Finite Group Databasebeta

← C2×#23774733 #35161070

Subgroups — C2×#23774733 #35161070

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×#19099095C211yesmaximal, characteristic
3#23774733C2116yesmaximal, direct product factor
4C2×#19098868C212yesmaximal, complemented
5C2×#19098849C211yesmaximal, characteristic, complemented
6C2×#19098843C211yesmaximal, characteristic, complemented
7C2×#19098803C212yesmaximal, complemented
8C2×#19098789C212yesmaximal
9C2×#19098783C212yesmaximal, complemented
10C2×#19098770C211yesmaximal, characteristic, complemented
11C2×C4⋊C8×D79C211yesmaximal, characteristic, complemented
12C2×#461×C79C211yesmaximal, characteristic, complemented
13C22×#11929614C2212yes—
14C22×C8×D79C2212yes—
15#19099118—24no—
16#19099118C2218yescomplemented
17C2×#14579161C2211yescharacteristic
18#19099095—24no—
19#19099095C2218yes—
20C2×#14579154—21no—
21C2×#14579152—21no—
22C2×#14579152C2211yescharacteristic
23C2×#14579148C2211yescharacteristic
24C2×#14579143—21no—
25C2×#14579141—21no—
26C2×#14579141C2211yescharacteristic
27C2×#14579137C2212yes—
28C2×#14579132C2211yescharacteristic
29C2×Q8×Dic79C2212yes—
30C2×C4×Dic158C2211yescharacteristic
31C2×#14579126—21no—
32C2×C4×C22⋊D79—21no—
33C2×#14579120—21no—
34C2×#14579119—21no—
35C2×C42×D79C2211yescharacteristic
36#19098868C22116yescomplemented
37#19098849C2218yescomplemented
38#19098843C2218yescomplemented
39#19098803C22116yescomplemented
40#19098789C22116yes—
41#19098783C22116yescomplemented
42#19098770C2218yescomplemented
43C2×#14579094C2212yes—
44C2×#14579093C2212yes—
45C2×C4⋊(C79⋊C8)C2211yescharacteristic
46C2×Dic79⋊C8C2212yes—
47C4⋊C8×D79C2218yescomplemented
48C2×#14579086C2212yes—
49C2×#14579085C2212yes—
50C2×#14579082C2212yes—
51C22×SD16×C79C2212yes—
52C2×C4⋊Q8×C79C2211yescharacteristic
53C2×C4×D4×C79C2211yescharacteristic
54#461×C79C2218yescomplemented
55C2×C4⋊C8×C79C2211yescharacteristic
56C2×#103×C79C2212yes—
57C22×Dic158—21no—
58C22×Dic158C2311yescharacteristic
59C22×C22⋊D79—21no—
60C22×C4×D79D411yescharacteristic
61C22×C4×D79D411yescharacteristic
62C22×C4×D79C2311yescharacteristic
63C2×#11929614—44no—
64C2×#11929614C2318yes—
65C22×C79⋊C8C2312yes—
66C2×C8×D79—44no—
67C2×C8×D79C2318yes—
68#14579161—22no—
69#14579161C2314yes—
70#14579154—28no—
71#14579152—28no—
72#14579152—22no—
73#14579152C2314yes—
74#14579148—22no—
75#14579148C2314yes—
76#14579143—28no—
77#14579141—28no—
78#14579141—22no—
79#14579141C2314yes—
80#14579137—24no—
81#14579137C2318yes—
82#14579132—22no—
83#14579132C2314yes—
84Q8×Dic79—24no—
85Q8×Dic79C2318yes—
86C4×Dic158—22no—
87C4×Dic158C2314yes—
88#14579126—28no—
89C4×C22⋊D79—28no—
90#14579120—28no—
91#14579119—28no—
92C2×C4⋊Dic79—21no—
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×C22⋊Dic79—21no—
102C2×C22⋊Dic79—21no—
103C2×C22⋊Dic79—21no—
104C2×(C2×D79)⋊C4—22no—
105C2×(C2×D79)⋊C4—21no—
106C2×C4×Dic79—21no—
107C2×C4×Dic79—22no—
108C2×C4×Dic79D411yescharacteristic
109C2×C4×Dic79D411yescharacteristic
110C2×C4×Dic79C2311yescharacteristic
111C42×D79D414yes—
112C42×D79C2314yes—
113#14579094C2318yes—
114#14579093C2318yes—
115C4⋊(C79⋊C8)C2314yes—
116Dic79⋊C8C2318yes—
117#14579086C2318yes—
118#14579085C2318yes—
119#14579082C2318yes—
120C22×Q8×C79C2312yes—
121C22×D4×C79C2311yescharacteristic
122C23×C4×C79—21no—
123C2×SD16×C79—44no—
124C2×SD16×C79C2318yes—
125C22×C8×C79C2312yes—
126C4⋊Q8×C79—22no—
127C4⋊Q8×C79C2314yes—
128C4×D4×C79—22no—
129C4×D4×C79C2314yes—
130C2×C4⋊C4×C79—21no—
131C2×C4⋊C4×C79—21no—
132C2×C4⋊C4×C79C2311yescharacteristic
133C2×C22⋊C4×C79—21no—
134C2×C42×C79C2311yescharacteristic
135C4⋊C8×C79C2314yes—
136#103×C79C2318yes—
137C23×D79C2×D411yescharacteristic
138C2×Dic158—24no—
139C2×Dic158—21no—
140C2×Dic158—22no—
141C2×Dic158—22no—
142C2×Dic158—44no—
143C2×Dic158C2412yes—
144C2×C22⋊D79—24no—
145C2×C22⋊D79—44no—
146C22×Dic79—22no—
147C22×Dic79—21no—
148C22×Dic79—21no—
149C22×Dic79C2×D411yescharacteristic
150C22×Dic79C2×D411yescharacteristic
151C22×Dic79C2×D411yescharacteristic
152C2×C4×D79—24no—
153C2×C4×D79—24no—
154C2×C4×D79—22no—
155C2×C4×D79—22no—
156C2×C4×D79C2×D412yes—
157C2×C4×D79C2×D414yes—
158C2×C4×D79C2×D414yes—
159C2×C4×D79C2412yes—
160#11929614—48no—
161C2×C79⋊C8—24no—
162C2×C79⋊C8C2414yes—
163C8×D79—48no—
164C4⋊Dic79—24no—
165C4⋊Dic79—24no—
166C4⋊Dic79—28no—
167C4⋊Dic79—24no—
168C4⋊Dic79—21no—
169C4⋊Dic79—22no—
170C4⋊Dic79C2414yes—
171C4⋊Dic79C2412yes—
172Dic79⋊C4—24no—
173Dic79⋊C4—28no—
174Dic79⋊C4—24no—
175C22⋊Dic79—24no—
176C22⋊Dic79—24no—
177C22⋊Dic79—24no—
178(C2×D79)⋊C4—28no—
179(C2×D79)⋊C4—24no—
180C4×Dic79—28no—
181C4×Dic79—24no—
182C4×Dic79C2×D412yes—
183C4×Dic79C2×D414yes—
184C4×Dic79C2×D414yes—
185C4×Dic79C2412yes—
186C24×C79—21no—
187C2×Q8×C79—22no—
188C2×Q8×C79—24no—
189C2×Q8×C79—24no—
190C2×Q8×C79C2414yes—
191C2×D4×C79—21no—
192C2×D4×C79—22no—
193C2×D4×C79—22no—
194C2×D4×C79C2412yes—
195C22×C4×C79—21no—
196C22×C4×C79—22no—
197C22×C4×C79—24no—
198C22×C4×C79—44no—
199C22×C4×C79C2×D411yescharacteristic
200C22×C4×C79C2×D411yescharacteristic
201C22×C4×C79C2411yescharacteristic
202SD16×C79—48no—
203C2×C8×C79—24no—
204C2×C8×C79C2414yes—
205C4⋊C4×C79—24no—
206C4⋊C4×C79—24no—
207C4⋊C4×C79—21no—
208C4⋊C4×C79C2412yes—
209C22⋊C4×C79—24no—
210C42×C79C2×D412yes—
211C42×C79C2412yes—
212C23×C79—22no—
213C23×C79—42no—
214C23×C79—42no—
215C23×C79—42no—
216C23×C79C22×D411yescharacteristic
217C22×D79—22no—
218C22×D79—22no—
219C22×D79#13714yes—
220C22×D79C22×D412yes—
221Q8×C79—24no—
222Q8×C79—24no—
223Q8×C79—44no—
224D4×C79—22no—
225D4×C79—22no—
226D4×C79—42no—
227C2×C4×C79—22no—
228C2×C4×C79—24no—
229C2×C4×C79—22no—
230C2×C4×C79—44no—
231C2×C4×C79—42no—
232C2×C4×C79—48no—
233C2×C4×C79#13514yes—
234C2×C4×C79C22×D412yes—
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C22×D412yes—
237C2×C4×C79H(5,𝔽2)12yes—
238C2×C4×C792−1+412yes—
239C2×C4×C79C2511yescharacteristic
240C22⋊D79—48no—
241C2×Dic79—22no—
242C2×Dic79—22no—
243C2×Dic79—22no—
244C2×Dic79—22no—
245C2×Dic79—24no—
246C2×Dic79—22no—
247C2×Dic79—42no—
248C2×Dic79—44no—
249C2×Dic79—42no—
250C2×Dic79C22×D412yes—
251C2×Dic79C22×D412yes—
252C2×Dic79C22×D412yes—
253C4×D79—28no—
254C4×D79—24no—
255C4×D79—28no—
256C4×D79—22no—
257C4×D79—22no—
258Dic158—22no—
259Dic158—22no—
260Dic158—42no—
261Dic158—48no—
262C8×C79—24no—
263C79⋊C8—24no—
264C22×C79—42no—
265C22×C79—42no—
266C22×C79—42no—
267C22×C79—84no—
268C22×C79#53712yes—
269C22×C79C2×#13512yes—
270C22×C79C2×#13712yes—
271C22×C79C23×D411yescharacteristic
272C2×D79—22no—
273C2×D79—28no—
274C2×D79—22no—
275C2×D79C2×#13714yes—
276C4×C79—22no—
277C4×C79—44no—
278C4×C79—42no—
279C4×C79#53712yes—
280C4×C79C2×#13514yes—
281C4×C79C2×H(5,𝔽2)11yescharacteristic
282C4×C79C2×2−1+411yescharacteristic
283Dic79—22no—
284Dic79—22no—
285Dic79—22no—
286Dic79—42no—
287Dic79—44no—
288Dic79—42no—
289C2×#3248—791nomaximal, Sylow
290C2×C79—82no—
291C2×C79#324814yescomplemented
292C2×C79C2×#53711yescharacteristic
293C2×C79C22×#13511yescharacteristic
294C2×C79C22×#13711yescharacteristic
295D79—24no—
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×#462—791no—
306C2×#461D7911yescharacteristic, complemented
307C22×C4⋊C8—791no—
308C79C2×#324811yescharacteristic, complemented, Sylow
309C22×Q16—792no—
310C22×SD16C2×D7912yes—
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—792no—
324C2×C22⋊Q8—791no—
325C2×C22⋊Q8—1581no—
326C2×#121—791no—
327C2×C4×Q8—792no—
328C2×C4×Q8—791no—
329C2×C4×Q8—1581no—
330C2×C4×D4—1581no—
331C2×C4×D4C2×D7911yescharacteristic
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#462—798no—
342#461C2×D7918yescomplemented
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—791no—
354C22×Q8—1581no—
355C22×Q8C22×D7912yes—
356C22×D4—1581no—
357C22×D4C22×D7911yescharacteristic
358C23×C4—21no—
359C23×C4—791no—
360C23×C4—791no—
361C23×C4—791no—
362C2×Q16—3164no—
363C2×Q16—798no—
364C2×SD16—44no—
365C2×SD16C22×D7918yes—
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—798no—
382C22⋊Q8—794no—
383C22⋊Q8—1582no—
384C22⋊Q8—1588no—
385C22⋊Q8—1584no—
386#121—794no—
387#121—1582no—
388C4×Q8—798no—
389C4×Q8—794no—
390C4×Q8—1588no—
391C4×Q8—1584no—
392C4×Q8—1582no—
393C4×D4—22no—
394C4×D4—1588no—
395C4×D4C22×D7914yes—
396#117—1588no—
397#117—1588no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—791no—
401C2×C4⋊C4—792no—
402C2×C4⋊C4—1581no—
403C2×C4⋊C4—1581no—
404C2×C4⋊C4—1581no—
405C2×C4⋊C4—1582no—
406C2×C4⋊C4—1582no—
407C2×C4⋊C4—1581no—
408C2×C4⋊C4—1581no—
409C2×C4⋊C4C22×D7911yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1582no—
412C2×C22⋊C4—1581no—
413C2×C22⋊C4—1581no—
414C2×C22⋊C4—1581no—
415C2×C22⋊C4—1581no—
416C2×C42—791no—
417C2×C42—791no—
418C2×C42—791no—
419C2×C42—794no—
420C2×C42—794no—
421C2×C42—1582no—
422C2×C42—1581no—
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—21no—
434C24—791no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—3164no—
439C2×Q8—792no—
440C2×Q8—1582no—
441C2×Q8—1582no—
442C2×Q8—1581no—
443C2×Q8—1584no—
444C2×Q8C23×D7914yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—3164no—
449C2×D4—1584no—
450C2×D4C23×D7912yes—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—44no—
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—1581no—
464C22×C4—1581no—
465C22×C4—1582no—
466C22×C4—1582no—
467C22×C4—1582no—
468C22×C4—1584no—
469C22×C4D4×D7911yescharacteristic
470C22×C4D4×D7911yescharacteristic
471C22×C4C23×D7911yescharacteristic
472Q16—3168no—
473SD16—48no—
474C2×C8—24no—
475C2×C8—3168no—
476C2×C8—794no—
477C2×C8—1584no—
478C2×C8C23×D7914yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—794no—
483C4⋊C4—792no—
484C4⋊C4—1584no—
485C4⋊C4—1588no—
486C4⋊C4—1584no—
487C4⋊C4—1584no—
488C4⋊C4—1588no—
489C4⋊C4—1584no—
490C4⋊C4—1581no—
491C4⋊C4—1582no—
492C4⋊C4—1584no—
493C4⋊C4C23×D7912yes—
494C22⋊C4—24no—
495C22⋊C4—1588no—
496C22⋊C4—1584no—
497C22⋊C4—1584no—
498C22⋊C4—1584no—
499C22⋊C4—1584no—
500C42—792no—
501C42—794no—
502C42—792no—
503C42—794no—
504C42—1588no—
505C42—1584no—
506C42D4×D7912yes—
507C42C23×D7912yes—
508C23—22no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—794no—
513C23—792no—
514C23—1582no—
515C23—1582no—
516C23C2×D4×D7911yescharacteristic
517Q8—24no—
518Q8—24no—
519Q8—44no—
520Q8—3162no—
521Q8—3168no—
522Q8—1582no—
523Q8—1582no—
524D4—22no—
525D4—22no—
526D4—42no—
527D4—3168no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—3162no—
535C2×C4—3164no—
536C2×C4—3162no—
537C2×C4—792no—
538C2×C4—792no—
539C2×C4—792no—
540C2×C4—1588no—
541C2×C4—1582no—
542C2×C4—1582no—
543C2×C4—1584no—
544C2×C4—1582no—
545C2×C4—1582no—
546C2×C4—1582no—
547C2×C4—1582no—
548C2×C4—1582no—
549C2×C4—1584no—
550C2×C4—1588no—
551C2×C4#1457918214yes—
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—42no—
561C22—42no—
562C22—42no—
563C22—84no—
564C22—794no—
565C22—1588no—
566C22—1582no—
567C22—1582no—
568C22#1909914212yes—
569C22C2×#1457918212yes—
570C22#137×D7912yes—
571C22C22×D4×D7911yescharacteristic
572C4—22no—
573C4—42no—
574C4—44no—
575C4—3164no—
576C4—3162no—
577C4—3162no—
578C4—1582no—
579C4—1582no—
580C4—1582no—
581C4#1909914212yes—
582C4C2×#1457918214yes—
583C4C2×#1457920611yescharacteristic
584C4C2×#1457920811yescharacteristic
585C2—82no—
586C2—1584no—
587C2#2377473314yesdirect product factor
588C2C2×#1909914211yescharacteristic
589C2C22×#1457918211yescharacteristic
590C2C2×#137×D7911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#461×C79C2
socleC23×C79C22×D4
prime radicalC2×#14579132C22
hypercenterC2×#461D79
hypocenterC79C2×#3248
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#19099118C2semidirect
3. #23774733C2direct (N × H)
4. C2×#19098868C2semidirect
5. C2×#19098849C2semidirect
6. C2×#19098843C2semidirect
7. C2×#19098803C2semidirect
9. C2×#19098783C2semidirect
10. C2×#19098770C2semidirect
11. C2×C4⋊C8×D79C2semidirect
12. C2×#461×C79C2semidirect
16. #19099118C22semidirect
36. #19098868C22semidirect
37. #19098849C22semidirect
38. #19098843C22semidirect
39. #19098803C22semidirect
41. #19098783C22semidirect
42. #19098770C22semidirect
47. C4⋊C8×D79C22semidirect
54. #461×C79C22semidirect
291. C2×C79#3248semidirect
306. C2×#461D79semidirect
308. C79C2×#3248semidirect
342. #461C2×D79semidirect
587. C2#23774733direct (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×#190990952
#237747332
C2×#190988682
C2×#190988492
C2×#190988432
C2×#190988032
C2×#190987892
C2×#190987832
C2×#190987702
C2×C4⋊C8×D792
C2×#461×C792
C2×#324879