Subgroups — C₅₃×#9708 #27217220

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#332	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×#97×C₅₃	—	2	1	no	—
12	C₅₃×#391	—	2	1	no	—
13	C₅₃×#391	—	2	1	no	—
14	C₅₃×#391	—	2	2	no	—
15	C₅₃×#382	C₄	1	4	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#332	—	2	4	no	—
19	C₅₃×#332	—	2	4	no	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×Q₈×C₅₃	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
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	#97×C₅₃	—	4	2	no	—
43	#95×C₅₃	—	2	4	no	—
44	#95×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂×Q₈×C₅₃	—	4	1	no	—
47	C₂×Q₈×C₅₃	—	4	1	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	—	4	1	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	4	4	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	4	1	no	—
70	C₄²×C₅₃	—	2	4	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	C₂×D₄	1	2	yes	—
77	C₄²×C₅₃	#41	1	2	yes	—
78	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	Q₈×C₅₃	—	8	2	no	—
80	Q₈×C₅₃	—	8	2	no	—
81	Q₈×C₅₃	—	8	2	no	—
82	Q₈×C₅₃	—	8	1	no	—
83	Q₈×C₅₃	—	8	1	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	#101	1	4	yes	—
96	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	#121	1	2	yes	—
98	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#135	1	2	yes	—
101	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#137	1	2	yes	—
105	C₈×C₅₃	—	4	4	no	—
106	C₈×C₅₃	—	8	2	no	—
107	#9708	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₅₃	#390	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×#101	1	2	yes	—
110	C₂²×C₅₃	#416	1	1	yes	characteristic
111	C₂²×C₅₃	#417	1	2	yes	—
112	C₂²×C₅₃	#417	1	1	yes	characteristic
113	C₄×C₅₃	—	2	4	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	4	2	no	—
116	C₄×C₅₃	—	8	2	no	—
117	C₄×C₅₃	—	8	2	no	—
118	C₄×C₅₃	#463	1	2	yes	—
119	C₄×C₅₃	#464	1	2	yes	—
120	C₄×C₅₃	#489	1	2	yes	—
121	C₄×C₅₃	#496	1	1	yes	characteristic
122	C₄×C₅₃	#497	1	1	yes	characteristic
123	#2862	C₂×C₅₃	1	1	yes	characteristic
124	#1842	C₂×C₅₃	1	1	yes	characteristic
125	#1767	C₂×C₅₃	1	1	yes	characteristic
126	C₂×#332	C₂×C₅₃	1	2	yes	—
127	C₂×#332	C₂×C₅₃	1	2	yes	—
128	C₂×C₅₃	#1552	1	2	yes	—
129	C₂×C₅₃	#1561	1	2	yes	—
130	C₂×C₅₃	#1959	1	1	yes	characteristic
131	C₂×C₅₃	#1966	1	1	yes	characteristic
132	C₂×C₅₃	#1983	1	1	yes	characteristic
133	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
134	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	2	2	no	—
136	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
138	C₂×#97	—	2	1	no	—
139	#391	—	2	1	no	—
140	#391	—	2	1	no	—
141	#391	—	2	2	no	—
142	#382	C₄×C₅₃	1	4	yes	—
143	C₄³	C₂²×C₅₃	1	1	yes	characteristic
144	#336	—	2	1	no	—
145	#332	—	2	4	no	—
146	#332	—	2	4	no	—
147	C₅₃	#9708	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×Q₈	—	2	1	no	—
149	C₂²×Q₈	—	2	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
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	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
161	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
162	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
163	C₂×C₄²	D₄×C₅₃	1	2	yes	—
164	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	4	4	no	—
169	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
170	#97	—	4	2	no	—
171	#95	—	2	4	no	—
172	#95	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	1	no	—
204	C₄²	C₂×D₄×C₅₃	1	2	yes	—
205	C₄²	#41×C₅₃	1	2	yes	—
206	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
207	Q₈	—	8	2	no	—
208	Q₈	—	8	1	no	—
209	Q₈	—	8	1	no	—
210	Q₈	—	8	2	no	—
211	Q₈	—	8	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	#101×C₅₃	1	4	yes	—
224	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
225	C₂×C₄	#121×C₅₃	1	2	yes	—
226	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
227	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
228	C₂×C₄	#135×C₅₃	1	2	yes	—
229	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
230	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
231	C₂×C₄	#137×C₅₃	1	2	yes	—
232	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
233	C₈	—	4	4	no	—
234	C₈	—	8	2	no	—
235	C₂²	C₅₃×#390	1	1	yes	characteristic
236	C₂²	C₂×#101×C₅₃	1	2	yes	—
237	C₂²	C₅₃×#416	1	1	yes	characteristic
238	C₂²	C₅₃×#417	1	2	yes	—
239	C₂²	C₅₃×#417	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	C₅₃×#463	1	2	yes	—
246	C₄	C₅₃×#464	1	2	yes	—
247	C₄	C₅₃×#489	1	2	yes	—
248	C₄	C₅₃×#496	1	1	yes	characteristic
249	C₄	C₅₃×#497	1	1	yes	characteristic
250	C₂	C₅₃×#1552	1	2	yes	—
251	C₂	C₅₃×#1561	1	2	yes	—
252	C₂	C₅₃×#1959	1	1	yes	characteristic
253	C₂	C₅₃×#1966	1	1	yes	characteristic
254	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
107. #9708	C₅₃	direct (N × H)
147. C₅₃	#9708	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
C₅₃×#1767	2
C₂×C₅₃×#332	2
#9708	53

Subgroups — C53×#9708 #27217220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9708 #27217220