Subgroups — #36448454

Order 22784.

420 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	#26028814	C₂	1	1	yes	maximal, characteristic
2	#26020233	C₂	1	1	yes	maximal, characteristic
3	#26019024	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019023	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018564	C₂	1	1	yes	maximal, characteristic
6	#26018149	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₈₉	—	2	1	no	—
9	C₂×#15667969	C₂²	1	1	yes	characteristic
10	#19151651	C₂²	1	1	yes	characteristic
11	#19151623	—	2	2	no	—
12	#19151623	—	2	2	no	—
13	#19151622	C₂²	1	1	yes	characteristic
14	#19151621	—	2	1	no	—
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	C₂×#15667924	—	2	1	no	—
17	#19151352	—	2	1	no	—
18	#19151349	—	2	2	no	—
19	#19151348	—	2	2	no	—
20	#19151323	C₄	1	2	yes	—
21	#19151321	C₄	1	2	yes	—
22	#431×C₈₉	C₂²	1	1	yes	characteristic
23	#423×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
26	C₂×#13083558	—	2	1	no	—
27	C₂×#13083557	—	2	2	no	—
28	M₁₆×D₈₉	—	4	2	no	—
29	M₁₆×D₈₉	—	4	2	no	—
30	C₂²×#10623494	—	2	1	no	—
31	C₂×C₈×D₈₉	—	2	2	no	—
32	#15667969	—	2	1	no	—
33	#15667969	—	2	1	no	—
34	#15667969	—	2	1	no	—
35	#15667969	—	2	1	no	—
36	C₂×#13083553	C₂×C₄	1	1	yes	characteristic
37	C₂×#13083553	Q₈	1	1	yes	characteristic
38	C₂×#13083551	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083551	Q₈	1	1	yes	characteristic
40	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
41	C₄⋊#10623494	—	2	1	no	—
42	C₄⋊#10623494	—	2	1	no	—
43	C₄⋊#10623494	—	2	1	no	—
44	C₄⋊#10623494	—	2	1	no	—
45	C₄⋊#10623494	D₄	1	2	yes	—
46	#15667941	—	2	4	no	—
47	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
48	#15667924	—	4	2	no	—
49	C₄×#10623494	—	2	1	no	—
50	C₂×M₁₆×C₈₉	—	2	1	no	—
51	C₂²×C₈×C₈₉	—	2	1	no	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	—	2	1	no	—
54	C₄⋊C₈×C₈₉	—	2	1	no	—
55	C₄⋊C₈×C₈₉	—	2	1	no	—
56	C₄⋊C₈×C₈₉	—	2	1	no	—
57	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
58	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₈₉	—	2	1	no	—
60	C₂³×D₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
69	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
70	#13083558	—	4	1	no	—
71	#13083558	—	4	1	no	—
72	#13083557	—	4	4	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	4	1	no	—
76	C₂×#10623494	—	4	1	no	—
77	C₂×#10623494	—	4	1	no	—
78	C₈×D₈₉	—	4	4	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	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
92	M₁₆×C₈₉	—	4	1	no	—
93	M₁₆×C₈₉	—	4	1	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	4	1	no	—
97	C₂×C₈×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	—	4	1	no	—
99	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₈₉	#41	1	1	yes	characteristic
102	C₄²×C₈₉	#41	1	1	yes	characteristic
103	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×D₈₉	—	2	1	no	—
105	C₂²×D₈₉	—	2	1	no	—
106	C₂²×D₈₉	—	4	1	no	—
107	C₂²×D₈₉	—	4	1	no	—
108	C₂²×D₈₉	#106	1	2	yes	—
109	C₂×C₄×C₈₉	—	4	1	no	—
110	C₂×C₄×C₈₉	—	4	1	no	—
111	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	4	1	no	—
126	C₂×Dic₈₉	—	4	1	no	—
127	C₂×Dic₈₉	#106	1	2	yes	—
128	C₄×D₈₉	—	2	2	no	—
129	C₄×D₈₉	—	2	2	no	—
130	C₄×D₈₉	—	2	2	no	—
131	C₄×D₈₉	—	2	2	no	—
132	C₈×C₈₉	—	4	2	no	—
133	C₈×C₈₉	—	4	2	no	—
134	C₈×C₈₉	—	8	1	no	—
135	#10623494	—	4	2	no	—
136	#10623494	—	4	2	no	—
137	#10623494	—	8	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	1	no	—
140	C₂²×C₈₉	#385	1	1	yes	characteristic
141	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
142	C₂²×C₈₉	#428	1	1	yes	characteristic
143	C₂×D₈₉	—	2	2	no	—
144	C₂×D₈₉	—	4	1	no	—
145	C₂×D₈₉	—	4	1	no	—
146	C₂×D₈₉	—	4	2	no	—
147	C₂×D₈₉	—	8	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	#448	1	1	yes	characteristic
153	C₄×C₈₉	#450	1	1	yes	characteristic
154	C₄×C₈₉	#481	1	1	yes	characteristic
155	C₄×C₈₉	#483	1	1	yes	characteristic
156	Dic₈₉	—	2	2	no	—
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	—	4	1	no	—
159	#8862	—	89	1	no	maximal, Sylow
160	C₂×C₈₉	—	2	1	no	—
161	C₂×C₈₉	—	2	1	no	—
162	C₂×C₈₉	#1853	1	1	yes	characteristic
163	C₂×C₈₉	#1858	1	1	yes	characteristic
164	C₂×C₈₉	#1868	1	1	yes	characteristic
165	D₈₉	—	4	2	no	—
166	#2905	—	89	1	no	—
167	#1798	—	89	1	no	—
168	#1570	—	89	1	no	—
169	#1570	—	89	1	no	—
170	#1458	—	89	1	no	—
171	#1301	—	89	1	no	—
172	#1301	D₈₉	1	1	yes	characteristic, complemented
173	C₈₉	#8862	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	—	178	1	no	—
175	C₂×#117	—	89	1	no	—
176	#431	—	89	1	no	—
177	#431	C₂×D₈₉	1	1	yes	characteristic
178	#424	—	178	2	no	—
179	#424	—	178	2	no	—
180	#423	—	89	1	no	—
181	#423	—	178	1	no	—
182	#423	C₂×D₈₉	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	89	1	no	—
184	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
185	C₂×C₂²⋊C₈	—	178	1	no	—
186	#336	—	178	1	no	—
187	#334	—	178	2	no	—
188	#334	—	178	2	no	—
189	#324	—	89	2	no	—
190	#323	—	89	2	no	—
191	C₂³×C₄	—	89	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	356	2	no	—
194	C₂×M₁₆	—	356	2	no	—
195	C₂×M₁₆	—	178	2	no	—
196	C₂×M₁₆	—	178	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	178	2	no	—
199	C₂²×C₈	—	178	1	no	—
200	#117	—	178	1	no	—
201	#117	—	178	1	no	—
202	#117	—	178	1	no	—
203	#117	—	178	1	no	—
204	C₂×C₄⋊C₄	—	89	1	no	—
205	C₂×C₄⋊C₄	—	89	1	no	—
206	C₂×C₂²⋊C₄	—	89	1	no	—
207	C₂×C₂²⋊C₄	—	89	1	no	—
208	C₂×C₄²	—	89	1	no	—
209	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	89	2	no	—
215	C₄⋊C₈	—	178	4	no	—
216	C₄⋊C₈	—	178	1	no	—
217	C₄⋊C₈	—	178	1	no	—
218	C₄⋊C₈	—	178	1	no	—
219	C₄⋊C₈	—	178	1	no	—
220	C₄⋊C₈	D₃₅₆	1	2	yes	—
221	C₂²⋊C₈	—	89	2	no	—
222	C₂²⋊C₈	—	356	2	no	—
223	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	178	1	no	—
226	C₂⁴	—	89	1	no	—
227	C₂²×C₄	—	89	1	no	—
228	C₂²×C₄	—	89	2	no	—
229	C₂²×C₄	—	89	2	no	—
230	C₂²×C₄	—	89	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	—	178	1	no	—
236	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
239	M₁₆	—	4	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	356	4	no	—
242	M₁₆	—	356	1	no	—
243	M₁₆	—	356	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	356	4	no	—
250	C₂×C₈	—	356	1	no	—
251	C₂×C₈	—	356	1	no	—
252	C₂×C₈	—	356	1	no	—
253	C₂×C₈	—	178	2	no	—
254	C₂×C₈	—	178	2	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	C₂²⋊C₄	—	178	1	no	—
260	C₂²⋊C₄	—	178	1	no	—
261	C₂²⋊C₄	—	178	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₄²	—	178	1	no	—
264	C₄²	—	178	1	no	—
265	C₄²	D₄×D₈₉	1	1	yes	characteristic
266	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
267	C₄²	#13083573	1	1	yes	characteristic
268	C₄²	#13083574	1	1	yes	characteristic
269	C₂³	—	89	2	no	—
270	C₂³	—	356	1	no	—
271	C₂³	—	356	1	no	—
272	C₂³	—	178	1	no	—
273	C₂³	—	178	1	no	—
274	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	89	2	no	—
278	C₂×C₄	—	356	1	no	—
279	C₂×C₄	—	356	1	no	—
280	C₂×C₄	—	178	1	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	—	178	1	no	—
283	C₂×C₄	—	178	1	no	—
284	C₂×C₄	—	178	2	no	—
285	C₂×C₄	—	178	2	no	—
286	C₂×C₄	—	178	2	no	—
287	C₂×C₄	—	178	2	no	—
288	C₂×C₄	#15667972	1	1	yes	characteristic
289	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
290	C₂×C₄	#15667982	1	1	yes	characteristic
291	C₂×C₄	#15667990	1	1	yes	characteristic
292	C₂×C₄	#15667992	1	1	yes	characteristic
293	C₂×C₄	#15667994	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
295	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
296	C₂×C₄	#15668036	1	1	yes	characteristic
297	C₂×C₄	#15668044	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	356	2	no	—
302	C₈	—	356	2	no	—
303	C₈	—	712	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	356	2	no	—
307	C₂²	—	356	1	no	—
308	C₂²	—	356	1	no	—
309	C₂²	—	178	2	no	—
310	C₂²	—	712	1	no	—
311	C₂²	#19151472	1	1	yes	characteristic
312	C₂²	#106×D₈₉	1	1	yes	characteristic
313	C₂²	#19151639	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	356	1	no	—
319	C₄	—	356	1	no	—
320	C₄	—	178	2	no	—
321	C₄	#19151738	1	1	yes	characteristic
322	C₄	#19151750	1	1	yes	characteristic
323	C₄	#19151915	1	1	yes	characteristic
324	C₄	#19151927	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	356	2	no	—
328	C₂	#26020446	1	1	yes	characteristic
329	C₂	#26020473	1	1	yes	characteristic
330	C₂	#26020539	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667982
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1301	C₂
socle	C₂²×C₈₉	#385
prime radical	C₂³×D₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1301	D₈₉
hypocenter	C₈₉	#8862
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
3. #26019024	C₂	semidirect
4. #26019023	C₂	semidirect
6. #26018149	C₂	semidirect
7. C₈₉×#1301	C₂	semidirect
172. #1301	D₈₉	semidirect
173. C₈₉	#8862	semidirect

Maximal subgroups

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

subgroup	index
#26028814	2
#26020233	2
#26019024	2
#26019023	2
#26018564	2
#26018149	2
C₈₉×#1301	2
#8862	89