Subgroups — #26032617

Order 11392.

342 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152289	C₂	1	1	yes	maximal, characteristic
2	#19152286	C₂	1	1	yes	maximal, characteristic
3	#19151997	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151938	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151930	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151906	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151663	C₂	1	1	yes	maximal, characteristic
13	#19151633	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#479×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
19	#15668007	—	2	1	no	—
20	#15668006	—	2	1	no	—
21	#15668004	C₂²	1	1	yes	characteristic
22	#15668003	—	2	1	no	—
23	#15668002	—	2	1	no	—
24	#15667997	—	2	1	no	—
25	#15667997	—	2	1	no	—
26	#15667992	C₂²	1	1	yes	characteristic
27	#15667992	C₂²	1	1	yes	characteristic
28	#15667988	C₂²	1	1	yes	characteristic
29	#15667988	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
31	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
32	#15667970	—	2	1	no	—
33	#15667970	C₂²	1	1	yes	characteristic
34	#15667970	C₂²	1	1	yes	characteristic
35	#15667969	—	2	1	no	—
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	#15667945	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667945	C₂²	1	1	yes	characteristic
42	#15667944	C₂²	1	1	yes	characteristic
43	#15667943	C₂²	1	1	yes	characteristic
44	#15667943	C₂²	1	1	yes	characteristic
45	#15667937	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667935	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₄×#10623494	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×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	C₂³	1	1	yes	characteristic
60	C₂×D₃₅₆	D₄	1	1	yes	characteristic
61	C₂×C₄×D₈₉	—	2	1	no	—
62	C₂×C₄×D₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
65	#13083564	—	2	2	no	—
66	C₂×#10623494	C₂³	1	1	yes	characteristic
67	C₂×#10623494	C₂³	1	1	yes	characteristic
68	C₈×D₈₉	—	2	2	no	—
69	C₈×D₈₉	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
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	#13083553	—	2	1	no	—
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	—	2	1	no	—
83	#13083551	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	D₄	1	1	yes	characteristic
90	#13083551	D₄	1	1	yes	characteristic
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₄²×C₈₉	—	2	1	no	—
104	C₄²×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²×D₈₉	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	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
122	D₃₅₆	—	2	2	no	—
123	C₄×D₈₉	—	2	1	no	—
124	C₄×D₈₉	—	2	1	no	—
125	C₄×D₈₉	—	2	1	no	—
126	C₄×D₈₉	—	2	1	no	—
127	C₄×D₈₉	—	4	2	no	—
128	C₄×D₈₉	—	4	2	no	—
129	Dic₁₇₈	—	2	1	no	—
130	Dic₁₇₈	—	2	1	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	—	2	1	no	—
133	#10623494	—	2	1	no	—
134	#10623494	#41	1	2	yes	—
135	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₈₉	—	2	1	no	—
137	C₂×D₈₉	—	2	1	no	—
138	C₂×D₈₉	—	2	1	no	—
139	C₂×D₈₉	—	2	1	no	—
140	C₂×D₈₉	—	2	1	no	—
141	C₂×D₈₉	#135	1	2	yes	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	4	1	no	—
145	C₄×C₈₉	—	4	1	no	—
146	C₄×C₈₉	—	4	1	no	—
147	C₄×C₈₉	—	4	1	no	—
148	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₈₉	—	2	1	no	—
151	Dic₈₉	—	2	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	—	4	1	no	—
154	Dic₈₉	—	4	1	no	—
155	Dic₈₉	—	4	1	no	—
156	C₂×C₈₉	#547	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
158	C₂×C₈₉	#578	1	1	yes	characteristic
159	D₈₉	—	2	2	no	—
160	D₈₉	—	4	1	no	—
161	#3336	—	89	1	no	maximal, Sylow
162	C₈₉	#3336	1	1	yes	characteristic, complemented, Sylow
163	#552	—	89	1	no	—
164	#552	—	89	1	no	—
165	#499	—	89	1	no	—
166	#485	—	89	1	no	—
167	#485	—	89	1	no	—
168	#483	—	89	1	no	—
169	#483	—	89	1	no	—
170	#479	—	89	1	no	—
171	#479	D₈₉	1	1	yes	characteristic, complemented
172	#467	—	89	1	no	—
173	#467	—	89	1	no	—
174	C₄×Q₁₆	—	89	1	no	—
175	C₈×D₄	—	89	1	no	—
176	#427	—	89	1	no	—
177	C₂×#107	—	89	1	no	—
178	C₂×Q₁₆	—	89	1	no	—
179	C₂²×C₈	—	89	1	no	—
180	C₂²×C₈	—	89	1	no	—
181	#126	—	178	1	no	—
182	#126	—	178	1	no	—
183	#125	—	89	1	no	—
184	#125	—	178	1	no	—
185	#125	—	178	1	no	—
186	#125	C₂×D₈₉	1	1	yes	characteristic
187	#123	—	178	1	no	—
188	#123	—	178	1	no	—
189	C₂²⋊Q₈	—	89	1	no	—
190	C₂²⋊Q₈	—	89	1	no	—
191	C₂²⋊Q₈	—	89	1	no	—
192	C₂²⋊Q₈	—	89	1	no	—
193	C₄×Q₈	—	89	1	no	—
194	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
195	C₄×D₄	—	89	1	no	—
196	#117	—	89	1	no	—
197	#117	—	89	1	no	—
198	#117	—	178	1	no	—
199	#117	—	178	1	no	—
200	C₂×C₄⋊C₄	—	89	1	no	—
201	C₂×C₄⋊C₄	—	89	1	no	—
202	#107	—	89	1	no	—
203	#107	—	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	C₂×D₈₉	1	1	yes	characteristic
211	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
212	#103	—	89	1	no	—
213	#103	—	89	1	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#103	C₂×D₈₉	1	1	yes	characteristic
217	#103	C₂×D₈₉	1	1	yes	characteristic
218	C₂²⋊C₈	—	89	1	no	—
219	C₂²⋊C₈	—	89	1	no	—
220	C₄×C₈	—	89	1	no	—
221	C₂×Q₈	—	89	1	no	—
222	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
223	C₂×D₄	—	89	1	no	—
224	C₂²×C₄	—	89	1	no	—
225	C₂²×C₄	—	89	1	no	—
226	C₂²×C₄	—	178	1	no	—
227	C₂²×C₄	—	178	1	no	—
228	Q₁₆	—	178	2	no	—
229	C₂×C₈	—	89	1	no	—
230	C₂×C₈	—	89	1	no	—
231	C₂×C₈	—	178	2	no	—
232	C₂×C₈	—	178	2	no	—
233	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	89	1	no	—
239	C₄⋊C₄	—	89	1	no	—
240	C₄⋊C₄	—	89	1	no	—
241	C₄⋊C₄	—	89	1	no	—
242	C₄⋊C₄	—	178	1	no	—
243	C₄⋊C₄	—	178	1	no	—
244	C₄⋊C₄	—	178	1	no	—
245	C₄⋊C₄	—	178	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	C₂²×D₈₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	89	1	no	—
255	C₂²⋊C₄	—	89	1	no	—
256	C₂²⋊C₄	—	178	1	no	—
257	C₂²⋊C₄	—	178	1	no	—
258	C₂²⋊C₄	—	178	1	no	—
259	C₂²⋊C₄	—	178	1	no	—
260	C₂²⋊C₄	—	178	1	no	—
261	C₂²⋊C₄	—	178	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	178	1	no	—
264	C₄²	—	178	1	no	—
265	C₄²	—	178	1	no	—
266	C₄²	C₂²×D₈₉	1	1	yes	characteristic
267	C₂³	—	89	1	no	—
268	C₂³	—	89	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	178	1	no	—
272	Q₈	—	178	1	no	—
273	D₄	—	178	2	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	—	89	1	no	—
279	C₂×C₄	—	89	1	no	—
280	C₂×C₄	—	356	2	no	—
281	C₂×C₄	—	356	2	no	—
282	C₂×C₄	—	178	1	no	—
283	C₂×C₄	—	178	1	no	—
284	C₂×C₄	—	178	1	no	—
285	C₂×C₄	—	178	1	no	—
286	C₂×C₄	—	178	1	no	—
287	C₂×C₄	—	178	1	no	—
288	C₂×C₄	—	178	1	no	—
289	C₂×C₄	—	178	1	no	—
290	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
291	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	89	2	no	—
296	C₈	—	178	1	no	—
297	C₂²	—	89	2	no	—
298	C₂²	—	178	1	no	—
299	C₂²	—	178	1	no	—
300	C₂²	—	178	1	no	—
301	C₂²	—	178	1	no	—
302	C₂²	—	178	1	no	—
303	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	356	1	no	—
311	C₄	—	356	1	no	—
312	C₄	—	356	1	no	—
313	C₄	—	356	1	no	—
314	C₄	—	178	1	no	—
315	C₄	—	178	1	no	—
316	C₄	#41×D₈₉	1	1	yes	characteristic
317	C₄	#15668060	1	1	yes	characteristic
318	C₂	—	356	1	no	—
319	C₂	—	178	2	no	—
320	C₂	#19152250	1	1	yes	characteristic
321	C₂	#135×D₈₉	1	1	yes	characteristic
322	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#479×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₃₅₆	D₄
hypercenter	#479	D₈₉
hypocenter	C₈₉	#3336
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #19151997	C₂	semidirect
4. #19151941	C₂	semidirect
5. #19151938	C₂	semidirect
6. #19151930	C₂	semidirect
7. #19151927	C₂	semidirect
8. #19151906	C₂	semidirect
9. #19151838	C₂	semidirect
10. #19151838	C₂	semidirect
11. C₄×#13083564	C₂	semidirect
13. #19151633	C₂	semidirect
14. #107×D₈₉	C₂	semidirect
15. #479×C₈₉	C₂	semidirect
162. C₈₉	#3336	semidirect
171. #479	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152289	2
#19152286	2
#19151997	2
#19151941	2
#19151938	2
#19151930	2
#19151927	2
#19151906	2
#19151838	2
C₄×#13083564	2
#19151663	2
#19151633	2
#107×D₈₉	2
#479×C₈₉	2
#3336	89