Subgroups — #26031415

Order 11392.

334 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152127	C₂	1	1	yes	maximal, characteristic
2	#125×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152000	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151997	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151952	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151932	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151915	C₂	1	1	yes	maximal, characteristic
9	#19151910	C₂	1	1	yes	maximal, characteristic
10	#19151907	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151903	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151900	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151896	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151649	C₂	1	1	yes	maximal, characteristic, complemented
15	#488×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	#15668005	—	2	1	no	—
17	#15668004	C₂²	1	1	yes	characteristic
18	#15668004	C₂²	1	1	yes	characteristic
19	#15668003	C₂²	1	1	yes	characteristic
20	#15668003	C₂²	1	1	yes	characteristic
21	#15668002	—	2	1	no	—
22	#15668002	—	2	1	no	—
23	#15667994	—	2	1	no	—
24	#15667988	C₂²	1	1	yes	characteristic
25	#15667983	C₂²	1	1	yes	characteristic
26	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
27	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
28	#15667972	—	2	1	no	—
29	#15667968	—	2	1	no	—
30	C₄⋊C₄×D₈₉	—	2	1	no	—
31	C₄⋊C₄×D₈₉	—	2	1	no	—
32	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
34	C₄²×D₈₉	C₂²	1	1	yes	characteristic
35	#15667946	C₂²	1	1	yes	characteristic
36	#15667946	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667944	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₄×#10623494	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	#124×C₈₉	C₂²	1	1	yes	characteristic
54	#103×C₈₉	C₂²	1	1	yes	characteristic
55	#103×C₈₉	C₂²	1	1	yes	characteristic
56	#102×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×C₂²⋊D₈₉	—	2	1	no	—
60	C₂²×Dic₈₉	—	2	1	no	—
61	C₂×C₄×D₈₉	—	2	1	no	—
62	C₂×C₄×D₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
64	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
66	C₂×#10623494	C₂³	1	1	yes	characteristic
67	C₂×#10623494	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₈₉	—	2	1	no	—
69	C₄⋊Dic₈₉	—	2	1	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	#13083553	—	2	1	no	—
76	#13083553	—	2	1	no	—
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
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	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	D₄	1	1	yes	characteristic
89	C₄×Dic₈₉	D₄	1	1	yes	characteristic
90	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₄⋊C₄×C₈₉	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
102	C₂³×C₈₉	—	2	1	no	—
103	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
104	Q₈×C₈₉	—	2	1	no	—
105	Q₈×C₈₉	—	2	1	no	—
106	D₄×C₈₉	—	2	1	no	—
107	D₄×C₈₉	—	2	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	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₈₉	—	4	2	no	—
115	C₂×Dic₈₉	—	2	1	no	—
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	4	2	no	—
120	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄×D₈₉	—	2	2	no	—
124	C₄×D₈₉	—	2	1	no	—
125	C₄×D₈₉	—	2	1	no	—
126	C₄×D₈₉	—	4	1	no	—
127	C₄×D₈₉	—	4	2	no	—
128	C₄×D₈₉	—	4	2	no	—
129	C₈×C₈₉	—	2	1	no	—
130	C₈×C₈₉	—	2	1	no	—
131	#10623494	—	2	1	no	—
132	#10623494	—	2	1	no	—
133	C₂²×C₈₉	—	4	1	no	—
134	C₂²×C₈₉	—	4	1	no	—
135	C₂²×C₈₉	—	4	1	no	—
136	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₈₉	—	2	1	no	—
138	C₂×D₈₉	—	2	1	no	—
139	C₂×D₈₉	#135	1	2	yes	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	4	1	no	—
143	C₄×C₈₉	—	4	1	no	—
144	C₄×C₈₉	—	4	1	no	—
145	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
146	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₈₉	—	2	1	no	—
148	Dic₈₉	—	2	1	no	—
149	Dic₈₉	—	2	1	no	—
150	Dic₈₉	—	4	1	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	—	4	1	no	—
154	C₂×C₈₉	—	8	1	no	—
155	C₂×C₈₉	#536	1	1	yes	characteristic
156	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
157	C₂×C₈₉	#575	1	1	yes	characteristic
158	D₈₉	—	2	2	no	—
159	#3232	—	89	1	no	maximal, Sylow
160	C₈₉	#3232	1	1	yes	characteristic, complemented, Sylow
161	#529	—	89	1	no	—
162	C₂×#125	—	89	1	no	—
163	#499	—	89	1	no	—
164	#499	—	89	1	no	—
165	#488	—	89	1	no	—
166	#488	D₈₉	1	1	yes	characteristic, complemented
167	#484	—	89	1	no	—
168	#483	—	89	1	no	—
169	#481	—	89	1	no	—
170	#480	—	89	1	no	—
171	#479	—	89	1	no	—
172	#479	—	89	1	no	—
173	#478	—	89	1	no	—
174	#478	—	89	1	no	—
175	#431	—	89	1	no	—
176	#126	—	178	1	no	—
177	#125	—	89	1	no	—
178	#125	—	89	1	no	—
179	#125	—	89	1	no	—
180	#125	—	89	1	no	—
181	#125	—	178	1	no	—
182	#125	—	178	1	no	—
183	#125	C₂×D₈₉	1	1	yes	characteristic
184	#124	C₂×D₈₉	1	1	yes	characteristic
185	#123	—	178	1	no	—
186	C₂²⋊Q₈	—	89	1	no	—
187	#121	—	89	1	no	—
188	C₄×Q₈	—	89	1	no	—
189	C₄×D₄	—	89	1	no	—
190	C₄×D₄	—	178	1	no	—
191	#117	—	178	1	no	—
192	C₂×C₄⋊C₄	—	89	1	no	—
193	C₂×C₄⋊C₄	—	89	1	no	—
194	C₂×C₄⋊C₄	—	178	1	no	—
195	C₂×C₄⋊C₄	—	178	1	no	—
196	C₂×C₄²	—	89	1	no	—
197	#107	—	89	1	no	—
198	#107	—	89	1	no	—
199	#107	—	89	1	no	—
200	#107	—	89	1	no	—
201	#106	—	89	1	no	—
202	#106	—	89	1	no	—
203	#106	—	89	1	no	—
204	#106	—	89	1	no	—
205	C₄⋊C₈	—	89	1	no	—
206	C₄⋊C₈	—	89	1	no	—
207	#103	—	89	1	no	—
208	#103	—	89	1	no	—
209	#103	C₂×D₈₉	1	1	yes	characteristic
210	#103	C₂×D₈₉	1	1	yes	characteristic
211	#102	—	89	1	no	—
212	#102	—	89	1	no	—
213	#102	C₂×D₈₉	1	1	yes	characteristic
214	#102	C₂×D₈₉	1	1	yes	characteristic
215	C₂²⋊C₈	—	89	1	no	—
216	C₂²⋊C₈	—	89	1	no	—
217	C₄×C₈	—	89	1	no	—
218	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
219	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
220	C₂×D₄	—	178	1	no	—
221	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
222	C₂²×C₄	—	89	1	no	—
223	C₂²×C₄	—	89	1	no	—
224	C₂²×C₄	—	89	1	no	—
225	C₂²×C₄	—	178	1	no	—
226	C₂²×C₄	—	178	1	no	—
227	C₂²×C₄	—	178	1	no	—
228	C₂×C₈	—	89	1	no	—
229	C₂×C₈	—	89	1	no	—
230	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	89	1	no	—
235	C₄⋊C₄	—	89	1	no	—
236	C₄⋊C₄	—	89	1	no	—
237	C₄⋊C₄	—	89	1	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₄⋊C₄	—	178	1	no	—
241	C₄⋊C₄	—	178	1	no	—
242	C₄⋊C₄	—	178	1	no	—
243	C₄⋊C₄	—	178	1	no	—
244	C₄⋊C₄	—	178	1	no	—
245	C₄⋊C₄	—	178	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	C₂²×D₈₉	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	178	1	no	—
255	C₂²⋊C₄	—	178	1	no	—
256	C₂²⋊C₄	—	178	1	no	—
257	C₄²	—	89	1	no	—
258	C₄²	—	89	1	no	—
259	C₄²	—	89	1	no	—
260	C₄²	—	178	1	no	—
261	C₄²	C₂²×D₈₉	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	—	89	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	D₄	—	2	1	no	—
267	D₄	—	2	1	no	—
268	D₄	—	356	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	89	1	no	—
273	C₂×C₄	—	89	1	no	—
274	C₂×C₄	—	89	1	no	—
275	C₂×C₄	—	356	1	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	178	2	no	—
280	C₂×C₄	—	178	1	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	—	178	1	no	—
283	C₂×C₄	—	178	1	no	—
284	C₂×C₄	—	178	1	no	—
285	C₂×C₄	—	178	1	no	—
286	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
287	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	178	1	no	—
292	C₈	—	178	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	89	2	no	—
297	C₂²	—	178	1	no	—
298	C₂²	—	178	1	no	—
299	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	356	1	no	—
306	C₄	—	356	1	no	—
307	C₄	—	356	1	no	—
308	C₄	—	356	1	no	—
309	C₄	—	178	1	no	—
310	C₄	—	178	1	no	—
311	C₄	—	178	1	no	—
312	C₄	#15668059	1	1	yes	characteristic
313	C₄	#15668059	1	1	yes	characteristic
314	C₂	—	8	1	no	—
315	C₂	—	178	2	no	—
316	C₂	#19152168	1	1	yes	characteristic
317	C₂	#135×D₈₉	1	1	yes	characteristic
318	C₂	#19152395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#488×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#15667983	C₂²
hypercenter	#488	D₈₉
hypocenter	C₈₉	#3232
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
2. #125×D₈₉	C₂	semidirect
3. #19152000	C₂	semidirect
4. #19151997	C₂	semidirect
5. #19151952	C₂	semidirect
6. #19151932	C₂	semidirect
7. #19151927	C₂	semidirect
10. #19151907	C₂	semidirect
11. #19151903	C₂	semidirect
12. #19151900	C₂	semidirect
13. #19151896	C₂	semidirect
14. #19151649	C₂	semidirect
15. #488×C₈₉	C₂	semidirect
160. C₈₉	#3232	semidirect
166. #488	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152127	2
#125×D₈₉	2
#19152000	2
#19151997	2
#19151952	2
#19151932	2
#19151927	2
#19151915	2
#19151910	2
#19151907	2
#19151903	2
#19151900	2
#19151896	2
#19151649	2
#488×C₈₉	2
#3232	89