Subgroups — #36459471

Order 22784.

510 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021058	C₂	1	1	yes	maximal, characteristic
2	#26021058	C₂	1	1	yes	maximal, characteristic
3	#26021058	C₂	1	1	yes	maximal, characteristic
4	#26020412	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020409	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668009	C₂²	1	1	yes	characteristic
9	C₂×#15667946	—	2	2	no	—
10	C₂×#15667944	—	2	2	no	—
11	C₂×#15667934	—	2	2	no	—
12	C₂×#15667934	—	2	2	no	—
13	C₂×#15667934	—	2	2	no	—
14	#19151468	C₂²	1	1	yes	characteristic
15	#19151468	C₂²	1	1	yes	characteristic
16	#19151468	C₂²	1	1	yes	characteristic
17	#19151361	—	2	2	no	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×D₃₅₆	—	2	1	no	—
22	C₂²×D₃₅₆	—	2	1	no	—
23	C₂²×D₃₅₆	—	2	1	no	—
24	#15668009	—	2	1	no	—
25	#15668009	—	2	1	no	—
26	#15668009	—	2	1	no	—
27	#15668009	Q₈	1	2	yes	—
28	C₂×C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×C₄⋊Dic₈₉	—	2	1	no	—
33	C₂×C₄⋊Dic₈₉	—	2	1	no	—
34	C₂×C₄⋊Dic₈₉	—	2	1	no	—
35	C₂×C₄⋊Dic₈₉	—	2	1	no	—
36	C₂×C₄⋊Dic₈₉	—	2	1	no	—
37	#15667946	—	4	4	no	—
38	#15667944	—	4	4	no	—
39	#15667934	—	4	4	no	—
40	#15667934	—	4	4	no	—
41	#15667934	—	4	4	no	—
42	C₂²×C₈×C₈₉	D₄	1	2	yes	—
43	C₂²×C₈×C₈₉	D₄	1	2	yes	—
44	C₂²×C₈×C₈₉	D₄	1	2	yes	—
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₈₉	—	2	1	no	—
47	C₄×C₈×C₈₉	—	2	1	no	—
48	C₄×C₈×C₈₉	—	2	1	no	—
49	C₄×C₈×C₈₉	—	2	1	no	—
50	C₄×C₈×C₈₉	—	2	1	no	—
51	C₄×C₈×C₈₉	—	2	1	no	—
52	C₂³×D₈₉	—	4	1	no	—
53	C₂×D₃₅₆	—	2	2	no	—
54	C₂×D₃₅₆	—	2	2	no	—
55	C₂×D₃₅₆	—	2	2	no	—
56	C₂×D₃₅₆	—	4	2	no	—
57	C₂×D₃₅₆	—	4	2	no	—
58	C₂×D₃₅₆	—	4	2	no	—
59	C₂×D₃₅₆	—	4	2	no	—
60	C₂×D₃₅₆	—	4	2	no	—
61	C₂×D₃₅₆	—	4	2	no	—
62	C₂×D₃₅₆	—	4	1	no	—
63	C₂×D₃₅₆	—	4	1	no	—
64	C₂×D₃₅₆	—	4	1	no	—
65	C₂²×Dic₈₉	—	4	1	no	—
66	C₂²×Dic₈₉	—	4	1	no	—
67	C₂²×Dic₈₉	—	4	1	no	—
68	C₄⋊Dic₈₉	—	4	2	no	—
69	C₄⋊Dic₈₉	—	4	2	no	—
70	C₄⋊Dic₈₉	—	4	2	no	—
71	C₄⋊Dic₈₉	—	4	2	no	—
72	C₄⋊Dic₈₉	—	4	2	no	—
73	C₄⋊Dic₈₉	—	4	2	no	—
74	C₄⋊Dic₈₉	—	4	2	no	—
75	C₄⋊Dic₈₉	—	4	2	no	—
76	C₄⋊Dic₈₉	—	4	2	no	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	—	2	2	no	—
85	C₂×C₈×C₈₉	—	2	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	D₈	1	4	yes	—
88	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
89	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₈₉	#41	1	1	yes	characteristic
91	C₄²×C₈₉	#41	1	1	yes	characteristic
92	C₄²×C₈₉	#41	1	1	yes	characteristic
93	C₂³×C₈₉	#127	1	1	yes	characteristic
94	C₂²×D₈₉	—	4	2	no	—
95	C₂²×D₈₉	—	8	1	no	—
96	C₂²×D₈₉	—	8	1	no	—
97	C₂²×D₈₉	—	8	1	no	—
98	C₂²×D₈₉	—	8	1	no	—
99	C₂²×D₈₉	—	8	1	no	—
100	C₂²×D₈₉	—	8	1	no	—
101	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₈₉	#135	1	2	yes	—
110	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₈₉	#137	1	2	yes	—
113	C₂×Dic₈₉	—	8	1	no	—
114	C₂×Dic₈₉	—	8	1	no	—
115	C₂×Dic₈₉	—	8	1	no	—
116	C₂×Dic₈₉	—	8	1	no	—
117	C₂×Dic₈₉	—	8	1	no	—
118	C₂×Dic₈₉	—	8	1	no	—
119	C₂×Dic₈₉	—	8	1	no	—
120	C₂×Dic₈₉	—	8	1	no	—
121	C₂×Dic₈₉	—	8	1	no	—
122	D₃₅₆	—	4	4	no	—
123	D₃₅₆	—	4	4	no	—
124	D₃₅₆	—	4	4	no	—
125	D₃₅₆	—	8	2	no	—
126	D₃₅₆	—	8	2	no	—
127	D₃₅₆	—	8	2	no	—
128	C₈×C₈₉	—	2	4	no	—
129	C₈×C₈₉	—	2	4	no	—
130	C₈×C₈₉	—	4	2	no	—
131	C₂²×C₈₉	#399	1	1	yes	characteristic
132	C₂²×C₈₉	#492	1	1	yes	characteristic
133	C₂²×C₈₉	#493	1	1	yes	characteristic
134	C₂²×C₈₉	#495	1	1	yes	characteristic
135	C₂²×C₈₉	#496	1	1	yes	characteristic
136	C₂²×C₈₉	#496	1	1	yes	characteristic
137	C₂²×C₈₉	#497	1	1	yes	characteristic
138	C₂×D₈₉	—	8	2	no	—
139	C₂×D₈₉	—	8	2	no	—
140	C₂×D₈₉	—	8	2	no	—
141	C₂×D₈₉	—	16	1	no	—
142	C₂×D₈₉	—	16	1	no	—
143	C₂×D₈₉	—	16	1	no	—
144	C₂×D₈₉	—	16	1	no	—
145	C₄×C₈₉	#474	1	2	yes	—
146	C₄×C₈₉	#476	1	2	yes	—
147	C₄×C₈₉	#478	1	2	yes	—
148	C₄×C₈₉	#480	1	2	yes	—
149	C₄×C₈₉	#481	1	2	yes	—
150	C₄×C₈₉	#482	1	2	yes	—
151	Dic₈₉	—	16	1	no	—
152	Dic₈₉	—	16	1	no	—
153	Dic₈₉	—	16	1	no	—
154	#10672	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	#1706	1	1	yes	characteristic
156	C₂×C₈₉	#1714	1	1	yes	characteristic
157	C₂×C₈₉	#1720	1	1	yes	characteristic
158	C₂×C₈₉	#1721	1	1	yes	characteristic
159	C₂×C₈₉	#2092	1	1	yes	characteristic
160	C₂×C₈₉	#2093	1	1	yes	characteristic
161	C₂×C₈₉	#2094	1	1	yes	characteristic
162	D₈₉	—	16	2	no	—
163	#1957	—	89	1	no	—
164	#1957	—	89	1	no	—
165	#1957	—	89	1	no	—
166	#1845	—	89	1	no	—
167	#1844	—	89	1	no	—
168	#1834	—	89	1	no	—
169	#1273	D₈₉	1	1	yes	characteristic, complemented
170	C₈₉	#10672	1	1	yes	characteristic, complemented, Sylow
171	C₂×#127	—	89	1	no	—
172	C₂×#107	—	178	2	no	—
173	C₂×#106	—	178	2	no	—
174	C₂×#102	—	178	2	no	—
175	C₂×#102	—	178	2	no	—
176	C₂×#102	—	178	2	no	—
177	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
180	#384	—	89	1	no	—
181	#384	—	89	1	no	—
182	#384	—	89	1	no	—
183	#340	—	178	2	no	—
184	C₂²×D₄	—	178	1	no	—
185	C₂²×D₄	—	178	1	no	—
186	C₂²×D₄	—	178	1	no	—
187	C₂²×C₈	D₃₅₆	1	2	yes	—
188	C₂²×C₈	D₃₅₆	1	2	yes	—
189	C₂²×C₈	D₃₅₆	1	2	yes	—
190	#127	—	89	2	no	—
191	#127	—	178	1	no	—
192	#127	—	178	1	no	—
193	#127	—	178	1	no	—
194	C₂×C₄⋊C₄	—	178	1	no	—
195	C₂×C₄⋊C₄	—	178	1	no	—
196	C₂×C₄⋊C₄	—	178	1	no	—
197	C₂×C₄⋊C₄	—	178	1	no	—
198	C₂×C₄⋊C₄	—	178	1	no	—
199	C₂×C₄⋊C₄	—	178	1	no	—
200	C₂×C₄⋊C₄	—	178	1	no	—
201	C₂×C₄⋊C₄	—	178	1	no	—
202	C₂×C₄⋊C₄	—	178	1	no	—
203	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
204	#107	—	356	4	no	—
205	#106	—	356	4	no	—
206	#102	—	356	4	no	—
207	#102	—	356	4	no	—
208	#102	—	356	4	no	—
209	C₄×C₈	—	2	1	no	—
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	—	2	1	no	—
215	C₂⁴	—	356	1	no	—
216	C₂×D₄	—	356	1	no	—
217	C₂×D₄	—	356	1	no	—
218	C₂×D₄	—	356	1	no	—
219	C₂×D₄	—	356	2	no	—
220	C₂×D₄	—	356	2	no	—
221	C₂×D₄	—	356	2	no	—
222	C₂×D₄	—	356	2	no	—
223	C₂×D₄	—	356	2	no	—
224	C₂×D₄	—	356	2	no	—
225	C₂×D₄	—	178	2	no	—
226	C₂×D₄	—	178	2	no	—
227	C₂×D₄	—	178	2	no	—
228	C₂²×C₄	—	356	1	no	—
229	C₂²×C₄	—	356	1	no	—
230	C₂²×C₄	—	356	1	no	—
231	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	D₇₁₂	1	4	yes	—
242	C₂×C₈	#13083563	1	4	yes	—
243	C₄⋊C₄	—	356	2	no	—
244	C₄⋊C₄	—	356	2	no	—
245	C₄⋊C₄	—	356	2	no	—
246	C₄⋊C₄	—	356	2	no	—
247	C₄⋊C₄	—	356	2	no	—
248	C₄⋊C₄	—	356	2	no	—
249	C₄⋊C₄	—	356	2	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄²	Q₈×D₈₉	1	1	yes	characteristic
253	C₄²	#13083573	1	1	yes	characteristic
254	C₄²	#13083573	1	1	yes	characteristic
255	C₄²	#13083573	1	1	yes	characteristic
256	C₂³	—	356	2	no	—
257	C₂³	—	712	1	no	—
258	C₂³	—	712	1	no	—
259	C₂³	—	712	1	no	—
260	C₂³	—	712	1	no	—
261	C₂³	—	712	1	no	—
262	C₂³	—	712	1	no	—
263	C₂³	#15668009	1	1	yes	characteristic
264	D₄	—	356	4	no	—
265	D₄	—	356	4	no	—
266	D₄	—	356	4	no	—
267	D₄	—	712	2	no	—
268	D₄	—	712	2	no	—
269	D₄	—	712	2	no	—
270	C₂×C₄	—	712	1	no	—
271	C₂×C₄	—	712	1	no	—
272	C₂×C₄	—	712	1	no	—
273	C₂×C₄	—	712	1	no	—
274	C₂×C₄	—	712	1	no	—
275	C₂×C₄	—	712	1	no	—
276	C₂×C₄	—	712	1	no	—
277	C₂×C₄	—	712	1	no	—
278	C₂×C₄	—	712	1	no	—
279	C₂×C₄	#15667992	1	1	yes	characteristic
280	C₂×C₄	#15667992	1	1	yes	characteristic
281	C₂×C₄	#15667992	1	1	yes	characteristic
282	C₂×C₄	#15667996	1	1	yes	characteristic
283	C₂×C₄	#15667996	1	1	yes	characteristic
284	C₂×C₄	#15667996	1	1	yes	characteristic
285	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
286	C₂×C₄	C₂×#13083563	1	2	yes	—
287	C₂×C₄	#15668034	1	2	yes	—
288	C₂×C₄	#15668040	1	2	yes	—
289	C₂×C₄	#15668040	1	2	yes	—
290	C₂×C₄	#15668045	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₈	—	4	2	no	—
294	C₂²	—	1424	1	no	—
295	C₂²	—	1424	1	no	—
296	C₂²	—	1424	1	no	—
297	C₂²	—	1424	1	no	—
298	C₂²	—	712	2	no	—
299	C₂²	—	712	2	no	—
300	C₂²	—	712	2	no	—
301	C₂²	#19151535	1	1	yes	characteristic
302	C₂²	#19151971	1	1	yes	characteristic
303	C₂²	#19151975	1	1	yes	characteristic
304	C₂²	#19151982	1	1	yes	characteristic
305	C₂²	#19151987	1	1	yes	characteristic
306	C₂²	#19151987	1	1	yes	characteristic
307	C₂²	#19151992	1	1	yes	characteristic
308	C₄	—	1424	1	no	—
309	C₄	—	1424	1	no	—
310	C₄	—	1424	1	no	—
311	C₄	#19151879	1	2	yes	—
312	C₄	#19151889	1	2	yes	—
313	C₄	#19151899	1	2	yes	—
314	C₄	#19151912	1	2	yes	—
315	C₄	#19151917	1	2	yes	—
316	C₄	#19151923	1	2	yes	—
317	C₂	—	1424	2	no	—
318	C₂	#26019844	1	1	yes	characteristic
319	C₂	#26019874	1	1	yes	characteristic
320	C₂	#26019912	1	1	yes	characteristic
321	C₂	#26019918	1	1	yes	characteristic
322	C₂	#26021772	1	1	yes	characteristic
323	C₂	#26021779	1	1	yes	characteristic
324	C₂	#26021784	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668009
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1273	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#15668009	C₂²
hypercenter	#1273	D₈₉
hypocenter	C₈₉	#10672
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
4. #26020412	C₂	semidirect
5. #26020409	C₂	semidirect
6. #26020380	C₂	semidirect
7. C₈₉×#1273	C₂	semidirect
169. #1273	D₈₉	semidirect
170. C₈₉	#10672	semidirect

Maximal subgroups

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

subgroup	index
#26021058	2
#26020412	2
#26020409	2
#26020380	2
C₈₉×#1273	2
#10672	89