Subgroups — #26031535

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	#19152219	C₂	1	1	yes	maximal, characteristic
2	#19152008	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152001	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151960	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151955	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151939	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151933	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151922	C₂	1	1	yes	maximal, characteristic
9	#19151911	C₂	1	1	yes	maximal, characteristic
10	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151890	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151875	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151659	C₂	1	1	yes	maximal, characteristic, complemented
15	#544×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	#15668012	C₂²	1	1	yes	characteristic
17	#15668007	—	2	1	no	—
18	#15668000	C₂²	1	1	yes	characteristic
19	#15667998	—	2	1	no	—
20	#15667994	—	2	1	no	—
21	#15667991	C₂²	1	1	yes	characteristic
22	#15667990	—	2	1	no	—
23	#15667986	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
25	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
26	#15667945	C₂²	1	1	yes	characteristic
27	#15667945	C₂²	1	1	yes	characteristic
28	#15667945	C₂²	1	1	yes	characteristic
29	#15667945	C₂²	1	1	yes	characteristic
30	#15667945	C₂²	1	1	yes	characteristic
31	#15667945	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	#15667935	C₂²	1	1	yes	characteristic
37	#15667935	C₂²	1	1	yes	characteristic
38	#15667935	C₂²	1	1	yes	characteristic
39	#15667935	C₂²	1	1	yes	characteristic
40	#15667932	C₂²	1	1	yes	characteristic
41	#15667932	C₂²	1	1	yes	characteristic
42	#15667932	C₂²	1	1	yes	characteristic
43	#15667932	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	—	2	1	no	—
49	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
50	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
51	#125×C₈₉	—	2	1	no	—
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	#125×C₈₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₉	—	2	1	no	—
55	C₂²⋊Q₈×C₈₉	—	2	1	no	—
56	#117×C₈₉	C₂²	1	1	yes	characteristic
57	#117×C₈₉	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	—	2	1	no	—
64	C₂×#10623494	C₂³	1	1	yes	characteristic
65	C₂×#10623494	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	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
71	#13083553	—	2	1	no	—
72	#13083553	—	2	1	no	—
73	C₂²⋊Dic₈₉	—	2	1	no	—
74	#13083551	—	2	1	no	—
75	#13083551	—	2	1	no	—
76	#13083551	—	2	1	no	—
77	#13083551	—	2	1	no	—
78	C₄×Dic₈₉	—	2	1	no	—
79	C₂×Q₈×C₈₉	—	2	1	no	—
80	C₂×Q₈×C₈₉	—	2	1	no	—
81	C₂²×C₄×C₈₉	—	2	1	no	—
82	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₈₉	—	2	1	no	—
84	C₄⋊C₄×C₈₉	—	2	1	no	—
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	—	2	1	no	—
89	C₄⋊C₄×C₈₉	—	2	1	no	—
90	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
91	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
103	C₂²×D₈₉	—	2	1	no	—
104	Q₈×C₈₉	—	8	1	no	—
105	Q₈×C₈₉	—	8	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×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	—	2	1	no	—
112	C₂×C₄×C₈₉	—	8	1	no	—
113	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₈₉	—	8	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	D₃₅₆	—	2	1	no	—
124	D₃₅₆	—	2	1	no	—
125	C₄×D₈₉	—	8	1	no	—
126	Dic₁₇₈	—	2	1	no	—
127	Dic₁₇₈	—	2	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	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₈₉	—	4	1	no	—
138	C₂×D₈₉	—	4	1	no	—
139	C₂×D₈₉	—	4	1	no	—
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	—	4	1	no	—
149	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	—	4	1	no	—
154	C₂×C₈₉	—	4	1	no	—
155	C₂×C₈₉	#536	1	1	yes	characteristic
156	C₂×C₈₉	#576	1	1	yes	characteristic
157	C₂×C₈₉	#578	1	1	yes	characteristic
158	D₈₉	—	8	1	no	—
159	#3243	—	89	1	no	maximal, Sylow
160	C₈₉	#3243	1	1	yes	characteristic, complemented, Sylow
161	#544	D₈₉	1	1	yes	characteristic, complemented
162	#542	—	89	1	no	—
163	#501	—	89	1	no	—
164	#500	—	89	1	no	—
165	#489	—	89	1	no	—
166	#488	—	89	1	no	—
167	#485	—	89	1	no	—
168	#484	—	89	1	no	—
169	#482	—	89	1	no	—
170	#480	—	89	1	no	—
171	#479	—	89	1	no	—
172	#478	—	89	1	no	—
173	#477	—	89	1	no	—
174	#474	—	89	1	no	—
175	#433	—	89	1	no	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	—	89	1	no	—
178	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
179	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
180	#126	—	178	1	no	—
181	#125	—	2	1	no	—
182	#125	C₂×D₈₉	1	1	yes	characteristic
183	#125	C₂×D₈₉	1	1	yes	characteristic
184	#124	—	89	1	no	—
185	#124	—	178	1	no	—
186	#123	—	178	1	no	—
187	C₂²⋊Q₈	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₂²⋊Q₈	—	89	1	no	—
190	C₂²⋊Q₈	—	178	1	no	—
191	#121	—	89	1	no	—
192	C₄×Q₈	—	89	1	no	—
193	C₄×D₄	—	89	1	no	—
194	#117	C₂×D₈₉	1	1	yes	characteristic
195	#117	C₂×D₈₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
197	#107	—	89	1	no	—
198	#107	—	89	1	no	—
199	#107	—	89	1	no	—
200	#107	—	89	1	no	—
201	#107	—	89	1	no	—
202	#107	—	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	—	2	1	no	—
221	C₂×Q₈	—	89	1	no	—
222	C₂×Q₈	—	178	1	no	—
223	C₂×D₄	—	89	1	no	—
224	C₂×D₄	—	178	1	no	—
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	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	89	1	no	—
240	C₄⋊C₄	—	89	1	no	—
241	C₄⋊C₄	—	178	1	no	—
242	C₄⋊C₄	—	178	1	no	—
243	C₄⋊C₄	—	178	1	no	—
244	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
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	—	178	1	no	—
252	C₂²⋊C₄	—	178	1	no	—
253	C₂²⋊C₄	—	178	1	no	—
254	C₂²⋊C₄	—	178	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	—	178	1	no	—
263	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
264	Q₈	—	8	1	no	—
265	Q₈	—	8	1	no	—
266	Q₈	—	178	1	no	—
267	Q₈	—	178	1	no	—
268	Q₈	—	712	1	no	—
269	D₄	—	178	1	no	—
270	D₄	—	178	1	no	—
271	D₄	—	712	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	—	2	1	no	—
278	C₂×C₄	—	8	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	—	356	1	no	—
297	C₂²	—	356	1	no	—
298	C₂²	—	356	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	—	4	1	no	—
309	C₄	—	356	1	no	—
310	C₄	—	356	1	no	—
311	C₄	—	356	1	no	—
312	C₄	#15668060	1	1	yes	characteristic
313	C₄	#15668060	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	712	1	no	—
316	C₂	#19152172	1	1	yes	characteristic
317	C₂	#19152401	1	1	yes	characteristic
318	C₂	#19152413	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#544×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#15667986	C₂²
hypercenter	#544	D₈₉
hypocenter	C₈₉	#3243
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
2. #19152008	C₂	semidirect
3. #19152001	C₂	semidirect
4. #19151960	C₂	semidirect
5. #19151955	C₂	semidirect
6. #19151939	C₂	semidirect
7. #19151933	C₂	semidirect
10. #19151902	C₂	semidirect
11. #19151895	C₂	semidirect
12. #19151890	C₂	semidirect
13. #19151875	C₂	semidirect
14. #19151659	C₂	semidirect
15. #544×C₈₉	C₂	semidirect
160. C₈₉	#3243	semidirect
161. #544	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152219	2
#19152008	2
#19152001	2
#19151960	2
#19151955	2
#19151939	2
#19151933	2
#19151922	2
#19151911	2
#19151902	2
#19151895	2
#19151890	2
#19151875	2
#19151659	2
#544×C₈₉	2
#3243	89