C₅₃	1	1	yes	characteristic
132	#494	C₄×C₅₃	1	1	yes	characteristic
133	#487	C₄×C₅₃	1	2	yes	—
134	#467	C₄×C₅₃	1	4	yes	—
135	#446	—	2	2	no	—
136	#446	C₂²×C₅₃	1	4	yes	—
137	C₈×Q₈	—	2	2	no	—
138	C₈×Q₈	C₂²×C₅₃	1	4	yes	—
139	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	—
140	#424	—	2	2	no	—
141	#423	C₂²×C₅₃	1	2	yes	—
142	#405	—	2	2	no	—
143	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
146	#335	C₂²×C₅₃	1	4	yes	—
147	Q₈⋊C₈	C₂²×C₅₃	1	8	yes	—
148	#322	C₂²×C₅₃	1	4	yes	—
149	C₅₃	#17977	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
154	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
155	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
156	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
157	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
158	#125	—	2	2	no	—
159	C₂²⋊Q₈	—	2	2	no	—
160	C₂²⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
163	#117	—	2	2	no	—
164	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
165	#107	C₂×C₄×C₅₃	1	4	yes	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
170	#103	C₂×C₄×C₅₃	1	8	yes	—
171	#97	—	2	4	no	—
172	C₄×C₈	—	2	4	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
175	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
176	C₄×C₈	D₄×C₅₃	1	2	yes	—
177	C₄×C₈	C₂³×C₅₃	1	2	yes	—
178	C₄×C₈	C₂³×C₅₃	1	2	yes	—
179	C₄×C₈	C₂³×C₅₃	1	2	yes	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
182	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
185	Q₁₆	—	4	4	no	—
186	M₁₆	—	4	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	Q₁₆×C₅₃	1	4	yes	complemented
193	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
194	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
206	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₄²	#41×C₅₃	1	1	yes	characteristic
208	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
209	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
210	Q₈	—	2	8	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
218	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
220	C₂×C₄	#135×C₅₃	1	2	yes	—
221	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	4	4	no	—
229	C₈	C₂×Q₁₆×C₅₃	1	4	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₄×Q₁₆×C₅₃	1	2	yes	—
233	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	4	no	—
237	C₄	—	4	4	no	—
238	C₄	C₄×Q₁₆×C₅₃	1	2	yes	—
239	C₄	C₅₃×#444	1	2	yes	—
240	C₄	C₅₃×#568	1	2	yes	—
241	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
242	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	C₅₃×#2932	1	1	yes	characteristic
245	C₂	C₂×C₄×Q₁₆×C₅₃	1	1	yes	characteristic
246	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ 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×#1584	C₂	semidirect
68. C₂×C₈×C₅₃	Q₁₆	semidirect
106. #17977	C₅₃	direct (N × H)
122. #1584	C₂×C₅₃	semidirect
149. C₅₃	#17977	direct (N × H)
192. C₂×C₈	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×#2960	2
C₅₃×#1584	2
C₅₃×#1567	2
C₅₃×#1493	2
C₂×C₅₃×#322	2
#17977	53

Subgroups — C53×#17977 #27225489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17977 #27225489