Subgroups — C₅₃×#11958 #27219470

Order 13568.

404 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×#3147	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1360	C₂	1	4	yes	maximal, complemented
5	C₂×#135×C₅₃	—	2	1	no	—
6	C₂×#135×C₅₃	C₄	1	2	yes	—
7	C₅₃×#532	—	2	1	no	—
8	C₅₃×#495	—	2	1	no	—
9	C₅₃×#495	C₂²	1	2	yes	—
10	C₅₃×#494	C₄	1	1	yes	characteristic
11	C₅₃×#493	C₄	1	1	yes	characteristic
12	C₅₃×#492	—	2	1	no	—
13	C₅₃×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₁₆×C₅₃	—	2	1	no	—
18	C₅₃×#334	—	2	1	no	—
19	C₅₃×#334	C₂²	1	2	yes	—
20	C₂×#41×C₅₃	—	2	2	no	—
21	#135×C₅₃	—	4	4	no	—
22	#135×C₅₃	—	4	4	no	—
23	C₂×Q₁₆×C₅₃	—	2	1	no	—
24	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×SD₁₆×C₅₃	—	2	2	no	—
26	C₂×SD₁₆×C₅₃	—	2	2	no	—
27	C₂×D₈×C₅₃	—	2	1	no	—
28	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₅₃	—	4	1	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₃	—	2	1	no	—
33	#127×C₅₃	—	2	1	no	—
34	#124×C₅₃	—	2	2	no	—
35	#121×C₅₃	—	4	1	no	—
36	#121×C₅₃	—	4	1	no	—
37	C₄×D₄×C₅₃	—	4	1	no	—
38	C₄×D₄×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	4	1	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	#41×C₅₃	—	4	4	no	—
48	#41×C₅₃	—	4	4	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	4	1	no	—
53	C₂×D₄×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
58	Q₁₆×C₅₃	—	4	2	no	—
59	Q₁₆×C₅₃	—	4	2	no	—
60	SD₁₆×C₅₃	—	4	4	no	—
61	SD₁₆×C₅₃	—	4	4	no	—
62	D₈×C₅₃	—	4	2	no	—
63	D₈×C₅₃	—	4	2	no	—
64	M₁₆×C₅₃	—	8	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	—	4	1	no	—
73	C₄⋊C₄×C₅₃	—	4	1	no	—
74	C₂²⋊C₄×C₅₃	—	4	2	no	—
75	C₂²⋊C₄×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₅₃	D₈	1	1	yes	characteristic
78	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₁₆×C₅₃	—	4	2	no	—
81	C₂³×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	#102	1	1	yes	characteristic
83	Q₈×C₅₃	—	4	2	no	—
84	Q₈×C₅₃	—	4	2	no	—
85	D₄×C₅₃	—	4	2	no	—
86	D₄×C₅₃	—	4	2	no	—
87	D₄×C₅₃	—	8	4	no	—
88	D₄×C₅₃	—	8	2	no	—
89	D₄×C₅₃	—	8	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	8	4	no	—
96	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₈×C₅₃	—	2	1	no	—
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	—	8	2	no	—
105	C₈×C₅₃	#100	1	2	yes	—
106	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
107	#11958	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	8	2	no	—
112	C₂²×C₅₃	—	8	2	no	—
113	C₂²×C₅₃	—	8	2	no	—
114	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	8	2	no	—
119	C₄×C₅₃	#359	1	2	yes	—
120	C₄×C₅₃	#360	1	2	yes	—
121	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
122	C₄×C₅₃	#418	1	1	yes	characteristic
123	#3147	C₂×C₅₃	1	1	yes	characteristic
124	#2162	C₂×C₅₃	1	1	yes	characteristic, complemented
125	#1566	C₂×C₅₃	1	1	yes	characteristic, complemented
126	#1360	C₂×C₅₃	1	4	yes	complemented
127	C₂×C₅₃	—	4	1	no	—
128	C₂×C₅₃	—	16	2	no	—
129	C₂×C₅₃	#1545	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#360	1	1	yes	characteristic
132	C₂×#135	—	2	1	no	—
133	C₂×#135	C₄×C₅₃	1	2	yes	—
134	#532	—	2	1	no	—
135	#495	—	2	1	no	—
136	#495	C₂²×C₅₃	1	2	yes	—
137	#494	C₄×C₅₃	1	1	yes	characteristic
138	#493	C₄×C₅₃	1	1	yes	characteristic
139	#492	—	2	1	no	—
140	#423	—	2	1	no	—
141	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
142	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
143	C₂²⋊C₁₆	—	2	1	no	—
144	C₄×C₁₆	—	2	1	no	—
145	#334	—	2	1	no	—
146	#334	C₂²×C₅₃	1	2	yes	—
147	C₅₃	#11958	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×#41	—	2	2	no	—
149	#135	—	4	4	no	—
150	#135	—	4	4	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
153	C₂×SD₁₆	—	2	2	no	—
154	C₂×SD₁₆	—	2	2	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
157	C₂×M₁₆	—	4	1	no	—
158	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	1	no	—
161	#127	—	2	1	no	—
162	#124	—	2	2	no	—
163	#121	—	4	1	no	—
164	#121	—	4	1	no	—
165	C₄×D₄	—	4	1	no	—
166	C₄×D₄	—	4	1	no	—
167	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
168	C₂×C₁₆	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	4	1	no	—
171	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
172	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
173	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
174	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
175	#41	—	4	4	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
186	Q₁₆	—	4	2	no	—
187	Q₁₆	—	4	2	no	—
188	SD₁₆	—	4	4	no	—
189	SD₁₆	—	4	4	no	—
190	D₈	—	4	2	no	—
191	D₈	—	4	2	no	—
192	M₁₆	—	8	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
205	C₄²	D₈×C₅₃	1	1	yes	characteristic
206	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₁₆	—	4	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	#102×C₅₃	1	1	yes	characteristic
211	Q₈	—	4	2	no	—
212	Q₈	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
225	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
226	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	8	2	no	—
233	C₈	#100×C₅₃	1	2	yes	—
234	C₈	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	8	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	8	2	no	—
246	C₄	C₅₃×#359	1	2	yes	—
247	C₄	C₅₃×#360	1	2	yes	—
248	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
249	C₄	C₅₃×#418	1	1	yes	characteristic
250	C₂	—	4	1	no	—
251	C₂	—	16	2	no	—
252	C₂	C₅₃×#1545	1	1	yes	characteristic
253	C₂	C₂×C₅₃×#359	1	1	yes	characteristic
254	C₂	C₂×C₅₃×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₅₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2162	C₂	semidirect
3. C₅₃×#1566	C₂	semidirect
4. C₅₃×#1360	C₂	semidirect
107. #11958	C₅₃	direct (N × H)
124. #2162	C₂×C₅₃	semidirect
125. #1566	C₂×C₅₃	semidirect
126. #1360	C₂×C₅₃	semidirect
147. C₅₃	#11958	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×#3147	2
C₅₃×#2162	2
C₅₃×#1566	2
C₅₃×#1360	2
#11958	53

Subgroups — C53×#11958 #27219470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11958 #27219470