Subgroups — #26032713

Order 11392.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 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	#19152290	C₂	1	1	yes	maximal, characteristic
3	#19152006	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151937	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151931	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151926	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151903	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151881	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151853	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151705	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151669	C₂	1	1	yes	maximal, characteristic
13	#19151639	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15667944	C₂	1	1	yes	maximal, characteristic, complemented
15	#467×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083557	C₂²	1	1	yes	characteristic
17	#15668011	C₂²	1	1	yes	characteristic
18	#15668007	—	2	1	no	—
19	#15668003	—	2	1	no	—
20	#15668003	C₂²	1	1	yes	characteristic
21	#15668002	—	2	1	no	—
22	#15667996	—	2	1	no	—
23	#15667992	—	2	1	no	—
24	#15667989	—	2	1	no	—
25	#15667988	C₂²	1	1	yes	characteristic
26	#15667988	C₂²	1	1	yes	characteristic
27	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
28	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
30	#15667972	—	2	1	no	—
31	#15667970	C₂²	1	1	yes	characteristic
32	#15667968	—	2	1	no	—
33	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
36	#15667945	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667944	C₂²	1	1	yes	characteristic
39	#15667944	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667944	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	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667936	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×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	#107×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
61	C₂²×Dic₈₉	—	2	1	no	—
62	C₂²×Dic₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	—	2	1	no	—
64	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
65	#13083557	—	2	2	no	—
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	—	2	1	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	D₄	1	1	yes	characteristic
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
95	Q₁₆×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	2	2	no	—
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	C₂²⋊C₄×C₈₉	—	2	1	no	—
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	Q₈×C₈₉	—	2	1	no	—
110	Q₈×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	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₈₉	—	2	2	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	8	1	no	—
126	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	C₄×D₈₉	—	2	1	no	—
130	C₄×D₈₉	—	2	1	no	—
131	C₄×D₈₉	—	8	1	no	—
132	Dic₁₇₈	—	8	1	no	—
133	C₈×C₈₉	—	2	1	no	—
134	C₈×C₈₉	#41	1	2	yes	—
135	#10623494	—	2	1	no	—
136	#10623494	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	#137	1	2	yes	—
140	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₈₉	—	2	1	no	—
142	C₂×D₈₉	—	2	1	no	—
143	C₂×D₈₉	—	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	C₂×#41	1	1	yes	characteristic
150	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₈₉	—	2	1	no	—
152	Dic₈₉	—	2	1	no	—
153	Dic₈₉	—	2	1	no	—
154	Dic₈₉	—	4	1	no	—
155	Dic₈₉	—	4	1	no	—
156	Dic₈₉	—	4	1	no	—
157	Dic₈₉	—	4	1	no	—
158	C₂×C₈₉	—	2	2	no	—
159	C₂×C₈₉	#547	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
161	C₂×C₈₉	#577	1	1	yes	characteristic
162	D₈₉	—	4	1	no	—
163	#3344	—	89	1	no	maximal, Sylow
164	C₈₉	#3344	1	1	yes	characteristic, complemented, Sylow
165	#554	—	89	1	no	—
166	#552	—	89	1	no	—
167	#501	—	89	1	no	—
168	#485	—	89	1	no	—
169	#484	—	89	1	no	—
170	#483	—	89	1	no	—
171	#483	—	89	1	no	—
172	#479	—	89	1	no	—
173	#475	—	89	1	no	—
174	#470	—	89	1	no	—
175	#467	D₈₉	1	1	yes	characteristic, complemented
176	#441	—	89	1	no	—
177	#435	—	89	1	no	—
178	#428	—	89	1	no	—
179	C₂×#106	—	89	1	no	—
180	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
181	C₂×M₁₆	—	89	1	no	—
182	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
183	C₄⋊Q₈	—	89	1	no	—
184	#126	—	178	1	no	—
185	#125	—	89	1	no	—
186	#125	—	178	1	no	—
187	#125	—	178	1	no	—
188	#123	—	178	1	no	—
189	C₂²⋊Q₈	—	89	1	no	—
190	C₂²⋊Q₈	—	89	1	no	—
191	C₂²⋊Q₈	—	178	1	no	—
192	C₂²⋊Q₈	—	178	1	no	—
193	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
194	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
195	C₄×Q₈	—	89	1	no	—
196	C₄×Q₈	—	89	1	no	—
197	C₄×D₄	—	89	1	no	—
198	C₄×D₄	—	178	1	no	—
199	#117	—	89	1	no	—
200	#117	—	178	1	no	—
201	C₂×C₄⋊C₄	—	89	1	no	—
202	C₂×C₄⋊C₄	—	89	1	no	—
203	C₂×C₄⋊C₄	—	89	1	no	—
204	#107	—	89	1	no	—
205	#107	—	89	1	no	—
206	#107	C₂×D₈₉	1	1	yes	characteristic
207	#106	—	89	1	no	—
208	#106	—	89	1	no	—
209	#106	—	89	1	no	—
210	#106	—	89	1	no	—
211	#106	—	89	1	no	—
212	#106	—	89	1	no	—
213	C₄⋊C₈	—	89	1	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#103	—	89	1	no	—
217	#103	—	89	1	no	—
218	#103	C₂×D₈₉	1	1	yes	characteristic
219	#103	C₂×D₈₉	1	1	yes	characteristic
220	C₂²⋊C₈	—	89	1	no	—
221	C₂²⋊C₈	—	89	1	no	—
222	#97	—	89	1	no	—
223	C₂×Q₈	—	178	1	no	—
224	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
225	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
226	C₂×D₄	—	89	1	no	—
227	C₂²×C₄	—	89	1	no	—
228	C₂²×C₄	—	178	1	no	—
229	C₂²×C₄	—	178	1	no	—
230	C₂²×C₄	—	178	1	no	—
231	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	M₁₆	—	178	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	89	1	no	—
236	C₂×C₈	—	89	1	no	—
237	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	89	1	no	—
242	C₄⋊C₄	—	89	1	no	—
243	C₄⋊C₄	—	89	1	no	—
244	C₄⋊C₄	—	89	1	no	—
245	C₄⋊C₄	—	89	1	no	—
246	C₄⋊C₄	—	178	1	no	—
247	C₄⋊C₄	—	178	1	no	—
248	C₄⋊C₄	—	178	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	C₂²×D₈₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	89	1	no	—
261	C₂²⋊C₄	—	89	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₂²⋊C₄	—	178	1	no	—
264	C₂²⋊C₄	—	178	1	no	—
265	C₄²	—	89	1	no	—
266	C₄²	—	178	1	no	—
267	C₄²	—	178	1	no	—
268	C₄²	—	178	1	no	—
269	C₂³	—	89	1	no	—
270	C₂³	D₄×D₈₉	1	1	yes	characteristic
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	712	1	no	—
276	D₄	—	178	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	89	1	no	—
284	C₂×C₄	—	89	1	no	—
285	C₂×C₄	—	89	1	no	—
286	C₂×C₄	—	356	2	no	—
287	C₂×C₄	—	178	1	no	—
288	C₂×C₄	—	178	1	no	—
289	C₂×C₄	—	178	1	no	—
290	C₂×C₄	—	178	1	no	—
291	C₂×C₄	—	178	1	no	—
292	C₂×C₄	—	178	1	no	—
293	C₂×C₄	—	712	1	no	—
294	C₂×C₄	—	712	1	no	—
295	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	178	1	no	—
299	C₈	—	178	1	no	—
300	C₈	#13083573	1	2	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	178	1	no	—
304	C₂²	—	178	1	no	—
305	C₂²	—	178	1	no	—
306	C₂²	#15668044	1	2	yes	—
307	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	356	1	no	—
314	C₄	—	356	1	no	—
315	C₄	—	356	1	no	—
316	C₄	—	356	1	no	—
317	C₄	—	178	1	no	—
318	C₄	—	178	1	no	—
319	C₄	—	178	1	no	—
320	C₄	C₂×#13083573	1	1	yes	characteristic
321	C₄	#15668059	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	356	1	no	—
324	C₂	#19152248	1	1	yes	characteristic
325	C₂	C₂×#15668044	1	1	yes	characteristic
326	C₂	#19152405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#467×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×C₂²⋊D₈₉	D₄
hypercenter	#467	D₈₉
hypocenter	C₈₉	#3344
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
3. #19152006	C₂	semidirect
4. #19151937	C₂	semidirect
5. #19151931	C₂	semidirect
6. #19151927	C₂	semidirect
7. #19151926	C₂	semidirect
8. #19151903	C₂	semidirect
9. #19151881	C₂	semidirect
10. #19151853	C₂	semidirect
11. #19151705	C₂	semidirect
13. #19151639	C₂	semidirect
14. C₂×#15667944	C₂	semidirect
15. #467×C₈₉	C₂	semidirect
164. C₈₉	#3344	semidirect
175. #467	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
#19152290	2
#19152006	2
#19151937	2
#19151931	2
#19151927	2
#19151926	2
#19151903	2
#19151881	2
#19151853	2
#19151705	2
#19151669	2
#19151639	2
C₂×#15667944	2
#467×C₈₉	2
#3344	89