Subgroups — C₅₃×#9697 #27217209

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#333	C₂	1	2	yes	maximal
5	C₂×C₅₃×#332	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	—	2	1	no	—
12	C₅₃×#391	—	2	1	no	—
13	C₅₃×#391	—	2	1	no	—
14	C₅₃×#391	C₄	1	4	yes	—
15	C₅₃×#382	—	2	2	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#333	—	2	4	no	—
19	C₅₃×#332	—	2	4	no	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	4	4	no	—
41	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
42	C₄×C₈×C₅₃	—	4	2	no	—
43	#95×C₅₃	—	2	4	no	—
44	#95×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	2	4	no	—
46	C₂×Q₈×C₅₃	—	4	4	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂×Q₈×C₅₃	—	4	1	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	4	4	no	—
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	1	no	—
66	C₄⋊C₄×C₅₃	—	4	1	no	—
67	C₄⋊C₄×C₅₃	—	4	1	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	SD₁₆	1	4	yes	—
74	C₄²×C₅₃	Q₁₆	1	4	yes	—
75	C₄²×C₅₃	C₂×D₄	1	2	yes	—
76	C₄²×C₅₃	#41	1	2	yes	—
77	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₅₃	—	8	4	no	—
79	Q₈×C₅₃	—	8	2	no	—
80	Q₈×C₅₃	—	8	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	#103	1	4	yes	—
92	C₂×C₄×C₅₃	#103	1	4	yes	—
93	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	#121	1	2	yes	—
95	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	#135	1	2	yes	—
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
102	C₈×C₅₃	—	4	4	no	—
103	C₈×C₅₃	—	8	2	no	—
104	#9697	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₅₃	#390	1	1	yes	characteristic
106	C₂²×C₅₃	#413	1	2	yes	—
107	C₂²×C₅₃	C₂×#103	1	2	yes	—
108	C₂²×C₅₃	#416	1	1	yes	characteristic
109	C₂²×C₅₃	#417	1	1	yes	characteristic
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	—	2	4	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	—	8	2	no	—
115	C₄×C₅₃	#463	1	2	yes	—
116	C₄×C₅₃	#464	1	2	yes	—
117	C₄×C₅₃	#487	1	2	yes	—
118	C₄×C₅₃	#492	1	1	yes	characteristic
119	C₄×C₅₃	#494	1	1	yes	characteristic
120	#2862	C₂×C₅₃	1	1	yes	characteristic
121	#1842	C₂×C₅₃	1	1	yes	characteristic
122	C₄×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
123	C₂×#333	C₂×C₅₃	1	2	yes	—
124	C₂×#332	C₂×C₅₃	1	2	yes	—
125	C₂×C₅₃	#1542	1	2	yes	—
126	C₂×C₅₃	#1550	1	2	yes	—
127	C₂×C₅₃	#1956	1	1	yes	characteristic
128	C₂×C₅₃	#1967	1	1	yes	characteristic
129	C₂×C₅₃	#1983	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
132	C₂×C₄⋊C₈	—	2	2	no	—
133	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
134	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
135	C₂×C₄×C₈	—	2	1	no	—
136	#391	—	2	1	no	—
137	#391	—	2	1	no	—
138	#391	C₄×C₅₃	1	4	yes	—
139	#382	—	2	2	no	—
140	C₄³	C₂²×C₅₃	1	1	yes	characteristic
141	#336	—	2	1	no	—
142	#333	—	2	4	no	—
143	#332	—	2	4	no	—
144	C₅₃	#9697	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
157	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
158	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
159	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
160	C₂×C₄²	D₄×C₅₃	1	2	yes	—
161	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	4	4	no	—
166	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
167	C₄×C₈	—	4	2	no	—
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
179	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	4	2	no	—
198	C₄²	SD₁₆×C₅₃	1	4	yes	—
199	C₄²	Q₁₆×C₅₃	1	4	yes	—
200	C₄²	C₂×D₄×C₅₃	1	2	yes	—
201	C₄²	#41×C₅₃	1	2	yes	—
202	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
203	Q₈	—	8	4	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	#103×C₅₃	1	4	yes	—
217	C₂×C₄	#103×C₅₃	1	4	yes	—
218	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
219	C₂×C₄	#121×C₅₃	1	2	yes	—
220	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
221	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
223	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
224	C₂×C₄	#135×C₅₃	1	2	yes	—
225	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
226	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
227	C₈	—	4	4	no	—
228	C₈	—	8	2	no	—
229	C₂²	C₅₃×#390	1	1	yes	characteristic
230	C₂²	C₅₃×#413	1	2	yes	—
231	C₂²	C₂×#103×C₅₃	1	2	yes	—
232	C₂²	C₅₃×#416	1	1	yes	characteristic
233	C₂²	C₅₃×#417	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	2	no	—
239	C₄	C₅₃×#463	1	2	yes	—
240	C₄	C₅₃×#464	1	2	yes	—
241	C₄	C₅₃×#487	1	2	yes	—
242	C₄	C₅₃×#492	1	1	yes	characteristic
243	C₄	C₅₃×#494	1	1	yes	characteristic
244	C₂	C₅₃×#1542	1	2	yes	—
245	C₂	C₅₃×#1550	1	2	yes	—
246	C₂	C₅₃×#1956	1	1	yes	characteristic
247	C₂	C₅₃×#1967	1	1	yes	characteristic
248	C₂	C₅₃×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
104. #9697	C₅₃	direct (N × H)
144. C₅₃	#9697	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×#2862	2
C₅₃×#1842	2
C₄×C₄⋊C₈×C₅₃	2
C₂×C₅₃×#333	2
C₂×C₅₃×#332	2
#9697	53

Subgroups — C53×#9697 #27217209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9697 #27217209