Subgroups — #26032691

Order 11392.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152286	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152251	C₂	1	1	yes	maximal, characteristic
3	#19151935	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151930	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151924	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151913	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151890	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151845	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151828	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151671	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151637	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#501×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	—	2	1	no	—
17	C₂×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083557	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	#15668007	—	2	1	no	—
21	#15668003	—	2	1	no	—
22	#15667997	—	2	1	no	—
23	#15667997	—	2	1	no	—
24	#15667997	—	2	1	no	—
25	#15667992	C₂²	1	1	yes	characteristic
26	#15667992	C₂²	1	1	yes	characteristic
27	#15667988	—	2	1	no	—
28	#15667987	C₂²	1	1	yes	characteristic
29	#15667987	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
31	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
32	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
33	#15667970	—	2	1	no	—
34	#15667970	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	#15667946	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667945	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	#125×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#106×C₈₉	C₂²	1	1	yes	characteristic
57	#106×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	#13083575	—	4	4	no	—
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	—	2	1	no	—
63	C₂×D₃₅₆	D₄	1	1	yes	characteristic
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	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	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
71	#13083563	—	2	2	no	—
72	#13083557	—	2	2	no	—
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₈×D₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	D₄	1	1	yes	characteristic
92	#13083551	D₄	1	1	yes	characteristic
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₂×Q₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
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	C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²×D₈₉	—	2	1	no	—
107	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₈₉	—	4	2	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
122	D₃₅₆	—	2	2	no	—
123	D₃₅₆	—	2	1	no	—
124	D₃₅₆	—	2	1	no	—
125	D₃₅₆	—	4	2	no	—
126	D₃₅₆	—	4	2	no	—
127	C₄×D₈₉	—	2	1	no	—
128	C₄×D₈₉	—	2	1	no	—
129	C₄×D₈₉	—	2	1	no	—
130	C₄×D₈₉	—	2	1	no	—
131	C₄×D₈₉	—	4	2	no	—
132	C₄×D₈₉	—	4	2	no	—
133	C₄×D₈₉	—	4	2	no	—
134	C₄×D₈₉	—	4	2	no	—
135	Dic₁₇₈	—	2	1	no	—
136	Dic₁₇₈	—	2	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	#41	1	2	yes	complemented
139	#10623494	—	2	1	no	—
140	#10623494	—	2	1	no	—
141	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₈₉	—	2	1	no	—
143	C₂×D₈₉	—	2	1	no	—
144	C₂×D₈₉	—	2	1	no	—
145	C₂×D₈₉	—	2	1	no	—
146	C₂×D₈₉	—	2	1	no	—
147	C₂×D₈₉	—	4	1	no	—
148	C₂×D₈₉	—	4	1	no	—
149	C₂×D₈₉	—	4	1	no	—
150	C₂×D₈₉	AGL(1,ℤ/8)	1	2	yes	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	4	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	C₂×C₈₉	#547	1	1	yes	characteristic
165	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₈₉	#578	1	1	yes	characteristic
167	D₈₉	—	2	2	no	—
168	D₈₉	—	4	1	no	—
169	D₈₉	—	8	1	no	—
170	#3343	—	89	1	no	maximal, Sylow
171	C₈₉	#3343	1	1	yes	characteristic, complemented, Sylow
172	#552	—	89	1	no	—
173	#547	—	89	1	no	—
174	#501	D₈₉	1	1	yes	characteristic, complemented
175	#484	—	89	1	no	—
176	#483	—	89	1	no	—
177	#482	—	89	1	no	—
178	#480	—	89	1	no	—
179	#478	—	89	1	no	—
180	#477	—	89	1	no	—
181	#468	—	89	1	no	—
182	#465	—	89	1	no	—
183	#440	—	89	1	no	—
184	#435	—	89	1	no	—
185	#428	—	89	1	no	—
186	C₂×#107	—	89	1	no	—
187	C₂×#41	—	178	1	no	—
188	C₂×SD₁₆	—	89	1	no	—
189	C₂×M₁₆	—	89	1	no	—
190	C₂²×C₈	—	89	1	no	—
191	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
192	#126	—	178	1	no	—
193	#125	—	178	1	no	—
194	#125	C₂×D₈₉	1	1	yes	characteristic
195	#123	—	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	#121	—	89	1	no	—
202	#121	—	89	1	no	—
203	C₄×Q₈	—	89	1	no	—
204	C₄×D₄	—	89	1	no	—
205	C₄×D₄	—	89	1	no	—
206	#117	—	89	1	no	—
207	#117	—	178	1	no	—
208	C₂×C₄⋊C₄	—	89	1	no	—
209	C₂×C₄⋊C₄	—	89	1	no	—
210	C₂×C₄⋊C₄	—	89	1	no	—
211	#107	—	89	1	no	—
212	#107	—	89	1	no	—
213	#107	—	89	1	no	—
214	#107	—	89	1	no	—
215	#107	C₂×D₈₉	1	1	yes	characteristic
216	#107	C₂×D₈₉	1	1	yes	characteristic
217	#106	—	89	1	no	—
218	#106	C₂×D₈₉	1	1	yes	characteristic
219	#106	C₂×D₈₉	1	1	yes	characteristic
220	C₄⋊C₈	—	89	1	no	—
221	#103	—	89	1	no	—
222	#103	—	89	1	no	—
223	#103	—	89	1	no	—
224	#102	—	89	1	no	—
225	#102	—	89	1	no	—
226	#102	—	89	1	no	—
227	C₂²⋊C₈	—	89	1	no	—
228	C₂²⋊C₈	—	89	1	no	—
229	#97	C₂×D₈₉	1	1	yes	characteristic
230	#41	—	356	4	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	89	1	no	—
233	C₂×D₄	—	89	1	no	—
234	C₂×D₄	—	89	1	no	—
235	C₂×D₄	—	178	1	no	—
236	C₂×D₄	—	178	1	no	—
237	C₂²×C₄	—	89	1	no	—
238	C₂²×C₄	—	89	1	no	—
239	C₂²×C₄	—	178	1	no	—
240	C₂²×C₄	—	178	1	no	—
241	C₂²×C₄	—	178	1	no	—
242	C₂²×C₄	—	178	1	no	—
243	SD₁₆	—	178	2	no	—
244	M₁₆	—	178	2	no	—
245	C₂×C₈	—	89	1	no	—
246	C₂×C₈	—	89	1	no	—
247	C₂×C₈	—	178	2	no	—
248	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	89	1	no	—
254	C₄⋊C₄	—	89	1	no	—
255	C₄⋊C₄	—	89	1	no	—
256	C₄⋊C₄	—	178	1	no	—
257	C₄⋊C₄	—	178	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	C₂²×D₈₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	89	1	no	—
267	C₂²⋊C₄	—	89	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₂²⋊C₄	—	178	1	no	—
271	C₂²⋊C₄	—	178	1	no	—
272	C₂²⋊C₄	—	178	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₄²	—	178	1	no	—
275	C₄²	—	178	1	no	—
276	C₄²	C₂²×D₈₉	1	1	yes	characteristic
277	C₂³	—	89	1	no	—
278	C₂³	—	89	1	no	—
279	C₂³	—	178	1	no	—
280	Q₈	—	4	2	no	—
281	Q₈	—	178	1	no	—
282	Q₈	—	178	1	no	—
283	D₄	—	356	2	no	—
284	D₄	—	356	2	no	—
285	D₄	—	178	2	no	—
286	D₄	—	178	1	no	—
287	D₄	—	178	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	89	1	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	356	2	no	—
295	C₂×C₄	—	356	2	no	—
296	C₂×C₄	—	356	2	no	—
297	C₂×C₄	—	356	2	no	—
298	C₂×C₄	—	178	1	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	178	1	no	—
301	C₂×C₄	—	178	1	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
306	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	178	1	no	—
310	C₈	—	178	1	no	—
311	C₈	#13083575	1	2	yes	complemented
312	C₂²	—	89	2	no	—
313	C₂²	—	356	1	no	—
314	C₂²	—	356	1	no	—
315	C₂²	—	356	1	no	—
316	C₂²	—	178	1	no	—
317	C₂²	—	178	1	no	—
318	C₂²	—	178	1	no	—
319	C₂²	—	178	1	no	—
320	C₂²	—	178	1	no	—
321	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	356	1	no	—
329	C₄	—	356	1	no	—
330	C₄	—	356	1	no	—
331	C₄	—	178	1	no	—
332	C₄	—	178	1	no	—
333	C₄	C₂×#13083575	1	1	yes	characteristic
334	C₄	#15668060	1	1	yes	characteristic
335	C₂	—	356	1	no	—
336	C₂	—	178	2	no	—
337	C₂	—	712	1	no	—
338	C₂	#19152251	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
340	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#501×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152251	C₂
hypercenter	#501	D₈₉
hypocenter	C₈₉	#3343
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
1. #19152286	C₂	semidirect
3. #19151935	C₂	semidirect
4. #19151930	C₂	semidirect
5. #19151924	C₂	semidirect
6. #19151913	C₂	semidirect
7. #19151895	C₂	semidirect
8. #19151890	C₂	semidirect
9. #19151845	C₂	semidirect
10. #19151828	C₂	semidirect
11. #19151700	C₂	semidirect
12. #19151671	C₂	semidirect
13. #19151637	C₂	semidirect
14. #107×D₈₉	C₂	semidirect
15. #501×C₈₉	C₂	semidirect
138. C₈×C₈₉	#41	semidirect
171. C₈₉	#3343	semidirect
174. #501	D₈₉	semidirect
311. C₈	#13083575	semidirect

Maximal subgroups

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

subgroup	index
#19152286	2
#19152251	2
#19151935	2
#19151930	2
#19151924	2
#19151913	2
#19151895	2
#19151890	2
#19151845	2
#19151828	2
#19151700	2
#19151671	2
#19151637	2
#107×D₈₉	2
#501×C₈₉	2
#3343	89