Subgroups — #26032639

Order 11392.

356 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	#19152306	C₂	1	1	yes	maximal, characteristic
2	#19152303	C₂	1	1	yes	maximal, characteristic
3	#19152002	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151935	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151934	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151932	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151892	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151891	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151837	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151690	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151661	C₂	1	1	yes	maximal, characteristic
13	C₂×#15667945	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
19	#15668011	C₂²	1	1	yes	characteristic
20	#15668011	C₂²	1	1	yes	characteristic
21	#15668002	—	2	1	no	—
22	#15668002	—	2	1	no	—
23	#15667992	—	2	1	no	—
24	#15667992	C₂²	1	1	yes	characteristic
25	#15667991	—	2	1	no	—
26	#15667991	C₂²	1	1	yes	characteristic
27	#15667990	—	2	1	no	—
28	#15667989	—	2	1	no	—
29	#15667988	C₂²	1	1	yes	characteristic
30	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
31	#15667977	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₉	—	2	1	no	—
33	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
34	#15667972	—	2	1	no	—
35	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	#15667946	C₂²	1	1	yes	characteristic
39	#15667946	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667945	C₂²	1	1	yes	characteristic
42	#15667945	C₂²	1	1	yes	characteristic
43	#15667945	C₂²	1	1	yes	characteristic
44	#15667945	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667937	C₂²	1	1	yes	characteristic
48	#15667936	C₂²	1	1	yes	characteristic
49	#15667935	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
63	C₂²×Dic₈₉	—	2	1	no	—
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
67	#13083564	—	2	2	no	—
68	C₂×#10623494	—	2	2	no	—
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₈×D₈₉	—	2	2	no	—
72	C₄⋊Dic₈₉	—	2	1	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	#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	#13083553	—	2	1	no	—
85	#13083553	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	D₄	1	1	yes	characteristic
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₄⋊C₄×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	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₈₉	—	2	1	no	—
108	Q₈×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	—	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	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₈₉	Q₁₆	1	4	yes	complemented
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	8	1	no	—
124	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄×D₈₉	—	2	1	no	—
128	C₄×D₈₉	—	2	1	no	—
129	C₄×D₈₉	—	8	1	no	—
130	C₄×D₈₉	—	8	1	no	—
131	Dic₁₇₈	—	2	1	no	—
132	Dic₁₇₈	—	2	1	no	—
133	Dic₁₇₈	—	8	1	no	—
134	Dic₁₇₈	—	8	1	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	#10623494	—	2	1	no	—
138	#10623494	#41	1	2	yes	—
139	C₂²×C₈₉	—	2	1	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
142	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₈₉	—	2	1	no	—
144	C₂×D₈₉	—	2	1	no	—
145	C₂×D₈₉	C₂×Q₁₆	1	2	yes	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	4	1	no	—
148	C₄×C₈₉	—	4	1	no	—
149	C₄×C₈₉	—	4	1	no	—
150	C₄×C₈₉	—	4	1	no	—
151	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₈₉	—	2	1	no	—
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	4	1	no	—
156	Dic₈₉	—	4	1	no	—
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	C₂×Q₁₆	1	2	yes	—
160	C₂×C₈₉	—	2	2	no	—
161	C₂×C₈₉	#547	1	1	yes	characteristic
162	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
163	C₂×C₈₉	#576	1	1	yes	characteristic
164	D₈₉	—	2	2	no	—
165	#3338	—	89	1	no	maximal, Sylow
166	C₈₉	#3338	1	1	yes	characteristic, complemented, Sylow
167	#554	—	89	1	no	—
168	#554	—	89	1	no	—
169	#500	—	89	1	no	—
170	#484	—	89	1	no	—
171	#484	—	89	1	no	—
172	#484	—	89	1	no	—
173	#484	D₈₉	1	1	yes	characteristic, complemented
174	#477	—	89	1	no	—
175	#477	—	89	1	no	—
176	#467	—	89	1	no	—
177	#467	—	89	1	no	—
178	C₄×Q₁₆	—	89	1	no	—
179	C₈×D₄	—	89	1	no	—
180	C₂×#107	—	89	1	no	—
181	C₂×#107	—	89	1	no	—
182	C₂×Q₁₆	—	89	1	no	—
183	C₂²×C₈	—	89	1	no	—
184	C₂²×C₈	—	89	1	no	—
185	C₄⋊Q₈	—	89	1	no	—
186	C₄⋊Q₈	—	89	1	no	—
187	#125	—	178	1	no	—
188	#125	—	178	1	no	—
189	C₂²⋊Q₈	—	89	1	no	—
190	C₂²⋊Q₈	—	89	1	no	—
191	C₂²⋊Q₈	—	89	1	no	—
192	C₂²⋊Q₈	—	178	1	no	—
193	C₂²⋊Q₈	—	178	1	no	—
194	C₂²⋊Q₈	—	178	1	no	—
195	C₂²⋊Q₈	—	178	1	no	—
196	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
197	C₄×Q₈	—	89	1	no	—
198	C₄×Q₈	—	89	1	no	—
199	C₄×D₄	—	89	1	no	—
200	C₄×D₄	—	178	1	no	—
201	C₄×D₄	—	178	1	no	—
202	C₂×C₄⋊C₄	—	89	1	no	—
203	C₂×C₄⋊C₄	—	89	1	no	—
204	C₂×C₄⋊C₄	—	89	1	no	—
205	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
206	#107	—	89	1	no	—
207	#107	—	89	1	no	—
208	#107	—	89	1	no	—
209	#107	—	89	1	no	—
210	#107	—	89	1	no	—
211	#107	—	89	1	no	—
212	#107	—	89	1	no	—
213	#107	C₂×D₈₉	1	1	yes	characteristic
214	#107	C₂×D₈₉	1	1	yes	characteristic
215	C₄⋊C₈	—	89	1	no	—
216	#103	—	89	1	no	—
217	#103	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	—	89	1	no	—
220	#103	C₂×D₈₉	1	1	yes	characteristic
221	#103	C₂×D₈₉	1	1	yes	characteristic
222	C₂²⋊C₈	—	89	1	no	—
223	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
224	C₄×C₈	—	89	1	no	—
225	C₂×Q₈	—	89	1	no	—
226	C₂×Q₈	—	178	1	no	—
227	C₂×Q₈	—	178	1	no	—
228	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
229	C₂×D₄	—	89	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	89	1	no	—
232	C₂²×C₄	—	178	1	no	—
233	C₂²×C₄	—	178	1	no	—
234	C₂²×C₄	—	178	1	no	—
235	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
236	Q₁₆	—	178	2	no	—
237	C₂×C₈	—	89	1	no	—
238	C₂×C₈	—	89	1	no	—
239	C₂×C₈	—	178	2	no	—
240	C₂×C₈	—	178	2	no	—
241	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	89	1	no	—
246	C₄⋊C₄	—	89	1	no	—
247	C₄⋊C₄	—	89	1	no	—
248	C₄⋊C₄	—	89	1	no	—
249	C₄⋊C₄	—	178	1	no	—
250	C₄⋊C₄	—	178	1	no	—
251	C₄⋊C₄	—	178	1	no	—
252	C₄⋊C₄	—	178	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	—	178	1	no	—
256	C₄⋊C₄	—	178	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	89	1	no	—
264	C₂²⋊C₄	—	89	1	no	—
265	C₂²⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	178	1	no	—
267	C₂²⋊C₄	—	178	1	no	—
268	C₄²	—	89	1	no	—
269	C₄²	—	178	1	no	—
270	C₄²	—	178	1	no	—
271	C₂³	—	89	1	no	—
272	C₂³	D₄×D₈₉	1	1	yes	characteristic
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	178	1	no	—
276	Q₈	—	178	1	no	—
277	Q₈	—	712	1	no	—
278	Q₈	—	712	1	no	—
279	D₄	—	89	4	no	—
280	C₂×C₄	—	2	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	—	8	1	no	—
287	C₂×C₄	—	89	1	no	—
288	C₂×C₄	—	89	1	no	—
289	C₂×C₄	—	89	1	no	—
290	C₂×C₄	—	178	1	no	—
291	C₂×C₄	—	178	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	178	1	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	—	712	1	no	—
297	C₂×C₄	—	712	1	no	—
298	C₂×C₄	—	712	1	no	—
299	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	89	2	no	—
304	C₈	—	178	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	89	2	no	—
308	C₂²	—	178	1	no	—
309	C₂²	—	178	1	no	—
310	C₂²	Q₁₆×D₈₉	1	2	yes	—
311	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	89	2	no	—
318	C₄	—	356	1	no	—
319	C₄	—	356	1	no	—
320	C₄	—	356	1	no	—
321	C₄	—	356	1	no	—
322	C₄	—	178	1	no	—
323	C₄	—	178	1	no	—
324	C₄	#41×D₈₉	1	1	yes	characteristic
325	C₄	#15668059	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	178	2	no	—
328	C₂	#19152246	1	1	yes	characteristic
329	C₂	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
330	C₂	#19152402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#484×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×C₂²⋊D₈₉	D₄
hypercenter	#484	D₈₉
hypocenter	C₈₉	#3338
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
3. #19152002	C₂	semidirect
4. #19151935	C₂	semidirect
5. #19151934	C₂	semidirect
6. #19151932	C₂	semidirect
7. #19151892	C₂	semidirect
8. #19151891	C₂	semidirect
9. #19151838	C₂	semidirect
10. #19151837	C₂	semidirect
11. #19151690	C₂	semidirect
13. C₂×#15667945	C₂	semidirect
14. #107×D₈₉	C₂	semidirect
15. #484×C₈₉	C₂	semidirect
118. C₂²⋊D₈₉	Q₁₆	semidirect
166. C₈₉	#3338	semidirect
173. #484	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152306	2
#19152303	2
#19152002	2
#19151935	2
#19151934	2
#19151932	2
#19151892	2
#19151891	2
#19151838	2
#19151837	2
#19151690	2
#19151661	2
C₂×#15667945	2
#107×D₈₉	2
#484×C₈₉	2
#3338	89