Finite Group Databasebeta

← C2×#23761937 #34255510

Subgroups — C2×#23761937 #34255510

Order 20224.

1098 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099099C211yesmaximal, characteristic
2C2×#19098617C211yesmaximal, characteristic, complemented
3C2×#19098427C211yesmaximal, characteristic, complemented
4#23761937C218yesmaximal, direct product factor
5C2×#19098300C211yesmaximal, characteristic, complemented
6C2×#19098298C211yesmaximal, characteristic
7C2×#19098296C211yesmaximal, characteristic
8C2×#328×C79C211yesmaximal, characteristic, complemented
9#19099099—22no—
10#19099099—22no—
11#19099099C414yescomplemented
12#19099099C2214yes—
13C2×#14579156—21no—
14C2×#14579152C411yescharacteristic, complemented
15C2×#14579150C411yescharacteristic, complemented
16C2×#14579145—21no—
17C2×#14579135C2211yescharacteristic
18C2×Q8×Dic79C411yescharacteristic, complemented
19C2×#14579123C411yescharacteristic, complemented
20C2×(C22⋊D79)⋊C4—21no—
21C2×#14579119—21no—
22C22×Dic79⋊C4—21no—
23C22×C4×Dic79C2211yescharacteristic
24#19098617—22no—
25#19098617C2214yescomplemented
26C2×Dic79⋊C8—21no—
27C2×C22⋊(C79⋊C8)C2211yescharacteristic
28C2×#14579071—21no—
29#19098427—22no—
30#19098427C2214yescomplemented
31C2×C79⋊#95—21no—
32C2×C79⋊#95C2211yescharacteristic
33#19098300C2214yescomplemented
34#19098298C2214yes—
35#19098296C2214yes—
36C2×C22⋊Q8×C79C2211yescharacteristic
37C2×C22⋊C8×C79C2211yescharacteristic
38C2×#95×C79C2211yescharacteristic
39#328×C79C2214yescomplemented
40C22×D316C2×C411yescharacteristic
41C22×C22⋊D79—21no—
42C23×Dic79—21no—
43C23×Dic79D411yescharacteristic
44C23×Dic79D411yescharacteristic
45C22×C4×D79—21no—
46C22×C79⋊C8—21no—
47#14579156—24no—
48#14579156—24no—
49#14579152—22no—
50#14579152C2×C414yescomplemented
51#14579150—22no—
52#14579150C2×C414yescomplemented
53#14579145—24no—
54#14579145—24no—
55#14579135—21no—
56#14579135—21no—
57#14579135C2×C412yes—
58#14579135C2312yes—
59Q8×Dic79—22no—
60Q8×Dic79C2×C414yescomplemented
61#14579123—22no—
62#14579123C2×C414yescomplemented
63(C22⋊D79)⋊C4—24no—
64(C22⋊D79)⋊C4—24no—
65#14579119—24no—
66#14579119—24no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79C2×C411yescharacteristic
70C2×Dic79⋊C4—21no—
71C2×Dic79⋊C4—24no—
72C2×Dic79⋊C4—44no—
73C2×(C2×D79)⋊C4—21no—
74C2×(C2×D79)⋊C4—21no—
75C2×(C2×D79)⋊C4—21no—
76C2×(C2×D79)⋊C4—21no—
77C2×C4×Dic79—21no—
78C2×C4×Dic79C2×C411yescharacteristic
79C2×C4×Dic79C2×C411yescharacteristic
80C2×C4×Dic79C2×C412yes—
81C2×C4×Dic79D412yes—
82C2×C4×Dic79D411yescharacteristic
83C2×C4×Dic79D411yescharacteristic
84C2×C4×Dic79D412yes—
85C2×C4×Dic79C2312yes—
86Dic79⋊C8—24no—
87C22⋊(C79⋊C8)—21no—
88C22⋊(C79⋊C8)C2312yes—
89#14579071—24no—
90C79⋊#95—21no—
91C79⋊#95—24no—
92C79⋊#95C2312yes—
93C22×Q8×C79C2×C411yescharacteristic
94C23×C4×C79—21no—
95C23×C4×C79C2311yescharacteristic
96C22×C8×C79—21no—
97C22⋊Q8×C79—21no—
98C22⋊Q8×C79—21no—
99C22⋊Q8×C79C2×C412yes—
100C22⋊Q8×C79C2312yes—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79C2×C411yescharacteristic
103C2×C22⋊C4×C79—21no—
104C22⋊C8×C79—21no—
105C22⋊C8×C79C2312yes—
106#95×C79—21no—
107#95×C79C2312yes—
108C23×D79—21no—
109C2×D316—21no—
110C2×D316—42no—
111C2×D316C22×C412yes—
112C2×C22⋊D79—22no—
113C2×C22⋊D79—22no—
114C2×C22⋊D79—42no—
115C2×C22⋊D79—42no—
116C22×Dic79—21no—
117C22×Dic79—22no—
118C22×Dic79—22no—
119C22×Dic79—22no—
120C22×Dic79—42no—
121C22×Dic79—42no—
122C22×Dic79—42no—
123C22×Dic79—42no—
124C22×Dic79C22⋊C411yescharacteristic
125C22×Dic79C22⋊C411yescharacteristic
126C22×Dic79C22⋊C411yescharacteristic
127C22×Dic79C22⋊C411yescharacteristic
128C22×Dic79C22⋊C412yes—
129C22×Dic79C22⋊C412yes—
130C22×Dic79C2×D412yes—
131C22×Dic79C2×D412yes—
132C2×C4×D79—24no—
133C2×C4×D79—44no—
134C2×C79⋊C8—22no—
135C2×C79⋊C8—42no—
136C4⋊Dic79—24no—
137C4⋊Dic79—24no—
138C4⋊Dic79—21no—
139C4⋊Dic79C22×C412yes—
140Dic79⋊C4—24no—
141Dic79⋊C4—48no—
142(C2×D79)⋊C4—24no—
143(C2×D79)⋊C4—24no—
144(C2×D79)⋊C4—24no—
145(C2×D79)⋊C4—22no—
146(C2×D79)⋊C4—22no—
147C4×Dic79—24no—
148C4×Dic79C22⋊C412yes—
149C4×Dic79C22⋊C412yes—
150C4×Dic79C22⋊C412yes—
151C4×Dic79C22⋊C412yes—
152C4×Dic79C22×C412yes—
153C4×Dic79C22×C412yes—
154C4×Dic79C2×D412yes—
155C4×Dic79C2×D412yes—
156C24×C79C2×D411yescharacteristic
157C2×Q8×C79—21no—
158C2×Q8×C79—42no—
159C2×Q8×C79C22×C412yes—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—22no—
163C22×C4×C79—42no—
164C22×C4×C79—42no—
165C22×C4×C79—42no—
166C22×C4×C79—42no—
167C22×C4×C79C22×C411yescharacteristic
168C22×C4×C79C22×C411yescharacteristic
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2411yescharacteristic
173C2×C8×C79—22no—
174C2×C8×C79—42no—
175C4⋊C4×C79—22no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—22no—
178C4⋊C4×C79C22×C412yes—
179C22⋊C4×C79—22no—
180C22⋊C4×C79—22no—
181C23×C79—22no—
182C23×C79—22no—
183C23×C79C2×C22⋊C411yescharacteristic
184C23×C79C2×C22⋊C411yescharacteristic
185C23×C79AGL(1,ℤ/8)12yes—
186C23×C79#13712yes—
187C23×C79C22×D411yescharacteristic
188C22×D79—22no—
189C22×D79—42no—
190C22×D79—42no—
191C22×D79—42no—
192Q8×C79—42no—
193Q8×C79—42no—
194C2×C4×C79—22no—
195C2×C4×C79—22no—
196C2×C4×C79—22no—
197C2×C4×C79—22no—
198C2×C4×C79—41no—
199C2×C4×C79—41no—
200C2×C4×C79—41no—
201C2×C4×C79—41no—
202C2×C4×C79—42no—
203C2×C4×C79—42no—
204C2×C4×C79—42no—
205C2×C4×C79—42no—
206C2×C4×C79—42no—
207C2×C4×C79—84no—
208C2×C4×C79C2×C22⋊C411yescharacteristic
209C2×C4×C79C2×C22⋊C411yescharacteristic
210C2×C4×C79C23×C411yescharacteristic
211C2×C4×C79C22×D411yescharacteristic
212C22⋊D79—42no—
213C22⋊D79—42no—
214C22⋊D79—44no—
215C2×Dic79—22no—
216C2×Dic79—22no—
217C2×Dic79—24no—
218C2×Dic79—22no—
219C2×Dic79—22no—
220C2×Dic79—22no—
221C2×Dic79—22no—
222C2×Dic79—22no—
223C2×Dic79—42no—
224C2×Dic79—42no—
225C2×Dic79—42no—
226C2×Dic79—42no—
227C2×Dic79—42no—
228C2×Dic79—42no—
229C2×Dic79—84no—
230C2×Dic79C2×C22⋊C412yes—
231C2×Dic79C2×C22⋊C412yes—
232C2×Dic79C2×C22⋊C412yes—
233C2×Dic79C2×C22⋊C412yes—
234D316—42no—
235D316—42no—
236C4×D79—48no—
237C8×C79—42no—
238C79⋊C8—42no—
239C22×C79—21no—
240C22×C79—21no—
241C22×C79—21no—
242C22×C79—21no—
243C22×C79—24no—
244C22×C79—24no—
245C22×C79#41012yes—
246C22×C79#41712yes—
247C22×C79#41712yes—
248C22×C79#42112yes—
249C22×C79C22×C22⋊C411yescharacteristic
250C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C79C2×#13711yescharacteristic
252C2×D79—42no—
253C2×D79—42no—
254C2×D79—42no—
255C2×D79—84no—
256C4×C79—21no—
257C4×C79—21no—
258C4×C79—21no—
259C4×C79—21no—
260C4×C79—42no—
261C4×C79—42no—
262C4×C79—82no—
263Dic79—22no—
264Dic79—22no—
265Dic79—22no—
266Dic79—22no—
267Dic79—42no—
268Dic79—82no—
269C2×#1512—791nomaximal, Sylow
270C2×C79—21no—
271C2×C79—21no—
272C2×C79—22no—
273C2×C79#151214yescomplemented
274C2×C79C2×#41011yescharacteristic
275C2×C79C2×#41711yescharacteristic
276C2×C79C2×#42111yescharacteristic
277D79—82no—
278C2×#529—791no—
279C2×#432—791no—
280C2×#382—791no—
281#1512—798no—
282C2×#328—791no—
283C2×#328D7911yescharacteristic, complemented
284C2×#327—791no—
285C2×#327—791no—
286C79C2×#151211yescharacteristic, complemented, Sylow
287#529—794no—
288#529—794no—
289#529—1582no—
290#529—1582no—
291C2×#126—1581no—
292C2×#125—791no—
293C2×#124—791no—
294C2×#123—1581no—
295C2×C22⋊Q8C2×D7911yescharacteristic
296C2×#121—791no—
297C2×C4×Q8—791no—
298C2×C4×D4—791no—
299C2×C4×D4—1581no—
300C2×#117—1581no—
301C22×C4⋊C4—1581no—
302C22×C42—791no—
303#432—794no—
304#432—1582no—
305C2×C4⋊C8—1581no—
306C2×C22⋊C8—791no—
307C2×C22⋊C8C2×D7911yescharacteristic
308C2×#97—1581no—
309#382—794no—
310#382—1582no—
311C2×#95—791no—
312C2×#95—1581no—
313C2×#95C2×D7911yescharacteristic
314#328—794no—
315#328C2×D7914yescomplemented
316#327—794no—
317#327—794no—
318C22×Q8C4×D7911yescharacteristic
319C22×D4—791no—
320C22×D4—1581no—
321C23×C4—21no—
322C23×C4—791no—
323C23×C4—791no—
324C23×C4—1581no—
325C23×C4—1581no—
326C23×C4C22×D7911yescharacteristic
327C22×C8—21no—
328C22×C8—1581no—
329#126—1584no—
330#126—1584no—
331#125—794no—
332#125—1582no—
333#124—794no—
334#124—1582no—
335#123—1584no—
336#123—1584no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8C4×D7912yes—
340C22⋊Q8C22×D7912yes—
341#121—792no—
342#121—792no—
343#121—1581no—
344#121—1581no—
345C4×Q8—794no—
346C4×Q8—1582no—
347C4×D4—794no—
348C4×D4—1582no—
349C4×D4—1584no—
350C4×D4—1584no—
351#117—1584no—
352#117—1584no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—3164no—
355C2×C4⋊C4—791no—
356C2×C4⋊C4—1584no—
357C2×C4⋊C4—1581no—
358C2×C4⋊C4—1581no—
359C2×C4⋊C4—1581no—
360C2×C4⋊C4C4×D7911yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—1581no—
363C2×C22⋊C4—1581no—
364C2×C22⋊C4—1581no—
365C2×C22⋊C4—1581no—
366C2×C42—791no—
367C2×C42—791no—
368C2×C42—792no—
369C2×C42—791no—
370C2×C42—791no—
371C2×C42—792no—
372C2×C42—792no—
373C2×C42—792no—
374C2×C42—1581no—
375C4⋊C8—1584no—
376C22⋊C8—21no—
377C22⋊C8—792no—
378C22⋊C8—1581no—
379C22⋊C8C22×D7912yes—
380#97—1584no—
381#95—21no—
382#95—792no—
383#95—1584no—
384#95—1581no—
385#95C22×D7912yes—
386C24—1581no—
387C24D4×D7911yescharacteristic
388C2×Q8—21no—
389C2×Q8—42no—
390C2×Q8C2×C4×D7912yes—
391C2×D4—3162no—
392C2×D4—3162no—
393C2×D4—3162no—
394C2×D4—792no—
395C2×D4—1582no—
396C2×D4—1582no—
397C2×D4—1581no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—3164no—
406C22×C4—3162no—
407C22×C4—3162no—
408C22×C4—3162no—
409C22×C4—3162no—
410C22×C4—792no—
411C22×C4—792no—
412C22×C4—792no—
413C22×C4—791no—
414C22×C4—791no—
415C22×C4—791no—
416C22×C4—791no—
417C22×C4—792no—
418C22×C4—1584no—
419C22×C4—1582no—
420C22×C4—1581no—
421C22×C4—1582no—
422C22×C4—1582no—
423C22×C4C2×C4×D7911yescharacteristic
424C22×C4C2×C4×D7911yescharacteristic
425C22×C4D4×D7911yescharacteristic
426C22×C4D4×D7911yescharacteristic
427C22×C4D4×D7911yescharacteristic
428C22×C4C23×D7911yescharacteristic
429C2×C8—22no—
430C2×C8—42no—
431C2×C8—3162no—
432C2×C8—1582no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—3168no—
437C4⋊C4—792no—
438C4⋊C4—1584no—
439C4⋊C4—1584no—
440C4⋊C4—1584no—
441C4⋊C4—1581no—
442C4⋊C4C2×C4×D7912yes—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—1584no—
446C22⋊C4—1584no—
447C22⋊C4—1584no—
448C22⋊C4—1582no—
449C22⋊C4—1582no—
450C42—792no—
451C42—792no—
452C42—792no—
453C42—792no—
454C42—792no—
455C42—792no—
456C42—792no—
457C42—792no—
458C42—1584no—
459C23—22no—
460C23—22no—
461C23—3162no—
462C23—3162no—
463C23—3162no—
464C23—1582no—
465C23C22⋊C4×D7911yescharacteristic
466C23C22⋊C4×D7911yescharacteristic
467C23#1457918812yes—
468C23#1457919312yes—
469C23C2×D4×D7911yescharacteristic
470Q8—42no—
471Q8—42no—
472D4—3162no—
473D4—3162no—
474D4—3164no—
475D4—3162no—
476D4—3162no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—84no—
491C2×C4—3168no—
492C2×C4—3162no—
493C2×C4—3162no—
494C2×C4—3162no—
495C2×C4—3162no—
496C2×C4—3162no—
497C2×C4—3162no—
498C2×C4—792no—
499C2×C4—792no—
500C2×C4—792no—
501C2×C4—792no—
502C2×C4—6324no—
503C2×C4—1582no—
504C2×C4—1584no—
505C2×C4—1582no—
506C2×C4—1582no—
507C2×C4—1582no—
508C2×C4—1582no—
509C2×C4—1582no—
510C2×C4—1582no—
511C2×C4C22⋊C4×D7911yescharacteristic
512C2×C4C22⋊C4×D7911yescharacteristic
513C2×C4C22×C4×D7911yescharacteristic
514C2×C4C2×D4×D7911yescharacteristic
515C8—42no—
516C8—3162no—
517C22—24no—
518C22—24no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—3162no—
524C22—3162no—
525C22—3162no—
526C22—6324no—
527C22#1909853512yes—
528C22#1909856312yes—
529C22#1909856312yes—
530C22#1909857812yes—
531C22C2×C22⋊C4×D7911yescharacteristic
532C22C2×#1457918811yescharacteristic
533C22C2×#1457919311yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—3162no—
542C4—6322no—
543C4—1582no—
544C4—1582no—
545C4—1582no—
546C4—1582no—
547C2—21no—
548C2—22no—
549C2—21no—
550C2—6322no—
551C2#2376193714yesdirect product factor
552C2C2×#1909853511yescharacteristic
553C2C2×#1909856311yescharacteristic
554C2C2×#1909857811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#328×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC22×C4×C79C24
hypercenterC2×#328D79
hypocenterC79C2×#1512
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×#19098617C2semidirect
3. C2×#19098427C2semidirect
4. #23761937C2direct (N × H)
5. C2×#19098300C2semidirect
8. C2×#328×C79C2semidirect
11. #19099099C4semidirect
14. C2×#14579152C4semidirect
15. C2×#14579150C4semidirect
18. C2×Q8×Dic79C4semidirect
19. C2×#14579123C4semidirect
25. #19098617C22semidirect
30. #19098427C22semidirect
33. #19098300C22semidirect
39. #328×C79C22semidirect
50. #14579152C2×C4semidirect
52. #14579150C2×C4semidirect
60. Q8×Dic79C2×C4semidirect
62. #14579123C2×C4semidirect
273. C2×C79#1512semidirect
283. C2×#328D79semidirect
286. C79C2×#1512semidirect
315. #328C2×D79semidirect
551. C2#23761937direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190990992
C2×#190986172
C2×#190984272
#237619372
C2×#190983002
C2×#190982982
C2×#190982962
C2×#328×C792
C2×#151279