Subgroups — #27300144

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810425	C₂	1	2	yes	maximal
3	#20274879	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272807	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₂₁₂	—	2	1	no	—
8	C₄×D₄₂₄	—	2	2	no	—
9	C₄×D₄₂₄	C₄	1	4	yes	complemented
10	C₂×#13133151	C₂²	1	2	yes	—
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133149	—	2	1	no	—
13	C₂×#13133139	—	2	1	no	—
14	#16810489	—	2	1	no	—
15	#16810425	—	2	4	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×C₄⋊C₁₆	—	2	2	no	—
19	C₂²×D₂₁₂	—	2	1	no	—
20	C₂²×C₄×D₅₃	—	4	1	no	—
21	C₂×D₄₂₄	—	2	1	no	—
22	C₂×D₄₂₄	—	2	1	no	—
23	C₂×D₄₂₄	—	2	1	no	—
24	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
25	C₂×D₄₂₄	D₄	1	2	yes	—
26	C₂×D₄₂₄	Q₈	1	2	yes	—
27	C₄×D₂₁₂	—	2	4	no	—
28	C₄×D₂₁₂	—	4	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×#11719921	—	4	1	no	—
32	#13133151	—	2	1	no	—
33	#13133151	—	2	2	no	—
34	#13133151	—	2	2	no	—
35	#13133151	C₂×C₄	1	2	yes	—
36	#13133149	—	4	2	no	—
37	#13133139	—	2	4	no	—
38	#13133124	—	4	1	no	—
39	#13133124	—	4	1	no	—
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₂³×D₅₃	—	4	1	no	—
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	4	1	no	—
53	C₂²×Dic₅₃	—	4	2	no	—
54	C₂×D₂₁₂	—	2	2	no	—
55	C₂×D₂₁₂	—	2	2	no	—
56	C₂×D₂₁₂	—	2	2	no	—
57	C₂×D₂₁₂	—	4	1	no	—
58	C₂×D₂₁₂	—	4	1	no	—
59	C₂×D₂₁₂	—	4	1	no	—
60	C₂×C₄×D₅₃	—	4	4	no	—
61	C₂×C₄×D₅₃	—	8	4	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
70	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	SD₁₆	1	2	yes	—
73	#11719921	—	4	4	no	—
74	C₄⋊Dic₅₃	—	2	2	no	—
75	C₄⋊Dic₅₃	—	4	1	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄⋊Dic₅₃	—	4	2	no	—
78	D₄₂₄	—	2	2	no	—
79	D₄₂₄	—	2	2	no	—
80	D₄₂₄	—	2	2	no	—
81	D₄₂₄	C₄⋊C₄	1	4	yes	—
82	C₁₆×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₅₃	—	4	2	no	—
85	C₂²×D₅₃	—	4	2	no	—
86	C₂²×D₅₃	—	4	2	no	—
87	C₂²×D₅₃	—	8	2	no	—
88	C₂²×D₅₃	—	8	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	#102	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
99	C₂×Dic₅₃	—	4	2	no	—
100	C₂×Dic₅₃	—	8	2	no	—
101	C₂×Dic₅₃	—	8	2	no	—
102	C₂×Dic₅₃	—	8	2	no	—
103	C₂×Dic₅₃	—	8	2	no	—
104	D₂₁₂	—	2	4	no	—
105	D₂₁₂	—	4	2	no	—
106	D₂₁₂	—	4	2	no	—
107	D₂₁₂	—	4	2	no	—
108	C₄×D₅₃	—	8	8	no	—
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₈×C₅₃	#117	1	1	yes	characteristic
112	C₈×C₅₃	#121	1	1	yes	characteristic
113	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
114	#12813	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
116	C₂²×C₅₃	#459	1	1	yes	characteristic
117	C₂²×C₅₃	#474	1	1	yes	characteristic
118	C₂²×C₅₃	#508	1	2	yes	—
119	C₂²×C₅₃	#509	1	2	yes	—
120	C₂×D₅₃	—	4	4	no	—
121	C₂×D₅₃	—	8	2	no	—
122	C₂×D₅₃	—	8	2	no	—
123	C₂×D₅₃	—	8	4	no	—
124	C₂×D₅₃	—	8	4	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	#380	1	1	yes	characteristic
128	C₄×C₅₃	#417	1	1	yes	characteristic
129	C₄×C₅₃	#461	1	1	yes	characteristic
130	C₄×C₅₃	#476	1	1	yes	characteristic
131	Dic₅₃	—	8	2	no	—
132	Dic₅₃	—	16	2	no	—
133	C₂×C₄×D₈	—	53	1	no	—
134	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
135	C₂×#357	—	53	2	no	—
136	#1846	—	53	1	no	—
137	#1380	—	53	2	no	—
138	C₂×C₅₃	#1802	1	1	yes	characteristic
139	C₂×C₅₃	#2138	1	1	yes	characteristic
140	C₂×C₅₃	#2140	1	1	yes	characteristic
141	C₂×C₅₃	#2210	1	2	yes	—
142	C₂×C₅₃	#2228	1	2	yes	—
143	D₅₃	—	8	4	no	—
144	C₂²×D₈	—	53	1	no	—
145	C₂×C₄×D₄	—	106	1	no	—
146	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
147	C₄×D₈	—	53	4	no	—
148	C₄×D₈	—	106	2	no	—
149	C₂×#107	—	53	1	no	—
150	C₂×#107	—	53	2	no	—
151	C₂×#106	—	106	1	no	—
152	C₂×#102	—	106	1	no	—
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	#382	—	106	1	no	—
155	C₄⋊C₁₆	—	2	2	no	—
156	#357	—	106	4	no	—
157	C₅₃	#12813	1	1	yes	characteristic, complemented, Sylow
158	C₂²×D₄	—	106	1	no	—
159	C₂³×C₄	—	212	1	no	—
160	C₂×D₈	—	53	2	no	—
161	C₂×D₈	—	53	2	no	—
162	C₂×D₈	—	53	2	no	—
163	C₂×D₈	—	106	1	no	—
164	C₂×D₈	—	106	1	no	—
165	C₂×D₈	—	106	1	no	—
166	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
168	C₄×D₄	—	106	4	no	—
169	C₄×D₄	—	212	2	no	—
170	C₂×C₄⋊C₄	—	106	1	no	—
171	C₂×C₄⋊C₄	—	106	2	no	—
172	C₂×C₂²⋊C₄	—	212	1	no	—
173	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	2	no	—
177	#107	—	53	2	no	—
178	#107	—	106	1	no	—
179	#107	—	106	2	no	—
180	#107	—	106	2	no	—
181	#106	—	212	2	no	—
182	#102	—	106	4	no	—
183	C₄×C₈	C₄×D₅₃	1	2	yes	—
184	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
185	#95	—	212	1	no	—
186	#95	—	212	1	no	—
187	C₂⁴	—	212	1	no	—
188	C₂×D₄	—	106	2	no	—
189	C₂×D₄	—	106	2	no	—
190	C₂×D₄	—	106	2	no	—
191	C₂×D₄	—	212	1	no	—
192	C₂×D₄	—	212	1	no	—
193	C₂×D₄	—	212	1	no	—
194	C₂²×C₄	—	424	4	no	—
195	C₂²×C₄	—	212	2	no	—
196	C₂²×C₄	—	212	1	no	—
197	C₂²×C₄	—	212	4	no	—
198	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
199	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
201	D₈	—	53	4	no	—
202	D₈	—	106	2	no	—
203	D₈	—	106	2	no	—
204	D₈	—	106	2	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	#11719921	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
209	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
210	C₂×C₈	#11719947	1	1	yes	characteristic
211	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
212	C₂×C₈	#11719949	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	C₄⋊C₄	—	106	2	no	—
215	C₄⋊C₄	—	212	2	no	—
216	C₄⋊C₄	—	212	2	no	—
217	C₄⋊C₄	—	212	1	no	—
218	C₂²⋊C₄	—	212	4	no	—
219	C₄²	#11719913	1	2	yes	—
220	C₄²	#11719921	1	2	yes	—
221	C₁₆	—	4	2	no	—
222	C₂³	—	424	2	no	—
223	C₂³	—	424	2	no	—
224	C₂³	—	212	2	no	—
225	C₂³	—	212	2	no	—
226	C₂³	—	212	2	no	—
227	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
228	D₄	—	106	4	no	—
229	D₄	—	212	2	no	—
230	D₄	—	212	2	no	—
231	D₄	—	212	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	424	8	no	—
235	C₂×C₄	—	424	2	no	—
236	C₂×C₄	—	424	2	no	—
237	C₂×C₄	—	424	2	no	—
238	C₂×C₄	—	424	2	no	—
239	C₂×C₄	—	212	2	no	—
240	C₂×C₄	#13133139	1	2	yes	—
241	C₂×C₄	#13133139	1	2	yes	—
242	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
243	C₂×C₄	#13133192	1	1	yes	characteristic
244	C₂×C₄	#13133197	1	1	yes	characteristic
245	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
246	C₂×C₄	#13133245	1	1	yes	characteristic
247	C₂×C₄	#13133250	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
250	C₈	#13133175	1	1	yes	characteristic
251	C₈	#13133188	1	1	yes	characteristic
252	C₈	#13133193	1	1	yes	characteristic
253	C₂²	—	424	4	no	—
254	C₂²	—	424	4	no	—
255	C₂²	—	424	2	no	—
256	C₂²	—	424	2	no	—
257	C₂²	—	212	4	no	—
258	C₂²	C₂×#13133139	1	1	yes	characteristic
259	C₂²	#16810819	1	1	yes	characteristic
260	C₂²	#16810904	1	1	yes	characteristic
261	C₂²	#16811055	1	2	yes	—
262	C₂²	#16811061	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	848	2	no	—
266	C₄	—	424	2	no	—
267	C₄	#16810480	1	1	yes	characteristic
268	C₄	#16810625	1	1	yes	characteristic
269	C₄	#16810829	1	1	yes	characteristic
270	C₄	#16810914	1	1	yes	characteristic
271	C₂	—	424	4	no	—
272	C₂	#20274716	1	1	yes	characteristic
273	C₂	#20276423	1	1	yes	characteristic
274	C₂	#20276432	1	1	yes	characteristic
275	C₂	#20276771	1	2	yes	—
276	C₂	#20276873	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₄₂₄	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12813
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20274879	C₂	semidirect
4. #20272807	C₂	semidirect
5. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
9. C₄×D₄₂₄	C₄	semidirect
134. C₂×C₄⋊C₁₆	D₅₃	semidirect
157. C₅₃	#12813	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄₂₄	2
C₂×#16810425	2
#20274879	2
#20272807	2
C₂×C₅₃×C₄⋊C₁₆	2
#12813	53