Finite Group Databasebeta

← C2×#26020466 #36555236

Subgroups — C2×#26020466 #36555236

Order 22784.

1402 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152076C211yesmaximal, characteristic
2C2×#19151638C212yesmaximal
3C22×#15667931C211yesmaximal, characteristic
4#26020466C218yesmaximal, direct product factor
5C2×#19151358C212yesmaximal
6C2×#391×C89C211yesmaximal, characteristic
7C22×#13083558C2212yes—
8#19152076—22no—
9#19152076—24no—
10#19152076C2214yes—
11C2×Q8×Dic89—22no—
12C2×Q8×Dic89—22no—
13C2×#15667971—22no—
14C2×#15667971C2212yes—
15#19151638—24no—
16#19151638C2218yes—
17C2×#15667945—22no—
18C2×#15667943—22no—
19C2×#15667931C418yescomplemented
20C2×#15667931C2214yes—
21#19151358C2218yes—
22C23×Q8×C89C2211yescharacteristic
23C22×C4⋊C4×C89C2211yescharacteristic
24C22×C42×C89C2211yescharacteristic
25#391×C89—22no—
26#391×C89C2214yes—
27C2×#95×C89—22no—
28C2×#13083558—22no—
29C2×#13083558—24no—
30C2×#13083558C2×C418yes—
31C2×#13083558C2314yes—
32C22×#10623494—22no—
33Q8×Dic89—28no—
34Q8×Dic89—28no—
35Q8×Dic89—28no—
36Q8×Dic89—28no—
37#15667971—28no—
38#15667971—28no—
39#15667971—22no—
40#15667971—22no—
41#15667971D414yes—
42#15667971C2314yes—
43C2×C4⋊Dic89—22no—
44C2×C4⋊Dic89—22no—
45C2×C4⋊Dic89D412yes—
46C2×C4⋊Dic89D412yes—
47C2×C22⋊Dic89—22no—
48C2×C4×Dic89—24no—
49C2×C4×Dic89—22no—
50#15667945—28no—
51#15667943—28no—
52#15667931C2×C4116yescomplemented
53C22×Q8×C89—22no—
54C22×Q8×C89—22no—
55C22×Q8×C89—24no—
56C22×Q8×C89C2×C414yes—
57C22×Q8×C89D412yes—
58C22×Q8×C89Q814yes—
59C22×Q8×C89C2312yes—
60C23×C4×C89—21no—
61C23×C4×C89—21no—
62C23×C4×C89D412yes—
63C23×C4×C89C2311yescharacteristic
64C2×C4⋊C4×C89—22no—
65C2×C4⋊C4×C89—22no—
66C2×C4⋊C4×C89—21no—
67C2×C4⋊C4×C89C2312yes—
68C2×C42×C89—24no—
69C2×C42×C89—21no—
70C2×C42×C89C2312yes—
71#95×C89—28no—
72C22×Dic89—22no—
73C22×Dic89—22no—
74#13083558—28no—
75#13083558—24no—
76#13083558C22×C418yes—
77C2×#10623494—28no—
78C2×#10623494—24no—
79C4⋊Dic89—28no—
80C4⋊Dic89—28no—
81C4⋊Dic89—22no—
82C4⋊Dic89—22no—
83C4⋊Dic89C2×D414yes—
84C4⋊Dic89C2×D414yes—
85C22⋊Dic89—24no—
86C22⋊Dic89—24no—
87C4×Dic89—216no—
88C4×Dic89—24no—
89C4×Dic89—24no—
90C24×C89C2×D411yescharacteristic
91C2×Q8×C89—28no—
92C2×Q8×C89—28no—
93C2×Q8×C89—24no—
94C2×Q8×C89—28no—
95C2×Q8×C89—28no—
96C2×Q8×C89—24no—
97C2×Q8×C89—24no—
98C2×Q8×C89—24no—
99C2×Q8×C89C4⋊C418yes—
100C2×Q8×C89C22×C414yes—
101C2×Q8×C89C2×Q814yes—
102C22×C4×C89—22no—
103C22×C4×C89—22no—
104C22×C4×C89—24no—
105C22×C4×C89—24no—
106C22×C4×C89—22no—
107C22×C4×C89—22no—
108C22×C4×C89—42no—
109C22×C4×C89—42no—
110C22×C4×C89—44no—
111C22×C4×C89—41no—
112C22×C4×C89—41no—
113C22×C4×C89C4⋊C418yes—
114C22×C4×C89C22×C412yes—
115C22×C4×C89C2×D414yes—
116C22×C4×C89C2×D411yescharacteristic
117C22×C4×C89C2×D412yes—
118C22×C4×C89C2×D412yes—
119C22×C4×C89C2×Q812yes—
120C22×C4×C89#4114yes—
121C22×C4×C89C2411yescharacteristic
122C4⋊C4×C89—24no—
123C4⋊C4×C89—42no—
124C42×C89—24no—
125C42×C89—42no—
126C23×C89—22no—
127C23×C89—22no—
128C23×C89C2×C4⋊C412yes—
129C23×C89#12011yescharacteristic
130C23×C89#12011yescharacteristic
131C23×C89#12312yes—
132C23×C89C22×D411yescharacteristic
133Q8×C89—28no—
134Q8×C89—28no—
135Q8×C89—216no—
136C2×C4×C89—24no—
137C2×C4×C89—24no—
138C2×C4×C89—28no—
139C2×C4×C89—28no—
140C2×C4×C89—24no—
141C2×C4×C89—48no—
142C2×C4×C89—42no—
143C2×C4×C89—42no—
144C2×C4×C89—44no—
145C2×C4×C89—42no—
146C2×C4×C89—82no—
147C2×C4×C89—82no—
148C2×C4×C89C2×C4⋊C418yes—
149C2×C4×C89C2×C4⋊C412yes—
150C2×C4×C89C4×D414yes—
151C2×C4×C89C4×D414yes—
152C2×C4×C89#12012yes—
153C2×C4×C89C22⋊Q814yes—
154C2×C4×C89C22⋊Q814yes—
155C2×C4×C89#12312yes—
156C2×C4×C89C23×C411yescharacteristic
157C2×C4×C89C22×D412yes—
158C2×C4×C89C22×Q811yescharacteristic
159C2×C4×C89C2×#4112yes—
160C2×Dic89—24no—
161C2×Dic89—24no—
162C2×Dic89—44no—
163C2×Dic89—44no—
164#10623494—28no—
165C22×C89—24no—
166C22×C89—24no—
167C22×C89—21no—
168C22×C89—21no—
169C22×C89—21no—
170C22×C89—21no—
171C22×C89#38512yes—
172C22×C89#38512yes—
173C22×C89#45412yes—
174C22×C89#45612yes—
175C22×C89C22×C4⋊C411yescharacteristic
176C22×C89C2×#12011yescharacteristic
177C22×C89C2×#12311yescharacteristic
178C4×C89—28no—
179C4×C89—44no—
180C4×C89—82no—
181C4×C89#38514yes—
182C4×C89C2×C4×D412yes—
183C4×C89C2×C22⋊Q812yes—
184Dic89—44no—
185Dic89—44no—
186C2×#1857—891nomaximal, Sylow
187C2×C89—21no—
188C2×C89—21no—
189C2×C89—22no—
190C2×C89#185714yescomplemented
191C2×C89C2×#38511yescharacteristic
192C2×C89C2×#45411yescharacteristic
193C2×C89C2×#45611yescharacteristic
194C2×#519—891no—
195C2×#428—892no—
196C22×#101—891no—
197C2×#391D8911yescharacteristic
198#1857—898no—
199C2×#339—892no—
200C89C2×#185711yescharacteristic, complemented, Sylow
201C23×Q8C2×D8911yescharacteristic
202C22×M16—892no—
203#519—894no—
204#519—1784no—
205#519—1782no—
206C2×C4×Q8—1782no—
207C2×C4×Q8—1782no—
208C2×#117—892no—
209C2×#117—1782no—
210C22×C4⋊C4C2×D8911yescharacteristic
211C22×C42C2×D8911yescharacteristic
212#428—898no—
213#428—1784no—
214C2×#107—1782no—
215C2×#106—1782no—
216C2×#101—898no—
217C2×#101—894no—
218#391—22no—
219#391C2×D8914yes—
220C2×#95—22no—
221#339—898no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—24no—
225C22×Q8Dic17814yes—
226C22×Q8C4×D8914yes—
227C22×Q8C22⋊D8912yes—
228C22×Q8C22×D8912yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4C22⋊D8912yes—
232C23×C4C22×D8911yescharacteristic
233C2×M16—898no—
234C2×M16—894no—
235C2×M16—1782no—
236C2×M16—1784no—
237C22×C8—1782no—
238C4×Q8—1788no—
239C4×Q8—1788no—
240C4×Q8—1788no—
241C4×Q8—1788no—
242#117—894no—
243#117—894no—
244#117—1788no—
245#117—1782no—
246#117—1782no—
247#117—1788no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—892no—
252C2×C4⋊C4—892no—
253C2×C4⋊C4—1782no—
254C2×C4⋊C4—1782no—
255C2×C4⋊C4C22×D8912yes—
256C2×C22⋊C4—1782no—
257C2×C42—24no—
258C2×C42—21no—
259C2×C42—1784no—
260C2×C42—1782no—
261C2×C42C22×D8912yes—
262#107—1788no—
263#106—1788no—
264#101—8916no—
265#95—28no—
266C24C2×C22⋊D8911yescharacteristic
267C2×Q8—28no—
268C2×Q8—28no—
269C2×Q8—24no—
270C2×Q8—28no—
271C2×Q8—28no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8#1308355318yes—
276C2×Q8C2×C4×D8914yes—
277C2×Q8C2×Dic17814yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—1782no—
290C22×C4—1782no—
291C22×C4#1308355318yes—
292C22×C4C2×C4×D8912yes—
293C22×C4D4×D8912yes—
294C22×C4D4×D8912yes—
295C22×C4C2×C22⋊D8914yes—
296C22×C4C2×C22⋊D8911yescharacteristic
297C22×C4C2×Dic17812yes—
298C22×C4#1308357314yes—
299C22×C4C23×D8911yescharacteristic
300M16—898no—
301M16—1788no—
302M16—1784no—
303C2×C8—1788no—
304C2×C8—1784no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—894no—
308C4⋊C4—894no—
309C4⋊C4—1788no—
310C4⋊C4—1788no—
311C4⋊C4—1782no—
312C4⋊C4—1782no—
313C22⋊C4—1784no—
314C22⋊C4—1784no—
315C42—24no—
316C42—42no—
317C42—17816no—
318C42—1784no—
319C42—1784no—
320C23—22no—
321C23—22no—
322C23C2×#1308355312yes—
323C23#1566798011yescharacteristic
324C23#1566798011yescharacteristic
325C23#1566799312yes—
326C23C22×C22⋊D8911yescharacteristic
327Q8—216no—
328Q8—28no—
329Q8—28no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—48no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—3564no—
343C2×C4—3564no—
344C2×C4—1784no—
345C2×C4—1784no—
346C2×C4C2×#1308355318yes—
347C2×C4C2×#1308355312yes—
348C2×C4#1566797214yes—
349C2×C4#1566797214yes—
350C2×C4#1566798012yes—
351C2×C4#1566798914yes—
352C2×C4#1566798914yes—
353C2×C4#1566799312yes—
354C2×C4C22×C4×D8911yescharacteristic
355C2×C4C2×D4×D8912yes—
356C2×C4C22×Dic17811yescharacteristic
357C2×C4C2×#1308357312yes—
358C8—1788no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—24no—
365C22#1915147012yes—
366C22#1915147012yes—
367C22#1915177012yes—
368C22#1915178012yes—
369C22C22×#1308355311yescharacteristic
370C22C2×#1566798011yescharacteristic
371C22C2×#1566799311yescharacteristic
372C4—28no—
373C4—44no—
374C4—82no—
375C4—3564no—
376C4—3564no—
377C4#1915147014yes—
378C4C2×#1566797212yes—
379C4C2×#1566798912yes—
380C2—21no—
381C2—22no—
382C2—21no—
383C2#2602046614yesdirect product factor
384C2C2×#1915147011yescharacteristic
385C2C2×#1915177011yescharacteristic
386C2C2×#1915178011yescharacteristic
subgroupG/H
centerC23#15667980
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC2×#391×C89C2
socleC23×C89#120
prime radicalC24×C89C2×D4
cosocleC22×C4×C89C24
hypercenterC2×#391D89
hypocenterC89C2×#1857
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
4. #26020466C2direct (N × H)
19. C2×#15667931C4semidirect
52. #15667931C2×C4semidirect
190. C2×C89#1857semidirect
200. C89C2×#1857semidirect
383. C2#26020466direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191520762
C2×#191516382
C22×#156679312
#260204662
C2×#191513582
C2×#391×C892
C2×#185789