Subgroups — #26032607

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152303	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152248	C₂	1	1	yes	maximal, characteristic
3	#19151995	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151930	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151929	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151915	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151886	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151882	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151826	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151825	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151689	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151661	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#15667943	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#481×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	—	2	1	no	—
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	C₂²×#10623494	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	#15668011	C₂²	1	1	yes	characteristic
21	#15668011	C₂²	1	1	yes	characteristic
22	#15668002	—	2	1	no	—
23	#15667994	—	2	1	no	—
24	#15667993	—	2	1	no	—
25	#15667992	C₂²	1	1	yes	characteristic
26	#15667991	—	2	1	no	—
27	#15667991	C₂²	1	1	yes	characteristic
28	#15667990	—	2	1	no	—
29	#15667989	—	2	1	no	—
30	#15667983	C₂²	1	1	yes	characteristic
31	#15667977	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	—	2	1	no	—
34	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	#15667944	C₂²	1	1	yes	characteristic
39	#15667944	C₂²	1	1	yes	characteristic
40	#15667943	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667943	C₂²	1	1	yes	characteristic
44	#15667943	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667937	C₂²	1	1	yes	characteristic
48	#15667935	C₂²	1	1	yes	characteristic
49	#15667933	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	#121×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
54	#106×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	#102×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
70	#13083562	—	2	2	no	—
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₈×D₈₉	—	2	2	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
91	#13083551	—	2	1	no	—
92	#13083551	D₄	1	1	yes	characteristic
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₈₉	—	2	1	no	—
96	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	—	2	1	no	—
98	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂³×C₈₉	—	2	1	no	—
107	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
109	D₄×C₈₉	—	2	1	no	—
110	D₄×C₈₉	—	2	1	no	—
111	D₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	8	1	no	—
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₈₉	—	4	2	no	—
121	C₂²⋊D₈₉	SD₁₆	1	4	yes	complemented
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
131	C₄×D₈₉	—	2	1	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	4	2	no	—
134	C₄×D₈₉	—	8	1	no	—
135	Dic₁₇₈	—	2	1	no	—
136	Dic₁₇₈	—	2	1	no	—
137	Dic₁₇₈	—	4	2	no	—
138	Dic₁₇₈	—	8	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	#10623494	—	2	1	no	—
142	#10623494	#41	1	2	yes	complemented
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	C₂×SD₁₆	1	2	yes	—
149	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₈₉	—	2	1	no	—
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	C₂×SD₁₆	1	2	yes	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	—	4	1	no	—
157	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₈₉	—	2	1	no	—
160	Dic₈₉	—	2	1	no	—
161	Dic₈₉	—	4	1	no	—
162	Dic₈₉	—	4	1	no	—
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	C₂×SD₁₆	1	2	yes	—
166	C₂×C₈₉	—	2	2	no	—
167	C₂×C₈₉	—	8	1	no	—
168	C₂×C₈₉	#547	1	1	yes	characteristic
169	C₂×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
170	C₂×C₈₉	#577	1	1	yes	characteristic
171	D₈₉	—	2	2	no	—
172	#3335	—	89	1	no	maximal, Sylow
173	C₈₉	#3335	1	1	yes	characteristic, complemented, Sylow
174	#554	—	89	1	no	—
175	#547	—	89	1	no	—
176	#498	—	89	1	no	—
177	#483	—	89	1	no	—
178	#483	—	89	1	no	—
179	#481	—	89	1	no	—
180	#481	D₈₉	1	1	yes	characteristic, complemented
181	#476	—	89	1	no	—
182	#475	—	89	1	no	—
183	#465	—	89	1	no	—
184	#465	—	89	1	no	—
185	C₄×SD₁₆	—	89	1	no	—
186	C₈×D₄	—	89	1	no	—
187	C₂×#106	—	89	1	no	—
188	C₂×#106	—	89	1	no	—
189	C₂×#41	—	178	1	no	—
190	C₂×SD₁₆	—	89	1	no	—
191	C₂²×C₈	—	89	1	no	—
192	C₂²×C₈	—	89	1	no	—
193	C₄⋊Q₈	—	89	1	no	—
194	C₄⋊Q₈	—	89	1	no	—
195	#125	—	178	1	no	—
196	#123	—	178	1	no	—
197	#123	—	178	1	no	—
198	C₂²⋊Q₈	—	89	1	no	—
199	C₂²⋊Q₈	—	89	1	no	—
200	C₂²⋊Q₈	—	178	1	no	—
201	C₂²⋊Q₈	—	178	1	no	—
202	C₂²⋊Q₈	—	178	1	no	—
203	#121	—	89	1	no	—
204	#121	C₂×D₈₉	1	1	yes	characteristic
205	C₄×Q₈	—	89	1	no	—
206	C₄×D₄	—	89	1	no	—
207	C₄×D₄	—	89	1	no	—
208	C₄×D₄	—	178	1	no	—
209	C₂×C₄⋊C₄	—	89	1	no	—
210	C₂×C₄⋊C₄	—	89	1	no	—
211	C₂×C₄⋊C₄	—	89	1	no	—
212	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
213	#106	—	89	1	no	—
214	#106	—	89	1	no	—
215	#106	—	89	1	no	—
216	#106	—	89	1	no	—
217	#106	—	89	1	no	—
218	#106	—	89	1	no	—
219	#106	—	89	1	no	—
220	#106	C₂×D₈₉	1	1	yes	characteristic
221	#106	C₂×D₈₉	1	1	yes	characteristic
222	C₄⋊C₈	—	89	1	no	—
223	#103	—	89	1	no	—
224	#103	—	89	1	no	—
225	#103	—	89	1	no	—
226	#102	—	89	1	no	—
227	#102	C₂×D₈₉	1	1	yes	characteristic
228	#102	C₂×D₈₉	1	1	yes	characteristic
229	C₂²⋊C₈	—	89	1	no	—
230	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
231	C₄×C₈	—	89	1	no	—
232	#41	—	356	4	no	—
233	C₂×Q₈	—	89	1	no	—
234	C₂×Q₈	—	178	1	no	—
235	C₂×Q₈	—	178	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	89	1	no	—
238	C₂×D₄	—	178	1	no	—
239	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	89	1	no	—
242	C₂²×C₄	—	178	1	no	—
243	C₂²×C₄	—	178	1	no	—
244	C₂²×C₄	—	178	1	no	—
245	C₂²×C₄	—	178	1	no	—
246	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
247	SD₁₆	—	178	2	no	—
248	C₂×C₈	—	89	1	no	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	—	178	2	no	—
251	C₂×C₈	—	178	2	no	—
252	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	89	1	no	—
256	C₄⋊C₄	—	89	1	no	—
257	C₄⋊C₄	—	89	1	no	—
258	C₄⋊C₄	—	89	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	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	89	1	no	—
272	C₂²⋊C₄	—	89	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₄²	—	89	1	no	—
278	C₄²	—	178	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	89	1	no	—
281	C₂³	D₄×D₈₉	1	1	yes	characteristic
282	Q₈	—	356	2	no	—
283	Q₈	—	178	1	no	—
284	Q₈	—	178	1	no	—
285	Q₈	—	712	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	4	2	no	—
289	D₄	—	89	4	no	—
290	D₄	—	356	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	89	1	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	356	2	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	712	1	no	—
310	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	89	2	no	—
315	C₈	—	178	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	89	2	no	—
322	C₂²	—	178	1	no	—
323	C₂²	—	178	1	no	—
324	C₂²	SD₁₆×D₈₉	1	2	yes	—
325	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	89	2	no	—
331	C₄	—	356	1	no	—
332	C₄	—	356	1	no	—
333	C₄	—	356	1	no	—
334	C₄	—	356	1	no	—
335	C₄	—	178	1	no	—
336	C₄	—	178	1	no	—
337	C₄	#41×D₈₉	1	1	yes	characteristic
338	C₄	#15668059	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	8	1	no	—
341	C₂	—	178	2	no	—
342	C₂	#19152246	1	1	yes	characteristic
343	C₂	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
344	C₂	#19152408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#481×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152248	C₂
hypercenter	#481	D₈₉
hypocenter	C₈₉	#3335
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #481

Complements

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

N	H ≅ G/N	type
1. #19152303	C₂	semidirect
3. #19151995	C₂	semidirect
4. #19151930	C₂	semidirect
5. #19151929	C₂	semidirect
6. #19151915	C₂	semidirect
7. #19151886	C₂	semidirect
8. #19151882	C₂	semidirect
9. #19151826	C₂	semidirect
10. #19151825	C₂	semidirect
11. #19151689	C₂	semidirect
12. #19151661	C₂	semidirect
13. C₂×#15667943	C₂	semidirect
14. #106×D₈₉	C₂	semidirect
15. #481×C₈₉	C₂	semidirect
121. C₂²⋊D₈₉	SD₁₆	semidirect
142. #10623494	#41	semidirect
173. C₈₉	#3335	semidirect
180. #481	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152303	2
#19152248	2
#19151995	2
#19151930	2
#19151929	2
#19151915	2
#19151886	2
#19151882	2
#19151826	2
#19151825	2
#19151689	2
#19151661	2
C₂×#15667943	2
#106×D₈₉	2
#481×C₈₉	2
#3335	89