Subgroups — #36452939

Order 22784.

326 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021197	C₂	1	1	yes	maximal, characteristic
2	#26021028	C₂	1	1	yes	maximal, characteristic
3	#26018707	C₂	1	1	yes	maximal, characteristic
4	#26018533	C₂	1	1	yes	maximal, characteristic
5	#26018154	C₂	1	1	yes	maximal, characteristic
6	#26018091	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1542	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667948	—	2	1	no	—
9	#19151622	C₂²	1	1	yes	characteristic
10	C₄×#13083558	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
13	#19151363	—	2	1	no	—
14	#19151357	—	2	1	no	—
15	#19151357	—	2	1	no	—
16	Q₈⋊#10623494	—	2	1	no	—
17	Q₈⋊#10623494	—	2	1	no	—
18	#19151320	—	2	1	no	—
19	C₈×#10623494	—	2	1	no	—
20	#533×C₈₉	C₂²	1	1	yes	characteristic
21	#431×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#333×C₈₉	C₄	1	1	yes	characteristic, complemented
24	#333×C₈₉	C₄	1	1	yes	characteristic, complemented
25	#332×C₈₉	C₄	1	1	yes	characteristic, complemented
26	#332×C₈₉	C₄	1	1	yes	characteristic, complemented
27	C₂×#13083558	—	2	1	no	—
28	C₂×#13083558	D₄	1	1	yes	characteristic
29	C₂×#13083558	D₄	1	1	yes	characteristic
30	C₂²×#10623494	—	2	1	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×C₄×Dic₈₉	—	2	1	no	—
33	#15667948	—	4	2	no	—
34	C₄⋊#10623494	—	2	1	no	—
35	C₄⋊#10623494	—	2	1	no	—
36	#15667923	—	2	1	no	—
37	C₄×#10623494	—	2	1	no	—
38	C₄×#10623494	—	2	1	no	—
39	C₄×#10623494	D₄	1	1	yes	characteristic
40	C₄×#10623494	D₄	1	1	yes	characteristic
41	#15667919	—	2	1	no	—
42	C₂²×C₈×C₈₉	—	2	1	no	—
43	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	#125×C₈₉	—	2	1	no	—
46	C₂²⋊Q₈×C₈₉	—	2	1	no	—
47	C₂²⋊Q₈×C₈₉	—	2	1	no	—
48	C₄×Q₈×C₈₉	—	2	1	no	—
49	C₄×Q₈×C₈₉	—	2	1	no	—
50	#117×C₈₉	—	2	1	no	—
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₈₉	—	2	1	no	—
53	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂²⋊C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₂²×Dic₈₉	—	4	1	no	—
60	#13083558	—	2	1	no	—
61	#13083558	—	2	1	no	—
62	#13083558	—	4	1	no	—
63	#13083558	—	4	1	no	—
64	#13083558	SD₁₆	1	2	yes	—
65	#13083558	Q₁₆	1	2	yes	—
66	C₂×#10623494	—	2	1	no	—
67	C₂×#10623494	—	2	1	no	—
68	C₂×#10623494	—	2	1	no	—
69	C₂×#10623494	—	2	1	no	—
70	C₂×#10623494	—	4	2	no	—
71	C₂×#10623494	—	4	1	no	—
72	C₄⋊Dic₈₉	—	2	2	no	—
73	C₄⋊Dic₈₉	—	4	1	no	—
74	C₄×Dic₈₉	—	4	1	no	—
75	C₄×Dic₈₉	—	4	1	no	—
76	C₂×Q₈×C₈₉	—	2	1	no	—
77	C₂×Q₈×C₈₉	—	2	1	no	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	—	2	1	no	—
83	C₂×C₈×C₈₉	—	2	1	no	—
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	—	4	1	no	—
86	C₂×C₈×C₈₉	—	4	1	no	—
87	C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₄⋊C₄×C₈₉	—	2	1	no	—
89	C₄⋊C₄×C₈₉	—	2	1	no	—
90	C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₄⋊C₄×C₈₉	—	4	1	no	—
92	C₄⋊C₄×C₈₉	—	4	1	no	—
93	C₂²⋊C₄×C₈₉	—	4	1	no	—
94	C₄²×C₈₉	—	4	1	no	—
95	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₉	#41	1	1	yes	characteristic
98	C₄²×C₈₉	#41	1	1	yes	characteristic
99	C₂³×C₈₉	#124	1	1	yes	characteristic
100	Q₈×C₈₉	—	4	1	no	—
101	Q₈×C₈₉	—	4	1	no	—
102	Q₈×C₈₉	—	8	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	—	4	1	no	—
107	C₂×C₄×C₈₉	—	4	1	no	—
108	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#126	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
118	C₂×Dic₈₉	—	4	2	no	—
119	C₂×Dic₈₉	—	8	1	no	—
120	C₂×Dic₈₉	—	8	1	no	—
121	C₈×C₈₉	—	4	1	no	—
122	C₈×C₈₉	—	4	1	no	—
123	C₈×C₈₉	—	4	1	no	—
124	C₈×C₈₉	—	4	1	no	—
125	#10623494	—	2	2	no	—
126	#10623494	—	2	2	no	—
127	#10623494	—	2	2	no	—
128	#10623494	—	4	1	no	—
129	#10623494	—	8	1	no	—
130	C₂²×C₈₉	—	2	1	no	—
131	C₂²×C₈₉	—	2	1	no	—
132	C₂²×C₈₉	#388	1	1	yes	characteristic
133	C₂²×C₈₉	#487	1	1	yes	characteristic
134	C₂²×C₈₉	#488	1	1	yes	characteristic
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	2	1	no	—
138	C₄×C₈₉	—	8	1	no	—
139	C₄×C₈₉	—	8	1	no	—
140	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
141	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
142	C₄×C₈₉	#443	1	1	yes	characteristic
143	C₄×C₈₉	#444	1	1	yes	characteristic
144	C₄×C₈₉	#465	1	1	yes	characteristic
145	C₄×C₈₉	#467	1	1	yes	characteristic
146	Dic₈₉	—	8	2	no	—
147	#9589	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	#1929	1	1	yes	characteristic
151	C₂×C₈₉	#1932	1	1	yes	characteristic
152	C₂×C₈₉	#1933	1	1	yes	characteristic
153	#1986	—	89	1	no	—
154	#1950	—	89	1	no	—
155	#1542	D₈₉	1	1	yes	characteristic
156	#1493	—	89	1	no	—
157	C₈×M₁₆	—	89	1	no	—
158	#1303	—	89	1	no	—
159	#1279	—	89	1	no	—
160	C₈₉	#9589	1	1	yes	characteristic, complemented, Sylow
161	#533	C₂×D₈₉	1	1	yes	characteristic
162	#431	C₂×D₈₉	1	1	yes	characteristic
163	C₂×#108	—	178	1	no	—
164	#423	—	89	1	no	—
165	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
166	C₄×M₁₆	—	89	1	no	—
167	C₂×C₄×C₈	—	89	1	no	—
168	C₄×C₄⋊C₄	—	89	1	no	—
169	#341	—	178	1	no	—
170	#339	—	178	1	no	—
171	#339	—	178	1	no	—
172	#333	Dic₈₉	1	1	yes	characteristic, complemented
173	#333	Dic₈₉	1	1	yes	characteristic, complemented
174	#332	Dic₈₉	1	1	yes	characteristic, complemented
175	#332	Dic₈₉	1	1	yes	characteristic, complemented
176	Q₈⋊C₈	—	178	1	no	—
177	Q₈⋊C₈	—	178	1	no	—
178	#322	—	178	1	no	—
179	C₈²	—	178	1	no	—
180	C₂×M₁₆	—	89	1	no	—
181	C₂×M₁₆	—	89	1	no	—
182	C₂×M₁₆	—	178	1	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	178	1	no	—
185	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
186	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
187	#125	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₂²⋊Q₈	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	—	2	1	no	—
192	#117	—	2	1	no	—
193	C₂×C₄⋊C₄	—	178	1	no	—
194	C₂×C₄²	—	178	1	no	—
195	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
196	#108	—	356	2	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	178	1	no	—
199	C₄⋊C₈	—	178	1	no	—
200	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
202	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
203	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
204	C₂²⋊C₈	—	2	1	no	—
205	#97	—	178	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	89	1	no	—
208	C₄×C₈	—	89	1	no	—
209	C₄×C₈	—	178	1	no	—
210	C₄×C₈	—	178	1	no	—
211	#95	—	178	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂²×C₄	—	356	1	no	—
215	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	#13083573	1	1	yes	characteristic
217	C₂²×C₄	#13083574	1	1	yes	characteristic
218	M₁₆	—	89	2	no	—
219	M₁₆	—	89	2	no	—
220	M₁₆	—	356	1	no	—
221	M₁₆	—	356	1	no	—
222	M₁₆	—	178	1	no	—
223	M₁₆	—	178	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	356	2	no	—
231	C₂×C₈	—	356	1	no	—
232	C₂×C₈	—	178	1	no	—
233	C₂×C₈	—	178	1	no	—
234	C₂×C₈	—	178	1	no	—
235	C₂×C₈	—	178	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	356	1	no	—
243	C₄⋊C₄	—	178	2	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	356	1	no	—
247	C₄²	—	356	1	no	—
248	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
249	C₄²	D₄×D₈₉	1	1	yes	characteristic
250	C₄²	#13083573	1	1	yes	characteristic
251	C₄²	#13083574	1	1	yes	characteristic
252	C₂³	#15667998	1	1	yes	characteristic
253	Q₈	—	4	1	no	—
254	Q₈	—	4	1	no	—
255	Q₈	—	8	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	356	2	no	—
262	C₂×C₄	—	712	1	no	—
263	C₂×C₄	—	712	1	no	—
264	C₂×C₄	#15667971	1	1	yes	characteristic
265	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
266	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
267	C₂×C₄	#15667985	1	1	yes	characteristic
268	C₂×C₄	#15667994	1	1	yes	characteristic
269	C₂×C₄	#15668005	1	1	yes	characteristic
270	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
271	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
272	C₂×C₄	#15668033	1	1	yes	characteristic
273	C₂×C₄	#15668035	1	1	yes	characteristic
274	C₈	—	4	1	no	—
275	C₈	—	4	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	356	1	no	—
279	C₈	—	178	2	no	—
280	C₈	—	178	2	no	—
281	C₈	—	178	2	no	—
282	C₈	—	712	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#19151491	1	1	yes	characteristic
286	C₂²	#19151947	1	1	yes	characteristic
287	C₂²	#19151954	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	—	712	2	no	—
294	C₄	SD₁₆×Dic₈₉	1	1	yes	characteristic
295	C₄	Q₁₆×Dic₈₉	1	1	yes	characteristic
296	C₄	#19151715	1	1	yes	characteristic
297	C₄	#19151720	1	1	yes	characteristic
298	C₄	#19151830	1	1	yes	characteristic
299	C₄	#19151838	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#26020902	1	1	yes	characteristic
303	C₂	#26020923	1	1	yes	characteristic
304	C₂	#26020930	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667985
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1542	C₂
socle	C₂²×C₈₉	#388
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1542	D₈₉
hypocenter	C₈₉	#9589
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1542

Complements

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

N	H ≅ G/N	type
23. #333×C₈₉	C₄	semidirect
24. #333×C₈₉	C₄	semidirect
25. #332×C₈₉	C₄	semidirect
26. #332×C₈₉	C₄	semidirect
160. C₈₉	#9589	semidirect
172. #333	Dic₈₉	semidirect
173. #333	Dic₈₉	semidirect
174. #332	Dic₈₉	semidirect
175. #332	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26021197	2
#26021028	2
#26018707	2
#26018533	2
#26018154	2
#26018091	2
C₈₉×#1542	2
#9589	89