Subgroups — C₅₃×#9662 #27217174

Order 13568.

482 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	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1984	C₂	1	2	yes	maximal
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₄×Q₁₆×C₅₃	—	2	2	no	—
9	C₄×Q₁₆×C₅₃	—	2	2	no	—
10	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#103×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	4	no	—
17	C₅₃×#335	—	2	2	no	—
18	C₅₃×Q₈⋊C₈	—	2	4	no	—
19	C₅₃×#322	—	2	2	no	—
20	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×Q₁₆×C₅₃	—	2	2	no	—
22	C₂×Q₁₆×C₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₄×Q₈×C₅₃	—	2	2	no	—
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	—	2	2	no	—
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	1	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
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	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
57	Q₁₆×C₅₃	—	2	8	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	4	4	no	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	4	2	no	—
72	C₄⋊C₄×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	—	4	2	no	—
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
80	Q₈×C₅₃	—	2	4	no	—
81	Q₈×C₅₃	—	2	4	no	—
82	Q₈×C₅₃	—	4	2	no	—
83	Q₈×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	#131	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	#135	1	2	yes	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	2	4	no	—
103	C₈×C₅₃	—	4	2	no	—
104	C₈×C₅₃	—	8	2	no	—
105	#9662	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₅₃	#390	1	1	yes	characteristic
107	C₂²×C₅₃	#435	1	2	yes	—
108	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
109	C₂²×C₅₃	#444	1	2	yes	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	4	4	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	#407	1	1	yes	characteristic
115	C₄×C₅₃	#408	1	1	yes	characteristic
116	C₄×C₅₃	#435	1	2	yes	—
117	C₄×C₅₃	#467	1	2	yes	—
118	C₄×C₅₃	#494	1	1	yes	characteristic
119	C₄×C₅₃	#495	1	1	yes	characteristic
120	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
121	#1984	C₂×C₅₃	1	2	yes	—
122	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
123	C₂×Q₈⋊C₈	C₂×C₅₃	1	2	yes	—
124	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
125	C₂×C₅₃	#1584	1	4	yes	—
126	C₂×C₅₃	#1955	1	1	yes	characteristic
127	C₂×C₅₃	#1969	1	1	yes	characteristic
128	C₂×C₅₃	#1975	1	1	yes	characteristic
129	C₂²×Q₁₆	C₄×C₅₃	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
131	C₄×Q₁₆	—	2	2	no	—
132	C₄×Q₁₆	—	2	2	no	—
133	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
135	C₂×#103	C₄×C₅₃	1	2	yes	—
136	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
139	#336	—	2	4	no	—
140	#335	—	2	2	no	—
141	Q₈⋊C₈	—	2	4	no	—
142	#322	—	2	2	no	—
143	C₅₃	#9662	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₅₃	1	2	yes	—
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
160	#107	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	#103	—	2	4	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	D₄×C₅₃	1	2	yes	—
168	C₄×C₈	D₄×C₅₃	1	2	yes	—
169	C₄×C₈	D₄×C₅₃	1	2	yes	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	M₁₆×C₅₃	1	4	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
181	Q₁₆	—	2	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	4	no	—
190	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
191	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
192	C₂×C₈	Q₁₆×C₅₃	1	4	yes	complemented
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₄²	#41×C₅₃	1	1	yes	characteristic
203	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
215	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
216	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
217	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
218	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
219	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
220	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
221	C₂×C₄	#131×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
223	C₂×C₄	#135×C₅₃	1	2	yes	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₈	—	8	2	no	—
229	C₂²	C₅₃×#390	1	1	yes	characteristic
230	C₂²	C₅₃×#435	1	2	yes	—
231	C₂²	C₄×Q₁₆×C₅₃	1	2	yes	—
232	C₂²	C₅₃×#444	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	C₅₃×#407	1	1	yes	characteristic
238	C₄	C₅₃×#408	1	1	yes	characteristic
239	C₄	C₅₃×#435	1	2	yes	—
240	C₄	C₅₃×#467	1	2	yes	—
241	C₄	C₅₃×#494	1	1	yes	characteristic
242	C₄	C₅₃×#495	1	1	yes	characteristic
243	C₂	C₅₃×#1584	1	4	yes	—
244	C₂	C₅₃×#1955	1	1	yes	characteristic
245	C₂	C₅₃×#1969	1	1	yes	characteristic
246	C₂	C₅₃×#1975	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
68. C₂×C₈×C₅₃	Q₁₆	semidirect
105. #9662	C₅₃	direct (N × H)
143. C₅₃	#9662	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
C₂×C₄×Q₁₆×C₅₃	2
C₅₃×#1984	2
C₂×C₅₃×#335	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×C₅₃×#322	2
#9662	53

Subgroups — C53×#9662 #27217174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9662 #27217174