Subgroups — #26031586

Order 11392.

354 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152152	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152127	C₂	1	1	yes	maximal, characteristic
3	#19151906	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151881	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151876	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151836	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151826	C₂	1	1	yes	maximal, characteristic
10	#19151824	C₂	1	1	yes	maximal, characteristic
11	#19151817	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151801	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	#15668011	C₂²	1	1	yes	characteristic
21	#15668010	C₂²	1	1	yes	characteristic
22	#15668005	—	2	1	no	—
23	#15668004	C₂²	1	1	yes	characteristic
24	#15668002	—	2	1	no	—
25	#15667994	—	2	1	no	—
26	#15667992	C₂²	1	1	yes	characteristic
27	#15667990	—	2	1	no	—
28	#15667988	C₂²	1	1	yes	characteristic
29	#15667988	C₂²	1	1	yes	characteristic
30	#15667983	C₂²	1	1	yes	characteristic
31	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₇₈	—	2	1	no	—
33	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
34	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
35	#15667972	—	2	1	no	—
36	#15667969	—	2	1	no	—
37	#15667968	—	2	1	no	—
38	C₄²×D₈₉	C₂²	1	1	yes	characteristic
39	#15667946	C₂²	1	1	yes	characteristic
40	#15667945	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	#15667936	C₂²	1	1	yes	characteristic
51	#15667933	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	#124×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₉	—	2	1	no	—
62	C₂²×Dic₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
64	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
66	#13083564	—	4	1	no	—
67	#13083562	—	4	1	no	—
68	C₂×#10623494	C₂³	1	1	yes	characteristic
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₈×D₈₉	—	4	1	no	—
71	C₈×D₈₉	—	4	1	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
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	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	D₄	1	1	yes	characteristic
93	C₄×Dic₈₉	D₄	1	1	yes	characteristic
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
98	Q₁₆×C₈₉	—	4	1	no	—
99	SD₁₆×C₈₉	—	4	1	no	—
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₄²×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	—	2	1	no	—
109	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₉	—	2	1	no	—
111	Q₈×C₈₉	—	2	1	no	—
112	Q₈×C₈₉	—	2	1	no	—
113	Q₈×C₈₉	—	2	1	no	—
114	D₄×C₈₉	—	2	1	no	—
115	D₄×C₈₉	—	2	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	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
131	C₄×D₈₉	—	2	2	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	2	2	no	—
135	Dic₁₇₈	—	2	1	no	—
136	Dic₁₇₈	—	2	1	no	—
137	Dic₁₇₈	—	4	2	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	#10623494	—	2	1	no	—
141	#10623494	—	2	1	no	—
142	C₂²×C₈₉	—	4	1	no	—
143	C₂²×C₈₉	—	4	1	no	—
144	C₂²×C₈₉	—	4	1	no	—
145	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₈₉	—	2	1	no	—
147	C₂×D₈₉	—	2	1	no	—
148	C₂×D₈₉	#137	1	2	yes	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	4	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₈₉	—	2	1	no	—
157	Dic₈₉	—	2	1	no	—
158	Dic₈₉	—	4	1	no	—
159	Dic₈₉	—	4	1	no	—
160	Dic₈₉	—	4	1	no	—
161	Dic₈₉	—	4	1	no	—
162	Dic₈₉	#135	1	2	yes	—
163	C₂×C₈₉	—	8	1	no	—
164	C₂×C₈₉	#537	1	1	yes	characteristic
165	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
166	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
167	D₈₉	—	2	2	no	—
168	#3248	—	89	1	no	maximal, Sylow
169	C₈₉	#3248	1	1	yes	characteristic, complemented, Sylow
170	#533	—	89	1	no	—
171	#529	—	89	1	no	—
172	#479	—	89	1	no	—
173	#479	—	89	1	no	—
174	#475	—	89	1	no	—
175	#474	—	89	1	no	—
176	#467	—	89	1	no	—
177	#467	—	89	1	no	—
178	#465	—	89	1	no	—
179	#465	—	89	1	no	—
180	#464	—	89	1	no	—
181	#464	D₈₉	1	1	yes	characteristic, complemented
182	#462	—	89	1	no	—
183	#461	—	89	1	no	—
184	C₂×C₄⋊C₈	—	89	1	no	—
185	C₂×Q₁₆	—	89	1	no	—
186	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
187	C₂×SD₁₆	—	89	1	no	—
188	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
189	C₂²×C₈	—	89	1	no	—
190	C₂²×C₈	—	89	1	no	—
191	C₄⋊Q₈	—	89	1	no	—
192	C₄⋊Q₈	—	89	1	no	—
193	#126	—	178	1	no	—
194	#125	—	89	1	no	—
195	#125	—	178	1	no	—
196	#124	C₂×D₈₉	1	1	yes	characteristic
197	#123	—	178	1	no	—
198	C₂²⋊Q₈	—	89	1	no	—
199	C₂²⋊Q₈	—	89	1	no	—
200	C₂²⋊Q₈	—	89	1	no	—
201	C₂²⋊Q₈	—	178	1	no	—
202	#121	—	89	1	no	—
203	C₄×Q₈	—	89	1	no	—
204	C₄×Q₈	—	89	1	no	—
205	C₄×Q₈	—	178	1	no	—
206	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
207	C₄×D₄	—	89	1	no	—
208	C₄×D₄	—	178	1	no	—
209	#117	—	178	1	no	—
210	#117	—	178	1	no	—
211	C₂×C₄²	—	89	1	no	—
212	#107	—	89	1	no	—
213	#107	—	89	1	no	—
214	#106	—	89	1	no	—
215	#106	—	89	1	no	—
216	C₄⋊C₈	—	89	1	no	—
217	C₄⋊C₈	—	89	1	no	—
218	C₄⋊C₈	—	89	1	no	—
219	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
220	#103	—	89	1	no	—
221	#103	—	89	1	no	—
222	#103	—	89	1	no	—
223	#103	—	89	1	no	—
224	#103	—	89	1	no	—
225	#103	C₂×D₈₉	1	1	yes	characteristic
226	#102	—	89	1	no	—
227	#102	C₂×D₈₉	1	1	yes	characteristic
228	C₂×Q₈	—	89	1	no	—
229	C₂×Q₈	—	178	1	no	—
230	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×D₄	—	178	1	no	—
233	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
234	C₂²×C₄	—	89	1	no	—
235	C₂²×C₄	—	89	1	no	—
236	C₂²×C₄	—	89	1	no	—
237	C₂²×C₄	—	178	1	no	—
238	Q₁₆	—	4	1	no	—
239	Q₁₆	—	356	1	no	—
240	SD₁₆	—	4	1	no	—
241	SD₁₆	—	356	1	no	—
242	C₂×C₈	—	89	1	no	—
243	C₂×C₈	—	89	1	no	—
244	C₂×C₈	—	356	1	no	—
245	C₂×C₈	—	356	1	no	—
246	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	89	1	no	—
251	C₄⋊C₄	—	89	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	—	178	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	178	1	no	—
267	C₂²⋊C₄	—	178	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	89	1	no	—
272	C₄²	—	89	1	no	—
273	C₄²	—	89	1	no	—
274	C₄²	—	178	1	no	—
275	C₄²	—	178	1	no	—
276	C₄²	C₂²×D₈₉	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	89	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	356	2	no	—
284	Q₈	—	178	1	no	—
285	Q₈	—	178	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	356	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	89	1	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	356	2	no	—
296	C₂×C₄	—	178	2	no	—
297	C₂×C₄	—	178	2	no	—
298	C₂×C₄	—	178	1	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	178	1	no	—
301	C₂×C₄	—	178	1	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
305	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	178	1	no	—
310	C₈	—	178	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	89	2	no	—
315	C₂²	—	178	1	no	—
316	C₂²	—	178	1	no	—
317	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	89	2	no	—
324	C₄	—	356	1	no	—
325	C₄	—	356	1	no	—
326	C₄	—	356	1	no	—
327	C₄	—	356	1	no	—
328	C₄	—	178	1	no	—
329	C₄	—	178	1	no	—
330	C₄	#15668057	1	1	yes	characteristic
331	C₄	#15668059	1	1	yes	characteristic
332	C₂	—	8	1	no	—
333	C₂	—	178	2	no	—
334	C₂	#19152177	1	1	yes	characteristic
335	C₂	#135×D₈₉	1	1	yes	characteristic
336	C₂	#137×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#464×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#15667983	C₂²
hypercenter	#464	D₈₉
hypocenter	C₈₉	#3248
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #19152152	C₂	semidirect
3. #19151906	C₂	semidirect
4. #19151905	C₂	semidirect
5. #19151881	C₂	semidirect
6. #19151876	C₂	semidirect
7. #19151838	C₂	semidirect
8. #19151836	C₂	semidirect
11. #19151817	C₂	semidirect
12. #19151806	C₂	semidirect
13. #19151801	C₂	semidirect
14. C₄⋊C₈×D₈₉	C₂	semidirect
15. #464×C₈₉	C₂	semidirect
169. C₈₉	#3248	semidirect
181. #464	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152152	2
#19152127	2
#19151906	2
#19151905	2
#19151881	2
#19151876	2
#19151838	2
#19151836	2
#19151826	2
#19151824	2
#19151817	2
#19151806	2
#19151801	2
C₄⋊C₈×D₈₉	2
#464×C₈₉	2
#3248	89