Finite Group Databasebeta

← C2×#23776025 #35167018

Subgroups — C2×#23776025 #35167018

Order 20224.

1930 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099279C211yesmaximal, characteristic
2C2×#19099274C211yesmaximal, characteristic, complemented
3#23776025C2116yesmaximal, direct product factor
4C2×#19098906C212yesmaximal, complemented
5C2×#19098878C212yesmaximal, complemented
6C2×#19098855C211yesmaximal, characteristic, complemented
7C2×#19098850C211yesmaximal, characteristic, complemented
8C2×#19098810C212yesmaximal, complemented
9C2×C4×#11929613C211yesmaximal, characteristic, complemented
10C2×#19098642C211yesmaximal, characteristic, complemented
11C2×#19098604C212yesmaximal, complemented
12C2×#500×C79C211yesmaximal, characteristic, complemented
13C22×#11929613C2211yescharacteristic
14C22×#11929607C2212yes—
15#19099279—24no—
16#19099279C2218yes—
17#19099274—24no—
18#19099274C2218yescomplemented
19C2×#14579152—21no—
20C2×#14579150—22no—
21C2×#14579141C2212yes—
22C2×#14579137—22no—
23C2×#14579136—21no—
24C2×#14579136C2212yes—
25C2×C4×Dic158C2211yescharacteristic
26C2×C4×D316C2211yescharacteristic
27C2×C4×D316C2211yescharacteristic
28C2×#14579123—21no—
29C2×#14579119—21no—
30C2×#14579119C2212yes—
31C2×C4⋊C4×D79C2212yes—
32#19098906C22116yescomplemented
33#19098878C22116yescomplemented
34#19098855C2218yescomplemented
35#19098850C2218yescomplemented
36#19098810C22116yescomplemented
37C4×#11929613C2218yescomplemented
38#19098642C2218yescomplemented
39#19098604C22116yescomplemented
40C2×#14579094C2212yes—
41C2×#14579093C2211yescharacteristic
42C2×#14579092C2212yes—
43C2×C4⋊(C79⋊C8)C2211yescharacteristic
44C2×#14579086C2211yescharacteristic
45C2×#14579084C2212yes—
46C2×#14579083C2211yescharacteristic
47C2×#14579081C2212yes—
48C2×(C2×D79)⋊C8C2212yes—
49C2×C4⋊Q8×C79C2211yescharacteristic
50C2×C4⋊Q8×C79C2211yescharacteristic
51#500×C79C2218yescomplemented
52C2×#107×C79C2212yes—
53C2×#107×C79C2212yes—
54C2×C4×C8×C79C2211yescharacteristic
55C22×Dic158C2311yescharacteristic
56C22×D316—22no—
57C22×D316D411yescharacteristic
58C22×D316C2311yescharacteristic
59C22×C4×D79—21no—
60C22×C4×D79—22no—
61C22×C4×D79C2312yes—
62C2×#11929613—24no—
63C2×#11929613C2314yes—
64C2×#11929607—28no—
65C2×#11929607C2318yes—
66C22×C79⋊C8C2312yes—
67#14579152—28no—
68#14579150—216no—
69#14579141—24no—
70#14579141C2318yes—
71#14579137—216no—
72#14579136—28no—
73#14579136—24no—
74#14579136C2318yes—
75C4×Dic158—22no—
76C4×Dic158C2314yes—
77C4×D316—22no—
78C4×D316D414yes—
79C4×D316C2314yes—
80C4×D316C2314yes—
81#14579123—28no—
82#14579119—24no—
83#14579119—28no—
84#14579119C2318yes—
85C2×C4⋊Dic79—22no—
86C2×C4⋊Dic79D411yescharacteristic
87C2×C4⋊Dic79C2311yescharacteristic
88C2×C4⋊Dic79C2311yescharacteristic
89C2×Dic79⋊C4—21no—
90C2×Dic79⋊C4—22no—
91C4⋊C4×D79—24no—
92C4⋊C4×D79C2318yes—
93C2×(C2×D79)⋊C4—21no—
94C2×(C2×D79)⋊C4—22no—
95C2×(C2×D79)⋊C4—22no—
96C2×(C2×D79)⋊C4—22no—
97C2×(C2×D79)⋊C4D412yes—
98C2×C4×Dic79—21no—
99C2×C4×Dic79—22no—
100#14579094C2318yes—
101#14579093C2314yes—
102#14579092C2318yes—
103C4⋊(C79⋊C8)C2314yes—
104#14579086C2314yes—
105#14579084C2318yes—
106#14579083C2314yes—
107#14579081C2318yes—
108(C2×D79)⋊C8C2318yes—
109C22×Q8×C79—21no—
110C22×Q8×C79—21no—
111C22×C8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C4⋊Q8×C79—22no—
114C4⋊Q8×C79—22no—
115C4⋊Q8×C79C2314yes—
116C4⋊Q8×C79C2314yes—
117C2×C4⋊C4×C79—21no—
118C2×C4⋊C4×C79—21no—
119C2×C4⋊C4×C79C2312yes—
120C2×C4⋊C4×C79C2312yes—
121C2×C42×C79C2311yescharacteristic
122#107×C79C2318yes—
123#107×C79C2318yes—
124C4×C8×C79C2314yes—
125C23×D79—21no—
126C23×D79C2×D412yes—
127C2×Dic158—21no—
128C2×Dic158—22no—
129C2×Dic158—22no—
130C2×Dic158C2412yes—
131C2×D316—28no—
132C2×D316—24no—
133C2×D316—21no—
134C2×D316—22no—
135C2×D316—22no—
136C2×D316—84no—
137C2×D316C2×D414yes—
138C2×D316C2412yes—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C22×Dic79—21no—
142C22×Dic79C2×D412yes—
143C2×C4×D79—28no—
144C2×C4×D79—24no—
145C2×C4×D79—24no—
146C2×C4×D79—24no—
147C2×C4×D79—44no—
148C2×C4×D79—84no—
149C2×C4×D79C2×D414yes—
150C2×C4×D79C2414yes—
151#11929613—28no—
152#11929607—216no—
153C2×C79⋊C8—24no—
154C2×C79⋊C8C2414yes—
155C4⋊Dic79—28no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79C2×D414yes—
159C4⋊Dic79C2412yes—
160C4⋊Dic79C2412yes—
161Dic79⋊C4—28no—
162Dic79⋊C4—24no—
163(C2×D79)⋊C4—28no—
164(C2×D79)⋊C4—28no—
165(C2×D79)⋊C4—28no—
166(C2×D79)⋊C4—24no—
167(C2×D79)⋊C4C2×D418yes—
168C4×Dic79—28no—
169C4×Dic79—24no—
170C2×Q8×C79—24no—
171C2×Q8×C79—24no—
172C2×Q8×C79—82no—
173C2×Q8×C79—82no—
174C22×C4×C79—22no—
175C22×C4×C79—22no—
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2411yescharacteristic
179C2×C8×C79—22no—
180C2×C8×C79#4114yes—
181C2×C8×C79C2412yes—
182C2×C8×C79C2412yes—
183C4⋊C4×C79—24no—
184C4⋊C4×C79—24no—
185C4⋊C4×C79—22no—
186C4⋊C4×C79—22no—
187C4⋊C4×C79C2414yes—
188C4⋊C4×C79C2414yes—
189C42×C79C2×D412yes—
190C42×C79C2412yes—
191C23×C79C22×D411yescharacteristic
192C22×D79—24no—
193C22×D79—24no—
194C22×D79—24no—
195C22×D79—22no—
196C22×D79—42no—
197C22×D79—42no—
198C22×D79—42no—
199C22×D79C22×D414yes—
200Q8×C79—84no—
201Q8×C79—84no—
202C2×C4×C79—22no—
203C2×C4×C79—24no—
204C2×C4×C79—24no—
205C2×C4×C79—44no—
206C2×C4×C79—44no—
207C2×C4×C79#13714yes—
208C2×C4×C79C22×D412yes—
209C2×C4×C79C22×D411yescharacteristic
210C2×C4×C79C22×D412yes—
211C2×C4×C79C2×#4112yes—
212C2×C4×C792−1+412yes—
213C2×C4×C79C2511yescharacteristic
214C2×Dic79—24no—
215C2×Dic79—22no—
216C2×Dic79—22no—
217C2×Dic79—22no—
218C2×Dic79—42no—
219C2×Dic79—42no—
220C2×Dic79—42no—
221C2×Dic79C22×D414yes—
222D316—28no—
223D316—22no—
224D316—22no—
225D316—42no—
226D316—88no—
227C4×D79—28no—
228C4×D79—24no—
229C4×D79—24no—
230C4×D79—48no—
231C4×D79—88no—
232Dic158—22no—
233Dic158—22no—
234Dic158—42no—
235C8×C79—22no—
236C8×C79C2×#4114yes—
237C79⋊C8—24no—
238C22×C79#54712yes—
239C22×C79C2×#13712yes—
240C22×C79#57612yes—
241C22×C79C23×D411yescharacteristic
242C2×D79—24no—
243C2×D79—24no—
244C2×D79—24no—
245C2×D79—48no—
246C2×D79—42no—
247C2×D79—42no—
248C2×D79—42no—
249C2×D79—84no—
250C4×C79—22no—
251C4×C79—44no—
252C4×C79—44no—
253C4×C79#54712yes—
254C4×C79C2×#13714yes—
255C4×C79C22×#4111yescharacteristic
256C4×C79C2×2−1+411yescharacteristic
257Dic79—24no—
258Dic79—42no—
259Dic79—42no—
260Dic79—42no—
261C2×#3357—791nomaximal, Sylow
262C2×C79#335714yescomplemented
263C2×C79C2×#54711yescharacteristic
264C2×C79C22×#13711yescharacteristic
265C2×C79C2×#57611yescharacteristic
266D79—44no—
267D79—82no—
268C2×#555—791no—
269C2×#554—791no—
270#3357—7916no—
271C2×#500D7911yescharacteristic, complemented
272C2×#485—792no—
273C2×#480—792no—
274C2×#477—791no—
275C2×#476—791no—
276C2×#468—792no—
277C2×C4×SD16—791no—
278C2×#436—791no—
279C2×#428—792no—
280C79C2×#335711yescharacteristic, complemented, Sylow
281C22×SD16—791no—
282C22×M16—792no—
283#555—798no—
284#555—1584no—
285#554—798no—
286#554—1584no—
287C2×C4⋊Q8C2×D7911yescharacteristic
288C2×C4⋊Q8C2×D7911yescharacteristic
289C2×#125—1581no—
290C2×#124—1582no—
291C2×C22⋊Q8—792no—
292C2×C22⋊Q8—1582no—
293C2×#121—792no—
294C2×#121—1581no—
295C2×C4×Q8—791no—
296C2×C4×D4—791no—
297C2×C4×D4—791no—
298C2×C4×D4—1581no—
299C2×#117—792no—
300C2×#117—1581no—
301C22×C4⋊C4—792no—
302#500C2×D7918yescomplemented
303#485—7916no—
304#480—7916no—
305#477—798no—
306#476—798no—
307#468—7916no—
308C4×SD16—798no—
309#436—798no—
310#428—7916no—
311C2×#107—792no—
312C2×#107C2×D7912yes—
313C2×#107C2×D7912yes—
314C2×#106—791no—
315C2×#106—792no—
316C2×C4⋊C8—791no—
317C2×#103—791no—
318C2×#103—792no—
319C2×#102—791no—
320C2×#102—792no—
321C2×C22⋊C8—792no—
322C2×C4×C8C2×D7911yescharacteristic
323C22×Q8—21no—
324C22×Q8—21no—
325C22×Q8—791no—
326C22×D4—791no—
327C22×D4—791no—
328C22×D4—1582no—
329C23×C4—792no—
330C23×C4—1582no—
331C23×C4—1581no—
332C2×SD16—794no—
333C2×SD16—1584no—
334C2×M16—798no—
335C2×M16—1588no—
336C22×C8—792no—
337C22×C8C22×D7911yescharacteristic
338C22×C8C22×D7911yescharacteristic
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8C22×D7914yes—
342C4⋊Q8C22×D7914yes—
343#125—1588no—
344#124—15816no—
345C22⋊Q8—798no—
346C22⋊Q8—15816no—
347C22⋊Q8—1584no—
348#121—798no—
349#121—1584no—
350#121—1588no—
351C4×Q8—794no—
352C4×Q8—1582no—
353C4×D4—794no—
354C4×D4—794no—
355C4×D4—794no—
356C4×D4—1588no—
357C4×D4—1582no—
358#117—798no—
359#117—1588no—
360#117—1584no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—791no—
364C2×C4⋊C4—791no—
365C2×C4⋊C4—791no—
366C2×C4⋊C4—798no—
367C2×C4⋊C4—1584no—
368C2×C4⋊C4—1582no—
369C2×C4⋊C4—1582no—
370C2×C4⋊C4—1581no—
371C2×C4⋊C4C22×D7912yes—
372C2×C4⋊C4C22×D7912yes—
373C2×C22⋊C4—792no—
374C2×C22⋊C4—1582no—
375C2×C22⋊C4—1582no—
376C2×C22⋊C4—1582no—
377C2×C22⋊C4—1581no—
378C2×C42—1582no—
379C2×C42—1581no—
380C2×C42C22×D7911yescharacteristic
381#107—798no—
382#107C22×D7918yes—
383#107C22×D7918yes—
384#106—794no—
385#106—798no—
386C4⋊C8—794no—
387#103—794no—
388#103—798no—
389#102—794no—
390#102—798no—
391C22⋊C8—798no—
392C4×C8C22×D7914yes—
393C24—792no—
394C24—1581no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—792no—
400C2×Q8—1582no—
401C2×Q8—1582no—
402C2×Q8—1581no—
403C2×D4—792no—
404C2×D4—794no—
405C2×D4—6324no—
406C2×D4—1588no—
407C2×D4—1582no—
408C2×D4—1582no—
409C2×D4—1584no—
410C2×D4—1581no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—3164no—
414C22×C4—792no—
415C22×C4—794no—
416C22×C4—794no—
417C22×C4—6324no—
418C22×C4—1588no—
419C22×C4—1581no—
420C22×C4—1581no—
421C22×C4—1581no—
422C22×C4—1584no—
423C22×C4—1584no—
424C22×C4—1584no—
425C22×C4D4×D7911yescharacteristic
426C22×C4D4×D7911yescharacteristic
427C22×C4C23×D7911yescharacteristic
428SD16—1588no—
429M16—15816no—
430C2×C8—22no—
431C2×C8—794no—
432C2×C8—1584no—
433C2×C8#1192962514yes—
434C2×C8C23×D7912yes—
435C2×C8C23×D7912yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—794no—
441C4⋊C4—792no—
442C4⋊C4—792no—
443C4⋊C4—1588no—
444C4⋊C4—1588no—
445C4⋊C4—1584no—
446C4⋊C4—1581no—
447C4⋊C4—1581no—
448C4⋊C4C23×D7914yes—
449C4⋊C4C23×D7914yes—
450C22⋊C4—798no—
451C22⋊C4—1588no—
452C22⋊C4—1588no—
453C22⋊C4—1588no—
454C22⋊C4—1584no—
455C42—1588no—
456C42—1584no—
457C42D4×D7912yes—
458C42C23×D7912yes—
459C23—3162no—
460C23—3162no—
461C23—3162no—
462C23—794no—
463C23—1582no—
464C23—1584no—
465C23—1584no—
466C23—1584no—
467C23C2×D4×D7911yescharacteristic
468Q8—84no—
469Q8—84no—
470Q8—3162no—
471Q8—1582no—
472Q8—1582no—
473D4—3162no—
474D4—6328no—
475D4—1582no—
476D4—1588no—
477D4—1582no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—3162no—
484C2×C4—3162no—
485C2×C4—3162no—
486C2×C4—3168no—
487C2×C4—794no—
488C2×C4—6328no—
489C2×C4—1582no—
490C2×C4—1582no—
491C2×C4—1582no—
492C2×C4—1584no—
493C2×C4—1584no—
494C2×C4—1588no—
495C2×C4—1584no—
496C2×C4#1457919314yes—
497C2×C4C2×D4×D7912yes—
498C2×C4C2×D4×D7911yescharacteristic
499C2×C4C2×D4×D7912yes—
500C2×C4C2×#1192962512yes—
501C2×C4#1457920912yes—
502C2×C4C24×D7911yescharacteristic
503C8—22no—
504C8—1584no—
505C8C2×#1192962514yes—
506C22—3168no—
507C22—3162no—
508C22—3162no—
509C22—3162no—
510C22—6324no—
511C22—1584no—
512C22—1584no—
513C22—1584no—
514C22#1909921612yes—
515C22C2×#1457919312yes—
516C22#1909936412yes—
517C22C22×D4×D7911yescharacteristic
518C4—22no—
519C4—44no—
520C4—44no—
521C4—3162no—
522C4—3162no—
523C4—3162no—
524C4—1584no—
525C4#1909921612yes—
526C4C2×#1457919314yes—
527C4C22×#1192962511yescharacteristic
528C4C2×#1457920911yescharacteristic
529C2—3164no—
530C2—6322no—
531C2#2377602514yesdirect product factor
532C2C2×#1909921611yescharacteristic
533C2C22×#1457919311yescharacteristic
534C2C2×#1909936411yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#500×C79C2
socleC23×C79C22×D4
prime radicalC2×#19099279C2
hypercenterC2×#500D79
hypocenterC79C2×#3357
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
2. C2×#19099274C2semidirect
3. #23776025C2direct (N × H)
4. C2×#19098906C2semidirect
5. C2×#19098878C2semidirect
6. C2×#19098855C2semidirect
7. C2×#19098850C2semidirect
8. C2×#19098810C2semidirect
9. C2×C4×#11929613C2semidirect
10. C2×#19098642C2semidirect
11. C2×#19098604C2semidirect
12. C2×#500×C79C2semidirect
18. #19099274C22semidirect
32. #19098906C22semidirect
33. #19098878C22semidirect
34. #19098855C22semidirect
35. #19098850C22semidirect
36. #19098810C22semidirect
37. C4×#11929613C22semidirect
38. #19098642C22semidirect
39. #19098604C22semidirect
51. #500×C79C22semidirect
262. C2×C79#3357semidirect
271. C2×#500D79semidirect
280. C79C2×#3357semidirect
302. #500C2×D79semidirect
531. C2#23776025direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190992792
C2×#190992742
#237760252
C2×#190989062
C2×#190988782
C2×#190988552
C2×#190988502
C2×#190988102
C2×C4×#119296132
C2×#190986422
C2×#190986042
C2×#500×C792
C2×#335779