Finite Group Databasebeta

← C2×#23774631 #35160402

Subgroups — C2×#23774631 #35160402

Order 20224.

2018 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D79C211yesmaximal, characteristic, complemented
2C2×#19099098C211yesmaximal, characteristic
3#23774631C2116yesmaximal, direct product factor
4C2×#19098973C211yesmaximal, characteristic, complemented
5C2×#19098925C211yesmaximal, characteristic, complemented
6C2×#19098923C211yesmaximal, characteristic, complemented
7C2×#19098900C212yesmaximal, complemented
8C2×#19098883C212yesmaximal
9C2×#19098847C212yesmaximal, complemented
10C2×#19098842C212yesmaximal, complemented
11C2×#19098618C211yesmaximal, characteristic, complemented
12C2×#501×C79C211yesmaximal, characteristic, complemented
13C22×Q8×D79—21no—
14C2×#14579161C2211yescharacteristic
15C2×#14579160—21no—
16C2×#14579160C2212yes—
17C2×#14579159—21no—
18C4⋊Q8×D79—24no—
19C4⋊Q8×D79C2218yescomplemented
20#19099098—24no—
21#19099098C2218yes—
22C2×#14579156—21no—
23C2×#14579149C2211yescharacteristic
24C2×#14579146—21no—
25C2×#14579141C2211yescharacteristic
26C2×#14579136C2211yescharacteristic
27C2×#14579126C2211yescharacteristic
28C2×#14579123—21no—
29C2×#14579123C2211yescharacteristic
30C2×#14579119—21no—
31C2×C4⋊C4×D79—21no—
32C2×C4⋊C4×D79C2212yes—
33C2×C42×D79C2211yescharacteristic
34#19098973C2218yescomplemented
35#19098925C2218yescomplemented
36#19098923C2218yescomplemented
37#19098900C22116yescomplemented
38#19098883C22116yes—
39#19098847C22116yescomplemented
40#19098842C22116yescomplemented
41#19098618C2218yescomplemented
42C2×#14579094C2212yes—
43C2×#14579092C2212yes—
44C2×Dic79⋊C8C2212yes—
45C2×#14579086C2212yes—
46C2×#14579084C2212yes—
47C2×#14579083C2212yes—
48C2×#14579081C2212yes—
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×Dic158—21no—
58C22×Dic158—22no—
59C22×Dic158C2311yescharacteristic
60C2×Q8×D79—28no—
61C2×Q8×D79—48no—
62C22×D316—21no—
63C22×D316C2311yescharacteristic
64C22×C4×D79—22no—
65C22×C4×D79—21no—
66C22×C4×D79D411yescharacteristic
67C22×C4×D79D411yescharacteristic
68C22×C4×D79C2311yescharacteristic
69C22×C79⋊C8C2312yes—
70#14579161—22no—
71#14579161C2314yes—
72#14579160—24no—
73#14579160—28no—
74#14579160C2318yes—
75#14579159—28no—
76#14579156—28no—
77#14579149—22no—
78#14579149C2314yes—
79#14579146—28no—
80#14579141—22no—
81#14579141C2314yes—
82#14579136—22no—
83#14579136C2314yes—
84#14579126—22no—
85#14579126C2314yes—
86#14579123—28no—
87#14579123—22no—
88#14579123C2314yes—
89#14579119—28no—
90C2×C4⋊Dic79—21no—
91C2×C4⋊Dic79—21no—
92C2×C4⋊Dic79C2312yes—
93C2×Dic79⋊C4—22no—
94C2×Dic79⋊C4—22no—
95C2×Dic79⋊C4—21no—
96C4⋊C4×D79—28no—
97C4⋊C4×D79—24no—
98C4⋊C4×D79C2318yes—
99C2×(C2×D79)⋊C4—21no—
100C2×(C2×D79)⋊C4—21no—
101C2×(C2×D79)⋊C4—21no—
102C2×(C2×D79)⋊C4—21no—
103C2×(C2×D79)⋊C4—21no—
104C2×C4×Dic79—21no—
105C2×C4×Dic79D411yescharacteristic
106C2×C4×Dic79D411yescharacteristic
107C2×C4×Dic79C2311yescharacteristic
108C42×D79D414yes—
109C42×D79C2314yes—
110#14579094C2318yes—
111#14579092C2318yes—
112Dic79⋊C8C2318yes—
113#14579086C2318yes—
114#14579084C2318yes—
115#14579083C2318yes—
116#14579081C2318yes—
117(C2×D79)⋊C8C2318yes—
118#14579072C2314yes—
119C22×Q8×C79—21no—
120C22×C8×C79C2312yes—
121C4⋊Q8×C79—22no—
122C4⋊Q8×C79C2314yes—
123#125×C79—22no—
124#125×C79C2314yes—
125C2×C4⋊C4×C79—21no—
126C2×C4⋊C4×C79—21no—
127C2×C4⋊C4×C79—21no—
128C2×C4⋊C4×C79C2311yescharacteristic
129C2×C4⋊C4×C79C2312yes—
130C2×C4⋊C4×C79C2311yescharacteristic
131C2×C42×C79C2311yescharacteristic
132#107×C79C2318yes—
133#106×C79C2318yes—
134#97×C79C2314yes—
135C23×D79—21no—
136C23×D79C2×D411yescharacteristic
137C2×Dic158—28no—
138C2×Dic158—24no—
139C2×Dic158—21no—
140C2×Dic158—24no—
141C2×Dic158—44no—
142C2×Dic158—48no—
143C2×Dic158C2412yes—
144Q8×D79—432no—
145C2×D316—24no—
146C2×D316—21no—
147C2×D316—24no—
148C2×D316—44no—
149C2×D316C2412yes—
150C22×Dic79—22no—
151C22×Dic79—21no—
152C22×Dic79C2×D411yescharacteristic
153C22×Dic79C2×D411yescharacteristic
154C22×Dic79C2×D411yescharacteristic
155C2×C4×D79—24no—
156C2×C4×D79—24no—
157C2×C4×D79—24no—
158C2×C4×D79—28no—
159C2×C4×D79—24no—
160C2×C4×D79—48no—
161C2×C4×D79—44no—
162C2×C4×D79C2×D414yes—
163C2×C4×D79C2×D412yes—
164C2×C4×D79C2×D414yes—
165C2×C4×D79C2412yes—
166C2×C79⋊C8—24no—
167C2×C79⋊C8C2414yes—
168C4⋊Dic79—24no—
169C4⋊Dic79—24no—
170C4⋊Dic79—22no—
171C4⋊Dic79C2414yes—
172Dic79⋊C4—28no—
173Dic79⋊C4—28no—
174Dic79⋊C4—24no—
175(C2×D79)⋊C4—24no—
176(C2×D79)⋊C4—24no—
177(C2×D79)⋊C4—24no—
178(C2×D79)⋊C4—24no—
179(C2×D79)⋊C4—24no—
180C4×Dic79—24no—
181C4×Dic79C2×D412yes—
182C4×Dic79C2×D414yes—
183C4×Dic79C2×D414yes—
184C4×Dic79C2412yes—
185C2×Q8×C79—24no—
186C2×Q8×C79—44no—
187C22×C4×C79—22no—
188C22×C4×C79—21no—
189C22×C4×C79—21no—
190C22×C4×C79C2×D411yescharacteristic
191C22×C4×C79C2×D411yescharacteristic
192C22×C4×C79C2411yescharacteristic
193C2×C8×C79—24no—
194C2×C8×C79C2414yes—
195C4⋊C4×C79—24no—
196C4⋊C4×C79—24no—
197C4⋊C4×C79—24no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—22no—
201C4⋊C4×C79C2412yes—
202C4⋊C4×C79C2414yes—
203C4⋊C4×C79C2412yes—
204C42×C79C2×D412yes—
205C42×C79C2412yes—
206C23×C79C22×D411yescharacteristic
207C22×D79—22no—
208C22×D79—22no—
209C22×D79—22no—
210C22×D79—42no—
211C22×D79—42no—
212C22×D79—42no—
213C22×D79#13714yes—
214C22×D79C22×D412yes—
215Q8×C79—48no—
216C2×C4×C79—22no—
217C2×C4×C79—22no—
218C2×C4×C79—22no—
219C2×C4×C79—22no—
220C2×C4×C79—24no—
221C2×C4×C79—44no—
222C2×C4×C79—42no—
223C2×C4×C79—42no—
224C2×C4×C79C22×D412yes—
225C2×C4×C79C22×D412yes—
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C792−1+414yes—
228C2×C4×C79C2511yescharacteristic
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—24no—
233C2×Dic79—42no—
234C2×Dic79—44no—
235C2×Dic79AGL(1,ℤ/8)14yes—
236C2×Dic79C22×D412yes—
237C2×Dic79C22×D412yes—
238C2×Dic79C22×D412yes—
239D316—24no—
240D316—48no—
241D316—42no—
242C4×D79—28no—
243C4×D79—24no—
244C4×D79—24no—
245C4×D79—28no—
246C4×D79—416no—
247C4×D79—48no—
248Dic158—24no—
249Dic158—48no—
250Dic158—42no—
251Dic158—416no—
252C8×C79—24no—
253C79⋊C8—24no—
254C22×C79#53612yes—
255C22×C79C2×AGL(1,ℤ/8)12yes—
256C22×C79C2×#13712yes—
257C22×C79C23×D411yescharacteristic
258C2×D79—22no—
259C2×D79—28no—
260C2×D79—22no—
261C2×D79—42no—
262C2×D79—42no—
263C2×D79—42no—
264C2×D79—84no—
265C2×D79C2×#13714yes—
266C4×C79—22no—
267C4×C79—22no—
268C4×C79—44no—
269C4×C79—42no—
270C4×C79—42no—
271C4×C79#53612yes—
272C4×C79C2×2−1+412yes—
273Dic79—22no—
274Dic79—22no—
275Dic79—42no—
276Dic79—44no—
277Dic79C2×AGL(1,ℤ/8)14yes—
278C2×#3239—791nomaximal, Sylow
279C2×C79#323914yescomplemented
280C2×C79C2×#53611yescharacteristic
281C2×C79C22×AGL(1,ℤ/8)11yescharacteristic
282C2×C79C22×#13711yescharacteristic
283D79—24no—
284D79—82no—
285C22×C4⋊Q8—791no—
286C2×#529—791no—
287#3239—7916no—
288C2×#501—791no—
289C2×#501D7911yescharacteristic, complemented
290C2×#489—791no—
291C2×#489—791no—
292C2×#484—792no—
293C2×#481—792no—
294C2×#475—792no—
295C2×#474—792no—
296C2×#432—791no—
297C79C2×#323911yescharacteristic, complemented, Sylow
298C23×Q8—1581no—
299C2×C4⋊Q8—798no—
300C2×C4⋊Q8—792no—
301C2×C4⋊Q8—791no—
302C2×C4⋊Q8—1581no—
303C2×C4⋊Q8—1584no—
304C2×C4⋊Q8—1581no—
305C2×C4⋊Q8C2×D7911yescharacteristic
306#529—798no—
307#529—1584no—
308C2×#126—1581no—
309C2×#125C2×D7911yescharacteristic
310C2×#124—791no—
311C2×#123—1581no—
312C2×C22⋊Q8—791no—
313C2×#121—791no—
314C2×C4×Q8—791no—
315C2×C4×D4—791no—
316C2×C4×D4—1581no—
317C2×#117—1581no—
318C22×C4⋊C4—792no—
319C22×C4⋊C4—1581no—
320C22×C42—791no—
321#501—798no—
322#501C2×D7918yescomplemented
323#489—798no—
324#489—798no—
325#484—7916no—
326#481—7916no—
327#475—7916no—
328#474—7916no—
329#432—798no—
330C2×#107—792no—
331C2×#107C2×D7912yes—
332C2×#106—792no—
333C2×#106C2×D7912yes—
334C2×C4⋊C8—792no—
335C2×#103—792no—
336C2×#103—792no—
337C2×#102—792no—
338C2×#102—792no—
339C2×C22⋊C8—792no—
340C2×#97—791no—
341C2×#97C2×D7911yescharacteristic
342C22×Q8—21no—
343C22×Q8—3168no—
344C22×Q8—791no—
345C22×Q8—1582no—
346C22×Q8—1588no—
347C22×Q8—1581no—
348C22×D4—791no—
349C22×D4—1581no—
350C23×C4—791no—
351C23×C4—791no—
352C23×C4—791no—
353C23×C4—1581no—
354C23×C4—1582no—
355C22×C8—792no—
356C22×C8C22×D7912yes—
357C4⋊Q8—22no—
358C4⋊Q8—798no—
359C4⋊Q8—794no—
360C4⋊Q8—1588no—
361C4⋊Q8—1584no—
362C4⋊Q8—1588no—
363C4⋊Q8—1582no—
364C4⋊Q8C22×D7914yes—
365#126—1588no—
366#125—22no—
367#125C22×D7914yes—
368#124—794no—
369#124—1582no—
370#123—1588no—
371C22⋊Q8—794no—
372C22⋊Q8—1582no—
373#121—794no—
374#121—1582no—
375C4×Q8—794no—
376C4×Q8—1582no—
377C4×D4—794no—
378C4×D4—1582no—
379C4×D4—1588no—
380#117—1588no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—792no—
385C2×C4⋊C4—798no—
386C2×C4⋊C4—1584no—
387C2×C4⋊C4—1588no—
388C2×C4⋊C4—1582no—
389C2×C4⋊C4—1582no—
390C2×C4⋊C4—1581no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4C22×D7912yes—
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C22⋊C4—1581no—
397C2×C22⋊C4—1581no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C22⋊C4—1581no—
401C2×C42—791no—
402C2×C42—791no—
403C2×C42—791no—
404C2×C42—794no—
405C2×C42—794no—
406C2×C42—1581no—
407C2×C42C22×D7911yescharacteristic
408#107—798no—
409#107C22×D7918yes—
410#106—798no—
411#106C22×D7918yes—
412C4⋊C8—798no—
413#103—798no—
414#103—798no—
415#102—798no—
416#102—798no—
417C22⋊C8—798no—
418#97—794no—
419#97C22×D7914yes—
420C24—791no—
421C24—1581no—
422C2×Q8—24no—
423C2×Q8—44no—
424C2×Q8—3164no—
425C2×Q8—3168no—
426C2×Q8—31632no—
427C2×Q8—792no—
428C2×Q8—1588no—
429C2×Q8—1584no—
430C2×Q8—1581no—
431C2×Q8—1584no—
432C2×D4—3164no—
433C2×D4—792no—
434C2×D4—1584no—
435C2×D4—1584no—
436C2×D4—1581no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—3168no—
441C22×C4—3164no—
442C22×C4—791no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—792no—
446C22×C4—792no—
447C22×C4—794no—
448C22×C4—794no—
449C22×C4—1584no—
450C22×C4—1588no—
451C22×C4—1584no—
452C22×C4—1584no—
453C22×C4—1581no—
454C22×C4—1582no—
455C22×C4—1584no—
456C22×C4D4×D7911yescharacteristic
457C22×C4D4×D7911yescharacteristic
458C22×C4C23×D7911yescharacteristic
459C2×C8—24no—
460C2×C8—794no—
461C2×C8—1584no—
462C2×C8C23×D7914yes—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—794no—
470C4⋊C4—1588no—
471C4⋊C4—1588no—
472C4⋊C4—1584no—
473C4⋊C4—1584no—
474C4⋊C4—1582no—
475C4⋊C4—1584no—
476C4⋊C4C23×D7914yes—
477C4⋊C4C23×D7912yes—
478C4⋊C4C23×D7912yes—
479C22⋊C4—1584no—
480C22⋊C4—1584no—
481C22⋊C4—1584no—
482C22⋊C4—1584no—
483C22⋊C4—1584no—
484C42—792no—
485C42—794no—
486C42—792no—
487C42—794no—
488C42—1584no—
489C42D4×D7912yes—
490C42C23×D7912yes—
491C23—3162no—
492C23—3162no—
493C23—3162no—
494C23—794no—
495C23—792no—
496C23—1582no—
497C23—1582no—
498C23—1582no—
499C23C2×D4×D7911yescharacteristic
500Q8—48no—
501Q8—3168no—
502Q8—31616no—
503Q8—3162no—
504Q8—1584no—
505D4—3168no—
506D4—3162no—
507D4—1584no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—31616no—
517C2×C4—3168no—
518C2×C4—3162no—
519C2×C4—3164no—
520C2×C4—794no—
521C2×C4—792no—
522C2×C4—792no—
523C2×C4—792no—
524C2×C4—1588no—
525C2×C4—1588no—
526C2×C4—1582no—
527C2×C4—1584no—
528C2×C4—1582no—
529C2×C4—1582no—
530C2×C4—1584no—
531C2×C4—1584no—
532C2×C4C2×D4×D7912yes—
533C2×C4C2×D4×D7912yes—
534C2×C4C2×D4×D7911yescharacteristic
535C2×C4#1457920914yes—
536C2×C4C24×D7911yescharacteristic
537C8—24no—
538C8—1584no—
539C22—3162no—
540C22—3162no—
541C22—3162no—
542C22—794no—
543C22—6324no—
544C22—1582no—
545C22—1588no—
546C22—1582no—
547C22#1909913812yes—
548C22AGL(1,ℤ/8)×D7912yes—
549C22#137×D7912yes—
550C22C22×D4×D7911yescharacteristic
551C4—22no—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—3164no—
557C4—3162no—
558C4—794no—
559C4—1582no—
560C4—1582no—
561C4#1909913812yes—
562C4C2×#1457920912yes—
563C2—6322no—
564C2—1584no—
565C2#2377463114yesdirect product factor
566C2C2×#1909913811yescharacteristic
567C2C2×AGL(1,ℤ/8)×D7911yescharacteristic
568C2C2×#137×D7911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#501×C79C2
socleC23×C79C22×D4
prime radicalC2×#14579136C22
hypercenterC2×#501D79
hypocenterC79C2×#3239
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
1. C2×C4⋊Q8×D79C2semidirect
3. #23774631C2direct (N × H)
4. C2×#19098973C2semidirect
5. C2×#19098925C2semidirect
6. C2×#19098923C2semidirect
7. C2×#19098900C2semidirect
9. C2×#19098847C2semidirect
10. C2×#19098842C2semidirect
11. C2×#19098618C2semidirect
12. C2×#501×C79C2semidirect
19. C4⋊Q8×D79C22semidirect
34. #19098973C22semidirect
35. #19098925C22semidirect
36. #19098923C22semidirect
37. #19098900C22semidirect
39. #19098847C22semidirect
40. #19098842C22semidirect
41. #19098618C22semidirect
53. #501×C79C22semidirect
279. C2×C79#3239semidirect
289. C2×#501D79semidirect
297. C79C2×#3239semidirect
322. #501C2×D79semidirect
565. C2#23774631direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D792
C2×#190990982
#237746312
C2×#190989732
C2×#190989252
C2×#190989232
C2×#190989002
C2×#190988832
C2×#190988472
C2×#190988422
C2×#190986182
C2×#501×C792
C2×#323979