Subgroups — #26032717

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152295	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152248	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152005	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151919	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151915	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151914	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151910	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151896	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151886	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151831	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151710	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151667	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151640	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#469×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	—	2	1	no	—
17	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
18	#15668011	C₂²	1	1	yes	characteristic
19	#15668005	—	2	1	no	—
20	#15668003	C₂²	1	1	yes	characteristic
21	#15667994	—	2	1	no	—
22	#15667993	—	2	1	no	—
23	#15667993	—	2	1	no	—
24	#15667989	—	2	1	no	—
25	#15667983	—	2	1	no	—
26	#15667983	C₂²	1	1	yes	characteristic
27	#15667983	C₂²	1	1	yes	characteristic
28	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
31	#15667972	—	2	1	no	—
32	#15667971	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
36	#15667946	C₂²	1	1	yes	characteristic
37	#15667946	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	#15667943	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667933	C₂²	1	1	yes	characteristic
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667922	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	#121×C₈₉	C₂²	1	1	yes	characteristic
55	#121×C₈₉	C₂²	1	1	yes	characteristic
56	#106×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×C₂²⋊D₈₉	—	2	1	no	—
62	C₂×C₂²⋊D₈₉	—	2	1	no	—
63	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₂×#10623494	C₂³	1	1	yes	characteristic
72	C₈×D₈₉	—	2	2	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	—	2	1	no	—
75	C₄⋊Dic₈₉	C₂³	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	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₈₉	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
90	#13083551	D₄	1	1	yes	characteristic
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₈₉	—	2	1	no	—
94	C₂×D₄×C₈₉	—	2	1	no	—
95	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
98	D₈×C₈₉	—	2	2	no	—
99	M₁₆×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂³×C₈₉	—	2	1	no	—
107	C₂³×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
110	D₄×C₈₉	—	2	1	no	—
111	D₄×C₈₉	—	2	1	no	—
112	D₄×C₈₉	—	2	1	no	—
113	D₄×C₈₉	—	2	1	no	—
114	D₄×C₈₉	—	4	2	no	—
115	D₄×C₈₉	—	8	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	2	2	no	—
123	C₂²⋊D₈₉	—	4	2	no	—
124	C₂²⋊D₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	8	1	no	—
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	C₄×D₈₉	—	2	1	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	4	2	no	—
138	C₄×D₈₉	—	8	1	no	—
139	Dic₁₇₈	—	4	2	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	#41	1	2	yes	complemented
142	#10623494	—	2	1	no	—
143	#10623494	—	2	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	—	4	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	AGL(1,ℤ/8)	1	2	yes	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	4	1	no	—
166	Dic₈₉	—	4	1	no	—
167	Dic₈₉	—	4	1	no	—
168	Dic₈₉	—	4	1	no	—
169	C₂×C₈₉	—	4	1	no	—
170	C₂×C₈₉	—	8	1	no	—
171	C₂×C₈₉	—	8	1	no	—
172	C₂×C₈₉	#547	1	1	yes	characteristic
173	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₈₉	#577	1	1	yes	characteristic
175	D₈₉	—	2	2	no	—
176	#3345	—	89	1	no	maximal, Sylow
177	C₈₉	#3345	1	1	yes	characteristic, complemented, Sylow
178	#553	—	89	1	no	—
179	#547	—	89	1	no	—
180	#501	—	89	1	no	—
181	#482	—	89	1	no	—
182	#481	—	89	1	no	—
183	#481	—	89	1	no	—
184	#480	—	89	1	no	—
185	#478	—	89	1	no	—
186	#476	—	89	1	no	—
187	#469	D₈₉	1	1	yes	characteristic, complemented
188	#466	—	89	1	no	—
189	#442	—	89	1	no	—
190	#435	—	89	1	no	—
191	#428	—	89	1	no	—
192	C₂×#106	—	89	1	no	—
193	C₂×#41	—	178	1	no	—
194	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
195	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂²×C₈	—	89	1	no	—
197	C₄⋊Q₈	—	89	1	no	—
198	#126	—	178	1	no	—
199	#125	—	89	1	no	—
200	#123	—	178	1	no	—
201	#123	—	178	1	no	—
202	#123	—	178	1	no	—
203	C₂²⋊Q₈	—	178	1	no	—
204	#121	—	89	1	no	—
205	#121	—	89	1	no	—
206	#121	—	178	1	no	—
207	#121	C₂×D₈₉	1	1	yes	characteristic
208	#121	C₂×D₈₉	1	1	yes	characteristic
209	C₄×D₄	—	89	1	no	—
210	C₄×D₄	—	89	1	no	—
211	C₄×D₄	—	89	1	no	—
212	C₄×D₄	—	178	1	no	—
213	#117	—	89	1	no	—
214	C₂×C₄⋊C₄	—	89	1	no	—
215	C₂×C₄⋊C₄	—	89	1	no	—
216	C₂×C₄⋊C₄	—	89	1	no	—
217	#107	—	89	1	no	—
218	#107	—	89	1	no	—
219	#107	—	89	1	no	—
220	#106	—	89	1	no	—
221	#106	—	89	1	no	—
222	#106	—	89	1	no	—
223	#106	—	89	1	no	—
224	#106	—	89	1	no	—
225	#106	C₂×D₈₉	1	1	yes	characteristic
226	C₄⋊C₈	—	89	1	no	—
227	#102	—	89	1	no	—
228	#102	—	89	1	no	—
229	#102	—	89	1	no	—
230	#102	—	89	1	no	—
231	#102	C₂×D₈₉	1	1	yes	characteristic
232	#102	C₂×D₈₉	1	1	yes	characteristic
233	C₂²⋊C₈	—	89	1	no	—
234	C₂²⋊C₈	—	89	1	no	—
235	#97	—	89	1	no	—
236	#41	—	356	4	no	—
237	C₂×Q₈	—	178	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	89	1	no	—
241	C₂×D₄	—	178	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	178	1	no	—
247	C₂²×C₄	—	178	1	no	—
248	C₂²×C₄	—	178	1	no	—
249	C₂²×C₄	—	178	1	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
252	D₈	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	C₂×C₈	—	89	1	no	—
255	C₂×C₈	—	89	1	no	—
256	C₂×C₈	—	178	2	no	—
257	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₄⋊C₄	—	89	1	no	—
260	C₄⋊C₄	—	89	1	no	—
261	C₄⋊C₄	—	89	1	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	—	89	1	no	—
264	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	89	1	no	—
275	C₂²⋊C₄	—	89	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₄²	—	89	1	no	—
282	C₄²	—	178	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	89	1	no	—
286	C₂³	D₄×D₈₉	1	1	yes	characteristic
287	Q₈	—	356	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	1	no	—
294	D₄	—	356	2	no	—
295	D₄	—	356	2	no	—
296	D₄	—	178	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	89	1	no	—
303	C₂×C₄	—	89	1	no	—
304	C₂×C₄	—	356	2	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	—	712	1	no	—
314	C₂×C₄	—	712	1	no	—
315	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	1	no	—
320	C₈	#13083573	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	89	2	no	—
331	C₂²	—	178	1	no	—
332	C₂²	—	178	1	no	—
333	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	356	1	no	—
338	C₄	—	356	1	no	—
339	C₄	—	356	1	no	—
340	C₄	—	356	1	no	—
341	C₄	—	178	1	no	—
342	C₄	—	178	1	no	—
343	C₄	—	178	1	no	—
344	C₄	C₂×#13083573	1	1	yes	characteristic
345	C₄	#15668059	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	8	1	no	—
348	C₂	—	8	1	no	—
349	C₂	—	178	2	no	—
350	C₂	#19152248	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
352	C₂	#19152408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#469×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#469	D₈₉
hypocenter	C₈₉	#3345
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #19152295	C₂	semidirect
2. #19152248	C₂	semidirect
3. #19152005	C₂	semidirect
4. #19151919	C₂	semidirect
5. #19151915	C₂	semidirect
6. #19151914	C₂	semidirect
7. #19151910	C₂	semidirect
8. #19151896	C₂	semidirect
9. #19151886	C₂	semidirect
10. #19151831	C₂	semidirect
11. #19151710	C₂	semidirect
12. #19151667	C₂	semidirect
13. #19151640	C₂	semidirect
14. #106×D₈₉	C₂	semidirect
15. #469×C₈₉	C₂	semidirect
141. C₈×C₈₉	#41	semidirect
177. C₈₉	#3345	semidirect
187. #469	D₈₉	semidirect
320. C₈	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#19152295	2
#19152248	2
#19152005	2
#19151919	2
#19151915	2
#19151914	2
#19151910	2
#19151896	2
#19151886	2
#19151831	2
#19151710	2
#19151667	2
#19151640	2
#106×D₈₉	2
#469×C₈₉	2
#3345	89