Finite Group Databasebeta

← #36445488

Subgroups — #36445488

Order 22784.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029227C211yesmaximal, characteristic
2#26018838C211yesmaximal, characteristic
3#26018736C211yesmaximal, characteristic
4C8×#13083558C211yesmaximal, characteristic, complemented
5#26018151C211yesmaximal, characteristic, complemented
6#26018145C211yesmaximal, characteristic, complemented
7C89×#1746C211yesmaximal, characteristic, complemented
8#19152132C2211yescharacteristic
9#19151728C411yescharacteristic, complemented
10C8×Dic178C411yescharacteristic, complemented
11#19151677C411yescharacteristic, complemented
12#19151669—21no—
13C8×D356C411yescharacteristic, complemented
14C8×C22⋊D89—21no—
15#19151657—21no—
16#19151655C2211yescharacteristic
17#19151651C2211yescharacteristic
18#19151552—21no—
19C4×#13083558C2211yescharacteristic
20#19151339C412yescomplemented
21#19151337C412yescomplemented
22#19151334C412yes—
23#19151331C412yes—
24#19151320—21no—
25C8×#10623494—21no—
26C2×#97×C89C2211yescharacteristic
27C2×C4×C8×C89C2211yescharacteristic
28#376×C89C2211yescharacteristic
29#336×C89—21no—
30#336×C89—21no—
31C2×#13083558—21no—
32C2×#13083557—21no—
33C2×C8×D89—21no—
34#15668006—21no—
35#15668004C2×C411yescharacteristic
36#15668000C2×C411yescharacteristic
37#15667995—21no—
38#15667991C2×C411yescharacteristic
39#15667986C2×C411yescharacteristic
40C4×Dic178C2×C411yescharacteristic
41C4×D356C2×C411yescharacteristic
42C4×C22⋊D89—21no—
43#15667969—21no—
44C4⋊#10623494—21no—
45C4⋊#10623494C2×C411yescharacteristic
46C4⋊#10623494C2×C411yescharacteristic
47#15667941—21no—
48#15667941—21no—
49C22⋊#10623494—21no—
50C22⋊#10623494C2×C412yes—
51#15667924—21no—
52#15667924—21no—
53#15667923—21no—
54#15667923C2×C412yes—
55#15667922—21no—
56C4×#10623494—21no—
57C4×#10623494—21no—
58C8×Dic89—21no—
59C22×C8×C89—21no—
60C22×C8×C89D411yescharacteristic
61C22×C8×C89D411yescharacteristic
62C2×C42×C89—21no—
63C2×C42×C89C2311yescharacteristic
64#97×C89—21no—
65#97×C89C2×C412yes—
66C4×C8×C89C2×C411yescharacteristic
67C4×C8×C89C2×C411yescharacteristic
68C4×C8×C89D411yescharacteristic
69C4×C8×C89D411yescharacteristic
70#95×C89—21no—
71#95×C89C2×C412yes—
72C2×Dic178C4211yescharacteristic
73C2×D356C4211yescharacteristic
74C2×C22⋊D89—21no—
75C2×C4×D89—21no—
76#13083558—24no—
77#13083557—42no—
78C2×#10623494—22no—
79C2×#10623494—21no—
80C2×#10623494—21no—
81C2×#10623494—22no—
82C8×D89—42no—
83C4⋊Dic89C4211yescharacteristic
84C4⋊Dic89C4211yescharacteristic
85#13083553—21no—
86#13083553—21no—
87#13083553—21no—
88#13083553—21no—
89C22⋊Dic89—21no—
90#13083551—21no—
91#13083551—21no—
92#13083551—21no—
93#13083551—21no—
94C4×Dic89—21no—
95C22×C4×C89—22no—
96C22×C4×C89C22×C411yescharacteristic
97C22×C4×C89C2×D411yescharacteristic
98C22×C4×C89#4111yescharacteristic
99C2×C8×C89—22no—
100C2×C8×C89—41no—
101C2×C8×C89—41no—
102C2×C8×C89—41no—
103C2×C8×C89—41no—
104C2×C8×C89C22⋊C411yescharacteristic
105C2×C8×C89C22⋊C411yescharacteristic
106C2×C8×C89C22⋊C411yescharacteristic
107C2×C8×C89C22⋊C411yescharacteristic
108C42×C89—41no—
109C42×C89—41no—
110C42×C89C22×C411yescharacteristic
111C42×C89C22×C411yescharacteristic
112C42×C89C2×D411yescharacteristic
113C42×C89#4111yescharacteristic
114C23×C89C4×D411yescharacteristic
115C22×D89—21no—
116C2×C4×C89—22no—
117C2×C4×C89—21no—
118C2×C4×C89—21no—
119C2×C4×C89—41no—
120C2×C4×C89—42no—
121C2×C4×C89—42no—
122C2×C4×C89—42no—
123C2×C4×C89C2×C4211yescharacteristic
124C2×C4×C89C2×C22⋊C411yescharacteristic
125C2×C4×C89#11711yescharacteristic
126C2×C4×C89C4×D411yescharacteristic
127C2×C4×C89C4×D411yescharacteristic
128C2×C4×C89C4×D411yescharacteristic
129C22⋊D89—42no—
130C2×Dic89—21no—
131C2×Dic89—21no—
132C2×Dic89—21no—
133D356—41no—
134C4×D89—41no—
135C4×D89—41no—
136Dic178—41no—
137C8×C89—21no—
138C8×C89—21no—
139C8×C89—21no—
140C8×C89—21no—
141C8×C89—42no—
142#10623494—24no—
143#10623494—42no—
144#10623494—41no—
145#10623494—41no—
146C22×C89—21no—
147C22×C89—21no—
148C22×C89C4×C22⋊C411yescharacteristic
149C22×C89#44312yes—
150C2×D89—41no—
151C2×D89—41no—
152C2×D89—41no—
153C4×C89—21no—
154C4×C89—21no—
155C4×C89—21no—
156C4×C89—21no—
157C4×C89—21no—
158C4×C89—82no—
159C4×C89#42111yescharacteristic
160C4×C89#42111yescharacteristic
161Dic89—41no—
162Dic89—41no—
163Dic89—41no—
164#8250—891nomaximal, Sylow
165C2×C89—22no—
166C2×C89#175711yescharacteristic
167C2×C89#175911yescharacteristic
168C2×C89#176411yescharacteristic
169D89—81no—
170#2966—891no—
171#1746D8911yescharacteristic, complemented
172#1524—891no—
173#1500—891no—
174C8×M16—891no—
175#1302—891no—
176#1300—891no—
177C89#825011yescharacteristic, complemented, Sylow
178#529—891no—
179#446—891no—
180C8×Q8—891no—
181#436—891no—
182#435—1781no—
183C8×D4—891no—
184C8×D4—1781no—
185#433—1781no—
186#432—891no—
187#431—891no—
188#405—1781no—
189C4×M16—891no—
190C2×#97C2×D8911yescharacteristic
191C2×C4×C8C2×D8911yescharacteristic
192#376C2×D8911yescharacteristic
193#336—21no—
194#336—21no—
195#330—892no—
196#329—892no—
197#328—892no—
198#327—892no—
199#322—1781no—
200C82—1781no—
201C2×M16—1781no—
202C2×M16—1781no—
203C22×C8—21no—
204C22×C8—1781no—
205C22×C8C22⋊D8911yescharacteristic
206C22×C8C22⋊D8911yescharacteristic
207#126—1781no—
208#125—891no—
209#124—891no—
210#123—1781no—
211C22⋊Q8—891no—
212#121—891no—
213C4×Q8—891no—
214C4×D4—891no—
215C4×D4—1781no—
216#117—1781no—
217C2×C42—21no—
218C2×C42C22×D8911yescharacteristic
219C4⋊C8—891no—
220C4⋊C8—891no—
221C4⋊C8—1781no—
222C4⋊C8—1781no—
223C4⋊C8—1781no—
224C22⋊C8—892no—
225C22⋊C8—1781no—
226C22⋊C8—1781no—
227C22⋊C8—1781no—
228#97—21no—
229#97—892no—
230#97—1781no—
231#97—1781no—
232#97C4×D8912yes—
233C4×C8—1781no—
234C4×C8—1781no—
235C4×C8—1781no—
236C4×C8C4×D8911yescharacteristic
237C4×C8C4×D8911yescharacteristic
238C4×C8D35611yescharacteristic
239C4×C8D35611yescharacteristic
240#95—21no—
241#95C4×D8912yes—
242C2×Q8—891no—
243C2×D4—891no—
244C2×D4—1781no—
245C22×C4—22no—
246C22×C4—1781no—
247C22×C4C2×C4×D8911yescharacteristic
248C22×C4C2×C22⋊D8911yescharacteristic
249C22×C4#1308357411yescharacteristic
250M16—3562no—
251M16—1784no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—3562no—
258C2×C8—1782no—
259C2×C8—1781no—
260C2×C8—1781no—
261C2×C8—1782no—
262C2×C8#1308355111yescharacteristic
263C2×C8#1308355111yescharacteristic
264C2×C8#1308355111yescharacteristic
265C2×C8#1308355111yescharacteristic
266C4⋊C4—891no—
267C4⋊C4—891no—
268C4⋊C4—1781no—
269C4⋊C4—1781no—
270C4⋊C4—1781no—
271C4⋊C4—1781no—
272C22⋊C4—1781no—
273C22⋊C4—1781no—
274C22⋊C4—1781no—
275C22⋊C4—1781no—
276C22⋊C4—1781no—
277C42—41no—
278C42—41no—
279C42—1781no—
280C42C2×C4×D8911yescharacteristic
281C42C2×C4×D8911yescharacteristic
282C42C2×D35611yescharacteristic
283C42#1308357411yescharacteristic
284C23—1781no—
285C23C4×C22⋊D8911yescharacteristic
286Q8—3561no—
287D4—3561no—
288D4—3562no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—3561no—
297C2×C4—3561no—
298C2×C4—1781no—
299C2×C4—1781no—
300C2×C4—1781no—
301C2×C4C42×D8911yescharacteristic
302C2×C4C2×#1308355111yescharacteristic
303C2×C4#1566796911yescharacteristic
304C2×C4C4×C22⋊D8911yescharacteristic
305C2×C4C4×D35611yescharacteristic
306C2×C4C4×D35611yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C8—42no—
312C8—3562no—
313C8—3561no—
314C8—3561no—
315C8—1784no—
316C22—21no—
317C22—21no—
318C22—3561no—
319C22—3561no—
320C22—3561no—
321C22C4×#1308355111yescharacteristic
322C22#1915171312yes—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—82no—
329C4—3561no—
330C4—3561no—
331C4—3561no—
332C4#1915161611yescharacteristic
333C4#1915161611yescharacteristic
334C2—22no—
335C2—7121no—
336C2#2602005311yescharacteristic
337C2#2602006311yescharacteristic
338C2#2602009111yescharacteristic
subgroupG/H
centerC2×C4C2×#13083551
commutatorC2×C4×C89C2×C42
FrattiniC2×C42C22×D89
FittingC89×#1746C2
socleC22×C89C4×C22⋊C4
prime radical#15667986C2×C4
cosocleC2×C42×C89C23
hypercenter#1746D89
hypocenterC89#8250
LCSG ▷ C2×C4×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #1746

Complements

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

NH ≅ G/Ntype
4. C8×#13083558C2semidirect
5. #26018151C2semidirect
6. #26018145C2semidirect
7. C89×#1746C2semidirect
9. #19151728C4semidirect
10. C8×Dic178C4semidirect
11. #19151677C4semidirect
13. C8×D356C4semidirect
20. #19151339C4semidirect
21. #19151337C4semidirect
171. #1746D89semidirect
177. C89#8250semidirect

Maximal subgroups

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

subgroupindex
#260292272
#260188382
#260187362
C8×#130835582
#260181512
#260181452
C89×#17462
#825089