Subgroups — #27295034

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	1	yes	maximal, characteristic, complemented
2	#20274843	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272724	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
5	#16810493	—	2	1	no	—
6	C₄⋊#11719908	—	2	1	no	—
7	C₂²⋊#11719908	C₂²	1	2	yes	—
8	C₄×#11719908	—	2	1	no	—
9	#16810386	—	2	1	no	—
10	#16810386	C₂²	1	2	yes	—
11	C₅₃×#532	—	2	1	no	—
12	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
13	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
14	C₅₃×#486	—	2	1	no	—
15	C₅₃×#466	—	2	1	no	—
16	C₅₃×#466	C₂²	1	2	yes	—
17	C₄×D₈×C₅₃	—	2	1	no	—
18	C₄×D₈×C₅₃	C₄	1	2	yes	complemented
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	4	1	no	—
22	C₂×C₄⋊Dic₅₃	—	4	1	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#41×C₅₃	—	4	1	no	—
25	C₂×D₈×C₅₃	—	2	1	no	—
26	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₅₃	—	2	1	no	—
30	#127×C₅₃	—	2	1	no	—
31	#124×C₅₃	—	4	1	no	—
32	#121×C₅₃	—	2	2	no	—
33	#121×C₅₃	—	4	1	no	—
34	C₄×D₄×C₅₃	—	2	2	no	—
35	C₄×D₄×C₅₃	—	4	1	no	—
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	#107×C₅₃	—	2	1	no	—
38	#107×C₅₃	C₂×C₄	1	2	yes	—
39	#102×C₅₃	—	2	2	no	—
40	#102×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	#41×C₅₃	—	8	2	no	—
46	#41×C₅₃	—	8	2	no	—
47	C₂×Q₈×C₅₃	—	4	1	no	—
48	C₂×D₄×C₅₃	—	2	2	no	—
49	C₂×D₄×C₅₃	—	4	2	no	—
50	C₂×D₄×C₅₃	—	4	1	no	—
51	C₂×D₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	4	2	no	—
57	D₈×C₅₃	—	4	2	no	—
58	D₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	1	no	—
60	C₂×C₈×C₅₃	—	4	1	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	4	1	no	—
67	C₂²⋊C₄×C₅₃	—	4	2	no	—
68	C₂²⋊C₄×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
72	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	8	2	no	—
76	C₄⋊Dic₅₃	—	8	2	no	—
77	#11719908	—	4	2	no	—
78	C₂³×C₅₃	—	4	2	no	—
79	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
80	Q₈×C₅₃	—	8	2	no	—
81	D₄×C₅₃	—	4	2	no	—
82	D₄×C₅₃	—	4	2	no	—
83	D₄×C₅₃	—	8	2	no	—
84	D₄×C₅₃	—	8	2	no	—
85	D₄×C₅₃	—	8	2	no	—
86	D₄×C₅₃	—	8	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	8	4	no	—
92	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×Dic₅₃	—	8	2	no	—
101	C₂×Dic₅₃	—	8	2	no	—
102	C₂×Dic₅₃	—	8	2	no	—
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	#11856	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	8	2	no	—
111	C₂²×C₅₃	—	8	2	no	—
112	C₂²×C₅₃	—	8	2	no	—
113	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	8	2	no	—
119	C₄×C₅₃	#357	1	2	yes	—
120	C₄×C₅₃	#357	1	2	yes	—
121	C₄×C₅₃	#360	1	2	yes	—
122	C₄×C₅₃	#410	1	1	yes	characteristic
123	C₄×C₅₃	#421	1	1	yes	characteristic
124	Dic₅₃	—	16	2	no	—
125	#3169	D₅₃	1	1	yes	characteristic
126	#2162	—	53	1	no	—
127	#1834	—	53	1	no	—
128	#1347	—	53	4	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	2	1	no	—
131	C₂×C₅₃	—	16	2	no	—
132	C₂×C₅₃	#1517	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#360	1	1	yes	characteristic
135	#532	—	2	1	no	—
136	#500	Dic₅₃	1	1	yes	characteristic, complemented
137	#493	Dic₅₃	1	1	yes	characteristic, complemented
138	#486	—	2	1	no	—
139	#466	—	2	1	no	—
140	#466	C₂×D₅₃	1	2	yes	—
141	C₄×D₈	—	2	1	no	—
142	C₄×D₈	Dic₅₃	1	2	yes	complemented
143	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
144	#384	—	106	1	no	—
145	C₄⋊C₁₆	—	106	1	no	—
146	C₂²⋊C₁₆	—	53	2	no	—
147	C₄×C₁₆	—	106	1	no	—
148	#340	—	53	2	no	—
149	#340	—	106	1	no	—
150	C₅₃	#11856	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	4	1	no	—
152	C₂×D₈	—	2	1	no	—
153	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
154	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	1	no	—
157	#127	—	2	1	no	—
158	#124	—	4	1	no	—
159	#121	—	2	2	no	—
160	#121	—	4	1	no	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	—	4	1	no	—
163	C₂×C₄⋊C₄	—	106	2	no	—
164	C₂×C₄⋊C₄	—	212	1	no	—
165	C₂×C₄⋊C₄	—	212	1	no	—
166	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₁₆	—	106	2	no	—
168	#107	—	2	1	no	—
169	#107	C₂×Dic₅₃	1	2	yes	—
170	#102	—	2	2	no	—
171	#102	—	2	2	no	—
172	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
173	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
176	#41	—	8	2	no	—
177	#41	—	8	2	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	212	2	no	—
185	C₂²×C₄	#11719922	1	1	yes	characteristic
186	C₂²×C₄	#11719922	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
188	D₈	—	4	2	no	—
189	D₈	—	4	2	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
194	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	424	2	no	—
199	C₄⋊C₄	—	424	2	no	—
200	C₄⋊C₄	—	212	2	no	—
201	C₄⋊C₄	—	212	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	#11719923	1	1	yes	characteristic
205	C₄²	#11719923	1	1	yes	characteristic
206	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₁₆	—	212	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×#11719922	1	1	yes	characteristic
211	Q₈	—	8	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	424	2	no	—
224	C₂×C₄	—	424	2	no	—
225	C₂×C₄	—	424	2	no	—
226	C₂×C₄	#13133138	1	1	yes	characteristic
227	C₂×C₄	#13133138	1	1	yes	characteristic
228	C₂×C₄	#13133138	1	1	yes	characteristic
229	C₂×C₄	#13133138	1	1	yes	characteristic
230	C₂×C₄	#13133158	1	2	yes	—
231	C₂×C₄	#13133161	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₂×#13133138	1	1	yes	characteristic
244	C₂²	C₂×#13133158	1	1	yes	characteristic
245	C₂²	C₂×#13133161	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	848	2	no	—
250	C₄	#16810424	1	2	yes	—
251	C₄	#16810424	1	2	yes	—
252	C₄	#16810433	1	2	yes	—
253	C₄	#16810597	1	1	yes	characteristic
254	C₄	#16810640	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	#20273275	1	1	yes	characteristic
259	C₂	C₂×#16810424	1	1	yes	characteristic
260	C₂	C₂×#16810433	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂×#102
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#11856
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20276514	C₂	semidirect
2. #20274843	C₂	semidirect
3. #20272724	C₂	semidirect
12. C₅₃×#500	C₄	semidirect
13. C₅₃×#493	C₄	semidirect
18. C₄×D₈×C₅₃	C₄	semidirect
136. #500	Dic₅₃	semidirect
137. #493	Dic₅₃	semidirect
142. C₄×D₈	Dic₅₃	semidirect
150. C₅₃	#11856	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20276514	2
#20274843	2
#20272724	2
C₅₃×#3169	2
#11856	53