Subgroups — #28302121

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133286	C₂	1	1	yes	maximal, characteristic
2	#20290512	C₂	1	2	yes	maximal
3	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20289549	C₂	1	2	yes	maximal
5	#335×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133286	C₄	1	2	yes	—
7	C₂×#13133271	—	2	2	no	—
8	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊#6157	C₂²	1	2	yes	—
11	#16811497	C₄	1	2	yes	—
12	(C₂×#6157)⋊C₈	—	2	2	no	—
13	C₈×#88324	C₄	1	2	yes	complemented
14	C₈×#88324	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810376	C₄	1	1	yes	characteristic, complemented
18	#16810375	C₄	1	2	yes	complemented
19	C₅₃×#335	C₄	1	1	yes	characteristic, complemented
20	C₂×D₅₃⋊C₈	—	2	1	no	—
21	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
22	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₄⋊#6157	C₂×C₄	1	4	yes	—
24	C₂×C₄×#6157	—	2	2	no	—
25	#13133286	—	2	4	no	—
26	#13133271	—	4	4	no	—
27	C₄×#88324	—	2	1	no	—
28	C₄×#88324	C₂×C₄	1	2	yes	—
29	C₄×#88324	C₂×C₄	1	2	yes	—
30	C₄×#88324	Q₈	1	2	yes	—
31	(C₂×#6157)⋊C₄	—	2	2	no	—
32	C₂×C₈×D₅₃	—	2	2	no	—
33	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊#88322	C₂×C₄	1	2	yes	—
37	#13133146	—	2	2	no	—
38	C₄×#88322	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#6157	—	2	4	no	—
44	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
47	C₂×#88324	—	2	2	no	—
48	C₂×#88324	C₄⋊C₄	1	2	yes	—
49	C₂×#88324	C₄⋊C₄	1	2	yes	—
50	C₄⋊#6157	—	2	4	no	—
51	C₄⋊#6157	—	2	4	no	—
52	C₄×#6157	—	4	2	no	—
53	C₄×#6157	—	4	2	no	—
54	D₅₃⋊C₈	—	2	4	no	—
55	D₅₃⋊C₈	—	2	1	no	—
56	D₅₃⋊C₈	—	2	1	no	—
57	D₅₃⋊C₈	D₈	1	2	yes	—
58	D₅₃⋊C₈	Q₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₅₃	#41	1	1	yes	characteristic
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	C₄²	1	1	yes	characteristic
68	C₂×#88322	C₄²	1	1	yes	characteristic
69	C₈×D₅₃	—	2	2	no	—
70	C₈×D₅₃	—	4	2	no	—
71	C₈×D₅₃	—	4	2	no	—
72	C₈×D₅₃	M₁₆	1	4	yes	—
73	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×#6157	—	4	4	no	—
75	C₂×#6157	—	4	4	no	—
76	C₂×#6157	—	4	4	no	—
77	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×Dic₅₃	#117	1	1	yes	characteristic
82	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
83	C₄×D₅₃	—	2	2	no	—
84	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
85	C₄×D₅₃	#131	1	2	yes	—
86	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
87	C₄×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₄×D₅₃	#135	1	1	yes	characteristic
89	C₄×D₅₃	#135	1	1	yes	characteristic
90	#88324	—	2	4	no	—
91	#88324	—	2	2	no	—
92	#88324	#107	1	4	yes	complemented
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	4	2	no	—
95	C₈×C₅₃	#97	1	2	yes	complemented
96	#88322	—	2	1	no	—
97	#88322	—	4	2	no	—
98	#88322	#97	1	2	yes	complemented
99	#8282	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂×D₅₃	#436	1	2	yes	—
102	C₂×D₅₃	C₄×D₈	1	1	yes	characteristic
103	C₂×D₅₃	C₄×Q₁₆	1	1	yes	characteristic
104	C₂×D₅₃	#443	1	2	yes	—
105	#6157	—	8	4	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
108	C₄×C₅₃	#405	1	1	yes	characteristic
109	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
110	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
111	Dic₅₃	—	2	1	no	—
112	Dic₅₃	C₂×#107	1	1	yes	characteristic
113	Dic₅₃	#427	1	1	yes	characteristic
114	Dic₅₃	#446	1	2	yes	—
115	C₄×#107	—	53	1	no	—
116	#1770	—	53	2	no	—
117	C₂×#335	—	53	1	no	—
118	C₂×C₈²	—	53	1	no	—
119	#1279	—	53	2	no	—
120	C₂×C₅₃	#1770	1	1	yes	characteristic
121	C₂×C₅₃	#1772	1	1	yes	characteristic
122	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
123	D₅₃	#1589	1	2	yes	—
124	D₅₃	#1591	1	2	yes	—
125	C₂×#107	—	53	2	no	—
126	C₂×C₄⋊C₈	—	53	2	no	—
127	C₂×#97	—	106	2	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	—	53	1	no	—
131	C₄×C₄⋊C₄	—	53	2	no	—
132	#340	—	53	2	no	—
133	#336	—	106	2	no	—
134	#335	—	53	2	no	—
135	#335	—	53	1	no	—
136	#335	#6157	1	1	yes	characteristic, complemented
137	C₈²	—	53	2	no	—
138	C₈²	—	53	2	no	—
139	C₅₃	#8282	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	53	1	no	—
144	C₂²×C₈	—	106	2	no	—
145	C₂²×C₈	—	106	1	no	—
146	C₂×C₄⋊C₄	—	53	4	no	—
147	C₂×C₄²	—	53	1	no	—
148	C₂×C₄²	—	106	2	no	—
149	#107	—	106	4	no	—
150	C₄⋊C₈	—	53	2	no	—
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	C₂×#6157	1	2	yes	—
153	#97	—	212	4	no	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	53	2	no	—
156	C₄×C₈	—	53	1	no	—
157	C₄×C₈	—	53	1	no	—
158	C₄×C₈	—	53	2	no	—
159	C₄×C₈	—	53	1	no	—
160	C₄×C₈	—	106	1	no	—
161	C₄×C₈	C₂×#6157	1	1	yes	characteristic
162	#95	—	106	2	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	106	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	53	1	no	—
172	C₂×C₈	—	53	4	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	53	1	no	—
175	C₂×C₈	—	106	2	no	—
176	C₂×C₈	—	106	2	no	—
177	C₂×C₈	—	106	1	no	—
178	C₂×C₈	—	106	1	no	—
179	C₂×C₈	—	106	4	no	—
180	C₂×C₈	—	106	2	no	—
181	C₂×C₈	—	212	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	C₄×#6157	1	1	yes	characteristic
184	C₂×C₈	C₄×#6157	1	1	yes	characteristic
185	C₄⋊C₄	—	106	4	no	—
186	C₄⋊C₄	—	106	4	no	—
187	C₄²	—	53	1	no	—
188	C₄²	—	53	1	no	—
189	C₄²	—	53	1	no	—
190	C₄²	—	212	2	no	—
191	C₄²	—	212	2	no	—
192	C₄²	C₂²×#6157	1	1	yes	characteristic
193	C₂³	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	2	no	—
198	C₂×C₄	—	53	1	no	—
199	C₂×C₄	—	53	1	no	—
200	C₂×C₄	—	53	1	no	—
201	C₂×C₄	—	53	1	no	—
202	C₂×C₄	—	53	1	no	—
203	C₂×C₄	—	106	2	no	—
204	C₂×C₄	—	212	4	no	—
205	C₂×C₄	—	212	4	no	—
206	C₂×C₄	—	212	4	no	—
207	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
208	C₂×C₄	D₄×#6157	1	1	yes	characteristic
209	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	53	4	no	—
213	C₈	—	53	2	no	—
214	C₈	—	106	2	no	—
215	C₈	—	106	4	no	—
216	C₈	—	106	1	no	—
217	C₈	—	212	2	no	—
218	C₈	#13133271	1	2	yes	complemented
219	C₂²	—	53	2	no	—
220	C₂²	—	53	2	no	—
221	C₂²	—	53	1	no	—
222	C₂²	—	53	1	no	—
223	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	53	2	no	—
226	C₄	—	53	1	no	—
227	C₄	—	53	1	no	—
228	C₄	—	106	1	no	—
229	C₄	—	424	4	no	—
230	C₄	C₂×#13133271	1	1	yes	characteristic
231	C₄	#16811589	1	1	yes	characteristic
232	C₄	D₈×#6157	1	1	yes	characteristic
233	C₄	Q₁₆×#6157	1	1	yes	characteristic
234	C₂	—	53	2	no	—
235	C₂	—	53	2	no	—
236	C₂	#20290512	1	1	yes	characteristic
237	C₂	#20290520	1	1	yes	characteristic
238	C₂	#107×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×D₅₃	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	#8282
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
13. C₈×#88324	C₄	semidirect
14. C₈×#88324	C₄	semidirect
17. #16810376	C₄	semidirect
18. #16810375	C₄	semidirect
19. C₅₃×#335	C₄	semidirect
92. #88324	#107	semidirect
95. C₈×C₅₃	#97	semidirect
98. #88322	#97	semidirect
136. #335	#6157	semidirect
139. C₅₃	#8282	semidirect
218. C₈	#13133271	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133286	2
#20290512	2
C₈×D₅₃⋊C₈	2
#20289549	2
#335×D₅₃	2
#8282	53