Subgroups — #26031544

Order 11392.

320 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	#19152226	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152004	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151994	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151957	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151952	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151936	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151926	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151919	C₂	1	1	yes	maximal, characteristic
9	#19151914	C₂	1	1	yes	maximal, characteristic
10	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151898	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151888	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151883	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151659	C₂	1	1	yes	maximal, characteristic, complemented
15	#542×C₈₉	C₂	1	1	yes	maximal, characteristic
16	#15668012	C₂²	1	1	yes	characteristic
17	#15668012	C₂²	1	1	yes	characteristic
18	#15668011	—	2	1	no	—
19	#15668004	C₂²	1	1	yes	characteristic
20	#15668004	C₂²	1	1	yes	characteristic
21	#15668003	—	2	1	no	—
22	#15667989	—	2	1	no	—
23	#15667989	—	2	1	no	—
24	#15667971	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
26	#15667945	C₂²	1	1	yes	characteristic
27	#15667945	C₂²	1	1	yes	characteristic
28	#15667943	C₂²	1	1	yes	characteristic
29	#15667943	C₂²	1	1	yes	characteristic
30	#15667943	C₂²	1	1	yes	characteristic
31	#15667943	C₂²	1	1	yes	characteristic
32	#15667943	C₂²	1	1	yes	characteristic
33	#15667943	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	C₄⋊#10623494	C₂²	1	1	yes	characteristic
36	#15667936	C₂²	1	1	yes	characteristic
37	#15667936	C₂²	1	1	yes	characteristic
38	#15667936	C₂²	1	1	yes	characteristic
39	#15667936	C₂²	1	1	yes	characteristic
40	#15667933	C₂²	1	1	yes	characteristic
41	#15667933	C₂²	1	1	yes	characteristic
42	#15667933	C₂²	1	1	yes	characteristic
43	#15667933	C₂²	1	1	yes	characteristic
44	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
45	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
46	#15667923	C₂²	1	1	yes	characteristic
47	C₄×#10623494	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
49	#126×C₈₉	—	2	1	no	—
50	#124×C₈₉	—	2	1	no	—
51	#124×C₈₉	C₂²	1	1	yes	characteristic
52	#123×C₈₉	—	2	1	no	—
53	C₂²⋊Q₈×C₈₉	—	2	1	no	—
54	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	#121×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
58	#117×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂²×Dic₈₉	—	2	1	no	—
62	C₂×#10623494	C₂³	1	1	yes	characteristic
63	C₂×#10623494	C₂³	1	1	yes	characteristic
64	C₂×#10623494	C₂³	1	1	yes	characteristic
65	C₂×#10623494	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₈₉	—	2	1	no	—
67	C₄⋊Dic₈₉	—	2	1	no	—
68	C₄⋊Dic₈₉	—	2	1	no	—
69	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
73	#13083553	—	2	1	no	—
74	#13083553	—	2	1	no	—
75	#13083553	—	2	1	no	—
76	#13083553	—	2	1	no	—
77	C₂²⋊Dic₈₉	—	2	1	no	—
78	C₄×Dic₈₉	—	2	1	no	—
79	C₂×Q₈×C₈₉	—	2	1	no	—
80	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₂×D₄×C₈₉	—	2	1	no	—
82	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	—	2	1	no	—
84	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₂²⋊C₄×C₈₉	—	2	1	no	—
93	C₂²⋊C₄×C₈₉	—	2	1	no	—
94	C₂²⋊C₄×C₈₉	—	2	1	no	—
95	C₂²⋊C₄×C₈₉	—	2	1	no	—
96	C₂²⋊C₄×C₈₉	—	2	1	no	—
97	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
98	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
99	C₄²×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	C₂³	1	1	yes	characteristic
101	C₄²×C₈₉	C₂³	1	1	yes	characteristic
102	C₂³×C₈₉	—	2	1	no	—
103	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
104	Q₈×C₈₉	—	2	1	no	—
105	Q₈×C₈₉	—	2	1	no	—
106	Q₈×C₈₉	—	8	1	no	—
107	D₄×C₈₉	—	2	1	no	—
108	D₄×C₈₉	—	2	1	no	—
109	D₄×C₈₉	—	8	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	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	8	1	no	—
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
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	—	8	1	no	—
127	Dic₁₇₈	—	8	1	no	—
128	Dic₁₇₈	—	8	1	no	—
129	#10623494	—	2	1	no	—
130	#10623494	—	2	1	no	—
131	#10623494	—	2	1	no	—
132	#10623494	—	2	1	no	—
133	C₂²×C₈₉	—	2	1	no	—
134	C₂²×C₈₉	—	2	1	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	—	4	1	no	—
137	C₂²×C₈₉	—	4	1	no	—
138	C₂²×C₈₉	—	4	1	no	—
139	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	4	1	no	—
146	C₄×C₈₉	—	4	1	no	—
147	C₄×C₈₉	—	4	1	no	—
148	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
149	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₈₉	—	4	1	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	—	4	1	no	—
154	C₂×C₈₉	—	4	1	no	—
155	C₂×C₈₉	—	8	1	no	—
156	C₂×C₈₉	#536	1	1	yes	characteristic
157	C₂×C₈₉	#577	1	1	yes	characteristic
158	C₂×C₈₉	#577	1	1	yes	characteristic
159	#3244	—	89	1	no	maximal, Sylow
160	C₈₉	#3244	1	1	yes	characteristic, complemented, Sylow
161	#544	—	89	1	no	—
162	#542	D₈₉	1	1	yes	characteristic
163	#501	—	89	1	no	—
164	#498	—	89	1	no	—
165	#489	—	89	1	no	—
166	#488	—	89	1	no	—
167	#485	—	89	1	no	—
168	#483	—	89	1	no	—
169	#482	—	89	1	no	—
170	#481	—	89	1	no	—
171	#479	—	89	1	no	—
172	#478	—	89	1	no	—
173	#476	—	89	1	no	—
174	#475	—	89	1	no	—
175	#433	—	89	1	no	—
176	C₄⋊Q₈	—	89	1	no	—
177	C₄⋊Q₈	—	89	1	no	—
178	C₄⋊Q₈	—	178	1	no	—
179	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
180	#126	—	2	1	no	—
181	#125	—	89	1	no	—
182	#125	—	89	1	no	—
183	#125	—	178	1	no	—
184	#124	—	2	1	no	—
185	#124	C₂×D₈₉	1	1	yes	characteristic
186	#123	—	2	1	no	—
187	C₂²⋊Q₈	—	2	1	no	—
188	C₂²⋊Q₈	—	178	1	no	—
189	C₂²⋊Q₈	—	178	1	no	—
190	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
191	#121	C₂×D₈₉	1	1	yes	characteristic
192	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
193	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
194	#117	—	89	1	no	—
195	#117	C₂×D₈₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	89	1	no	—
197	#107	—	89	1	no	—
198	#107	—	89	1	no	—
199	#106	—	89	1	no	—
200	#106	—	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	—	89	1	no	—
210	#103	—	89	1	no	—
211	#102	—	89	1	no	—
212	#102	—	89	1	no	—
213	#102	—	89	1	no	—
214	#102	—	89	1	no	—
215	C₂²⋊C₈	—	89	1	no	—
216	C₂²⋊C₈	—	89	1	no	—
217	#97	—	89	1	no	—
218	C₄×C₈	—	89	1	no	—
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	178	1	no	—
221	C₂×Q₈	—	178	1	no	—
222	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	178	1	no	—
227	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
228	C₂×C₈	—	89	1	no	—
229	C₂×C₈	—	89	1	no	—
230	C₂×C₈	—	89	1	no	—
231	C₂×C₈	—	89	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	89	1	no	—
236	C₄⋊C₄	—	89	1	no	—
237	C₄⋊C₄	—	89	1	no	—
238	C₄⋊C₄	—	89	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	C₂²⋊D₈₉	1	1	yes	characteristic
247	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	178	1	no	—
256	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
257	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
258	C₄²	—	2	1	no	—
259	C₄²	—	178	1	no	—
260	C₄²	C₂²×D₈₉	1	1	yes	characteristic
261	C₄²	C₂²×D₈₉	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	8	1	no	—
267	Q₈	—	712	1	no	—
268	Q₈	—	712	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	8	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	178	1	no	—
279	C₂×C₄	—	178	1	no	—
280	C₂×C₄	—	178	1	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	—	712	1	no	—
283	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
289	C₈	—	178	1	no	—
290	C₈	—	178	1	no	—
291	C₈	—	178	1	no	—
292	C₈	—	178	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	356	1	no	—
309	C₄	—	356	1	no	—
310	C₄	—	356	1	no	—
311	C₄	—	356	1	no	—
312	C₄	#15668059	1	1	yes	characteristic
313	C₄	#15668059	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	8	1	no	—
316	C₂	#19152169	1	1	yes	characteristic
317	C₂	#19152406	1	1	yes	characteristic
318	C₂	#19152409	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#542×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#121×C₈₉	C₂²
hypercenter	#542	D₈₉
hypocenter	C₈₉	#3244
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #542

Complements

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

N	H ≅ G/N	type
1. #19152226	C₂	semidirect
2. #19152004	C₂	semidirect
3. #19151994	C₂	semidirect
4. #19151957	C₂	semidirect
5. #19151952	C₂	semidirect
6. #19151936	C₂	semidirect
7. #19151926	C₂	semidirect
10. #19151905	C₂	semidirect
11. #19151898	C₂	semidirect
12. #19151888	C₂	semidirect
13. #19151883	C₂	semidirect
14. #19151659	C₂	semidirect
160. C₈₉	#3244	semidirect

Maximal subgroups

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

subgroup	index
#19152226	2
#19152004	2
#19151994	2
#19151957	2
#19151952	2
#19151936	2
#19151926	2
#19151919	2
#19151914	2
#19151905	2
#19151898	2
#19151888	2
#19151883	2
#19151659	2
#542×C₈₉	2
#3244	89