Finite Group Databasebeta

← C2×#11748077 #14417968

Subgroups — C2×#11748077 #14417968

Order 1920.

928 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91790C211yesmaximal, characteristic, complemented
2C2×C3×#4047C211yesmaximal, characteristic, complemented
3C2×#91311C211yesmaximal, characteristic
4C2×C5×#815C211yesmaximal, characteristic, complemented
5C2×#90993C211yesmaximal, characteristic, complemented
6C2×#90980C211yesmaximal, characteristic, complemented
7C2×#90977C211yesmaximal, characteristic, complemented
8#11748077C218yesmaximal, direct product factor
9C2×#65006—31nomaximal, Hall
10C2×#6817C2211yescharacteristic
11C2×C3×#709C2211yescharacteristic
12C22×C3×C5×C8C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14#91790—22no—
15#91790C2214yescomplemented
16C2×#6705—21no—
17C2×C8×D15—21no—
18C2×C4×C5×Dic3C2211yescharacteristic
19C22×C5×C3⋊C8C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21C3×#4047—22no—
22C3×#4047C2214yescomplemented
23C2×C3×#676—21no—
24C2×C3×C8×D5—21no—
25C2×#6662C411yescharacteristic, complemented
26C2×#6658C411yescharacteristic, complemented
27#91311C414yescomplemented
28#91311C2214yes—
29C2×#6657C411yescharacteristic, complemented
30C2×#6653C411yescharacteristic, complemented
31C5×#815C2214yescomplemented
32#90993C2214yescomplemented
33#90980C2214yescomplemented
34#90977C2214yescomplemented
35C2×#5145—51nomaximal, Hall
36C2×#4050—31no—
37C2×#4047—31no—
38C2×#4047D311yescharacteristic, complemented
39C2×C5×#97—31no—
40C2×#3951—31no—
41C2×#3937—31no—
42C2×#3937—31no—
43#65006—38no—
44C22×C3×C4×C5C2311yescharacteristic
45C2×#1220—21no—
46#6817—21no—
47#6817—21no—
48#6817C2×C412yes—
49#6817C2312yes—
50C2×D60C2×C411yescharacteristic
51C2×C4×D15—21no—
52C2×Dic30C2×C411yescharacteristic
53C22×C5×Dic3—21no—
54C2×C3×C22⋊D5—21no—
55C3×#709—21no—
56C3×#709—21no—
57C3×#709C2×C412yes—
58C3×#709C2312yes—
59C2×C3×D20C2×C411yescharacteristic
60C2×C3×C4×D5—21no—
61C2×C3×Dic10C2×C411yescharacteristic
62C2×C3×C5×C8D411yescharacteristic
63C2×C3×C5×C8D411yescharacteristic
64C2×C3×C5×C8D412yes—
65C2×C3×C5×C8C2312yes—
66#6710—21no—
67#6710C2312yes—
68C2×C3⋊#168—21no—
69#6705—24no—
70C8×D15—24no—
71C4×C5×Dic3—21no—
72C4×C5×Dic3C2312yes—
73C2×C5×C3⋊C8C2×C411yescharacteristic
74C2×C5×C3⋊C8C2×C411yescharacteristic
75C2×C5×C3⋊C8C2×C412yes—
76C2×C5×C3⋊C8C2312yes—
77C3×#681—21no—
78C3×#681C2312yes—
79C2×C3×#168—21no—
80C3×#676—24no—
81C3×C8×D5—24no—
82#6662C2×C414yescomplemented
83#6658C2×C414yescomplemented
84#6657C2×C414yescomplemented
85#6653C2×C414yescomplemented
86C2×C3×#131—51no—
87C2×#950—51no—
88C2×#901—51no—
89C2×#837—51no—
90#5145—58no—
91C2×#825—51no—
92C2×#822—51no—
93C2×#815D511yescharacteristic, complemented
94C2×#709—31no—
95C2×#709C2×D311yescharacteristic
96C22×C5×C8—31no—
97C22×C5×C8C2×D311yescharacteristic
98C2×C42×C5—31no—
99C2×#681—31no—
100C2×#681C2×D311yescharacteristic
101C2×Dic20—31no—
102#4050—34no—
103#4050—34no—
104C2×D40—31no—
105C2×#677—31no—
106C2×#677—31no—
107#4047—22no—
108#4047—34no—
109#4047—62no—
110#4047C2×D314yescomplemented
111C2×#676—21no—
112C2×#676—61no—
113C2×C8×D5—21no—
114C2×C8×D5—61no—
115C5×#97—34no—
116#3951—34no—
117#3937—34no—
118#3937—34no—
119C2×#444—151noSylow
120C23×C3×C5#4111yescharacteristic
121C22×D15—21no—
122C22×C3×D5—21no—
123C2×C3×C4×C5C22×C411yescharacteristic
124C2×C3×C4×C5C22×C411yescharacteristic
125C2×C3×C4×C5C2×D411yescharacteristic
126C2×C3×C4×C5C2×D411yescharacteristic
127C2×C3×C4×C5#4111yescharacteristic
128C2×C3×C4×C5C2411yescharacteristic
129#1220—22no—
130#1220—22no—
131C2×Dic15—21no—
132D60—21no—
133D60C22×C412yes—
134C4×D15—22no—
135C4×D15—22no—
136Dic30—21no—
137Dic30C22×C412yes—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—42no—
140C3×C22⋊D5—22no—
141C3×C22⋊D5—22no—
142C2×C3×Dic5—21no—
143C3×D20—21no—
144C3×D20C22×C412yes—
145C3×C4×D5—22no—
146C3×C4×D5—22no—
147C3×Dic10—21no—
148C3×Dic10C22×C412yes—
149C3×C5×C8C2×D412yes—
150C3×C5×C8C2×D412yes—
151C3⋊#168—22no—
152C3×#168—22no—
153C5×C3⋊C8C22×C412yes—
154C5×C3⋊C8C22×C412yes—
155C2×C3×#41—51no—
156C2×#235—51no—
157C3×#131—54no—
158C3×#131—102no—
159C2×C3×M16—51no—
160C2×C3×M16—101no—
161C22×C3×C8—101no—
162C22×C3×C8C2×D511yescharacteristic
163#950—54no—
164#950—54no—
165C2×#216—51no—
166C2×#215—51no—
167C2×#214—51no—
168C2×#213—51no—
169C2×C4×Dic3C2×D511yescharacteristic
170C2×#208—51no—
171C22×C3⋊C8C2×D511yescharacteristic
172#901—54no—
173#901—102no—
174C2×ΓL(1,𝔽25)—101no—
175C2×D3×C8—101no—
176#837—54no—
177#825—54no—
178#822—54no—
179#815C2×D514yescomplemented
180C22×C4×C5—61no—
181C22×C4×C5C22×D311yescharacteristic
182C2×C22⋊D5—21no—
183C2×C22⋊D5—61no—
184#709—21no—
185#709—21no—
186#709—32no—
187#709—32no—
188#709—61no—
189#709—61no—
190#709C4×D312yes—
191#709C22×D312yes—
192C2×D20—31no—
193C2×D20C4×D311yescharacteristic
194C2×C4×D5—21no—
195C2×C4×D5—61no—
196C2×Dic10—31no—
197C2×Dic10C4×D311yescharacteristic
198C2×C5×C8—31no—
199C2×C5×C8—31no—
200C2×C5×C8—32no—
201C2×C5×C8—32no—
202C2×C5×C8#6711yescharacteristic
203C2×C5×C8#6711yescharacteristic
204C2×C5×C8#6712yes—
205C2×C5×C8C22×D312yes—
206C42×C5—32no—
207C42×C5—61no—
208#681—21no—
209#681—32no—
210#681—61no—
211#681C22×D312yes—
212C2×#168—21no—
213C2×#168—61no—
214Dic20—34no—
215D40—34no—
216#677—34no—
217#677—34no—
218#676—24no—
219#676—64no—
220C8×D5—24no—
221C8×D5—64no—
222C2×#135—151no—
223C2×#131—51no—
224C2×#131—151no—
225#444—158no—
226C2×#108—151no—
227C2×#104—151no—
228C2×#104—151no—
229C2×#97—31no—
230C22×C3×C5—22no—
231C22×C3×C5C4×D411yescharacteristic
232C22×C3×C5C4×D411yescharacteristic
233C22×C3×C5C2×#4111yescharacteristic
234C2×D15—22no—
235C2×D15—42no—
236C2×C3×D5—22no—
237C2×C3×D5—42no—
238C3×C4×C5C4×D411yescharacteristic
239C3×C4×C5C4×D411yescharacteristic
240C3×C4×C5C23×C411yescharacteristic
241C3×C4×C5C22×D411yescharacteristic
242Dic15—22no—
243C3×Dic5—22no—
244C5×Dic3—42no—
245C3×#41—52no—
246C3×#41—52no—
247C3×#41—101no—
248C3×#41—101no—
249C2×C3×Q8—51no—
250C2×C3×D4—51no—
251C2×C3×D4—101no—
252C22×C3×C4—101no—
253C22×C3×C4C22×D511yescharacteristic
254C2×#67—101no—
255C22×Dic3—21no—
256#235—52no—
257#235—52no—
258#235—101no—
259#235—101no—
260C2×D12—51no—
261C2×C4×D3—101no—
262C2×Dic6—51no—
263C3×M16—52no—
264C3×M16—101no—
265C3×M16—104no—
266C2×C3×C8—101no—
267C2×C3×C8—104no—
268C2×C3×C8D2011yescharacteristic
269C2×C3×C8D2011yescharacteristic
270C2×C3×C8D2012yes—
271C2×C3×C8C22×D512yes—
272#216—54no—
273#215—54no—
274#214—54no—
275#213—54no—
276C4×Dic3—21no—
277C4×Dic3C22×D512yes—
278#208—52no—
279#208—101no—
280C2×C3⋊C8—101no—
281C2×C3⋊C8C4×D511yescharacteristic
282C2×C3⋊C8C4×D511yescharacteristic
283C2×C3⋊C8C4×D512yes—
284C2×C3⋊C8C22×D512yes—
285ΓL(1,𝔽25)—104no—
286D3×C8—104no—
287C23×C5#23511yescharacteristic
288C22×D5—21no—
289C22×D5—61no—
290C2×C4×C5—62no—
291C2×C4×C5—122no—
292C2×C4×C5C2×C4×D311yescharacteristic
293C2×C4×C5C2×C4×D311yescharacteristic
294C2×C4×C5#23511yescharacteristic
295C2×C4×C5C2×#6711yescharacteristic
296C2×C4×C5C2×#6711yescharacteristic
297C2×C4×C5C23×D311yescharacteristic
298C22⋊D5—22no—
299C22⋊D5—22no—
300C22⋊D5—62no—
301C22⋊D5—62no—
302C2×Dic5—21no—
303C2×Dic5—61no—
304D20—21no—
305D20—32no—
306D20—61no—
307D20C2×C4×D312yes—
308C4×D5—22no—
309C4×D5—22no—
310C4×D5—62no—
311C4×D5—62no—
312Dic10—21no—
313Dic10—32no—
314Dic10—61no—
315Dic10C2×C4×D312yes—
316C5×C8—32no—
317C5×C8—32no—
318C5×C8C2×#6712yes—
319C5×C8C2×#6712yes—
320#168—22no—
321#168—62no—
322C2×#41—51no—
323C2×#41—151no—
324#135—154no—
325#135—154no—
326C2×Q16—151no—
327C2×SD16—151no—
328C2×SD16—151no—
329C2×D8—151no—
330#131—54no—
331#131—102no—
332#131—154no—
333#131—302no—
334C2×M16—51no—
335C2×M16—101no—
336C2×M16—151no—
337C2×M16—301no—
338C22×C8—31no—
339C22×C8—101no—
340C22×C8—301no—
341C22×C8D3×D511yescharacteristic
342C2×C42—31no—
343#108—154no—
344#104—154no—
345#104—154no—
346#97—34no—
347C2×C3×C5—21no—
348C2×C3×C5—21no—
349C2×C3×C5#44412yescomplemented
350C2×C3×C5C2×C4×D411yescharacteristic
351D15—42no—
352C3×D5—42no—
353C23×C3—101no—
354C23×C3#70911yescharacteristic
355C22×D3—101no—
356C3×Q8—52no—
357C3×Q8—101no—
358C3×D4—52no—
359C3×D4—102no—
360C3×D4—102no—
361C3×D4—101no—
362C2×C3×C4—101no—
363C2×C3×C4—102no—
364C2×C3×C4—102no—
365C2×C3×C4C2×C4×D511yescharacteristic
366C2×C3×C4C2×C4×D511yescharacteristic
367C2×C3×C4C2×D2011yescharacteristic
368C2×C3×C4C2×D2011yescharacteristic
369C2×C3×C4#70911yescharacteristic
370C2×C3×C4C23×D511yescharacteristic
371#67—102no—
372#67—102no—
373C2×Dic3—22no—
374C2×Dic3—42no—
375C2×Dic3—101no—
376D12—52no—
377D12—101no—
378C4×D3—102no—
379C4×D3—102no—
380Dic6—52no—
381Dic6—101no—
382C3×C8—102no—
383C3×C8C2×D2012yes—
384C3×C8C2×D2012yes—
385C3⋊C8—102no—
386C3⋊C8C2×C4×D512yes—
387C3⋊C8C2×C4×D512yes—
388C22×C5—22no—
389C22×C5C4×#6711yescharacteristic
390C22×C5C4×#6711yescharacteristic
391C22×C5C2×#23511yescharacteristic
392C2×D5—22no—
393C2×D5—42no—
394C2×D5—62no—
395C2×D5—122no—
396C4×C5—122no—
397C4×C5C4×#6711yescharacteristic
398C4×C5C4×#6711yescharacteristic
399C4×C5C22×C4×D311yescharacteristic
400C4×C5C22×#6711yescharacteristic
401Dic5—22no—
402Dic5—62no—
403#41—52no—
404#41—52no—
405#41—101no—
406#41—101no—
407#41—152no—
408#41—152no—
409#41—301no—
410#41—301no—
411C2×Q8—51no—
412C2×Q8—151no—
413C2×D4—51no—
414C2×D4—101no—
415C2×D4—151no—
416C2×D4—301no—
417C22×C4—61no—
418C22×C4—101no—
419C22×C4—301no—
420C22×C4C2×D3×D511yescharacteristic
421Q16—154no—
422SD16—154no—
423SD16—154no—
424D8—154no—
425M16—52no—
426M16—101no—
427M16—104no—
428M16—152no—
429M16—301no—
430M16—304no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—32no—
434C2×C8—32no—
435C2×C8—101no—
436C2×C8—104no—
437C2×C8—301no—
438C2×C8—304no—
439C2×C8#120212yes—
440C2×C8#120211yescharacteristic
441C2×C8#120211yescharacteristic
442C2×C8C2×D3×D512yes—
443C42—32no—
444C42—61no—
445C3×C5C2×#44411yescharacteristic, complemented, Hall
446C22×C3—22no—
447C22×C3—102no—
448C22×C3—202no—
449C22×C3C4×D2011yescharacteristic
450C22×C3C4×D2011yescharacteristic
451C22×C3C2×#70911yescharacteristic
452C2×D3—102no—
453C2×D3—202no—
454C3×C4—102no—
455C3×C4C4×D2011yescharacteristic
456C3×C4C4×D2011yescharacteristic
457C3×C4C22×C4×D511yescharacteristic
458C3×C4C22×D2011yescharacteristic
459Dic3—42no—
460Dic3—102no—
461C2×C5—21no—
462C2×C5—21no—
463C2×C5#514512yescomplemented
464C2×C5C2×C4×#6711yescharacteristic
465D5—42no—
466D5—122no—
467C23—101no—
468C23—301no—
469C23#677111yescharacteristic
470Q8—52no—
471Q8—101no—
472Q8—152no—
473Q8—301no—
474D4—52no—
475D4—102no—
476D4—102no—
477D4—101no—
478D4—152no—
479D4—302no—
480D4—302no—
481D4—301no—
482C2×C4—62no—
483C2×C4—101no—
484C2×C4—102no—
485C2×C4—102no—
486C2×C4—122no—
487C2×C4—301no—
488C2×C4—302no—
489C2×C4—302no—
490C2×C4#677111yescharacteristic
491C2×C4C4×D3×D511yescharacteristic
492C2×C4C4×D3×D511yescharacteristic
493C2×C4C2×#120211yescharacteristic
494C2×C4C2×#120211yescharacteristic
495C2×C4C22×D3×D511yescharacteristic
496C8—32no—
497C8—32no—
498C8—102no—
499C8—302no—
500C8C2×#120212yes—
501C8C2×#120212yes—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—202no—
505C2×C3#6500612yescomplemented
506C2×C3C2×C4×D2011yescharacteristic
507D3—202no—
508C5C2×#514511yescharacteristic, complemented, Sylow
509C22—22no—
510C22—102no—
511C22—202no—
512C22—302no—
513C22—602no—
514C22C4×#120211yescharacteristic
515C22C4×#120211yescharacteristic
516C22C2×#677111yescharacteristic
517C4—102no—
518C4—122no—
519C4—302no—
520C4C4×#120211yescharacteristic
521C4C4×#120211yescharacteristic
522C4C2×C4×D3×D511yescharacteristic
523C4C22×#120211yescharacteristic
524C3C2×#6500611yescharacteristic, complemented, Sylow
525C2—21no—
526C2—21no—
527C2—202no—
528C2—602no—
529C2#1174807712yesdirect product factor
530C2C2×C4×#120211yescharacteristic
subgroupG/H
centerC2×C4C2×#1202
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC22×C3×C5C4×D4
prime radicalC2×#91311C2
cosocleC2×C3×C4×C5C24
hypercenterC22×C8D3×D5
hypocenterC3×C5C2×#444
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C2×#91790C2semidirect
2. C2×C3×#4047C2semidirect
4. C2×C5×#815C2semidirect
5. C2×#90993C2semidirect
6. C2×#90980C2semidirect
7. C2×#90977C2semidirect
8. #11748077C2direct (N × H)
15. #91790C22semidirect
22. C3×#4047C22semidirect
25. C2×#6662C4semidirect
26. C2×#6658C4semidirect
27. #91311C4semidirect
29. C2×#6657C4semidirect
30. C2×#6653C4semidirect
31. C5×#815C22semidirect
32. #90993C22semidirect
33. #90980C22semidirect
34. #90977C22semidirect
38. C2×#4047D3semidirect
82. #6662C2×C4semidirect
83. #6658C2×C4semidirect
84. #6657C2×C4semidirect
85. #6653C2×C4semidirect
93. C2×#815D5semidirect
110. #4047C2×D3semidirect
179. #815C2×D5semidirect
349. C2×C3×C5#444semidirect
445. C3×C5C2×#444semidirect
463. C2×C5#5145semidirect
505. C2×C3#65006semidirect
508. C5C2×#5145semidirect
524. C3C2×#65006semidirect
529. C2#11748077direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917902
C2×C3×#40472
C2×#913112
C2×C5×#8152
C2×#909932
C2×#909802
C2×#909772
#117480772
C2×#650063
C2×#51455