Subgroups — #27302305

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811000	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810423	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₂₄	—	2	1	no	—
6	C₂²×D₄₂₄	C₂²	1	2	yes	—
7	C₂×#13133214	—	2	1	no	—
8	C₂²×#11719908	C₂²	1	2	yes	—
9	#16811000	—	2	2	no	—
10	#16811000	C₄	1	4	yes	complemented
11	#16810423	—	2	8	no	—
12	C₄×#11719908	—	2	2	no	—
13	C₂×#107×C₅₃	—	2	1	no	—
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	—	2	1	no	—
17	C₂²×D₂₁₂	—	2	2	no	—
18	C₂²×D₂₁₂	—	4	2	no	—
19	C₂×D₄₂₄	—	2	4	no	—
20	C₂×D₄₂₄	—	2	2	no	—
21	C₂×D₄₂₄	—	2	4	no	—
22	C₂×D₄₂₄	—	2	4	no	—
23	C₂×D₄₂₄	—	4	2	no	—
24	C₂×D₄₂₄	—	4	2	no	—
25	C₂×D₄₂₄	C₂×C₄	1	4	yes	—
26	#13133214	—	2	4	no	—
27	#13133214	—	4	2	no	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₂×#11719908	D₄	1	4	yes	complemented
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	#107×C₅₃	—	2	4	no	—
37	#107×C₅₃	—	4	2	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂³×D₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	D₈	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×D₂₁₂	—	2	4	no	—
45	C₂×D₂₁₂	—	4	2	no	—
46	C₂×D₂₁₂	—	4	2	no	—
47	C₂×D₂₁₂	—	4	2	no	—
48	C₂×D₂₁₂	—	4	2	no	—
49	C₂×D₂₁₂	—	4	8	no	—
50	C₂×D₂₁₂	—	4	2	no	—
51	C₂×D₂₁₂	—	8	4	no	—
52	C₂×D₂₁₂	—	8	4	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	D₈	1	2	yes	—
55	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	8	4	no	—
63	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₅₃	SD₁₆	1	2	yes	—
65	D₄₂₄	—	2	8	no	—
66	D₄₂₄	—	4	8	no	—
67	D₄₂₄	—	4	4	no	—
68	#11719908	—	2	4	no	—
69	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
70	C₂²×D₅₃	—	4	4	no	—
71	C₂²×D₅₃	—	8	2	no	—
72	C₂²×D₅₃	—	8	2	no	—
73	C₂²×D₅₃	—	8	4	no	—
74	C₂²×D₅₃	—	8	4	no	—
75	C₂×C₄×C₅₃	—	8	2	no	—
76	C₂×C₄×C₅₃	—	8	4	no	—
77	C₂×C₄×C₅₃	#102	1	4	yes	—
78	C₂×C₄×C₅₃	D₁₆	1	4	yes	—
79	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
86	D₂₁₂	—	4	4	no	—
87	D₂₁₂	—	4	4	no	—
88	D₂₁₂	—	8	16	no	—
89	D₂₁₂	—	8	2	no	—
90	D₂₁₂	—	8	2	no	—
91	C₈×C₅₃	#102	1	4	yes	—
92	C₈×C₅₃	C₄×D₄	1	2	yes	—
93	C₈×C₅₃	#121	1	2	yes	—
94	#13202	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
96	C₂²×C₅₃	#486	1	1	yes	characteristic
97	C₂²×C₅₃	#493	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×D₁₆	1	2	yes	—
99	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
100	C₂×D₅₃	—	8	8	no	—
101	C₂×D₅₃	—	8	4	no	—
102	C₂×D₅₃	—	16	4	no	—
103	C₂×D₅₃	—	16	2	no	—
104	C₂×D₅₃	—	16	2	no	—
105	C₄×C₅₃	—	16	2	no	—
106	C₄×C₅₃	#357	1	8	yes	complemented
107	C₄×C₅₃	#390	1	1	yes	characteristic
108	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
109	C₄×C₅₃	#486	1	1	yes	characteristic
110	C₄×C₅₃	#492	1	1	yes	characteristic
111	C₂×#493	—	53	1	no	—
112	C₂×#357	—	53	4	no	—
113	C₂×C₄×C₁₆	—	53	1	no	—
114	#1845	D₅₃	1	1	yes	characteristic, complemented
115	C₂×C₅₃	#1957	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#357	1	2	yes	—
117	C₂×C₅₃	#2240	1	2	yes	—
118	C₂×C₅₃	#2246	1	1	yes	characteristic
119	C₂×C₅₃	#2248	1	1	yes	characteristic
120	D₅₃	—	16	4	no	—
121	C₂²×D₈	—	53	2	no	—
122	C₂²×D₈	—	106	1	no	—
123	C₂×#127	—	106	1	no	—
124	C₂²×C₁₆	—	53	2	no	—
125	#493	—	53	4	no	—
126	#493	—	106	2	no	—
127	C₂×#107	—	2	1	no	—
128	C₂×#107	C₂×D₅₃	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
130	#384	—	2	1	no	—
131	#357	—	106	8	no	—
132	C₄×C₁₆	—	106	2	no	—
133	C₅₃	#13202	1	1	yes	characteristic, complemented, Sylow
134	C₂²×D₄	—	106	2	no	—
135	C₂²×D₄	—	212	2	no	—
136	C₂×D₈	—	53	4	no	—
137	C₂×D₈	—	106	2	no	—
138	C₂×D₈	—	106	4	no	—
139	C₂×D₈	—	106	4	no	—
140	C₂×D₈	—	106	4	no	—
141	C₂×D₈	—	212	2	no	—
142	C₂×D₈	—	212	2	no	—
143	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
145	#127	—	106	4	no	—
146	#127	—	212	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	4	2	no	—
149	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂×C₁₆	—	53	4	no	—
151	C₂×C₁₆	—	106	2	no	—
152	C₂×C₁₆	—	106	2	no	—
153	#107	—	2	4	no	—
154	#107	—	4	2	no	—
155	C₄×C₈	C₄×D₅₃	1	2	yes	—
156	C₄×C₈	D₂₁₂	1	2	yes	—
157	C₂⁴	—	212	2	no	—
158	C₂×D₄	—	106	4	no	—
159	C₂×D₄	—	424	4	no	—
160	C₂×D₄	—	424	4	no	—
161	C₂×D₄	—	212	8	no	—
162	C₂×D₄	—	212	2	no	—
163	C₂×D₄	—	212	2	no	—
164	C₂×D₄	—	212	2	no	—
165	C₂×D₄	—	212	2	no	—
166	C₂×D₄	—	212	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	#11719922	1	2	yes	—
169	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
170	D₈	—	106	8	no	—
171	D₈	—	212	8	no	—
172	D₈	—	212	4	no	—
173	C₂×C₈	#11719921	1	2	yes	—
174	C₂×C₈	#11719922	1	2	yes	—
175	C₂×C₈	#11719924	1	2	yes	—
176	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
177	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
178	C₂×C₈	D₄×D₅₃	1	2	yes	—
179	C₂×C₈	#11719949	1	2	yes	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	8	4	no	—
183	C₄²	#11719921	1	2	yes	—
184	C₄²	#11719924	1	2	yes	—
185	C₁₆	—	106	4	no	—
186	C₂³	—	424	4	no	—
187	C₂³	—	424	4	no	—
188	C₂³	—	424	2	no	—
189	C₂³	—	424	2	no	—
190	C₂³	—	212	4	no	—
191	C₂³	C₂×#11719922	1	1	yes	characteristic
192	D₄	—	424	16	no	—
193	D₄	—	424	2	no	—
194	D₄	—	424	2	no	—
195	D₄	—	212	4	no	—
196	D₄	—	212	4	no	—
197	C₂×C₄	—	8	4	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	#13133137	1	4	yes	—
200	C₂×C₄	#13133158	1	4	yes	—
201	C₂×C₄	#13133160	1	4	yes	—
202	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
203	C₂×C₄	#13133206	1	1	yes	characteristic
204	C₂×C₄	#13133213	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
208	C₈	#13133137	1	4	yes	—
209	C₈	#13133179	1	2	yes	—
210	C₈	#13133192	1	2	yes	—
211	C₂²	—	848	4	no	—
212	C₂²	—	848	2	no	—
213	C₂²	—	848	2	no	—
214	C₂²	—	424	4	no	—
215	C₂²	—	424	8	no	—
216	C₂²	C₂×#13133137	1	1	yes	characteristic
217	C₂²	#16810971	1	1	yes	characteristic
218	C₂²	#16810999	1	1	yes	characteristic
219	C₂²	C₂×#13133158	1	2	yes	—
220	C₂²	C₂×#13133160	1	2	yes	—
221	C₄	—	16	2	no	—
222	C₄	#16810423	1	8	yes	complemented
223	C₄	#16810524	1	1	yes	characteristic
224	C₄	C₂×#13133137	1	1	yes	characteristic
225	C₄	#16810971	1	1	yes	characteristic
226	C₄	#16810997	1	1	yes	characteristic
227	C₂	—	848	4	no	—
228	C₂	#20275520	1	1	yes	characteristic
229	C₂	C₂×#16810423	1	2	yes	—
230	C₂	#20276940	1	2	yes	—
231	C₂	#20276964	1	1	yes	characteristic
232	C₂	#20276972	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1845	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂×#16811000	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#13202
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
2. C₂×#16810423	C₂	semidirect
3. C₂×C₄×#11719908	C₂	semidirect
4. C₅₃×#1845	C₂	semidirect
10. #16811000	C₄	semidirect
30. C₂×#11719908	D₄	semidirect
106. C₄×C₅₃	#357	semidirect
114. #1845	D₅₃	semidirect
133. C₅₃	#13202	semidirect
222. C₄	#16810423	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811000	2
C₂×#16810423	2
C₂×C₄×#11719908	2
C₅₃×#1845	2
#13202	53