Finite Group Databasebeta

← #36470448

Subgroups — #36470448

Order 22784.

420 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26021893C212yesmaximal
2#26020302C211yesmaximal, characteristic, complemented
3#26020298C211yesmaximal, characteristic, complemented
4#26018381C212yesmaximal, complemented
5C89×#2930C211yesmaximal, characteristic
6C2×#15667948—21no—
7#19151636—21no—
8#19151582C2212yes—
9#19151572C2212yes—
10#19151388—22no—
11#19151385—22no—
12#19151356—21no—
13#19151355—21no—
14C2×#131×C89C2211yescharacteristic
15C22×M16×C89C411yescharacteristic, complemented
16#525×C89C2211yescharacteristic
17#436×C89—21no—
18#436×C89—21no—
19#435×C89—21no—
20#435×C89—21no—
21#424×C89C411yescharacteristic, complemented
22#408×C89C2211yescharacteristic
23#407×C89C411yescharacteristic, complemented
24#405×C89C411yescharacteristic, complemented
25C2×#13083558—22no—
26#15667971—22no—
27#15667948—41no—
28#15667948—41no—
29#15667946—41no—
30#15667944—41no—
31#15667931—22no—
32#15667929—22no—
33#15667927—22no—
34#15667927—22no—
35C2×#41×C89C2311yescharacteristic
36C23×C4×C89C2×C411yescharacteristic
37#131×C89—41no—
38#131×C89—41no—
39C2×M16×C89—21no—
40C2×M16×C89—21no—
41C2×M16×C89—41no—
42C2×M16×C89—41no—
43C2×M16×C89C2×C411yescharacteristic
44C2×M16×C89C2×C411yescharacteristic
45C2×M16×C89D411yescharacteristic
46C2×M16×C89D411yescharacteristic
47C22×C8×C89C2×C411yescharacteristic
48C22×C8×C89C2×C411yescharacteristic
49C22×C8×C89D411yescharacteristic
50C22×C8×C89D411yescharacteristic
51#123×C89D411yescharacteristic
52#123×C89Q811yescharacteristic
53C22⋊Q8×C89—21no—
54#121×C89—21no—
55#120×C89D411yescharacteristic
56#120×C89Q811yescharacteristic
57C4×D4×C89—21no—
58C4×D4×C89—21no—
59#117×C89C2×C411yescharacteristic
60C4⋊C8×C89—21no—
61C4⋊C8×C89—21no—
62C22⋊C8×C89—21no—
63C22⋊C8×C89—21no—
64C22⋊C8×C89—21no—
65C22⋊C8×C89—21no—
66#97×C89—21no—
67C4×C8×C89—21no—
68#13083558—42no—
69#13083558—42no—
70C2×#10623494—42no—
71C4⋊Dic89—42no—
72C22⋊Dic89—42no—
73C4×Dic89—42no—
74C24×C89C4⋊C411yescharacteristic
75#41×C89—41no—
76#41×C89—41no—
77C2×Q8×C89#4111yescharacteristic
78C2×D4×C89—21no—
79C2×D4×C89—21no—
80C2×D4×C89C2×D411yescharacteristic
81C2×D4×C89C2×Q811yescharacteristic
82C2×D4×C89#4111yescharacteristic
83C22×C4×C89—21no—
84C22×C4×C89—21no—
85C22×C4×C89—41no—
86C22×C4×C89—41no—
87C22×C4×C89C4⋊C411yescharacteristic
88C22×C4×C89C22×C411yescharacteristic
89C22×C4×C89C2×D411yescharacteristic
90C22×C4×C89C2×D411yescharacteristic
91M16×C89—41no—
92M16×C89—41no—
93M16×C89—41no—
94M16×C89—42no—
95M16×C89—41no—
96M16×C89—41no—
97M16×C89—41no—
98C2×C8×C89—21no—
99C2×C8×C89—21no—
100C2×C8×C89—21no—
101C2×C8×C89—21no—
102C2×C8×C89—21no—
103C2×C8×C89—21no—
104C2×C8×C89—21no—
105C2×C8×C89—21no—
106C2×C8×C89—41no—
107C2×C8×C89C22⋊C411yescharacteristic
108C2×C8×C89C22⋊C411yescharacteristic
109C2×C8×C89C22⋊C411yescharacteristic
110C2×C8×C89C22⋊C411yescharacteristic
111C4⋊C4×C89—21no—
112C4⋊C4×C89—21no—
113C4⋊C4×C89—21no—
114C22⋊C4×C89—21no—
115C22⋊C4×C89—21no—
116C22⋊C4×C89—21no—
117C22⋊C4×C89—21no—
118C22⋊C4×C89C4⋊C411yescharacteristic
119C22⋊C4×C89C4⋊C411yescharacteristic
120C42×C89—21no—
121C23×C89—21no—
122C23×C89—41no—
123C23×C89—81no—
124C23×C89C2×C4⋊C411yescharacteristic
125C23×C89#12111yescharacteristic
126C23×C89C22⋊Q811yescharacteristic
127Q8×C89—41no—
128D4×C89—41no—
129D4×C89—41no—
130D4×C89—41no—
131D4×C89—81no—
132D4×C89—81no—
133C2×C4×C89—21no—
134C2×C4×C89—21no—
135C2×C4×C89—21no—
136C2×C4×C89—21no—
137C2×C4×C89—21no—
138C2×C4×C89—21no—
139C2×C4×C89—21no—
140C2×C4×C89—21no—
141C2×C4×C89—21no—
142C2×C4×C89—42no—
143C2×C4×C89—41no—
144C2×C4×C89—41no—
145C2×C4×C89—41no—
146C2×C4×C89C2×C22⋊C411yescharacteristic
147C2×C4×C89#11711yescharacteristic
148C2×C4×C89#12111yescharacteristic
149C2×C4×C89C22⋊Q811yescharacteristic
150C2×Dic89—82no—
151C8×C89—21no—
152C8×C89—21no—
153C8×C89—21no—
154C8×C89—41no—
155C8×C89—41no—
156C8×C89#9911yescharacteristic
157C8×C89#9911yescharacteristic
158#10623494—82no—
159C22×C89—21no—
160C22×C89—21no—
161C22×C89—21no—
162C22×C89—41no—
163C22×C89—41no—
164C22×C89—41no—
165C22×C89—41no—
166C22×C89—41no—
167C22×C89—41no—
168C22×C89—82no—
169C22×C89#38011yescharacteristic
170C4×C89—21no—
171C4×C89—21no—
172C4×C89—21no—
173C4×C89—41no—
174C4×C89—41no—
175C4×C89—41no—
176C4×C89C2×#9911yescharacteristic
177C4×C89#41011yescharacteristic
178Dic89—162no—
179#12806—891nomaximal, Sylow
180C2×C89—21no—
181C2×C89—41no—
182C2×C89—41no—
183C2×C89—41no—
184C2×C89—81no—
185C2×C89#179911yescharacteristic
186#2930D8911yescharacteristic
187#2120—892no—
188#1813—891no—
189#1812—891no—
190#1395—892no—
191C89#1280611yescharacteristic, complemented, Sylow
192C2×#131C2×D8911yescharacteristic
193C22×M16Dic8911yescharacteristic, complemented
194#525C2×D8911yescharacteristic
195#436—21no—
196#436—21no—
197#435—21no—
198#435—21no—
199C2×#108—1781no—
200#427—1781no—
201#424Dic8911yescharacteristic, complemented
202#413—892no—
203#410—892no—
204#408C2×D8911yescharacteristic
205#407Dic8911yescharacteristic, complemented
206#405Dic8911yescharacteristic, complemented
207#352—1782no—
208AΣL(1,𝔽16)—1782no—
209#338—1781no—
210H(3,ℤ/4)—1781no—
211C2×#41C22×D8911yescharacteristic
212C23×C4C2×Dic8911yescharacteristic
213#131—41no—
214#131—41no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—41no—
218C2×M16—41no—
219C2×M16—1782no—
220C2×M16C2×Dic8911yescharacteristic
221C2×M16C2×Dic8911yescharacteristic
222C2×M16C22⋊D8911yescharacteristic
223C2×M16C22⋊D8911yescharacteristic
224C22×C8C2×Dic8911yescharacteristic
225C22×C8C2×Dic8911yescharacteristic
226C22×C8C22⋊D8911yescharacteristic
227C22×C8C22⋊D8911yescharacteristic
228#123Dic17811yescharacteristic
229#123D35611yescharacteristic
230C22⋊Q8—21no—
231#121—21no—
232#120Dic17811yescharacteristic
233#120D35611yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236#117—1782no—
237#117C2×Dic8911yescharacteristic
238#108—3561no—
239#108—3561no—
240#107—3561no—
241#106—3561no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244#101—1782no—
245#100—1782no—
246#99—1782no—
247#99—1782no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252#97—21no—
253C4×C8—21no—
254C24C4⋊Dic8911yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8#1308357411yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4C2×D35611yescharacteristic
261C2×D4C2×Dic17811yescharacteristic
262C2×D4#1308357411yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C4⋊Dic8911yescharacteristic
268C22×C4C22×Dic8911yescharacteristic
269C22×C4C2×C22⋊D8911yescharacteristic
270C22×C4C2×C22⋊D8911yescharacteristic
271M16—41no—
272M16—41no—
273M16—41no—
274M16—42no—
275M16—41no—
276M16—41no—
277M16—41no—
278M16—3562no—
279M16—3562no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—3562no—
290C2×C8C22⋊Dic8911yescharacteristic
291C2×C8C22⋊Dic8911yescharacteristic
292C2×C8C22⋊Dic8911yescharacteristic
293C2×C8C22⋊Dic8911yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—3562no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3562no—
303C22⋊C4C4⋊Dic8911yescharacteristic
304C22⋊C4C4⋊Dic8911yescharacteristic
305C42—21no—
306C42—3562no—
307C23—21no—
308C23—41no—
309C23—81no—
310C23C2×C4⋊Dic8911yescharacteristic
311C23#1566798611yescharacteristic
312C23#1566799111yescharacteristic
313Q8—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—7122no—
333C2×C4C2×C22⋊Dic8911yescharacteristic
334C2×C4#1566797111yescharacteristic
335C2×C4#1566798611yescharacteristic
336C2×C4#1566799111yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—41no—
341C8—41no—
342C8—7122no—
343C8#1566792711yescharacteristic
344C8#1566792711yescharacteristic
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—82no—
355C22#1915145811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—14242no—
363C4C2×#1566792711yescharacteristic
364C4#1915157211yescharacteristic
365C2—21no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2—81no—
370C2#2602024011yescharacteristic
subgroupG/H
centerC4#19151572
commutatorC22×C4×C89C22×C4
FrattiniC2×#41C22×D89
FittingC89×#2930C2
socleC2×C89#1799
prime radical#525×C89C22
cosocleC2×#41×C89C23
hypercenter#2930D89
hypocenterC89#12806
LCSG ▷ C22×C4×C89 ▷ C22×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #2930

Complements

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

NH ≅ G/Ntype
2. #26020302C2semidirect
3. #26020298C2semidirect
4. #26018381C2semidirect
15. C22×M16×C89C4semidirect
21. #424×C89C4semidirect
23. #407×C89C4semidirect
24. #405×C89C4semidirect
191. C89#12806semidirect
193. C22×M16Dic89semidirect
201. #424Dic89semidirect
205. #407Dic89semidirect
206. #405Dic89semidirect

Maximal subgroups

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

subgroupindex
#260218932
#260203022
#260202982
#260183812
C89×#29302
#1280689