Finite Group Databasebeta

← C2×#23774607 #35160384

Subgroups — C2×#23774607 #35160384

Order 20224.

1770 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×#19099118C211yesmaximal, characteristic
2C2×#125×D79C211yesmaximal, characteristic
3#23774607C2116yesmaximal, direct product factor
4C2×#19098969C211yesmaximal, characteristic, complemented
5C2×#19098933C211yesmaximal, characteristic, complemented
6C2×#19098932C211yesmaximal, characteristic, complemented
7C2×#19098900C212yesmaximal
8C2×#19098895C212yesmaximal
9C2×#19098873C212yesmaximal, complemented
10C2×#19098869C212yesmaximal, complemented
11C2×#19098618C211yesmaximal, characteristic
12C2×#501×C79C211yesmaximal, characteristic, complemented
13#19099118—24no—
14#19099118C2218yes—
15C2×#14579161C2211yescharacteristic
16C2×#14579153C2211yescharacteristic
17C2×#14579152—21no—
18C2×#14579152C2211yescharacteristic
19C2×#14579152C2211yescharacteristic
20C2×#14579151—21no—
21C2×#14579151—21no—
22#125×D79—24no—
23#125×D79C2218yes—
24C2×#14579141C2212yes—
25C2×#14579137—21no—
26C2×Q8×Dic79—21no—
27C2×#14579126C2212yes—
28C2×#14579119—21no—
29C2×C4⋊C4×D79—21no—
30C2×C4⋊C4×D79—21no—
31C2×C4⋊C4×D79C2211yescharacteristic
32C2×C4⋊C4×D79C2211yescharacteristic
33C2×C42×D79C2211yescharacteristic
34#19098969C2218yescomplemented
35#19098933C2218yescomplemented
36#19098932C2218yescomplemented
37#19098900C22116yes—
38#19098895C22116yes—
39#19098873C22116yescomplemented
40#19098869C22116yescomplemented
41#19098618C2218yes—
42C2×#14579094C2212yes—
43C2×#14579092C2212yes—
44C2×Dic79⋊C8C2212yes—
45C2×#14579086C2212yes—
46C2×#14579086C2212yes—
47C2×#14579084C2212yes—
48C2×#14579084C2212yes—
49C2×(C2×D79)⋊C8C2212yes—
50C2×#14579072C2211yescharacteristic
51C2×C4⋊Q8×C79C2211yescharacteristic
52C2×#125×C79C2211yescharacteristic
53#501×C79C2218yescomplemented
54C2×#107×C79C2212yes—
55C2×#106×C79C2212yes—
56C2×#97×C79C2211yescharacteristic
57C22×Dic158C2312yes—
58C22×C4×D79—21no—
59C22×C4×D79—21no—
60C22×C4×D79D411yescharacteristic
61C22×C4×D79D411yescharacteristic
62C22×C4×D79C2311yescharacteristic
63C22×C79⋊C8C2312yes—
64#14579161—22no—
65#14579161C2314yes—
66#14579153—22no—
67#14579153C2314yes—
68#14579152—22no—
69#14579152—22no—
70#14579152—28no—
71#14579152C2314yes—
72#14579152C2314yes—
73#14579151—28no—
74#14579151—28no—
75#14579141—24no—
76#14579141C2318yes—
77#14579137—28no—
78Q8×Dic79—28no—
79#14579126—24no—
80#14579126C2318yes—
81#14579119—28no—
82C2×C4⋊Dic79—21no—
83C2×C4⋊Dic79—21no—
84C2×C4⋊Dic79—21no—
85C2×C4⋊Dic79—21no—
86C2×C4⋊Dic79—22no—
87C2×C4⋊Dic79C2311yescharacteristic
88C2×C4⋊Dic79C2311yescharacteristic
89C2×Dic79⋊C4—21no—
90C2×Dic79⋊C4—21no—
91C2×Dic79⋊C4—21no—
92C2×Dic79⋊C4—21no—
93C2×Dic79⋊C4—21no—
94C2×Dic79⋊C4—21no—
95C2×Dic79⋊C4—22no—
96C4⋊C4×D79—28no—
97C4⋊C4×D79—28no—
98C4⋊C4×D79—22no—
99C4⋊C4×D79—22no—
100C4⋊C4×D79C2314yes—
101C4⋊C4×D79C2314yes—
102C2×(C2×D79)⋊C4—22no—
103C2×C4×Dic79—22no—
104C2×C4×Dic79D411yescharacteristic
105C2×C4×Dic79D411yescharacteristic
106C2×C4×Dic79C2311yescharacteristic
107C42×D79D414yes—
108C42×D79C2314yes—
109#14579094C2318yes—
110#14579092C2318yes—
111Dic79⋊C8C2318yes—
112#14579086C2318yes—
113#14579086C2318yes—
114#14579084C2318yes—
115#14579084C2318yes—
116(C2×D79)⋊C8C2318yes—
117#14579072C2314yes—
118C22×Q8×C79—21no—
119C22×C8×C79C2312yes—
120C4⋊Q8×C79—22no—
121C4⋊Q8×C79C2314yes—
122#125×C79—22no—
123#125×C79C2314yes—
124C2×C4⋊C4×C79—21no—
125C2×C4⋊C4×C79—21no—
126C2×C4⋊C4×C79—21no—
127C2×C4⋊C4×C79C2311yescharacteristic
128C2×C4⋊C4×C79C2312yes—
129C2×C4⋊C4×C79C2311yescharacteristic
130C2×C42×C79C2311yescharacteristic
131#107×C79C2318yes—
132#106×C79C2318yes—
133#97×C79C2314yes—
134C23×D79C2×D411yescharacteristic
135C2×Dic158—22no—
136C2×Dic158—28no—
137C2×Dic158C2414yes—
138C22×Dic79—22no—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C22×Dic79C2×D411yescharacteristic
142C22×Dic79C2×D411yescharacteristic
143C22×Dic79C2×D411yescharacteristic
144C2×C4×D79—24no—
145C2×C4×D79—24no—
146C2×C4×D79—24no—
147C2×C4×D79—24no—
148C2×C4×D79—42no—
149C2×C4×D79—44no—
150C2×C4×D79—44no—
151C2×C4×D79C2×D412yes—
152C2×C4×D79C2×D414yes—
153C2×C4×D79C2×D414yes—
154C2×C4×D79C2412yes—
155C2×C79⋊C8—24no—
156C2×C79⋊C8C2414yes—
157C4⋊Dic79—24no—
158C4⋊Dic79—24no—
159C4⋊Dic79—28no—
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79—24no—
163C4⋊Dic79—24no—
164C4⋊Dic79C2412yes—
165C4⋊Dic79C2412yes—
166Dic79⋊C4—24no—
167Dic79⋊C4—24no—
168Dic79⋊C4—24no—
169Dic79⋊C4—24no—
170Dic79⋊C4—28no—
171Dic79⋊C4—24no—
172Dic79⋊C4—24no—
173(C2×D79)⋊C4—28no—
174C4×Dic79—28no—
175C4×Dic79C2×D412yes—
176C4×Dic79C2×D414yes—
177C4×Dic79C2×D414yes—
178C4×Dic79C2412yes—
179C2×Q8×C79—24no—
180C2×Q8×C79—44no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—22no—
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79C2411yescharacteristic
187C2×C8×C79—24no—
188C2×C8×C79C2414yes—
189C4⋊C4×C79—24no—
190C4⋊C4×C79—24no—
191C4⋊C4×C79—24no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—22no—
195C4⋊C4×C79C2412yes—
196C4⋊C4×C79C2414yes—
197C4⋊C4×C79C2412yes—
198C42×C79C2×D412yes—
199C42×C79C2412yes—
200C23×C79C22×D411yescharacteristic
201C22×D79—22no—
202C22×D79—22no—
203C22×D79#13714yes—
204C22×D79C22×D412yes—
205Q8×C79—48no—
206C2×C4×C79—22no—
207C2×C4×C79—22no—
208C2×C4×C79—24no—
209C2×C4×C79—22no—
210C2×C4×C79—22no—
211C2×C4×C79—42no—
212C2×C4×C79—42no—
213C2×C4×C79—44no—
214C2×C4×C79C22×D412yes—
215C2×C4×C79C22×D412yes—
216C2×C4×C79C22×D411yescharacteristic
217C2×C4×C792−1+414yes—
218C2×C4×C79C2511yescharacteristic
219C2×Dic79—22no—
220C2×Dic79—22no—
221C2×Dic79—22no—
222C2×Dic79—24no—
223C2×Dic79—22no—
224C2×Dic79—22no—
225C2×Dic79—44no—
226C2×Dic79—42no—
227C2×Dic79—42no—
228C2×Dic79C22×D412yes—
229C2×Dic79C22×D412yes—
230C2×Dic79C22×D412yes—
231C4×D79—28no—
232C4×D79—24no—
233C4×D79—24no—
234C4×D79—44no—
235C4×D79—48no—
236C4×D79—48no—
237Dic158—28no—
238Dic158—44no—
239C8×C79—24no—
240C79⋊C8—24no—
241C22×C79#53612yes—
242C22×C79C2×#13712yes—
243C22×C79#57512yes—
244C22×C79C23×D411yescharacteristic
245C2×D79—22no—
246C2×D79—28no—
247C2×D79—22no—
248C2×D79C2×#13714yes—
249C4×C79—22no—
250C4×C79—22no—
251C4×C79—42no—
252C4×C79—44no—
253C4×C79—42no—
254C4×C79#53612yes—
255C4×C79C2×2−1+412yes—
256Dic79—22no—
257Dic79—22no—
258Dic79—22no—
259Dic79—44no—
260Dic79—42no—
261Dic79—42no—
262C2×#3236—791nomaximal, Sylow
263C2×C79#323614yescomplemented
264C2×C79C2×#53611yescharacteristic
265C2×C79C22×#13711yescharacteristic
266C2×C79C2×#57511yescharacteristic
267D79—24no—
268C2×#533—791no—
269C22×#125—791no—
270#3236—7916no—
271C2×#501—791no—
272C2×#501D7911yescharacteristic, complemented
273C2×#491—791no—
274C2×#491—791no—
275C2×#484—792no—
276C2×#483—792no—
277C2×#479—792no—
278C2×#479—792no—
279C2×#432—791no—
280C79C2×#323611yescharacteristic, complemented, Sylow
281#533—798no—
282#533—1584no—
283C2×C4⋊Q8—791no—
284C2×C4⋊Q8C2×D7911yescharacteristic
285C2×#125—798no—
286C2×#125—791no—
287C2×#125—791no—
288C2×#125—791no—
289C2×#125—1581no—
290C2×#125—1581no—
291C2×#125—1581no—
292C2×#125—1584no—
293C2×#125C2×D7911yescharacteristic
294C2×C22⋊Q8—792no—
295C2×C22⋊Q8—1581no—
296C2×C4×Q8—792no—
297C2×C4×Q8—1581no—
298C2×#117—1581no—
299C22×C4⋊C4—791no—
300C22×C4⋊C4—791no—
301C22×C4⋊C4—1581no—
302C22×C4⋊C4—1581no—
303C22×C42—791no—
304#501—798no—
305#501C2×D7918yescomplemented
306#491—798no—
307#491—798no—
308#484—7916no—
309#483—7916no—
310#479—7916no—
311#479—7916no—
312#432—798no—
313C2×#107—792no—
314C2×#107C2×D7912yes—
315C2×#106—792no—
316C2×#106C2×D7912yes—
317C2×C4⋊C8—792no—
318C2×#103—792no—
319C2×#103—792no—
320C2×#103—792no—
321C2×#103—792no—
322C2×C22⋊C8—792no—
323C2×#97—791no—
324C2×#97C2×D7911yescharacteristic
325C22×Q8—21no—
326C22×Q8—792no—
327C23×C4—791no—
328C23×C4—791no—
329C23×C4—791no—
330C23×C4—1581no—
331C23×C4—1581no—
332C22×C8—792no—
333C22×C8C22×D7912yes—
334C4⋊Q8—22no—
335C4⋊Q8—794no—
336C4⋊Q8—1582no—
337C4⋊Q8C22×D7914yes—
338#125—22no—
339#125—794no—
340#125—794no—
341#125—794no—
342#125—1588no—
343#125—1588no—
344#125—1582no—
345#125—1582no—
346#125—1588no—
347#125—1582no—
348#125C22×D7914yes—
349C22⋊Q8—798no—
350C22⋊Q8—1584no—
351C22⋊Q8—1588no—
352C4×Q8—798no—
353C4×Q8—1584no—
354C4×Q8—1588no—
355#117—1588no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—791no—
360C2×C4⋊C4—791no—
361C2×C4⋊C4—794no—
362C2×C4⋊C4—794no—
363C2×C4⋊C4—1582no—
364C2×C4⋊C4—1582no—
365C2×C4⋊C4—1588no—
366C2×C4⋊C4—1588no—
367C2×C4⋊C4—1581no—
368C2×C4⋊C4—1581no—
369C2×C4⋊C4—1581no—
370C2×C4⋊C4—1581no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4—1582no—
374C2×C4⋊C4—1582no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4C22×D7911yescharacteristic
380C2×C4⋊C4C22×D7911yescharacteristic
381C2×C4⋊C4C22×D7912yes—
382C2×C22⋊C4—1582no—
383C2×C42—791no—
384C2×C42—791no—
385C2×C42—791no—
386C2×C42—794no—
387C2×C42—794no—
388C2×C42—1582no—
389C2×C42C22×D7911yescharacteristic
390#107—798no—
391#107C22×D7918yes—
392#106—798no—
393#106C22×D7918yes—
394C4⋊C8—798no—
395#103—798no—
396#103—798no—
397#103—798no—
398#103—798no—
399C22⋊C8—798no—
400#97—794no—
401#97C22×D7914yes—
402C24—791no—
403C2×Q8—24no—
404C2×Q8—44no—
405C2×Q8—794no—
406C2×Q8—1588no—
407C2×Q8—1582no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—3162no—
412C22×C4—3164no—
413C22×C4—3164no—
414C22×C4—794no—
415C22×C4—791no—
416C22×C4—791no—
417C22×C4—791no—
418C22×C4—792no—
419C22×C4—794no—
420C22×C4—792no—
421C22×C4—1584no—
422C22×C4—1584no—
423C22×C4—1584no—
424C22×C4—1582no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1584no—
428C22×C4D4×D7911yescharacteristic
429C22×C4D4×D7911yescharacteristic
430C22×C4C23×D7911yescharacteristic
431C2×C8—24no—
432C2×C8—794no—
433C2×C8—1584no—
434C2×C8C23×D7914yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—792no—
442C4⋊C4—792no—
443C4⋊C4—1584no—
444C4⋊C4—1584no—
445C4⋊C4—1584no—
446C4⋊C4—1584no—
447C4⋊C4—1588no—
448C4⋊C4—1584no—
449C4⋊C4—1584no—
450C4⋊C4—1588no—
451C4⋊C4—1584no—
452C4⋊C4—1584no—
453C4⋊C4—1581no—
454C4⋊C4—1581no—
455C4⋊C4—1584no—
456C4⋊C4—1584no—
457C4⋊C4C23×D7912yes—
458C4⋊C4C23×D7912yes—
459C4⋊C4C23×D7914yes—
460C22⋊C4—1588no—
461C42—792no—
462C42—794no—
463C42—792no—
464C42—794no—
465C42—1588no—
466C42D4×D7912yes—
467C42C23×D7912yes—
468C23—794no—
469C23—792no—
470C23—1582no—
471C23—1582no—
472C23C2×D4×D7911yescharacteristic
473Q8—48no—
474Q8—3164no—
475Q8—1588no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—3164no—
485C2×C4—3168no—
486C2×C4—3168no—
487C2×C4—3164no—
488C2×C4—3162no—
489C2×C4—3162no—
490C2×C4—792no—
491C2×C4—792no—
492C2×C4—792no—
493C2×C4—1588no—
494C2×C4—1584no—
495C2×C4—1582no—
496C2×C4—1582no—
497C2×C4—1582no—
498C2×C4—1582no—
499C2×C4—1582no—
500C2×C4—1584no—
501C2×C4—1584no—
502C2×C4C2×D4×D7912yes—
503C2×C4C2×D4×D7912yes—
504C2×C4C2×D4×D7911yescharacteristic
505C2×C4#1457920914yes—
506C2×C4C24×D7911yescharacteristic
507C8—24no—
508C8—1584no—
509C22—794no—
510C22—1582no—
511C22—1588no—
512C22—1582no—
513C22#1909913812yes—
514C22#137×D7912yes—
515C22#1909935912yes—
516C22C22×D4×D7911yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—44no—
522C4—3162no—
523C4—3162no—
524C4—3164no—
525C4—1582no—
526C4—1582no—
527C4—1582no—
528C4#1909913812yes—
529C4C2×#1457920912yes—
530C2—1584no—
531C2#2377460714yesdirect product factor
532C2C2×#1909913811yescharacteristic
533C2C2×#137×D7911yescharacteristic
534C2C2×#1909935911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#501×C79C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
hypercenterC2×#501D79
hypocenterC79C2×#3236
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #23774607C2direct (N × H)
4. C2×#19098969C2semidirect
5. C2×#19098933C2semidirect
6. C2×#19098932C2semidirect
9. C2×#19098873C2semidirect
10. C2×#19098869C2semidirect
12. C2×#501×C79C2semidirect
34. #19098969C22semidirect
35. #19098933C22semidirect
36. #19098932C22semidirect
39. #19098873C22semidirect
40. #19098869C22semidirect
53. #501×C79C22semidirect
263. C2×C79#3236semidirect
272. C2×#501D79semidirect
280. C79C2×#3236semidirect
305. #501C2×D79semidirect
531. C2#23774607direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190991182
C2×#125×D792
#237746072
C2×#190989692
C2×#190989332
C2×#190989322
C2×#190989002
C2×#190988952
C2×#190988732
C2×#190988692
C2×#190986182
C2×#501×C792
C2×#323679