Subgroups — #27537312

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811039	C₂	1	2	yes	maximal, complemented
2	#20283428	C₂	1	1	yes	maximal, characteristic
3	#20276546	C₂	1	2	yes	maximal, complemented
4	#20276539	C₂	1	4	yes	maximal, complemented
5	#20276534	C₂	1	2	yes	maximal, complemented
6	#20276328	C₂	1	2	yes	maximal, complemented
7	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133223	C₂²	1	2	yes	—
10	#16811173	C₄	1	1	yes	characteristic
11	#16811039	—	2	8	no	—
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	C₂×#13133156	C₂²	1	4	yes	—
15	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
16	#16810702	C₄	1	2	yes	—
17	#16810702	C₄	1	2	yes	—
18	C₈×D₂₁₂	C₂²	1	2	yes	—
19	C₈×D₂₁₂	C₂²	1	2	yes	—
20	#16810647	C₄	1	1	yes	characteristic
21	#16810589	C₄	1	2	yes	—
22	#16810402	C₂²	1	4	yes	—
23	#16810402	C₂²	1	4	yes	—
24	#16810399	C₂²	1	2	yes	—
25	C₄×#11719908	C₂²	1	2	yes	—
26	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₂×#11719945	C₂×C₄	1	2	yes	—
32	#13133223	—	2	4	no	—
33	#13133223	—	2	4	no	—
34	C₂×#11719917	C₂×C₄	1	2	yes	—
35	C₂×#11719912	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₅₃	C₂³	1	4	yes	—
37	#13133217	C₈	1	1	yes	characteristic
38	#13133214	C₈	1	1	yes	characteristic
39	#13133205	C₈	1	2	yes	—
40	#13133191	C₈	1	2	yes	—
41	#13133191	C₈	1	2	yes	—
42	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
44	#13133157	—	2	2	no	—
45	#13133157	D₄	1	4	yes	complemented
46	#13133156	—	2	8	no	—
47	C₂×#11719908	C₂³	1	2	yes	—
48	C₂×#11719908	C₂³	1	2	yes	—
49	C₁₆×D₅₃	—	2	8	no	—
50	C₄⋊#88322	C₂×C₄	1	2	yes	—
51	C₄⋊#88322	C₂×C₄	1	2	yes	—
52	#13133129	C₂×C₄	1	4	yes	—
53	#13133129	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₅₃	—	2	4	no	—
58	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
59	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₅₃	C₂×C₈	1	4	yes	—
68	#11719945	—	2	4	no	—
69	#11719945	—	2	4	no	—
70	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
71	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
72	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
74	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₃	#41	1	2	yes	—
81	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	#11719921	C₂×C₈	1	4	yes	—
87	#11719921	C₂×C₈	1	4	yes	—
88	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
89	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
90	#11719917	—	2	4	no	—
91	C₂×#88322	C₂²×C₄	1	4	yes	—
92	#11719912	—	2	8	no	—
93	C₈×D₅₃	—	2	4	no	—
94	C₈×D₅₃	—	2	4	no	—
95	C₁₆×C₅₃	—	2	2	no	—
96	C₁₆×C₅₃	—	2	2	no	—
97	#11719908	—	2	2	no	—
98	#11719908	C₂×D₄	1	4	yes	—
99	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂²×D₅₃	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#131	1	2	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₅₃	—	2	8	no	—
112	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
113	D₂₁₂	—	2	4	no	—
114	D₂₁₂	—	2	4	no	—
115	D₂₁₂	—	4	2	no	—
116	C₄×D₅₃	—	2	4	no	—
117	C₄×D₅₃	—	2	4	no	—
118	Dic₁₀₆	—	2	4	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	C₄×D₄	1	2	yes	—
122	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
124	#88322	—	2	4	no	—
125	#33438	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	C₈×D₄	1	2	yes	—
129	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
130	C₂×D₅₃	—	2	4	no	—
131	C₂×D₅₃	—	2	4	no	—
132	C₂×D₅₃	—	2	4	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	C₈×D₄	1	2	yes	—
137	C₄×C₅₃	#504	1	2	yes	complemented
138	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
140	Dic₅₃	—	2	4	no	—
141	C₂×#504	—	53	2	no	—
142	#2931	—	53	1	no	—
143	#2169	—	53	2	no	—
144	#2168	—	53	4	no	—
145	D₄×C₁₆	—	53	2	no	—
146	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
147	#2113	—	53	2	no	—
148	C₄×M₃₂	—	53	1	no	—
149	C₂×C₅₃	—	2	2	no	—
150	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
152	C₂×C₅₃	#3407	1	1	yes	characteristic
153	D₅₃	—	4	4	no	—
154	C₂×#131	—	53	2	no	—
155	#532	—	53	1	no	—
156	#504	—	106	8	no	—
157	C₂×M₃₂	—	53	1	no	—
158	C₂×M₃₂	—	53	1	no	—
159	C₂×M₃₂	—	53	4	no	—
160	C₂²×C₁₆	—	53	4	no	—
161	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
162	#436	—	53	2	no	—
163	#436	—	53	2	no	—
164	C₈×D₄	—	53	2	no	—
165	C₈×D₄	—	53	2	no	—
166	#423	—	53	1	no	—
167	#408	—	53	2	no	—
168	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
169	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
170	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
171	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
172	C₂²⋊C₁₆	—	53	4	no	—
173	C₂²⋊C₁₆	—	53	4	no	—
174	#346	—	53	2	no	—
175	C₄×C₁₆	—	53	2	no	—
176	C₅₃	#33438	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	53	2	no	—
178	#131	—	106	4	no	—
179	#131	—	106	4	no	—
180	C₂×M₁₆	—	53	2	no	—
181	C₂×M₁₆	—	53	4	no	—
182	C₂²×C₈	—	53	4	no	—
183	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₄⋊Q₈	—	53	1	no	—
186	#127	—	53	1	no	—
187	#124	—	53	2	no	—
188	#121	—	53	2	no	—
189	#121	—	53	2	no	—
190	C₄×D₄	—	53	2	no	—
191	C₄×D₄	—	53	2	no	—
192	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
193	M₃₂	—	53	4	no	—
194	M₃₂	—	106	8	no	—
195	M₃₂	—	106	2	no	—
196	C₂×C₁₆	—	2	4	no	—
197	C₂×C₁₆	—	53	2	no	—
198	C₂×C₁₆	—	53	2	no	—
199	C₂×C₁₆	—	106	8	no	—
200	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
201	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
202	C₄⋊C₈	—	53	2	no	—
203	C₄⋊C₈	—	53	2	no	—
204	C₂²⋊C₈	—	53	4	no	—
205	C₂²⋊C₈	—	53	4	no	—
206	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	#41	—	106	4	no	—
211	#41	—	106	4	no	—
212	C₂×Q₈	—	53	2	no	—
213	C₂×D₄	—	53	4	no	—
214	C₂×D₄	—	53	2	no	—
215	C₂×D₄	—	53	2	no	—
216	C₂×D₄	—	53	2	no	—
217	C₂²×C₄	—	53	4	no	—
218	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
221	M₁₆	—	106	4	no	—
222	M₁₆	—	106	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	53	4	no	—
225	C₂×C₈	—	106	4	no	—
226	C₂×C₈	—	106	4	no	—
227	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₈	D₄×D₅₃	1	2	yes	—
231	C₂×C₈	#11719949	1	2	yes	—
232	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
233	C₄⋊C₄	—	53	2	no	—
234	C₄⋊C₄	—	53	2	no	—
235	C₂²⋊C₄	—	53	4	no	—
236	C₂²⋊C₄	—	53	4	no	—
237	C₄²	C₈×D₅₃	1	1	yes	characteristic
238	C₄²	C₈×D₅₃	1	1	yes	characteristic
239	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	53	4	no	—
244	C₁₆	—	106	2	no	—
245	C₂³	—	53	4	no	—
246	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
247	Q₈	—	106	4	no	—
248	D₄	—	106	8	no	—
249	D₄	—	106	4	no	—
250	D₄	—	106	4	no	—
251	D₄	—	212	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	53	4	no	—
255	C₂×C₄	—	106	4	no	—
256	C₂×C₄	—	106	4	no	—
257	C₂×C₄	#13133179	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
263	C₂×C₄	#13133225	1	2	yes	—
264	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	106	4	no	—
268	C₈	#13133179	1	2	yes	—
269	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
270	C₈	C₂×#11719949	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	106	4	no	—
274	C₂²	—	106	4	no	—
275	C₂²	—	106	4	no	—
276	C₂²	#16810688	1	2	yes	—
277	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	106	4	no	—
282	C₄	#16810688	1	2	yes	—
283	C₄	#16811039	1	2	yes	complemented
284	C₄	C₂×#13133179	1	1	yes	characteristic
285	C₄	C₂×#13133225	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	212	4	no	—
288	C₂	C₂×#16810688	1	1	yes	characteristic
289	C₂	C₂×#16811039	1	1	yes	characteristic
290	C₂	#20287817	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
prime radical	#16811173	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#33438
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16811039	C₂	semidirect
3. #20276546	C₂	semidirect
4. #20276539	C₂	semidirect
5. #20276534	C₂	semidirect
6. #20276328	C₂	semidirect
7. C₄×#13133157	C₂	semidirect
8. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
45. #13133157	D₄	semidirect
137. C₄×C₅₃	#504	semidirect
146. C₂×C₄⋊C₁₆	D₅₃	semidirect
176. C₅₃	#33438	semidirect
283. C₄	#16811039	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811039	2
#20283428	2
#20276546	2
#20276539	2
#20276534	2
#20276328	2
C₄×#13133157	2
C₂×C₅₃×C₄⋊C₁₆	2
#33438	53