Subgroups — #26031505

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	#19152203	C₂	1	1	yes	maximal, characteristic
2	#19152004	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151997	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151964	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151946	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151922	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151922	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151916	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151911	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151885	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	#15668009	C₂²	1	1	yes	characteristic
17	#15668004	C₂²	1	1	yes	characteristic
18	#15667997	—	2	1	no	—
19	#15667997	—	2	1	no	—
20	#15667986	C₂²	1	1	yes	characteristic
21	#15667986	C₂²	1	1	yes	characteristic
22	#15667985	—	2	1	no	—
23	#15667985	—	2	1	no	—
24	C₄×D₃₅₆	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	#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	#15667932	C₂²	1	1	yes	characteristic
37	#15667932	C₂²	1	1	yes	characteristic
38	#15667932	C₂²	1	1	yes	characteristic
39	#15667932	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×D₃₅₆	—	2	1	no	—
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×#10623494	C₂³	1	1	yes	characteristic
68	C₂×#10623494	C₂³	1	1	yes	characteristic
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₂×#10623494	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	#13083551	—	2	1	no	—
76	#13083551	—	2	1	no	—
77	#13083551	—	2	1	no	—
78	#13083551	—	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	C₂²×D₈₉	—	2	1	no	—
105	Q₈×C₈₉	—	8	1	no	—
106	Q₈×C₈₉	—	8	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	—	2	1	no	—
113	C₂×C₄×C₈₉	—	8	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	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	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₈₉	—	8	1	no	—
121	C₂²⋊D₈₉	—	8	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	D₃₅₆	—	2	1	no	—
125	D₃₅₆	—	2	1	no	—
126	D₃₅₆	—	2	1	no	—
127	D₃₅₆	—	2	1	no	—
128	D₃₅₆	—	8	1	no	—
129	D₃₅₆	—	8	1	no	—
130	C₄×D₈₉	—	8	1	no	—
131	C₄×D₈₉	—	8	1	no	—
132	#10623494	—	2	1	no	—
133	#10623494	—	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	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₈₉	—	4	1	no	—
141	C₂×D₈₉	—	4	1	no	—
142	C₂×D₈₉	—	4	1	no	—
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	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	4	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
156	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	—	4	1	no	—
159	C₂×C₈₉	—	4	1	no	—
160	C₂×C₈₉	#536	1	1	yes	characteristic
161	C₂×C₈₉	#576	1	1	yes	characteristic
162	C₂×C₈₉	#577	1	1	yes	characteristic
163	D₈₉	—	8	1	no	—
164	D₈₉	—	8	1	no	—
165	#3241	—	89	1	no	maximal, Sylow
166	C₈₉	#3241	1	1	yes	characteristic, complemented, Sylow
167	#544	D₈₉	1	1	yes	characteristic, complemented
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	—	89	1	no	—
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	—	2	1	no	—
183	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
184	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
185	#127	—	89	1	no	—
186	#125	—	2	1	no	—
187	#125	—	89	1	no	—
188	#125	C₂×D₈₉	1	1	yes	characteristic
189	#125	C₂×D₈₉	1	1	yes	characteristic
190	#123	—	178	1	no	—
191	#123	—	178	1	no	—
192	C₂²⋊Q₈	—	2	1	no	—
193	C₂²⋊Q₈	—	2	1	no	—
194	#121	—	89	1	no	—
195	#121	—	89	1	no	—
196	#121	—	178	1	no	—
197	#121	—	178	1	no	—
198	C₄×D₄	—	89	1	no	—
199	C₄×D₄	—	89	1	no	—
200	#117	C₂×D₈₉	1	1	yes	characteristic
201	#117	C₂×D₈₉	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
203	#107	—	89	1	no	—
204	#107	—	89	1	no	—
205	#107	—	89	1	no	—
206	#107	—	89	1	no	—
207	#106	—	89	1	no	—
208	#106	—	89	1	no	—
209	#106	—	89	1	no	—
210	#106	—	89	1	no	—
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	—	89	1	no	—
220	#102	—	89	1	no	—
221	C₂²⋊C₈	—	89	1	no	—
222	C₂²⋊C₈	—	89	1	no	—
223	#97	—	89	1	no	—
224	C₄×C₈	—	89	1	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×D₄	—	89	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	—	178	1	no	—
233	C₂²×C₄	—	2	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	—	89	1	no	—
240	C₂×C₈	—	89	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	89	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	178	1	no	—
251	C₄⋊C₄	—	178	1	no	—
252	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
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	—	178	1	no	—
260	C₂²⋊C₄	—	178	1	no	—
261	C₂²⋊C₄	—	178	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂²⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
265	C₄²	—	2	1	no	—
266	C₄²	C₂²×D₈₉	1	1	yes	characteristic
267	C₄²	C₂²×D₈₉	1	1	yes	characteristic
268	C₂³	—	178	1	no	—
269	C₂³	—	178	1	no	—
270	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
271	Q₈	—	8	1	no	—
272	Q₈	—	8	1	no	—
273	D₄	—	178	1	no	—
274	D₄	—	178	1	no	—
275	D₄	—	178	1	no	—
276	D₄	—	178	1	no	—
277	D₄	—	712	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	—	2	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	178	1	no	—
289	C₂×C₄	—	178	1	no	—
290	C₂×C₄	—	712	1	no	—
291	C₂×C₄	—	712	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
298	C₈	—	178	1	no	—
299	C₈	—	178	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	—	356	1	no	—
306	C₂²	—	356	1	no	—
307	C₂²	—	356	1	no	—
308	C₂²	—	356	1	no	—
309	C₂²	—	356	1	no	—
310	C₂²	—	356	1	no	—
311	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	356	1	no	—
322	C₄	—	356	1	no	—
323	C₄	#15668060	1	1	yes	characteristic
324	C₄	#15668060	1	1	yes	characteristic
325	C₂	—	4	1	no	—
326	C₂	—	712	1	no	—
327	C₂	—	712	1	no	—
328	C₂	#19152172	1	1	yes	characteristic
329	C₂	#19152401	1	1	yes	characteristic
330	C₂	#19152406	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	#19152203	C₂
hypercenter	#544	D₈₉
hypocenter	C₈₉	#3241
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. #19152004	C₂	semidirect
3. #19151997	C₂	semidirect
4. #19151964	C₂	semidirect
5. #19151946	C₂	semidirect
6. #19151922	C₂	semidirect
7. #19151922	C₂	semidirect
8. #19151916	C₂	semidirect
9. #19151911	C₂	semidirect
10. #19151895	C₂	semidirect
11. #19151895	C₂	semidirect
12. #19151885	C₂	semidirect
13. #19151875	C₂	semidirect
14. #19151659	C₂	semidirect
15. #544×C₈₉	C₂	semidirect
166. C₈₉	#3241	semidirect
167. #544	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152203	2
#19152004	2
#19151997	2
#19151964	2
#19151946	2
#19151922	2
#19151916	2
#19151911	2
#19151895	2
#19151885	2
#19151875	2
#19151659	2
#544×C₈₉	2
#3241	89