Finite Group Databasebeta

← C2×#23773529 #35156471

Subgroups — C2×#23773529 #35156471

Order 20224.

2118 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099118C211yesmaximal, characteristic
2C2×C4×Q8×D79C211yesmaximal, characteristic
3#23773529C2116yesmaximal, direct product factor
4C2×#19098892C212yesmaximal
5C2×#19098849C211yesmaximal, characteristic, complemented
6C2×#19098848C211yesmaximal, characteristic, complemented
7C2×#19098770C211yesmaximal, characteristic, complemented
8C2×#19098717C212yesmaximal
9C2×#19098670C212yesmaximal, complemented
10C2×#19098667C212yesmaximal, complemented
11C2×#19098619C211yesmaximal, characteristic
12C2×#462×C79C211yesmaximal, characteristic, complemented
13C22×Q8×D79C2211yescharacteristic
14C22×#11929614C2212yes—
15#19099118—24no—
16#19099118C2218yes—
17C2×#14579161C2211yescharacteristic
18C2×#14579152—21no—
19C2×#14579141—21no—
20C2×#14579137C2212yes—
21C2×Q8×Dic79C2212yes—
22C2×Q8×Dic79C2211yescharacteristic
23C2×C4×Dic158—21no—
24C2×C4×Dic158C2211yescharacteristic
25C2×#14579126—21no—
26C2×#14579126—21no—
27C2×#14579126C2211yescharacteristic
28C4×Q8×D79—24no—
29C4×Q8×D79C2218yes—
30C2×#14579119—21no—
31C2×C4⋊C4×D79—21no—
32C2×C4⋊C4×D79C2211yescharacteristic
33C2×C42×D79—21no—
34C2×C42×D79C2211yescharacteristic
35#19098892C22116yes—
36#19098849C2218yescomplemented
37#19098848C2218yescomplemented
38#19098770C2218yescomplemented
39#19098717C22116yes—
40#19098670C22116yescomplemented
41#19098667C22116yescomplemented
42#19098619C2218yes—
43C2×#14579093C2212yes—
44C2×#14579092C2212yes—
45C2×C4⋊(C79⋊C8)C2211yescharacteristic
46C2×#14579086C2212yes—
47C2×#14579085C2212yes—
48C2×#14579085C2212yes—
49C2×(C2×D79)⋊C8C2212yes—
50C2×#14579071C2212yes—
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×Dic158—21no—
59C22×Dic158C2311yescharacteristic
60C2×Q8×D79—24no—
61C2×Q8×D79—24no—
62C2×Q8×D79—22no—
63C2×Q8×D79C2314yes—
64C22×C4×D79—21no—
65C22×C4×D79—21no—
66C22×C4×D79D411yescharacteristic
67C22×C4×D79D411yescharacteristic
68C22×C4×D79C2311yescharacteristic
69C2×#11929614—28no—
70C2×#11929614C2318yes—
71C22×C79⋊C8C2312yes—
72#14579161—22no—
73#14579161C2314yes—
74#14579152—28no—
75#14579141—28no—
76#14579137—24no—
77#14579137C2318yes—
78Q8×Dic79—22no—
79Q8×Dic79—24no—
80Q8×Dic79C2318yes—
81Q8×Dic79C2314yes—
82C4×Dic158—22no—
83C4×Dic158—28no—
84C4×Dic158C2314yes—
85#14579126—22no—
86#14579126—28no—
87#14579126—28no—
88#14579126C2314yes—
89#14579119—28no—
90C2×C4⋊Dic79—21no—
91C2×C4⋊Dic79—21no—
92C2×C4⋊Dic79—22no—
93C2×C4⋊Dic79C2311yescharacteristic
94C2×C4⋊Dic79C2312yes—
95C2×Dic79⋊C4—21no—
96C2×Dic79⋊C4—21no—
97C2×Dic79⋊C4—21no—
98C2×Dic79⋊C4—22no—
99C4⋊C4×D79—28no—
100C4⋊C4×D79—22no—
101C4⋊C4×D79C2314yes—
102C2×(C2×D79)⋊C4—22no—
103C2×C4×Dic79—21no—
104C2×C4×Dic79—21no—
105C2×C4×Dic79—21no—
106C2×C4×Dic79—22no—
107C2×C4×Dic79D411yescharacteristic
108C2×C4×Dic79D411yescharacteristic
109C2×C4×Dic79C2311yescharacteristic
110C42×D79—28no—
111C42×D79D414yes—
112C42×D79C2314yes—
113#14579093C2318yes—
114#14579092C2318yes—
115C4⋊(C79⋊C8)C2314yes—
116#14579086C2318yes—
117#14579085C2318yes—
118#14579085C2318yes—
119(C2×D79)⋊C8C2318yes—
120#14579071C2318yes—
121C22×Q8×C79C2312yes—
122C22×Q8×C79C2311yescharacteristic
123C2×Q16×C79—28no—
124C2×Q16×C79C2318yes—
125C22×C8×C79C2312yes—
126C4⋊Q8×C79—22no—
127C4⋊Q8×C79C2314yes—
128C4×Q8×C79—22no—
129C4×Q8×C79C2314yes—
130C2×C4⋊C4×C79—21no—
131C2×C4⋊C4×C79—21no—
132C2×C4⋊C4×C79—21no—
133C2×C4⋊C4×C79C2311yescharacteristic
134C2×C42×C79—21no—
135C2×C42×C79C2311yescharacteristic
136C4⋊C8×C79C2314yes—
137#103×C79C2318yes—
138C23×D79C2×D411yescharacteristic
139C2×Dic158—28no—
140C2×Dic158—24no—
141C2×Dic158—21no—
142C2×Dic158—24no—
143C2×Dic158—22no—
144C2×Dic158—44no—
145C2×Dic158#4114yes—
146C2×Dic158C2412yes—
147Q8×D79—216no—
148Q8×D79—28no—
149Q8×D79—28no—
150C22×Dic79—21no—
151C22×Dic79—22no—
152C22×Dic79—21no—
153C22×Dic79C2×D411yescharacteristic
154C22×Dic79C2×D411yescharacteristic
155C22×Dic79C2×D411yescharacteristic
156C2×C4×D79—28no—
157C2×C4×D79—24no—
158C2×C4×D79—24no—
159C2×C4×D79—24no—
160C2×C4×D79—24no—
161C2×C4×D79—22no—
162C2×C4×D79—22no—
163C2×C4×D79—44no—
164C2×C4×D79C2×D412yes—
165C2×C4×D79C2×D414yes—
166C2×C4×D79C2×D414yes—
167C2×C4×D79C2412yes—
168#11929614—216no—
169C2×C79⋊C8—24no—
170C2×C79⋊C8C2414yes—
171C4⋊Dic79—24no—
172C4⋊Dic79—28no—
173C4⋊Dic79—21no—
174C4⋊Dic79—24no—
175C4⋊Dic79—22no—
176C4⋊Dic79C2412yes—
177C4⋊Dic79C2414yes—
178Dic79⋊C4—24no—
179Dic79⋊C4—24no—
180Dic79⋊C4—28no—
181Dic79⋊C4—24no—
182(C2×D79)⋊C4—28no—
183C4×Dic79—24no—
184C4×Dic79—24no—
185C4×Dic79—28no—
186C4×Dic79—24no—
187C4×Dic79C2×D412yes—
188C4×Dic79C2×D414yes—
189C4×Dic79C2×D414yes—
190C4×Dic79C2412yes—
191C2×Q8×C79—21no—
192C2×Q8×C79—22no—
193C2×Q8×C79—22no—
194C2×Q8×C79—24no—
195C2×Q8×C79—24no—
196C2×Q8×C79#4114yes—
197C2×Q8×C79C2414yes—
198C2×Q8×C79C2412yes—
199C22×C4×C79—21no—
200C22×C4×C79—21no—
201C22×C4×C79—22no—
202C22×C4×C79C2×D411yescharacteristic
203C22×C4×C79C2×D411yescharacteristic
204C22×C4×C79C2411yescharacteristic
205Q16×C79—216no—
206C2×C8×C79—24no—
207C2×C8×C79C2414yes—
208C4⋊C4×C79—24no—
209C4⋊C4×C79—24no—
210C4⋊C4×C79—24no—
211C4⋊C4×C79—21no—
212C4⋊C4×C79C2412yes—
213C42×C79—24no—
214C42×C79C2×D412yes—
215C42×C79C2412yes—
216C23×C79C22×D411yescharacteristic
217C22×D79—22no—
218C22×D79—22no—
219C22×D79#13714yes—
220C22×D79C22×D412yes—
221Q8×C79—24no—
222Q8×C79—22no—
223Q8×C79—24no—
224Q8×C79—24no—
225Q8×C79—44no—
226Q8×C79C2×#4114yes—
227C2×C4×C79—24no—
228C2×C4×C79—22no—
229C2×C4×C79—22no—
230C2×C4×C79—24no—
231C2×C4×C79—22no—
232C2×C4×C79—44no—
233C2×C4×C79—42no—
234C2×C4×C79#13714yes—
235C2×C4×C79C22×D412yes—
236C2×C4×C79C22×D411yescharacteristic
237C2×C4×C79C22×D412yes—
238C2×C4×C79C2×#4112yes—
239C2×C4×C79C2×#4112yes—
240C2×C4×C79C2511yescharacteristic
241C2×Dic79—24no—
242C2×Dic79—22no—
243C2×Dic79—22no—
244C2×Dic79—22no—
245C2×Dic79—22no—
246C2×Dic79—22no—
247C2×Dic79—24no—
248C2×Dic79—42no—
249C2×Dic79—44no—
250C2×Dic79C22×D412yes—
251C2×Dic79C22×D412yes—
252C2×Dic79C22×D412yes—
253C4×D79—216no—
254C4×D79—28no—
255C4×D79—24no—
256C4×D79—28no—
257C4×D79—22no—
258C4×D79—22no—
259C4×D79—48no—
260Dic158—216no—
261Dic158—24no—
262Dic158—22no—
263Dic158—48no—
264Dic158C2×#4114yes—
265C8×C79—24no—
266C79⋊C8—24no—
267C22×C79#52512yes—
268C22×C79C2×#13712yes—
269C22×C79C2×#13712yes—
270C22×C79C23×D411yescharacteristic
271C2×D79—28no—
272C2×D79—22no—
273C2×D79—22no—
274C2×D79C2×#13714yes—
275C4×C79—24no—
276C4×C79—22no—
277C4×C79—44no—
278C4×C79—42no—
279C4×C79#52512yes—
280C4×C79C2×#13714yes—
281C4×C79C22×#4111yescharacteristic
282C4×C79C22×#4111yescharacteristic
283Dic79—22no—
284Dic79—24no—
285Dic79—22no—
286Dic79—22no—
287Dic79—44no—
288Dic79—42no—
289C2×#3113—791nomaximal, Sylow
290C2×C79#311314yescomplemented
291C2×C79C2×#52511yescharacteristic
292C2×C79C22×#13711yescharacteristic
293C2×C79C22×#13711yescharacteristic
294D79—24no—
295C2×#533—791no—
296C22×C4×Q8—791no—
297#3113—7916no—
298C2×#483—792no—
299C2×#475—791no—
300C2×#475—791no—
301C2×#462—791no—
302C2×#462D7911yescharacteristic, complemented
303C2×#451—792no—
304C2×#441—792no—
305C2×#441—792no—
306C2×#432—791no—
307C79C2×#311311yescharacteristic, complemented, Sylow
308C23×Q8—791no—
309C22×Q16—792no—
310C22×Q16C2×D7912yes—
311#533—798no—
312#533—1584no—
313C2×C4⋊Q8—791no—
314C2×C4⋊Q8C2×D7911yescharacteristic
315C2×#125—1581no—
316C2×C22⋊Q8—792no—
317C2×C22⋊Q8—1581no—
318C2×C4×Q8—791no—
319C2×C4×Q8—792no—
320C2×C4×Q8—798no—
321C2×C4×Q8—791no—
322C2×C4×Q8—791no—
323C2×C4×Q8—1581no—
324C2×C4×Q8—1581no—
325C2×C4×Q8—1581no—
326C2×C4×Q8—1584no—
327C2×C4×Q8C2×D7911yescharacteristic
328C2×#117—1581no—
329C22×C4⋊C4—791no—
330C22×C4⋊C4—1581no—
331C22×C42—791no—
332C22×C42—1581no—
333#483—7916no—
334#475—798no—
335#475—798no—
336#462—798no—
337#462C2×D7918yescomplemented
338#451—7916no—
339#441—7916no—
340#441—7916no—
341#432—798no—
342C2×#106—792no—
343C2×#106—792no—
344C2×C4⋊C8—791no—
345C2×C4⋊C8C2×D7911yescharacteristic
346C2×#103—792no—
347C2×#103—792no—
348C2×#103—792no—
349C2×#103C2×D7912yes—
350C2×C22⋊C8—792no—
351C2×#97—792no—
352C22×Q8—794no—
353C22×Q8—791no—
354C22×Q8—1581no—
355C22×Q8—1582no—
356C22×Q8—1584no—
357C22×Q8—1584no—
358C22×Q8—1581no—
359C22×Q8C22×D7912yes—
360C22×Q8C22×D7911yescharacteristic
361C23×C4—791no—
362C23×C4—791no—
363C23×C4—791no—
364C23×C4—1581no—
365C23×C4—1581no—
366C2×Q16—28no—
367C2×Q16—798no—
368C2×Q16—1588no—
369C2×Q16C22×D7918yes—
370C22×C8—792no—
371C22×C8C22×D7912yes—
372C4⋊Q8—22no—
373C4⋊Q8—794no—
374C4⋊Q8—1582no—
375C4⋊Q8C22×D7914yes—
376#125—1588no—
377C22⋊Q8—798no—
378C22⋊Q8—1588no—
379C22⋊Q8—1584no—
380C4×Q8—22no—
381C4×Q8—794no—
382C4×Q8—798no—
383C4×Q8—794no—
384C4×Q8—794no—
385C4×Q8—1588no—
386C4×Q8—1582no—
387C4×Q8—1588no—
388C4×Q8—1588no—
389C4×Q8—1582no—
390C4×Q8—1584no—
391C4×Q8—1582no—
392C4×Q8C22×D7914yes—
393#117—1588no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—791no—
398C2×C4⋊C4—792no—
399C2×C4⋊C4—794no—
400C2×C4⋊C4—1582no—
401C2×C4⋊C4—1588no—
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—1582no—
411C2×C42—21no—
412C2×C42—791no—
413C2×C42—791no—
414C2×C42—791no—
415C2×C42—794no—
416C2×C42—794no—
417C2×C42—1588no—
418C2×C42—1582no—
419C2×C42—1581no—
420C2×C42—1581no—
421C2×C42—1581no—
422C2×C42C22×D7911yescharacteristic
423#106—798no—
424#106—798no—
425C4⋊C8—794no—
426C4⋊C8C22×D7914yes—
427#103—798no—
428#103—798no—
429#103—798no—
430#103C22×D7918yes—
431C22⋊C8—798no—
432#97—798no—
433C24—791no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—3164no—
440C2×Q8—794no—
441C2×Q8—792no—
442C2×Q8—1584no—
443C2×Q8—1588no—
444C2×Q8—1588no—
445C2×Q8—15816no—
446C2×Q8—1588no—
447C2×Q8—1582no—
448C2×Q8—1584no—
449C2×Q8—1581no—
450C2×Q8#1192962314yes—
451C2×Q8C23×D7914yes—
452C2×Q8C23×D7912yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—3164no—
457C22×C4—791no—
458C22×C4—791no—
459C22×C4—791no—
460C22×C4—792no—
461C22×C4—794no—
462C22×C4—794no—
463C22×C4—792no—
464C22×C4—1584no—
465C22×C4—1584no—
466C22×C4—1584no—
467C22×C4—1588no—
468C22×C4—1581no—
469C22×C4—1582no—
470C22×C4—1581no—
471C22×C4—1582no—
472C22×C4—1582no—
473C22×C4—1584no—
474C22×C4D4×D7911yescharacteristic
475C22×C4D4×D7911yescharacteristic
476C22×C4C23×D7911yescharacteristic
477Q16—216no—
478Q16—15816no—
479C2×C8—24no—
480C2×C8—794no—
481C2×C8—1584no—
482C2×C8C23×D7914yes—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—792no—
488C4⋊C4—794no—
489C4⋊C4—1584no—
490C4⋊C4—1584no—
491C4⋊C4—1588no—
492C4⋊C4—1584no—
493C4⋊C4—1588no—
494C4⋊C4—1584no—
495C4⋊C4—1581no—
496C4⋊C4—1582no—
497C4⋊C4—1584no—
498C4⋊C4C23×D7912yes—
499C22⋊C4—1588no—
500C42—24no—
501C42—792no—
502C42—792no—
503C42—794no—
504C42—794no—
505C42—1584no—
506C42—1584no—
507C42—1588no—
508C42—1584no—
509C42D4×D7912yes—
510C42C23×D7912yes—
511C23—794no—
512C23—792no—
513C23—1582no—
514C23—1582no—
515C23C2×D4×D7911yescharacteristic
516Q8—24no—
517Q8—22no—
518Q8—24no—
519Q8—24no—
520Q8—44no—
521Q8—3168no—
522Q8—794no—
523Q8—15816no—
524Q8—1584no—
525Q8—1582no—
526Q8C2×#1192962314yes—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—3168no—
535C2×C4—3162no—
536C2×C4—3164no—
537C2×C4—792no—
538C2×C4—792no—
539C2×C4—792no—
540C2×C4—1588no—
541C2×C4—15816no—
542C2×C4—1582no—
543C2×C4—1584no—
544C2×C4—1582no—
545C2×C4—1584no—
546C2×C4—1582no—
547C2×C4—1582no—
548C2×C4—1582no—
549C2×C4—1584no—
550C2×C4—1588no—
551C2×C4—1582no—
552C2×C4—1582no—
553C2×C4#1457919314yes—
554C2×C4C2×D4×D7912yes—
555C2×C4C2×D4×D7911yescharacteristic
556C2×C4C2×D4×D7912yes—
557C2×C4#41×D7912yes—
558C2×C4C2×#1192962312yes—
559C2×C4C24×D7911yescharacteristic
560C8—24no—
561C8—1584no—
562C22—794no—
563C22—1588no—
564C22—1582no—
565C22—1582no—
566C22#1909907312yes—
567C22#137×D7912yes—
568C22C2×#1457919312yes—
569C22C22×D4×D7911yescharacteristic
570C4—22no—
571C4—24no—
572C4—42no—
573C4—44no—
574C4—3162no—
575C4—3164no—
576C4—1582no—
577C4—1582no—
578C4—1582no—
579C4—1584no—
580C4#1909907312yes—
581C4C2×#1457919314yes—
582C4C2×#41×D7911yescharacteristic
583C4C22×#1192962311yescharacteristic
584C2—1584no—
585C2#2377352914yesdirect product factor
586C2C2×#1909907311yescharacteristic
587C2C2×#137×D7911yescharacteristic
588C2C22×#1457919311yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#462×C79C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
hypercenterC2×#462D79
hypocenterC79C2×#3113
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
3. #23773529C2direct (N × H)
5. C2×#19098849C2semidirect
6. C2×#19098848C2semidirect
7. C2×#19098770C2semidirect
9. C2×#19098670C2semidirect
10. C2×#19098667C2semidirect
12. C2×#462×C79C2semidirect
36. #19098849C22semidirect
37. #19098848C22semidirect
38. #19098770C22semidirect
40. #19098670C22semidirect
41. #19098667C22semidirect
54. #462×C79C22semidirect
290. C2×C79#3113semidirect
302. C2×#462D79semidirect
307. C79C2×#3113semidirect
337. #462C2×D79semidirect
585. C2#23773529direct (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×C4×Q8×D792
#237735292
C2×#190988922
C2×#190988492
C2×#190988482
C2×#190987702
C2×#190987172
C2×#190986702
C2×#190986672
C2×#190986192
C2×#462×C792
C2×#311379