Finite Group Databasebeta

← C2×#23774735 #35161072

Subgroups — C2×#23774735 #35161072

Order 20224.

1930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099118C211yesmaximal, characteristic, complemented
2C2×#19099100C211yesmaximal, characteristic
3#23774735C2116yesmaximal, direct product factor
4C2×#19098872C212yesmaximal, complemented
5C2×#19098848C211yesmaximal, characteristic, complemented
6C2×#19098844C211yesmaximal, characteristic, complemented
7C2×#19098801C212yesmaximal, complemented
8C2×#19098795C212yesmaximal
9C2×#19098787C212yesmaximal, complemented
10C2×#19098765C211yesmaximal, characteristic, complemented
11C2×C4⋊C8×D79C211yesmaximal, characteristic, complemented
12C2×#462×C79C211yesmaximal, characteristic, complemented
13C22×#11929611C2212yes—
14C22×C8×D79C2212yes—
15#19099118—24no—
16#19099118C2218yescomplemented
17C2×#14579161C2211yescharacteristic
18#19099100—24no—
19#19099100C2218yes—
20C2×#14579156—21no—
21C2×#14579152—21no—
22C2×#14579152C2211yescharacteristic
23C2×#14579150C2211yescharacteristic
24C2×#14579146—21no—
25C2×#14579141—21no—
26C2×#14579137C2212yes—
27C2×#14579137C2211yescharacteristic
28C2×#14579136C2211yescharacteristic
29C2×Q8×Dic79C2212yes—
30C2×#14579126—21no—
31C2×C4×D316—21no—
32C2×C4×D316C2211yescharacteristic
33C2×#14579119—21no—
34C2×#14579118—21no—
35C2×C42×D79C2211yescharacteristic
36#19098872C22116yescomplemented
37#19098848C2218yescomplemented
38#19098844C2218yescomplemented
39#19098801C22116yescomplemented
40#19098795C22116yes—
41#19098787C22116yescomplemented
42#19098765C2218yescomplemented
43C2×#14579095C2212yes—
44C2×#14579092C2212yes—
45C2×C4⋊(C79⋊C8)C2211yescharacteristic
46C2×Dic79⋊C8C2212yes—
47C4⋊C8×D79C2218yescomplemented
48C2×#14579085C2212yes—
49C2×#14579085C2212yes—
50C2×#14579083C2212yes—
51C22×Q16×C79C2212yes—
52C2×C4⋊Q8×C79C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54#462×C79C2218yescomplemented
55C2×C4⋊C8×C79C2211yescharacteristic
56C2×#103×C79C2212yes—
57C22×Dic158—21no—
58C22×D316—21no—
59C22×D316C2311yescharacteristic
60C22×C4×D79—21no—
61C22×C4×D79D411yescharacteristic
62C22×C4×D79D411yescharacteristic
63C22×C4×D79C2311yescharacteristic
64C2×#11929611—44no—
65C2×#11929611C2318yes—
66C22×C79⋊C8C2312yes—
67C2×C8×D79—44no—
68C2×C8×D79C2318yes—
69#14579161—22no—
70#14579161C2314yes—
71#14579156—28no—
72#14579152—28no—
73#14579152—22no—
74#14579152C2314yes—
75#14579150—22no—
76#14579150C2314yes—
77#14579146—28no—
78#14579141—28no—
79#14579137—22no—
80#14579137—24no—
81#14579137C2318yes—
82#14579137C2314yes—
83#14579136—22no—
84#14579136C2314yes—
85Q8×Dic79—24no—
86Q8×Dic79C2318yes—
87#14579126—28no—
88C4×D316—22no—
89C4×D316—28no—
90C4×D316C2314yes—
91#14579119—28no—
92#14579118—28no—
93C2×C4⋊Dic79—21no—
94C2×C4⋊Dic79—21no—
95C2×C4⋊Dic79—22no—
96C2×C4⋊Dic79C2311yescharacteristic
97C2×C4⋊Dic79C2312yes—
98C2×Dic79⋊C4—21no—
99C2×Dic79⋊C4—21no—
100C2×Dic79⋊C4—22no—
101C2×(C2×D79)⋊C4—21no—
102C2×(C2×D79)⋊C4—22no—
103C2×(C2×D79)⋊C4—21no—
104C2×(C2×D79)⋊C4—21no—
105C2×(C2×D79)⋊C4—21no—
106C2×(C2×D79)⋊C4—21no—
107C2×C4×Dic79—22no—
108C2×C4×Dic79D411yescharacteristic
109C2×C4×Dic79D411yescharacteristic
110C2×C4×Dic79C2311yescharacteristic
111C42×D79D414yes—
112C42×D79C2314yes—
113#14579095C2318yes—
114#14579092C2318yes—
115C4⋊(C79⋊C8)C2314yes—
116Dic79⋊C8C2318yes—
117#14579085C2318yes—
118#14579085C2318yes—
119#14579083C2318yes—
120C22×Q8×C79C2311yescharacteristic
121C22×Q8×C79C2312yes—
122C2×Q16×C79—44no—
123C2×Q16×C79C2318yes—
124C22×C8×C79C2312yes—
125C4⋊Q8×C79—22no—
126C4⋊Q8×C79C2314yes—
127C4×Q8×C79—22no—
128C4×Q8×C79C2314yes—
129C2×C4⋊C4×C79—21no—
130C2×C4⋊C4×C79—21no—
131C2×C4⋊C4×C79—21no—
132C2×C4⋊C4×C79C2311yescharacteristic
133C2×C42×C79—21no—
134C2×C42×C79C2311yescharacteristic
135C4⋊C8×C79C2314yes—
136#103×C79C2318yes—
137C23×D79—21no—
138C23×D79C2×D411yescharacteristic
139C2×Dic158—24no—
140C2×Dic158—44no—
141C2×D316—24no—
142C2×D316—21no—
143C2×D316—22no—
144C2×D316—22no—
145C2×D316—44no—
146C2×D316C2412yes—
147C22×Dic79—22no—
148C22×Dic79—21no—
149C22×Dic79C2×D411yescharacteristic
150C22×Dic79C2×D411yescharacteristic
151C22×Dic79C2×D411yescharacteristic
152C2×C4×D79—24no—
153C2×C4×D79—24no—
154C2×C4×D79—24no—
155C2×C4×D79—22no—
156C2×C4×D79—22no—
157C2×C4×D79—44no—
158C2×C4×D79C2×D412yes—
159C2×C4×D79C2×D414yes—
160C2×C4×D79C2×D414yes—
161C2×C4×D79C2412yes—
162#11929611—48no—
163C2×C79⋊C8—24no—
164C2×C79⋊C8C2414yes—
165C8×D79—48no—
166C4⋊Dic79—24no—
167C4⋊Dic79—28no—
168C4⋊Dic79—24no—
169C4⋊Dic79—21no—
170C4⋊Dic79—22no—
171C4⋊Dic79C2414yes—
172C4⋊Dic79C2412yes—
173Dic79⋊C4—24no—
174Dic79⋊C4—24no—
175Dic79⋊C4—28no—
176(C2×D79)⋊C4—24no—
177(C2×D79)⋊C4—24no—
178(C2×D79)⋊C4—24no—
179(C2×D79)⋊C4—28no—
180(C2×D79)⋊C4—24no—
181(C2×D79)⋊C4—24no—
182C4×Dic79—28no—
183C4×Dic79C2×D412yes—
184C4×Dic79C2×D414yes—
185C4×Dic79C2×D414yes—
186C4×Dic79C2412yes—
187C2×Q8×C79—21no—
188C2×Q8×C79—22no—
189C2×Q8×C79—22no—
190C2×Q8×C79—24no—
191C2×Q8×C79—22no—
192C2×Q8×C79—24no—
193C2×Q8×C79C2414yes—
194C2×Q8×C79C2412yes—
195C22×C4×C79—21no—
196C22×C4×C79—21no—
197C22×C4×C79—22no—
198C22×C4×C79C2×D411yescharacteristic
199C22×C4×C79C2×D411yescharacteristic
200C22×C4×C79C2411yescharacteristic
201Q16×C79—48no—
202C2×C8×C79—24no—
203C2×C8×C79C2414yes—
204C4⋊C4×C79—24no—
205C4⋊C4×C79—24no—
206C4⋊C4×C79—24no—
207C4⋊C4×C79—21no—
208C4⋊C4×C79C2412yes—
209C42×C79—24no—
210C42×C79C2×D412yes—
211C42×C79C2412yes—
212C23×C79C22×D411yescharacteristic
213C22×D79—22no—
214C22×D79—22no—
215C22×D79—22no—
216C22×D79—42no—
217C22×D79—42no—
218C22×D79—42no—
219C22×D79#13714yes—
220C22×D79C22×D412yes—
221Q8×C79—24no—
222Q8×C79—22no—
223Q8×C79—24no—
224Q8×C79—22no—
225Q8×C79—42no—
226Q8×C79—44no—
227C2×C4×C79—22no—
228C2×C4×C79—24no—
229C2×C4×C79—22no—
230C2×C4×C79—22no—
231C2×C4×C79—42no—
232C2×C4×C79—44no—
233C2×C4×C79—42no—
234C2×C4×C79#13514yes—
235C2×C4×C79C22×D412yes—
236C2×C4×C79C22×D411yescharacteristic
237C2×C4×C79C22×D412yes—
238C2×C4×C79H(5,𝔽2)12yes—
239C2×C4×C792−1+412yes—
240C2×C4×C79C2511yescharacteristic
241C2×Dic79—22no—
242C2×Dic79—22no—
243C2×Dic79—22no—
244C2×Dic79—24no—
245C2×Dic79—22no—
246C2×Dic79—42no—
247C2×Dic79—44no—
248C2×Dic79C22×D412yes—
249C2×Dic79C22×D412yes—
250C2×Dic79C22×D412yes—
251D316—22no—
252D316—22no—
253D316—48no—
254D316—42no—
255C4×D79—28no—
256C4×D79—24no—
257C4×D79—28no—
258C4×D79—22no—
259C4×D79—22no—
260C4×D79—48no—
261Dic158—48no—
262C8×C79—24no—
263C79⋊C8—24no—
264C22×C79#53712yes—
265C22×C79C2×#13512yes—
266C22×C79C2×#13712yes—
267C22×C79C23×D411yescharacteristic
268C2×D79—22no—
269C2×D79—28no—
270C2×D79—22no—
271C2×D79—42no—
272C2×D79—42no—
273C2×D79—42no—
274C2×D79—84no—
275C2×D79C2×#13714yes—
276C4×C79—22no—
277C4×C79—44no—
278C4×C79—42no—
279C4×C79—42no—
280C4×C79#53712yes—
281C4×C79C2×#13514yes—
282C4×C79C2×H(5,𝔽2)11yescharacteristic
283C4×C79C2×2−1+411yescharacteristic
284Dic79—22no—
285Dic79—22no—
286Dic79—22no—
287Dic79—44no—
288Dic79—42no—
289C2×#3248—791nomaximal, Sylow
290C2×C79#324814yescomplemented
291C2×C79C2×#53711yescharacteristic
292C2×C79C22×#13511yescharacteristic
293C2×C79C22×#13711yescharacteristic
294D79—24no—
295D79—82no—
296C2×#533—791no—
297C2×#529—791no—
298#3248—7916no—
299C2×#479—792no—
300C2×#475—791no—
301C2×#474—791no—
302C2×#467—792no—
303C2×#465—792no—
304C2×#464—792no—
305C2×#462D7911yescharacteristic, complemented
306C2×#461—791no—
307C22×C4⋊C8—791no—
308C79C2×#324811yescharacteristic, complemented, Sylow
309C22×Q16C2×D7912yes—
310C22×SD16—792no—
311C23×C8—792no—
312#533—798no—
313#533—1584no—
314C2×C4⋊Q8—791no—
315C2×C4⋊Q8C2×D7911yescharacteristic
316#529—798no—
317#529—1584no—
318C2×#126—1581no—
319C2×#125—791no—
320C2×#125—1581no—
321C2×#124—791no—
322C2×#123—1581no—
323C2×C22⋊Q8—791no—
324C2×C22⋊Q8—792no—
325C2×C22⋊Q8—1581no—
326C2×#121—791no—
327C2×C4×Q8—792no—
328C2×C4×Q8—1581no—
329C2×C4×Q8C2×D7911yescharacteristic
330C2×C4×D4—791no—
331C2×C4×D4—1581no—
332C2×#117—1581no—
333C2×#117—1581no—
334C22×C42—791no—
335#479—7916no—
336#475—798no—
337#474—798no—
338#467—7916no—
339#465—7916no—
340#464—7916no—
341#462C2×D7918yescomplemented
342#461—798no—
343C2×#107—792no—
344C2×#106—792no—
345C2×C4⋊C8—792no—
346C2×C4⋊C8—791no—
347C2×C4⋊C8—798no—
348C2×C4⋊C8C2×D7911yescharacteristic
349C2×#103—792no—
350C2×#103—792no—
351C2×#103C2×D7912yes—
352C2×#102—792no—
353C22×Q8—1581no—
354C22×Q8C22×D7912yes—
355C22×Q8C22×D7911yescharacteristic
356C22×D4—791no—
357C22×D4—1581no—
358C23×C4—791no—
359C23×C4—791no—
360C23×C4—791no—
361C23×C4—1581no—
362C2×Q16—44no—
363C2×Q16C22×D7918yes—
364C2×SD16—3164no—
365C2×SD16—798no—
366C22×C8—3164no—
367C22×C8—798no—
368C22×C8—792no—
369C22×C8C22×D7912yes—
370C4⋊Q8—22no—
371C4⋊Q8—794no—
372C4⋊Q8—1582no—
373C4⋊Q8C22×D7914yes—
374#126—1588no—
375#125—794no—
376#125—1582no—
377#125—1588no—
378#124—794no—
379#124—1582no—
380#123—1588no—
381C22⋊Q8—794no—
382C22⋊Q8—798no—
383C22⋊Q8—1588no—
384C22⋊Q8—1582no—
385C22⋊Q8—1584no—
386#121—794no—
387#121—1582no—
388C4×Q8—22no—
389C4×Q8—798no—
390C4×Q8—1588no—
391C4×Q8—1584no—
392C4×Q8C22×D7914yes—
393C4×D4—794no—
394C4×D4—1588no—
395C4×D4—1582no—
396#117—1588no—
397#117—1588no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—791no—
402C2×C4⋊C4—792no—
403C2×C4⋊C4—1581no—
404C2×C4⋊C4—1581no—
405C2×C4⋊C4—1581no—
406C2×C4⋊C4—1582no—
407C2×C4⋊C4—1582no—
408C2×C4⋊C4—1581no—
409C2×C4⋊C4C22×D7911yescharacteristic
410C2×C22⋊C4—1581no—
411C2×C22⋊C4—1581no—
412C2×C22⋊C4—1581no—
413C2×C22⋊C4—1582no—
414C2×C22⋊C4—1581no—
415C2×C22⋊C4—1581no—
416C2×C42—21no—
417C2×C42—791no—
418C2×C42—791no—
419C2×C42—791no—
420C2×C42—794no—
421C2×C42—794no—
422C2×C42—1582no—
423C2×C42C22×D7911yescharacteristic
424#107—798no—
425#106—798no—
426C4⋊C8—798no—
427C4⋊C8—794no—
428C4⋊C8C22×D7914yes—
429#103—798no—
430#103—798no—
431#103C22×D7918yes—
432#102—798no—
433C24—791no—
434C24—1581no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—3164no—
442C2×Q8—1584no—
443C2×Q8C23×D7912yes—
444C2×Q8C23×D7914yes—
445C2×D4—3164no—
446C2×D4—792no—
447C2×D4—1584no—
448C2×D4—1582no—
449C2×D4—1582no—
450C2×D4—1581no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—3164no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—791no—
458C22×C4—792no—
459C22×C4—794no—
460C22×C4—794no—
461C22×C4—792no—
462C22×C4—1584no—
463C22×C4—1582no—
464C22×C4—1581no—
465C22×C4—1584no—
466C22×C4—1582no—
467C22×C4—1582no—
468C22×C4—1584no—
469C22×C4D4×D7911yescharacteristic
470C22×C4D4×D7911yescharacteristic
471C22×C4C23×D7911yescharacteristic
472Q16—48no—
473SD16—3168no—
474C2×C8—24no—
475C2×C8—3168no—
476C2×C8—794no—
477C2×C8—1584no—
478C2×C8C23×D7914yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—794no—
484C4⋊C4—792no—
485C4⋊C4—1584no—
486C4⋊C4—1584no—
487C4⋊C4—1588no—
488C4⋊C4—1584no—
489C4⋊C4—1588no—
490C4⋊C4—1581no—
491C4⋊C4—1582no—
492C4⋊C4—1584no—
493C4⋊C4C23×D7912yes—
494C22⋊C4—1584no—
495C22⋊C4—1584no—
496C22⋊C4—1584no—
497C22⋊C4—1588no—
498C22⋊C4—1584no—
499C22⋊C4—1584no—
500C42—24no—
501C42—792no—
502C42—794no—
503C42—792no—
504C42—794no—
505C42—1588no—
506C42D4×D7912yes—
507C42C23×D7912yes—
508C23—3162no—
509C23—3162no—
510C23—3162no—
511C23—794no—
512C23—792no—
513C23—1582no—
514C23—1582no—
515C23—1582no—
516C23C2×D4×D7911yescharacteristic
517Q8—24no—
518Q8—22no—
519Q8—22no—
520Q8—24no—
521Q8—42no—
522Q8—44no—
523Q8—3168no—
524D4—3168no—
525D4—3162no—
526D4—1582no—
527D4—1582no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—3162no—
536C2×C4—3164no—
537C2×C4—3168no—
538C2×C4—792no—
539C2×C4—792no—
540C2×C4—792no—
541C2×C4—1588no—
542C2×C4—1584no—
543C2×C4—1582no—
544C2×C4—1582no—
545C2×C4—1582no—
546C2×C4—1582no—
547C2×C4—1582no—
548C2×C4—1582no—
549C2×C4—1584no—
550C2×C4—1588no—
551C2×C4#1457918414yes—
552C2×C4C2×D4×D7912yes—
553C2×C4C2×D4×D7911yescharacteristic
554C2×C4C2×D4×D7912yes—
555C2×C4#1457920612yes—
556C2×C4#1457920812yes—
557C2×C4C24×D7911yescharacteristic
558C8—24no—
559C8—1584no—
560C22—3162no—
561C22—3162no—
562C22—3162no—
563C22—794no—
564C22—6324no—
565C22—1588no—
566C22—1582no—
567C22—1582no—
568C22#1909914212yes—
569C22C2×#1457918412yes—
570C22#137×D7912yes—
571C22C22×D4×D7911yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—44no—
576C4—3162no—
577C4—3164no—
578C4—1582no—
579C4—1582no—
580C4—1582no—
581C4#1909914212yes—
582C4C2×#1457918414yes—
583C4C2×#1457920611yescharacteristic
584C4C2×#1457920811yescharacteristic
585C2—6322no—
586C2—1584no—
587C2#2377473514yesdirect product factor
588C2C2×#1909914211yescharacteristic
589C2C22×#1457918411yescharacteristic
590C2C2×#137×D7911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#462×C79C2
socleC23×C79C22×D4
prime radicalC2×#14579136C22
hypercenterC2×#462D79
hypocenterC79C2×#3248
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×#19099118C2semidirect
3. #23774735C2direct (N × H)
4. C2×#19098872C2semidirect
5. C2×#19098848C2semidirect
6. C2×#19098844C2semidirect
7. C2×#19098801C2semidirect
9. C2×#19098787C2semidirect
10. C2×#19098765C2semidirect
11. C2×C4⋊C8×D79C2semidirect
12. C2×#462×C79C2semidirect
16. #19099118C22semidirect
36. #19098872C22semidirect
37. #19098848C22semidirect
38. #19098844C22semidirect
39. #19098801C22semidirect
41. #19098787C22semidirect
42. #19098765C22semidirect
47. C4⋊C8×D79C22semidirect
54. #462×C79C22semidirect
290. C2×C79#3248semidirect
305. C2×#462D79semidirect
308. C79C2×#3248semidirect
341. #462C2×D79semidirect
587. C2#23774735direct (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×#190991002
#237747352
C2×#190988722
C2×#190988482
C2×#190988442
C2×#190988012
C2×#190987952
C2×#190987872
C2×#190987652
C2×C4⋊C8×D792
C2×#462×C792
C2×#324879