Subgroups — #26031503

Order 11392.

332 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	#19152226	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152201	C₂	1	1	yes	maximal, characteristic
3	#19152006	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151998	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151961	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151943	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151922	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151917	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151909	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151900	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151897	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151886	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151877	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151656	C₂	1	1	yes	maximal, characteristic, complemented
15	#482×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	#15668012	—	2	1	no	—
17	#15668011	C₂²	1	1	yes	characteristic
18	#15668011	C₂²	1	1	yes	characteristic
19	#15668008	C₂²	1	1	yes	characteristic
20	#15668004	—	2	1	no	—
21	#15668003	C₂²	1	1	yes	characteristic
22	#15668003	C₂²	1	1	yes	characteristic
23	#15668003	C₂²	1	1	yes	characteristic
24	#15667996	—	2	1	no	—
25	#15667994	—	2	1	no	—
26	#15667989	—	2	1	no	—
27	#15667989	—	2	1	no	—
28	#15667986	C₂²	1	1	yes	characteristic
29	#15667984	—	2	1	no	—
30	#15667983	—	2	1	no	—
31	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
32	#15667974	C₂²	1	1	yes	characteristic
33	#15667971	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
35	#15667945	C₂²	1	1	yes	characteristic
36	#15667945	C₂²	1	1	yes	characteristic
37	#15667945	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	#15667941	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667934	C₂²	1	1	yes	characteristic
44	#15667934	C₂²	1	1	yes	characteristic
45	#15667933	C₂²	1	1	yes	characteristic
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	#121×C₈₉	C₂²	1	1	yes	characteristic
53	#117×C₈₉	C₂²	1	1	yes	characteristic
54	#107×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	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	—	2	1	no	—
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×#10623494	C₂³	1	1	yes	characteristic
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
75	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	C₂²⋊Dic₈₉	—	2	1	no	—
85	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
90	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
91	C₂×D₄×C₈₉	—	2	1	no	—
92	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²⋊C₄×C₈₉	—	2	1	no	—
100	C₂²⋊C₄×C₈₉	—	2	1	no	—
101	C₄²×C₈₉	—	2	1	no	—
102	C₂³×C₈₉	—	2	1	no	—
103	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×D₈₉	—	2	1	no	—
105	D₄×C₈₉	—	2	1	no	—
106	D₄×C₈₉	—	2	1	no	—
107	D₄×C₈₉	—	8	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	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	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₈₉	—	8	1	no	—
116	C₂²⋊D₈₉	—	8	1	no	—
117	C₂²⋊D₈₉	—	8	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	8	1	no	—
122	C₂×Dic₈₉	—	8	1	no	—
123	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	D₃₅₆	—	2	1	no	—
128	D₃₅₆	—	2	1	no	—
129	C₄×D₈₉	—	8	1	no	—
130	Dic₁₇₈	—	8	1	no	—
131	Dic₁₇₈	—	8	1	no	—
132	C₈×C₈₉	—	2	1	no	—
133	C₈×C₈₉	—	2	1	no	—
134	#10623494	—	2	1	no	—
135	#10623494	—	2	1	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	4	1	no	—
140	C₂²×C₈₉	—	4	1	no	—
141	C₂²×C₈₉	—	4	1	no	—
142	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₈₉	—	4	1	no	—
144	C₂×D₈₉	—	4	1	no	—
145	C₂×D₈₉	—	4	1	no	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	4	1	no	—
148	C₄×C₈₉	—	4	1	no	—
149	C₄×C₈₉	—	4	1	no	—
150	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
151	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₈₉	—	2	1	no	—
153	Dic₈₉	—	2	1	no	—
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	4	1	no	—
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	—	4	1	no	—
159	C₂×C₈₉	—	4	1	no	—
160	C₂×C₈₉	—	8	1	no	—
161	C₂×C₈₉	#536	1	1	yes	characteristic
162	C₂×C₈₉	#576	1	1	yes	characteristic
163	C₂×C₈₉	#577	1	1	yes	characteristic
164	D₈₉	—	8	1	no	—
165	#3241	—	89	1	no	maximal, Sylow
166	C₈₉	#3241	1	1	yes	characteristic, complemented, Sylow
167	#544	—	89	1	no	—
168	#540	—	89	1	no	—
169	#501	—	89	1	no	—
170	#499	—	89	1	no	—
171	#490	—	89	1	no	—
172	#486	—	89	1	no	—
173	#482	—	89	1	no	—
174	#482	D₈₉	1	1	yes	characteristic, complemented
175	#481	—	89	1	no	—
176	#480	—	89	1	no	—
177	#478	—	89	1	no	—
178	#478	—	89	1	no	—
179	#476	—	89	1	no	—
180	#474	—	89	1	no	—
181	#433	—	89	1	no	—
182	C₄⋊Q₈	—	89	1	no	—
183	C₄⋊Q₈	—	89	1	no	—
184	C₄⋊Q₈	—	178	1	no	—
185	#127	—	89	1	no	—
186	#125	—	89	1	no	—
187	#125	—	89	1	no	—
188	#125	—	89	1	no	—
189	#125	—	178	1	no	—
190	#123	—	178	1	no	—
191	#123	—	178	1	no	—
192	C₂²⋊Q₈	—	178	1	no	—
193	C₂²⋊Q₈	—	178	1	no	—
194	#121	—	89	1	no	—
195	#121	—	178	1	no	—
196	#121	—	178	1	no	—
197	#121	C₂×D₈₉	1	1	yes	characteristic
198	C₄×D₄	—	89	1	no	—
199	C₄×D₄	—	89	1	no	—
200	#117	—	89	1	no	—
201	#117	C₂×D₈₉	1	1	yes	characteristic
202	C₂×C₄⋊C₄	—	89	1	no	—
203	#107	—	89	1	no	—
204	#107	—	89	1	no	—
205	#107	—	89	1	no	—
206	#107	C₂×D₈₉	1	1	yes	characteristic
207	#106	—	89	1	no	—
208	#106	—	89	1	no	—
209	#106	—	89	1	no	—
210	#106	C₂×D₈₉	1	1	yes	characteristic
211	C₄⋊C₈	—	89	1	no	—
212	C₄⋊C₈	—	89	1	no	—
213	#102	—	89	1	no	—
214	#102	—	89	1	no	—
215	#102	—	89	1	no	—
216	#102	—	89	1	no	—
217	#102	—	89	1	no	—
218	#102	—	89	1	no	—
219	#102	C₂×D₈₉	1	1	yes	characteristic
220	#102	C₂×D₈₉	1	1	yes	characteristic
221	C₂²⋊C₈	—	89	1	no	—
222	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
223	#97	—	89	1	no	—
224	C₄×C₈	—	89	1	no	—
225	C₂×Q₈	—	178	1	no	—
226	C₂×Q₈	—	178	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	89	1	no	—
229	C₂×D₄	—	178	1	no	—
230	C₂×D₄	—	178	1	no	—
231	C₂×D₄	—	178	1	no	—
232	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
233	C₂²×C₄	—	178	1	no	—
234	C₂²×C₄	—	178	1	no	—
235	C₂²×C₄	—	178	1	no	—
236	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
237	C₂×C₈	—	89	1	no	—
238	C₂×C₈	—	89	1	no	—
239	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₄⋊C₄	—	89	1	no	—
242	C₄⋊C₄	—	89	1	no	—
243	C₄⋊C₄	—	89	1	no	—
244	C₄⋊C₄	—	89	1	no	—
245	C₄⋊C₄	—	89	1	no	—
246	C₄⋊C₄	—	178	1	no	—
247	C₄⋊C₄	—	178	1	no	—
248	C₄⋊C₄	—	178	1	no	—
249	C₄⋊C₄	—	178	1	no	—
250	C₄⋊C₄	—	178	1	no	—
251	C₄⋊C₄	—	178	1	no	—
252	C₄⋊C₄	—	178	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	89	1	no	—
261	C₂²⋊C₄	—	89	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₂²⋊C₄	—	178	1	no	—
264	C₂²⋊C₄	—	178	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	89	1	no	—
267	C₄²	—	89	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	178	1	no	—
270	C₂³	D₄×D₈₉	1	1	yes	characteristic
271	Q₈	—	712	1	no	—
272	Q₈	—	712	1	no	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	D₄	—	8	1	no	—
276	D₄	—	178	1	no	—
277	D₄	—	178	1	no	—
278	D₄	—	712	1	no	—
279	D₄	—	712	1	no	—
280	D₄	—	712	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	89	1	no	—
287	C₂×C₄	—	89	1	no	—
288	C₂×C₄	—	89	1	no	—
289	C₂×C₄	—	89	1	no	—
290	C₂×C₄	—	178	1	no	—
291	C₂×C₄	—	178	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	712	1	no	—
294	C₂×C₄	—	712	1	no	—
295	C₂×C₄	—	712	1	no	—
296	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	178	1	no	—
301	C₈	—	178	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	356	1	no	—
309	C₂²	—	356	1	no	—
310	C₂²	—	356	1	no	—
311	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	356	1	no	—
317	C₄	—	356	1	no	—
318	C₄	—	356	1	no	—
319	C₄	—	178	1	no	—
320	C₄	—	178	1	no	—
321	C₄	—	178	1	no	—
322	C₄	—	178	1	no	—
323	C₄	#15668059	1	1	yes	characteristic
324	C₄	#15668059	1	1	yes	characteristic
325	C₂	—	4	1	no	—
326	C₂	—	8	1	no	—
327	C₂	—	712	1	no	—
328	C₂	#19152170	1	1	yes	characteristic
329	C₂	#19152402	1	1	yes	characteristic
330	C₂	#19152405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#482×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152201	C₂
hypercenter	#482	D₈₉
hypocenter	C₈₉	#3241
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
1. #19152226	C₂	semidirect
3. #19152006	C₂	semidirect
4. #19151998	C₂	semidirect
5. #19151961	C₂	semidirect
6. #19151943	C₂	semidirect
7. #19151922	C₂	semidirect
8. #19151917	C₂	semidirect
9. #19151909	C₂	semidirect
10. #19151900	C₂	semidirect
11. #19151897	C₂	semidirect
12. #19151886	C₂	semidirect
13. #19151877	C₂	semidirect
14. #19151656	C₂	semidirect
15. #482×C₈₉	C₂	semidirect
166. C₈₉	#3241	semidirect
174. #482	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152226	2
#19152201	2
#19152006	2
#19151998	2
#19151961	2
#19151943	2
#19151922	2
#19151917	2
#19151909	2
#19151900	2
#19151897	2
#19151886	2
#19151877	2
#19151656	2
#482×C₈₉	2
#3241	89