Subgroups — #36709330

Order 22784.

566 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029896	C₂	1	1	yes	maximal, characteristic
2	#26028860	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022465	C₂	1	8	yes	maximal, complemented
4	#26022026	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022025	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151411	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021872	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#361×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#19152364	—	2	1	no	—
10	C₂×#15668018	C₂²	1	1	yes	characteristic
11	#19152165	—	2	1	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	#19151848	C₂²	1	4	yes	—
14	#19151831	—	2	2	no	—
15	#19151646	—	2	1	no	—
16	#19151645	C₂²	1	4	yes	complemented
17	C₂×#15667948	C₂²	1	1	yes	characteristic
18	#108×D₈₉	—	2	2	no	—
19	#19151636	C₂²	1	1	yes	characteristic
20	#19151603	—	2	1	no	—
21	#19151422	C₂²	1	4	yes	complemented
22	#19151421	C₂²	1	4	yes	complemented
23	#19151411	C₂²	1	4	yes	—
24	#19151379	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
28	#361×C₈₉	C₂²	1	4	yes	—
29	C₂×#13083574	D₄	1	1	yes	characteristic
30	#15668020	—	4	2	no	—
31	#15668018	—	2	1	no	—
32	#15668018	—	2	1	no	—
33	#15668018	—	2	1	no	—
34	#15668018	—	2	1	no	—
35	#15668018	—	4	2	no	—
36	C₂×#13083558	—	2	1	no	—
37	C₂×#13083558	D₄	1	1	yes	characteristic
38	C₂×#13083557	C₂³	1	2	yes	—
39	M₁₆×D₈₉	—	2	4	no	—
40	C₂×C₈×D₈₉	D₄	1	2	yes	—
41	#15668008	—	4	1	no	—
42	#15667999	—	4	1	no	—
43	#15667983	—	2	4	no	—
44	#15667981	—	4	2	no	—
45	#15667971	—	2	1	no	—
46	#15667952	—	2	1	no	—
47	#15667952	C₂³	1	2	yes	—
48	C₂×#13083549	—	2	1	no	—
49	#15667948	—	2	1	no	—
50	#15667948	C₂³	1	2	yes	—
51	#15667947	—	2	4	no	—
52	#15667946	—	2	1	no	—
53	#15667944	C₂³	1	2	yes	—
54	#15667933	—	2	4	no	—
55	C₂²×D₄×C₈₉	—	2	1	no	—
56	C₂×D₈×C₈₉	—	2	1	no	—
57	C₂×D₈×C₈₉	—	2	1	no	—
58	C₂×D₈×C₈₉	—	2	1	no	—
59	C₂×D₈×C₈₉	—	2	1	no	—
60	C₂×D₈×C₈₉	—	2	1	no	—
61	C₂×D₈×C₈₉	C₂³	1	2	yes	—
62	C₂×M₁₆×C₈₉	—	2	1	no	—
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	M₃₂×C₈₉	—	2	1	no	—
65	M₃₂×C₈₉	C₂³	1	2	yes	—
66	C₂×C₁₆×C₈₉	—	2	1	no	—
67	#108×C₈₉	—	2	1	no	—
68	#108×C₈₉	C₂³	1	2	yes	—
69	#13083574	—	2	1	no	—
70	#13083574	—	2	1	no	—
71	#13083574	—	2	1	no	—
72	#13083574	—	2	1	no	—
73	C₂×Dic₁₇₈	SD₁₆	1	1	yes	characteristic
74	C₂×D₃₅₆	SD₁₆	1	1	yes	characteristic
75	C₂×C₂²⋊D₈₉	—	4	4	no	—
76	C₂×C₂²⋊D₈₉	SD₁₆	1	2	yes	—
77	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
78	#13083558	—	2	1	no	—
79	#13083558	—	2	1	no	—
80	#13083558	—	2	2	no	—
81	#13083558	—	4	1	no	—
82	#13083557	—	2	2	no	—
83	#13083557	—	2	2	no	—
84	#13083557	—	4	4	no	—
85	C₂×#10623494	—	4	1	no	—
86	C₂×#10623494	C₂×D₄	1	2	yes	—
87	C₈×D₈₉	—	2	2	no	—
88	C₈×D₈₉	—	2	2	no	—
89	C₈×D₈₉	—	4	4	no	—
90	C₄⋊Dic₈₉	—	2	2	no	—
91	C₂²⋊Dic₈₉	—	4	1	no	—
92	C₂²⋊Dic₈₉	—	4	4	no	—
93	C₄×Dic₈₉	—	4	1	no	—
94	#13083549	—	2	2	no	—
95	C₂⁴×C₈₉	—	4	1	no	—
96	C₂×D₄×C₈₉	—	2	2	no	—
97	C₂×D₄×C₈₉	—	4	1	no	—
98	C₂×D₄×C₈₉	—	4	1	no	—
99	C₂×D₄×C₈₉	—	4	1	no	—
100	C₂×D₄×C₈₉	—	4	1	no	—
101	C₂×D₄×C₈₉	—	4	1	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	D₈×C₈₉	—	2	2	no	—
104	D₈×C₈₉	—	2	2	no	—
105	D₈×C₈₉	—	4	1	no	—
106	D₈×C₈₉	—	4	1	no	—
107	M₁₆×C₈₉	—	2	2	no	—
108	M₁₆×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	—	4	1	no	—
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₁₆×C₈₉	—	2	2	no	—
117	C₂³×C₈₉	—	4	2	no	—
118	C₂³×C₈₉	—	8	1	no	—
119	C₂³×C₈₉	—	8	1	no	—
120	C₂³×C₈₉	—	16	2	no	—
121	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×D₈₉	C₂×SD₁₆	1	2	yes	—
123	D₄×C₈₉	—	4	2	no	—
124	D₄×C₈₉	—	4	2	no	—
125	D₄×C₈₉	—	8	1	no	—
126	D₄×C₈₉	—	8	1	no	—
127	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₈₉	—	2	2	no	—
134	C₂²⋊D₈₉	—	2	2	no	—
135	C₂²⋊D₈₉	—	16	4	no	—
136	C₂×Dic₈₉	—	4	2	no	—
137	C₂×Dic₈₉	C₂×SD₁₆	1	2	yes	—
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	—	2	1	no	—
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	2	2	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	—	2	1	no	—
144	C₈×C₈₉	—	4	2	no	—
145	C₈×C₈₉	#123	1	1	yes	characteristic
146	C₈×C₈₉	#123	1	1	yes	characteristic
147	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
149	#10623494	—	2	2	no	—
150	#10623494	—	4	2	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	8	4	no	—
153	C₂²×C₈₉	—	8	2	no	—
154	C₂²×C₈₉	—	16	2	no	—
155	C₂²×C₈₉	—	16	2	no	—
156	C₂²×C₈₉	#481	1	1	yes	characteristic
157	C₂²×C₈₉	#481	1	1	yes	characteristic
158	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
159	C₂×D₈₉	—	2	2	no	—
160	C₂×D₈₉	—	4	2	no	—
161	C₄×C₈₉	#481	1	1	yes	characteristic
162	C₄×C₈₉	#481	1	1	yes	characteristic
163	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
164	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	Dic₈₉	—	2	2	no	—
166	Dic₈₉	—	8	2	no	—
167	#33657	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	—	2	1	no	—
171	C₂×C₈₉	—	16	2	no	—
172	C₂×C₈₉	C₂×#481	1	1	yes	characteristic
173	D₈₉	—	4	2	no	—
174	#3058	—	89	1	no	—
175	#2916	—	89	1	no	—
176	#2238	—	89	8	no	—
177	#2155	—	89	1	no	—
178	#2155	—	89	1	no	—
179	C₂×#361	—	89	1	no	—
180	C₂×#361	D₈₉	1	1	yes	characteristic, complemented
181	#2115	—	89	1	no	—
182	C₈₉	#33657	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₈	C₂×D₈₉	1	1	yes	characteristic
184	#568	—	178	1	no	—
185	C₂×#131	—	89	1	no	—
186	#535	—	178	1	no	—
187	C₂×M₃₂	—	89	1	no	—
188	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
189	#469	—	89	4	no	—
190	#466	—	178	2	no	—
191	#430	—	89	4	no	—
192	#430	—	178	1	no	—
193	C₂×#108	—	89	1	no	—
194	C₂×#108	—	178	2	no	—
195	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
196	#427	—	89	1	no	—
197	#418	—	178	1	no	—
198	#365	—	89	4	no	—
199	#365	—	89	4	no	—
200	#361	—	89	4	no	—
201	#361	C₂×D₈₉	1	4	yes	—
202	#349	—	89	4	no	—
203	C₂×#41	—	89	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	C₂²×D₈₉	1	2	yes	—
211	#131	—	356	2	no	—
212	#131	—	356	2	no	—
213	#131	—	178	1	no	—
214	#131	—	178	1	no	—
215	#131	—	178	1	no	—
216	#131	—	178	1	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	89	1	no	—
219	C₂×M₁₆	—	89	2	no	—
220	C₂×M₁₆	—	178	4	no	—
221	C₂×M₁₆	—	178	1	no	—
222	C₂²×C₈	—	89	2	no	—
223	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
224	#127	—	356	1	no	—
225	#124	—	356	1	no	—
226	#121	—	178	4	no	—
227	#120	—	356	2	no	—
228	#117	—	178	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	89	2	no	—
231	M₃₂	—	178	1	no	—
232	M₃₂	C₂²×D₈₉	1	2	yes	—
233	C₂×C₁₆	—	2	1	no	—
234	C₂×C₁₆	—	178	1	no	—
235	#108	—	2	1	no	—
236	#108	—	89	2	no	—
237	#108	—	178	4	no	—
238	#108	—	178	1	no	—
239	#108	C₂²×D₈₉	1	2	yes	—
240	#107	—	178	1	no	—
241	#106	—	89	2	no	—
242	#102	—	178	4	no	—
243	C₂⁴	—	4	1	no	—
244	#41	—	178	1	no	—
245	#41	—	178	1	no	—
246	#41	—	178	1	no	—
247	#41	—	178	1	no	—
248	C₂×Q₈	—	89	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	89	2	no	—
256	C₂×D₄	—	89	1	no	—
257	C₂×D₄	—	356	4	no	—
258	C₂²×C₄	—	89	2	no	—
259	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	4	1	no	—
263	D₈	—	4	1	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	356	4	no	—
267	M₁₆	—	356	1	no	—
268	M₁₆	—	178	2	no	—
269	M₁₆	—	178	1	no	—
270	M₁₆	—	178	1	no	—
271	M₁₆	—	178	2	no	—
272	M₁₆	—	178	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	89	2	no	—
275	C₂×C₈	—	356	4	no	—
276	C₂×C₈	—	356	1	no	—
277	C₂×C₈	—	178	2	no	—
278	C₂×C₈	—	178	2	no	—
279	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
280	C₂×C₈	#13083573	1	1	yes	characteristic
281	C₂×C₈	#13083573	1	1	yes	characteristic
282	C₂×C₈	#13083574	1	1	yes	characteristic
283	C₂×C₈	#13083574	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
285	C₄⋊C₄	—	178	2	no	—
286	C₂²⋊C₄	—	356	4	no	—
287	C₂²⋊C₄	—	356	1	no	—
288	C₄²	—	356	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	178	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	1	no	—
293	C₂³	—	8	1	no	—
294	C₂³	—	16	2	no	—
295	C₂³	—	89	2	no	—
296	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
297	Q₈	—	178	1	no	—
298	Q₈	—	178	1	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	1	no	—
302	D₄	—	8	1	no	—
303	D₄	—	1424	4	no	—
304	D₄	—	178	2	no	—
305	D₄	—	178	2	no	—
306	D₄	—	178	1	no	—
307	D₄	—	178	1	no	—
308	C₂×C₄	—	89	2	no	—
309	C₂×C₄	—	356	2	no	—
310	C₂×C₄	—	178	2	no	—
311	C₂×C₄	—	178	2	no	—
312	C₂×C₄	#15667994	1	1	yes	characteristic
313	C₂×C₄	#15667994	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
315	C₂×C₄	#15668036	1	1	yes	characteristic
316	C₂×C₄	#15668036	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	356	2	no	—
320	C₈	—	178	2	no	—
321	C₈	#15667994	1	1	yes	characteristic
322	C₈	#15667994	1	1	yes	characteristic
323	C₈	C₂×#13083573	1	1	yes	characteristic
324	C₈	C₂×#13083574	1	1	yes	characteristic
325	C₂²	—	4	1	no	—
326	C₂²	—	8	4	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	16	2	no	—
329	C₂²	—	16	2	no	—
330	C₂²	—	356	2	no	—
331	C₂²	—	178	2	no	—
332	C₂²	#19151915	1	1	yes	characteristic
333	C₂²	#19151915	1	1	yes	characteristic
334	C₂²	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
335	C₄	—	178	2	no	—
336	C₄	—	712	2	no	—
337	C₄	#19151915	1	1	yes	characteristic
338	C₄	#19151915	1	1	yes	characteristic
339	C₄	C₂×#15667994	1	1	yes	characteristic
340	C₄	C₂×#15668036	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	16	2	no	—
345	C₂	—	356	2	no	—
346	C₂	C₂×#19151915	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19151915
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#361×C₈₉	C₂
socle	C₂×C₈₉	C₂×#481
prime radical	#26029896	C₂
hypercenter	C₂×#361	D₈₉
hypocenter	C₈₉	#33657
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. #26028860	C₂	semidirect
3. #26022465	C₂	semidirect
4. #26022026	C₂	semidirect
5. #26022025	C₂	semidirect
6. C₂×#19151411	C₂	semidirect
7. #26021872	C₂	semidirect
8. C₂×#361×C₈₉	C₂	semidirect
16. #19151645	C₂²	semidirect
21. #19151422	C₂²	semidirect
22. #19151421	C₂²	semidirect
24. #19151379	C₂²	semidirect
180. C₂×#361	D₈₉	semidirect
182. C₈₉	#33657	semidirect

Maximal subgroups

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

subgroup	index
#26029896	2
#26028860	2
#26022465	2
#26022026	2
#26022025	2
C₂×#19151411	2
#26021872	2
C₂×#361×C₈₉	2
#33657	89