Subgroups — #36454297

Order 22784.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030811	C₂	1	1	yes	maximal, characteristic
2	#26021228	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021079	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018706	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018702	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152155	C₂²	1	1	yes	characteristic
9	#19152148	C₂²	1	1	yes	characteristic
10	#19151996	C₄	1	1	yes	characteristic, complemented
11	#19151971	C₄	1	1	yes	characteristic, complemented
12	#19151949	C₄	1	1	yes	characteristic, complemented
13	#19151945	C₄	1	1	yes	characteristic, complemented
14	#19151827	—	2	2	no	—
15	C₄×#13083563	—	2	2	no	—
16	C₂×#15667938	—	2	2	no	—
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₈₉	—	2	2	no	—
19	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
20	#19151359	—	2	2	no	—
21	#19151328	—	2	2	no	—
22	#19151325	—	2	2	no	—
23	#423×C₈₉	C₂²	1	1	yes	characteristic
24	#423×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#334×C₈₉	—	2	2	no	—
27	C₂×#13083574	—	2	1	no	—
28	C₂×#13083563	—	2	2	no	—
29	C₂²×#10623494	—	2	1	no	—
30	#15668012	C₂×C₄	1	1	yes	characteristic
31	#15668012	C₂×C₄	1	1	yes	characteristic
32	#15668012	C₂×C₄	1	1	yes	characteristic
33	#15668009	C₂×C₄	1	1	yes	characteristic
34	#15668004	—	2	1	no	—
35	#15668000	—	2	1	no	—
36	#15667991	—	2	1	no	—
37	#15667991	—	2	1	no	—
38	#15667986	—	2	1	no	—
39	#15667986	—	2	1	no	—
40	C₄×Dic₁₇₈	—	2	1	no	—
41	C₄×Dic₁₇₈	—	2	1	no	—
42	C₄×D₃₅₆	—	2	1	no	—
43	C₄×D₃₅₆	—	2	1	no	—
44	#15667971	—	2	1	no	—
45	C₂×C₄×Dic₈₉	—	2	1	no	—
46	C₂×C₄×Dic₈₉	—	2	1	no	—
47	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
48	#15667944	—	2	2	no	—
49	#15667938	—	4	4	no	—
50	#15667937	—	2	2	no	—
51	#15667934	—	2	2	no	—
52	C₄×#10623494	—	2	1	no	—
53	C₄×#10623494	D₄	1	2	yes	—
54	C₈×Dic₈₉	—	2	8	no	—
55	C₂×M₁₆×C₈₉	—	2	1	no	—
56	C₂×M₁₆×C₈₉	—	2	1	no	—
57	C₂²×C₈×C₈₉	D₄	1	2	yes	—
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₄⋊C₈×C₈₉	—	2	1	no	—
62	C₄⋊C₈×C₈₉	—	2	1	no	—
63	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₈₉	D₄	1	2	yes	—
66	#13083574	—	4	2	no	—
67	#13083574	—	4	2	no	—
68	C₂×Dic₁₇₈	—	2	1	no	—
69	C₂×Dic₁₇₈	—	2	1	no	—
70	C₂×Dic₁₇₈	—	2	1	no	—
71	C₂×D₃₅₆	—	2	1	no	—
72	C₂×D₃₅₆	—	2	1	no	—
73	C₂×D₃₅₆	—	2	1	no	—
74	C₂×C₂²⋊D₈₉	—	4	1	no	—
75	C₂²×Dic₈₉	—	2	2	no	—
76	C₂×C₄×D₈₉	—	4	1	no	—
77	#13083563	—	4	4	no	—
78	C₂×#10623494	—	2	4	no	—
79	C₂×#10623494	—	2	2	no	—
80	C₄⋊Dic₈₉	—	2	1	no	—
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	—	2	1	no	—
84	C₄⋊Dic₈₉	—	2	1	no	—
85	C₄⋊Dic₈₉	—	2	1	no	—
86	#13083553	—	4	1	no	—
87	#13083553	—	4	1	no	—
88	C₂²⋊Dic₈₉	—	4	1	no	—
89	#13083551	—	4	1	no	—
90	#13083551	—	4	1	no	—
91	C₄×Dic₈₉	—	2	4	no	—
92	C₄×Dic₈₉	—	2	4	no	—
93	C₄×Dic₈₉	—	2	2	no	—
94	C₄×Dic₈₉	—	4	1	no	—
95	C₄×Dic₈₉	SD₁₆	1	4	yes	complemented
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
99	M₁₆×C₈₉	—	4	2	no	—
100	M₁₆×C₈₉	—	4	2	no	—
101	C₂×C₈×C₈₉	—	2	4	no	—
102	C₂×C₈×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	—	4	1	no	—
104	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
105	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
106	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	#41	1	1	yes	characteristic
110	C₂³×C₈₉	#121	1	1	yes	characteristic
111	C₂²×D₈₉	—	4	1	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
119	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₈₉	#135	1	2	yes	—
122	C₂²⋊D₈₉	—	8	2	no	—
123	C₂×Dic₈₉	—	2	4	no	—
124	C₂×Dic₈₉	—	2	4	no	—
125	C₂×Dic₈₉	—	2	4	no	—
126	C₂×Dic₈₉	—	4	1	no	—
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	1	no	—
129	D₃₅₆	—	4	2	no	—
130	D₃₅₆	—	4	2	no	—
131	D₃₅₆	—	4	2	no	—
132	C₄×D₈₉	—	8	1	no	—
133	C₄×D₈₉	—	8	1	no	—
134	Dic₁₇₈	—	4	2	no	—
135	Dic₁₇₈	—	4	2	no	—
136	Dic₁₇₈	—	4	2	no	—
137	C₈×C₈₉	—	2	2	no	—
138	C₈×C₈₉	—	8	1	no	—
139	C₈×C₈₉	—	8	1	no	—
140	C₈×C₈₉	#104	1	4	yes	complemented
141	#10623494	—	2	4	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	#390	1	1	yes	characteristic
145	C₂²×C₈₉	#465	1	2	yes	—
146	C₂×D₈₉	—	8	1	no	—
147	C₂×D₈₉	—	8	1	no	—
148	C₂×D₈₉	—	8	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	C₂×#104	1	2	yes	—
152	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
153	C₄×C₈₉	#443	1	2	yes	—
154	C₄×C₈₉	#492	1	1	yes	characteristic
155	C₄×C₈₉	#495	1	1	yes	characteristic
156	Dic₈₉	—	2	8	no	—
157	Dic₈₉	—	8	1	no	—
158	Dic₈₉	—	8	1	no	—
159	Dic₈₉	—	8	1	no	—
160	#9843	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	2	2	no	—
162	C₂×C₈₉	#1962	1	1	yes	characteristic
163	C₂×C₈₉	#1968	1	1	yes	characteristic
164	C₂×C₈₉	#1974	1	1	yes	characteristic
165	D₈₉	—	16	1	no	—
166	#3167	—	89	1	no	—
167	#1994	—	89	1	no	—
168	#1962	—	89	1	no	—
169	C₄²×C₈	—	89	1	no	—
170	#1566	D₈₉	1	1	yes	characteristic, complemented
171	#1493	—	89	1	no	—
172	#1492	—	89	1	no	—
173	C₈₉	#9843	1	1	yes	characteristic, complemented, Sylow
174	#533	—	89	1	no	—
175	#532	—	89	1	no	—
176	#498	—	89	1	no	—
177	#492	—	89	1	no	—
178	#487	—	89	1	no	—
179	#486	—	89	1	no	—
180	#465	—	178	2	no	—
181	C₄×SD₁₆	—	178	2	no	—
182	#423	C₂×D₈₉	1	1	yes	characteristic
183	#423	C₂×D₈₉	1	1	yes	characteristic
184	C₂×#104	—	178	2	no	—
185	C₂×C₄×C₈	—	89	1	no	—
186	C₂×C₄×C₈	—	178	2	no	—
187	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
188	C₄³	—	89	1	no	—
189	#339	—	178	2	no	—
190	#334	—	2	2	no	—
191	Q₈⋊C₈	—	178	2	no	—
192	D₄⋊C₈	—	178	2	no	—
193	C₂×#41	—	178	1	no	—
194	C₂×SD₁₆	—	178	2	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂²×C₈	—	178	1	no	—
198	C₂²×C₈	D₃₅₆	1	2	yes	—
199	C₄⋊Q₈	—	89	1	no	—
200	C₄⋊Q₈	—	89	1	no	—
201	C₄⋊Q₈	—	89	1	no	—
202	#127	—	89	1	no	—
203	#125	—	178	1	no	—
204	#124	—	178	1	no	—
205	C₂²⋊Q₈	—	178	1	no	—
206	C₂²⋊Q₈	—	178	1	no	—
207	#121	—	178	1	no	—
208	#121	—	178	1	no	—
209	C₄×Q₈	—	178	1	no	—
210	C₄×Q₈	—	178	1	no	—
211	C₄×D₄	—	178	1	no	—
212	C₄×D₄	—	178	1	no	—
213	#117	—	178	1	no	—
214	C₂×C₄²	—	89	2	no	—
215	C₂×C₄²	—	178	1	no	—
216	C₂×C₄²	—	178	1	no	—
217	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
218	#106	—	178	2	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	#104	—	356	4	no	—
224	#103	—	178	2	no	—
225	#102	—	178	2	no	—
226	C₄×C₈	—	89	2	no	—
227	C₄×C₈	—	178	8	no	—
228	C₄×C₈	—	178	1	no	—
229	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
230	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
232	#41	—	356	2	no	—
233	#41	—	356	2	no	—
234	C₂×Q₈	—	178	1	no	—
235	C₂×Q₈	—	178	1	no	—
236	C₂×Q₈	—	178	1	no	—
237	C₂×D₄	—	356	1	no	—
238	C₂×D₄	—	178	1	no	—
239	C₂×D₄	—	178	1	no	—
240	C₂×D₄	—	178	1	no	—
241	C₂²×C₄	—	356	1	no	—
242	C₂²×C₄	—	178	2	no	—
243	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
245	C₂²×C₄	#13083575	1	1	yes	characteristic
246	SD₁₆	—	356	4	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	178	4	no	—
253	C₂×C₈	—	178	2	no	—
254	C₂×C₈	#13083551	1	2	yes	—
255	C₂×C₈	#13083551	1	2	yes	—
256	C₄⋊C₄	—	356	1	no	—
257	C₄⋊C₄	—	356	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	—	178	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	—	178	1	no	—
264	C₂²⋊C₄	—	356	1	no	—
265	C₂²⋊C₄	—	356	1	no	—
266	C₂²⋊C₄	—	356	1	no	—
267	C₄²	—	89	4	no	—
268	C₄²	—	356	1	no	—
269	C₄²	—	178	4	no	—
270	C₄²	—	178	4	no	—
271	C₄²	—	178	2	no	—
272	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
273	C₄²	D₄×D₈₉	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₄²	#13083575	1	1	yes	characteristic
276	C₂³	—	356	1	no	—
277	C₂³	#15667987	1	1	yes	characteristic
278	Q₈	—	356	2	no	—
279	Q₈	—	356	2	no	—
280	Q₈	—	356	2	no	—
281	D₄	—	356	2	no	—
282	D₄	—	356	2	no	—
283	D₄	—	356	2	no	—
284	D₄	—	712	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	356	1	no	—
288	C₂×C₄	—	356	1	no	—
289	C₂×C₄	—	356	1	no	—
290	C₂×C₄	—	178	4	no	—
291	C₂×C₄	—	178	4	no	—
292	C₂×C₄	—	178	4	no	—
293	C₂×C₄	—	712	1	no	—
294	C₂×C₄	—	712	1	no	—
295	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
296	C₂×C₄	#15667974	1	1	yes	characteristic
297	C₂×C₄	#15667974	1	1	yes	characteristic
298	C₂×C₄	#15667987	1	1	yes	characteristic
299	C₂×C₄	#15668001	1	1	yes	characteristic
300	C₂×C₄	#15668008	1	1	yes	characteristic
301	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
302	C₂×C₄	#15668033	1	2	yes	—
303	C₈	—	2	2	no	—
304	C₈	—	8	1	no	—
305	C₈	—	8	1	no	—
306	C₈	—	178	4	no	—
307	C₈	#15667938	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	712	1	no	—
311	C₂²	—	712	1	no	—
312	C₂²	—	712	1	no	—
313	C₂²	#19151499	1	1	yes	characteristic
314	C₂²	#19151828	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	178	8	no	—
318	C₄	—	712	1	no	—
319	C₄	—	712	1	no	—
320	C₄	—	712	1	no	—
321	C₄	C₂×#15667938	1	2	yes	—
322	C₄	#19151685	1	2	yes	—
323	C₄	#19151714	1	2	yes	—
324	C₄	#19151969	1	1	yes	characteristic
325	C₄	#19151979	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	1424	1	no	—
328	C₂	#26021079	1	1	yes	characteristic
329	C₂	#26021104	1	1	yes	characteristic
330	C₂	#26021138	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	#390
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#9843
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
2. #26021228	C₂	semidirect
4. C₄×C₈×Dic₈₉	C₂	semidirect
5. #26018706	C₂	semidirect
7. C₈₉×#1566	C₂	semidirect
10. #19151996	C₄	semidirect
11. #19151971	C₄	semidirect
12. #19151949	C₄	semidirect
13. #19151945	C₄	semidirect
95. C₄×Dic₈₉	SD₁₆	semidirect
140. C₈×C₈₉	#104	semidirect
170. #1566	D₈₉	semidirect
173. C₈₉	#9843	semidirect
307. C₈	#15667938	semidirect

Maximal subgroups

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

subgroup	index
#26030811	2
#26021228	2
#26021079	2
C₄×C₈×Dic₈₉	2
#26018706	2
#26018702	2
C₈₉×#1566	2
#9843	89