Subgroups — #26030091

Order 11392.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152228	C₂	1	1	yes	maximal, characteristic
2	#19152074	C₂	1	1	yes	maximal, characteristic
3	#19151908	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151903	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151893	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151892	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151884	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151881	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151639	C₂	1	1	yes	maximal, characteristic
11	#19151633	C₂	1	1	yes	maximal, characteristic
12	#19151623	C₂	1	1	yes	maximal, characteristic
13	#19151605	C₂	1	1	yes	maximal, characteristic
14	#103×D₈₉	C₂	1	1	yes	maximal, characteristic
15	#479×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083557	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
19	#15668012	C₂²	1	1	yes	characteristic
20	#15668011	—	2	1	no	—
21	#15668011	C₂²	1	1	yes	characteristic
22	#15668003	C₂²	1	1	yes	characteristic
23	#15668002	—	2	1	no	—
24	#15667992	—	2	1	no	—
25	#15667990	—	2	1	no	—
26	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₇₈	—	2	1	no	—
28	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
29	#15667977	—	2	1	no	—
30	#15667977	C₂²	1	1	yes	characteristic
31	#15667970	—	2	1	no	—
32	#15667970	C₂²	1	1	yes	characteristic
33	#15667970	C₂²	1	1	yes	characteristic
34	#15667969	—	2	1	no	—
35	#15667969	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667945	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	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	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	#15667936	C₂²	1	1	yes	characteristic
50	#15667935	C₂²	1	1	yes	characteristic
51	#15667935	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	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	Q₈×D₈₉	—	2	2	no	—
64	Q₈×D₈₉	Q₈	1	4	yes	complemented
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
68	#13083557	—	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	D₄	1	1	yes	characteristic
84	#13083553	D₄	1	1	yes	characteristic
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	D₄	1	1	yes	characteristic
88	#13083551	D₄	1	1	yes	characteristic
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	1	no	—
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	C₄²×C₈₉	—	2	1	no	—
105	C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₈₉	—	2	1	no	—
108	Q₈×C₈₉	C₂×Q₈	1	2	yes	—
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	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄×D₈₉	—	2	4	no	—
124	C₄×D₈₉	—	8	1	no	—
125	C₄×D₈₉	C₂×Q₈	1	2	yes	—
126	C₄×D₈₉	#41	1	2	yes	—
127	Dic₁₇₈	—	2	1	no	—
128	Dic₁₇₈	—	2	4	no	—
129	Dic₁₇₈	—	8	1	no	—
130	Dic₁₇₈	—	8	1	no	—
131	Dic₁₇₈	C₂×Q₈	1	2	yes	—
132	C₈×C₈₉	—	2	1	no	—
133	C₈×C₈₉	—	2	1	no	—
134	#10623494	—	2	1	no	—
135	#10623494	—	2	1	no	—
136	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₈₉	—	2	1	no	—
138	C₂×D₈₉	—	2	1	no	—
139	C₂×D₈₉	C₂²⋊Q₈	1	2	yes	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	4	1	no	—
144	C₄×C₈₉	—	4	1	no	—
145	C₄×C₈₉	—	4	1	no	—
146	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
148	Dic₈₉	—	2	2	no	—
149	Dic₈₉	—	2	1	no	—
150	Dic₈₉	—	2	1	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	—	4	1	no	—
154	Dic₈₉	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₈₉	#577	1	1	yes	characteristic
157	C₂×C₈₉	#578	1	1	yes	characteristic
158	D₈₉	—	2	2	no	—
159	#3082	—	89	1	no	maximal, Sylow
160	C₈₉	#3082	1	1	yes	characteristic, complemented, Sylow
161	#544	—	89	1	no	—
162	#519	—	89	1	no	—
163	#479	—	89	1	no	—
164	#479	—	89	1	no	—
165	#479	—	89	1	no	—
166	#479	D₈₉	1	1	yes	characteristic, complemented
167	#477	—	89	1	no	—
168	#477	—	89	1	no	—
169	#475	—	89	1	no	—
170	#475	—	89	1	no	—
171	#428	—	89	1	no	—
172	#427	—	89	1	no	—
173	#424	—	89	1	no	—
174	#419	—	89	1	no	—
175	C₂×#103	—	89	1	no	—
176	C₂²×Q₈	—	89	1	no	—
177	C₂×M₁₆	—	89	1	no	—
178	C₂²×C₈	—	89	1	no	—
179	C₄⋊Q₈	—	89	1	no	—
180	C₄⋊Q₈	—	89	1	no	—
181	C₄⋊Q₈	—	178	1	no	—
182	#125	—	89	1	no	—
183	#125	—	178	1	no	—
184	#125	C₂×D₈₉	1	1	yes	characteristic
185	C₂²⋊Q₈	—	178	1	no	—
186	C₂²⋊Q₈	—	178	1	no	—
187	C₄×Q₈	—	89	1	no	—
188	C₄×Q₈	—	89	1	no	—
189	C₄×Q₈	—	89	1	no	—
190	C₄×Q₈	—	178	1	no	—
191	C₄×Q₈	—	178	1	no	—
192	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
193	#117	—	89	1	no	—
194	#117	—	89	1	no	—
195	#117	—	89	1	no	—
196	#117	—	178	1	no	—
197	#117	—	178	1	no	—
198	C₂×C₄⋊C₄	—	89	1	no	—
199	#107	—	89	1	no	—
200	#107	—	89	1	no	—
201	#107	—	89	1	no	—
202	#107	C₂×D₈₉	1	1	yes	characteristic
203	#106	—	89	1	no	—
204	#106	—	89	1	no	—
205	#106	—	89	1	no	—
206	#106	C₂×D₈₉	1	1	yes	characteristic
207	C₄⋊C₈	—	89	1	no	—
208	C₄⋊C₈	—	89	1	no	—
209	C₄⋊C₈	—	89	1	no	—
210	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
211	#103	—	89	1	no	—
212	#103	—	89	1	no	—
213	#103	—	89	1	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#103	—	89	1	no	—
217	#103	C₂×D₈₉	1	1	yes	characteristic
218	#103	C₂×D₈₉	1	1	yes	characteristic
219	C₂×Q₈	—	89	4	no	—
220	C₂×Q₈	—	89	1	no	—
221	C₂×Q₈	—	178	1	no	—
222	C₂×Q₈	—	178	1	no	—
223	C₂×Q₈	—	178	2	no	—
224	C₂×Q₈	—	178	1	no	—
225	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
226	C₂²×C₄	—	89	1	no	—
227	C₂²×C₄	—	178	1	no	—
228	C₂²×C₄	—	178	1	no	—
229	M₁₆	—	178	2	no	—
230	C₂×C₈	—	89	1	no	—
231	C₂×C₈	—	89	1	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	—	89	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	C₄²	—	2	1	no	—
259	C₄²	—	89	1	no	—
260	C₄²	—	178	1	no	—
261	C₄²	—	178	1	no	—
262	C₄²	—	178	1	no	—
263	C₄²	—	178	1	no	—
264	C₄²	—	178	1	no	—
265	C₄²	C₂²×D₈₉	1	1	yes	characteristic
266	C₂³	—	89	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	89	2	no	—
269	Q₈	—	178	4	no	—
270	Q₈	—	178	1	no	—
271	Q₈	—	712	1	no	—
272	Q₈	—	712	1	no	—
273	Q₈	Q₈×D₈₉	1	2	yes	—
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	2	no	—
280	C₂×C₄	—	89	2	no	—
281	C₂×C₄	—	89	1	no	—
282	C₂×C₄	—	89	1	no	—
283	C₂×C₄	—	178	4	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	—	712	1	no	—
289	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
290	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	178	1	no	—
295	C₈	—	178	1	no	—
296	C₂²	—	89	2	no	—
297	C₂²	—	178	1	no	—
298	C₂²	—	178	1	no	—
299	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	89	2	no	—
307	C₄	—	356	1	no	—
308	C₄	—	356	1	no	—
309	C₄	—	356	1	no	—
310	C₄	—	178	1	no	—
311	C₄	—	178	1	no	—
312	C₄	—	178	2	no	—
313	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
314	C₄	#41×D₈₉	1	1	yes	characteristic
315	C₂	—	178	2	no	—
316	C₂	C₂²⋊Q₈×D₈₉	1	1	yes	characteristic
317	C₂	#19152405	1	1	yes	characteristic
318	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	C₂²×D₈₉	C₂×D₄
hypercenter	#479	D₈₉
hypocenter	C₈₉	#3082
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. #19151908	C₂	semidirect
4. #19151903	C₂	semidirect
5. #19151902	C₂	semidirect
6. #19151893	C₂	semidirect
7. #19151892	C₂	semidirect
8. #19151884	C₂	semidirect
9. #19151881	C₂	semidirect
15. #479×C₈₉	C₂	semidirect
64. Q₈×D₈₉	Q₈	semidirect
160. C₈₉	#3082	semidirect
166. #479	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152228	2
#19152074	2
#19151908	2
#19151903	2
#19151902	2
#19151893	2
#19151892	2
#19151884	2
#19151881	2
#19151639	2
#19151633	2
#19151623	2
#19151605	2
#103×D₈₉	2
#479×C₈₉	2
#3082	89