Finite Group Databasebeta

← C2×#23769141 #34454627

Subgroups — C2×#23769141 #34454627

Order 20224.

3718 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579143C212yesmaximal, complemented
2C22×#14579132C211yesmaximal, characteristic
3C22×C4⋊C4×D79C211yesmaximal, characteristic, complemented
4#23769141C2116yesmaximal, direct product factor
5C2×#19098495C211yesmaximal, characteristic, complemented
6C2×#19098493C211yesmaximal, characteristic, complemented
7C2×#19098486C212yesmaximal, complemented
8C2×#19098486C212yesmaximal, complemented
9C2×#19098477C211yesmaximal, characteristic, complemented
10C2×#19098439C212yesmaximal, complemented
11C2×#19098422C211yesmaximal, characteristic, complemented
12C2×#394×C79C211yesmaximal, characteristic, complemented
13C23×C22⋊D79C2212yes—
14C23×C4×D79C2211yescharacteristic
15C23×C4×D79C2211yescharacteristic
16C23×C4×D79C2211yescharacteristic
17C2×#14579143—216no—
18C2×#14579143C22116yescomplemented
19C2×#14579132—28no—
20C2×#14579132C2218yes—
21C22×C4⋊Dic79C2211yescharacteristic
22C22×C4⋊Dic79C2212yes—
23C22×C4⋊Dic79C2211yescharacteristic
24C22×Dic79⋊C4C2211yescharacteristic
25C22×Dic79⋊C4C2211yescharacteristic
26C22×Dic79⋊C4C2212yes—
27C22×Dic79⋊C4C2212yes—
28C2×C4⋊C4×D79—28no—
29C2×C4⋊C4×D79C2218yescomplemented
30C22×C22⋊Dic79C2212yes—
31C22×C22⋊Dic79C2212yes—
32C22×C22⋊Dic79C2212yes—
33C22×(C2×D79)⋊C4C2212yes—
34#19098495C2218yescomplemented
35#19098493C2218yescomplemented
36#19098486C22116yescomplemented
37#19098486C22116yescomplemented
38#19098477C2218yescomplemented
39#19098439C22116yescomplemented
40#19098422C2218yescomplemented
41C2×C79⋊#95C2211yescharacteristic
42C2×C79⋊#95C2212yes—
43C2×C79⋊#95C2212yes—
44C23×D4×C79C2211yescharacteristic
45C22×C4⋊C4×C79C2211yescharacteristic
46C22×C22⋊C4×C79C2212yes—
47C22×C22⋊C4×C79C2212yes—
48#394×C79C2218yescomplemented
49C2×#95×C79C2211yescharacteristic
50C24×D79C2311yescharacteristic
51C22×C22⋊D79—28no—
52C22×C22⋊D79—28no—
53C22×C22⋊D79—28no—
54C22×C22⋊D79C2318yes—
55C23×Dic79C2312yes—
56C23×Dic79C2311yescharacteristic
57C23×Dic79C2312yes—
58C23×Dic79C2311yescharacteristic
59C23×Dic79C2311yescharacteristic
60C22×C4×D79—24no—
61C22×C4×D79—24no—
62C22×C4×D79—24no—
63C22×C4×D79—44no—
64C22×C4×D79—44no—
65C22×C4×D79D418yes—
66C22×C4×D79D418yescomplemented
67C22×C4×D79C2314yes—
68C22×C4×D79C2314yes—
69C22×C4×D79C2314yes—
70#14579143—264no—
71#14579132—232no—
72C2×C4⋊Dic79—24no—
73C2×C4⋊Dic79—24no—
74C2×C4⋊Dic79—28no—
75C2×C4⋊Dic79C2314yes—
76C2×C4⋊Dic79C2318yes—
77C2×C4⋊Dic79C2314yes—
78C2×Dic79⋊C4—24no—
79C2×Dic79⋊C4—28no—
80C2×Dic79⋊C4—24no—
81C2×Dic79⋊C4—44no—
82C2×Dic79⋊C4C2314yes—
83C2×Dic79⋊C4C2314yes—
84C2×Dic79⋊C4C2318yes—
85C2×Dic79⋊C4C2318yes—
86C4⋊C4×D79—232no—
87C2×C22⋊Dic79—28no—
88C2×C22⋊Dic79—28no—
89C2×C22⋊Dic79—44no—
90C2×C22⋊Dic79C2318yes—
91C2×C22⋊Dic79C2318yes—
92C2×C22⋊Dic79C2318yes—
93C2×(C2×D79)⋊C4—28no—
94C2×(C2×D79)⋊C4C2318yes—
95C79⋊#95C2314yes—
96C79⋊#95C2318yes—
97C79⋊#95C2318yes—
98C25×C79C2312yes—
99C22×D4×C79—24no—
100C22×D4×C79—44no—
101C22×D4×C79C2314yes—
102C23×C4×C79C2311yescharacteristic
103C23×C4×C79C2312yes—
104C23×C4×C79C2311yescharacteristic
105C23×C4×C79C2311yescharacteristic
106C2×C4⋊C4×C79—24no—
107C2×C4⋊C4×C79C2314yes—
108C2×C22⋊C4×C79—28no—
109C2×C22⋊C4×C79—44no—
110C2×C22⋊C4×C79C2318yes—
111C2×C22⋊C4×C79C2318yes—
112#95×C79C2314yes—
113C23×D79—24no—
114C23×D79—24no—
115C23×D79C2×D414yes—
116C23×D79C2×D414yes—
117C23×D79#4114yes—
118C23×D79C2412yes—
119C2×C22⋊D79—216no—
120C2×C22⋊D79—216no—
121C2×C22⋊D79—216no—
122C2×C22⋊D79—432no—
123C22×Dic79—22no—
124C22×Dic79—24no—
125C22×Dic79—24no—
126C22×Dic79—22no—
127C22×Dic79—24no—
128C22×Dic79—22no—
129C22×Dic79—24no—
130C22×Dic79—24no—
131C22×Dic79—24no—
132C22×Dic79—24no—
133C22×Dic79—24no—
134C22×Dic79C2×D414yes—
135C22×Dic79C2×D414yes—
136C22×Dic79C2414yes—
137C22×Dic79C2412yes—
138C22×Dic79C2414yes—
139C22×Dic79C2412yes—
140C22×Dic79C2412yes—
141C2×C4×D79—232no—
142C2×C4×D79—28no—
143C2×C4×D79—28no—
144C2×C4×D79—28no—
145C2×C4×D79—416no—
146C2×C4×D79—48no—
147C2×C4×D79—48no—
148C2×C4×D79—416no—
149C2×C4×D79C2×D4116yes—
150C2×C4×D79C2×D4116yescomplemented
151C4⋊Dic79—28no—
152C4⋊Dic79—216no—
153C4⋊Dic79—28no—
154Dic79⋊C4—28no—
155Dic79⋊C4—28no—
156Dic79⋊C4—216no—
157Dic79⋊C4—48no—
158C22⋊Dic79—216no—
159C22⋊Dic79—216no—
160C22⋊Dic79—48no—
161(C2×D79)⋊C4—216no—
162C24×C79—24no—
163C24×C79—24no—
164C24×C79—24no—
165C24×C79—24no—
166C24×C79—24no—
167C24×C79—24no—
168C24×C79—24no—
169C24×C79C2414yes—
170C24×C79C2411yescharacteristic
171C2×D4×C79—28no—
172C2×D4×C79—48no—
173C2×D4×C79—416no—
174C22×C4×C79—22no—
175C22×C4×C79—22no—
176C22×C4×C79—24no—
177C22×C4×C79—24no—
178C22×C4×C79—22no—
179C22×C4×C79—24no—
180C22×C4×C79—24no—
181C22×C4×C79—24no—
182C22×C4×C79C2×D414yes—
183C22×C4×C79C2×D414yes—
184C22×C4×C79C2414yes—
185C22×C4×C79C2412yes—
186C22×C4×C79C2412yes—
187C22×C4×C79C2412yes—
188C4⋊C4×C79—28no—
189C22⋊C4×C79—216no—
190C22⋊C4×C79—48no—
191C23×C79—24no—
192C23×C79—24no—
193C23×C79—24no—
194C23×C79—24no—
195C23×C79—24no—
196C23×C79—24no—
197C23×C79—24no—
198C23×C79—48no—
199C23×C79—48no—
200C23×C79—48no—
201C23×C79—48no—
202C23×C79—48no—
203C23×C79—48no—
204C23×C79—48no—
205C23×C79C22×D412yes—
206C23×C79C22×D412yes—
207C23×C79C2×#4112yes—
208C23×C79H(5,𝔽2)14yes—
209C23×C792−1+414yes—
210C23×C79C2511yescharacteristic
211C22×D79—28no—
212C22×D79—28no—
213C22×D79—216no—
214C22×D79—216no—
215C22×D79—24no—
216C22×D79—24no—
217C22×D79#121116yes—
218C22×D79C22×D414yes—
219C22×D79C22×D414yes—
220C22×D79C2×#4114yes—
221D4×C79—416no—
222C2×C4×C79—28no—
223C2×C4×C79—28no—
224C2×C4×C79—22no—
225C2×C4×C79—22no—
226C2×C4×C79—24no—
227C2×C4×C79—24no—
228C2×C4×C79—22no—
229C2×C4×C79—24no—
230C2×C4×C79—24no—
231C2×C4×C79—24no—
232C2×C4×C79—44no—
233C2×C4×C79—48no—
234C2×C4×C79C22×D414yes—
235C2×C4×C79C22×D414yes—
236C22⋊D79—432no—
237C2×Dic79—28no—
238C2×Dic79—28no—
239C2×Dic79—22no—
240C2×Dic79—22no—
241C2×Dic79—24no—
242C2×Dic79—24no—
243C2×Dic79—24no—
244C2×Dic79—22no—
245C2×Dic79—24no—
246C2×Dic79—24no—
247C2×Dic79—24no—
248C2×Dic79—24no—
249C2×Dic79—24no—
250C2×Dic79—48no—
251C2×Dic79—44no—
252C2×Dic79—48no—
253C2×Dic79C22×D414yes—
254C2×Dic79C22×D414yes—
255C4×D79—232no—
256C4×D79—416no—
257C4×D79—416no—
258C22×C79—44no—
259C22×C79—44no—
260C22×C79—44no—
261C22×C79—44no—
262C22×C79—44no—
263C22×C79—44no—
264C22×C79—44no—
265C22×C79—816no—
266C22×C79C2×#12114yes—
267C22×C79#53414yes—
268C22×C79#53614yes—
269C22×C79#53718yes—
270C22×C79#53918yes—
271C22×C79C23×D411yescharacteristic
272C22×C79C23×D411yescharacteristic
273C22×C79C22×#4111yescharacteristic
274C22×C79C2×H(5,𝔽2)12yes—
275C22×C79C2×2−1+412yes—
276C2×D79—232no—
277C2×D79—24no—
278C2×D79—24no—
279C2×D79—28no—
280C2×D79—28no—
281C2×D79C2×#12118yes—
282C4×C79—24no—
283C4×C79—24no—
284C4×C79—42no—
285C4×C79—44no—
286Dic79—24no—
287Dic79—24no—
288Dic79—44no—
289Dic79—42no—
290Dic79—44no—
291C2×#2618—791nomaximal, Sylow
292C2×C79—84no—
293C2×C79#261818yescomplemented
294C2×C79C22×#12111yescharacteristic
295C2×C79C2×#53411yescharacteristic
296C2×C79C2×#53611yescharacteristic
297C2×C79C2×#53712yes—
298C2×C79C2×#53912yes—
299D79—28no—
300C22×#123—792no—
301C22×#121—791no—
302C23×C4⋊C4—791no—
303#2618—7916no—
304C2×#398—791no—
305C2×#398—791no—
306C2×#396—792no—
307C2×#396—792no—
308C2×#394—791no—
309C2×#394D7911yescharacteristic, complemented
310C2×#386—792no—
311C2×#380—791no—
312C79C2×#261811yescharacteristic, complemented, Sylow
313C23×D4—792no—
314C23×D4C2×D7911yescharacteristic
315C24×C4—791no—
316C24×C4—791no—
317C24×C4—791no—
318C2×#123—7916no—
319C2×#123—15816no—
320C2×#121—798no—
321C2×#121—1588no—
322C22×C4⋊C4—792no—
323C22×C4⋊C4—791no—
324C22×C4⋊C4—791no—
325C22×C4⋊C4—791no—
326C22×C4⋊C4—792no—
327C22×C4⋊C4—792no—
328C22×C4⋊C4—791no—
329C22×C4⋊C4—798no—
330C22×C4⋊C4—1588no—
331C22×C4⋊C4C2×D7911yescharacteristic
332C22×C22⋊C4—792no—
333C22×C22⋊C4—792no—
334C22×C22⋊C4—792no—
335C22×C22⋊C4—792no—
336C22×C22⋊C4C2×D7912yes—
337C22×C22⋊C4C2×D7912yes—
338#398—798no—
339#398—798no—
340#396—7916no—
341#396—7916no—
342#394—798no—
343#394C2×D7918yescomplemented
344#386—7916no—
345#380—798no—
346C2×#95—791no—
347C2×#95—792no—
348C2×#95—792no—
349C2×#95C2×D7911yescharacteristic
350C25—791no—
351C25C22×D7912yes—
352C22×D4—24no—
353C22×D4—44no—
354C22×D4—798no—
355C22×D4—1588no—
356C22×D4—1588no—
357C22×D4—1588no—
358C22×D4C22×D7914yes—
359C23×C4—3164no—
360C23×C4—3164no—
361C23×C4—798no—
362C23×C4—794no—
363C23×C4—794no—
364C23×C4—791no—
365C23×C4—791no—
366C23×C4—798no—
367C23×C4—794no—
368C23×C4—792no—
369C23×C4—791no—
370C23×C4—792no—
371C23×C4—1584no—
372C23×C4—1584no—
373C23×C4—1584no—
374C23×C4C22×D7912yes—
375C23×C4C22×D7911yescharacteristic
376C23×C4C22×D7911yescharacteristic
377C23×C4C22×D7911yescharacteristic
378#123—15864no—
379#121—15832no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—3164no—
382C2×C4⋊C4—798no—
383C2×C4⋊C4—794no—
384C2×C4⋊C4—794no—
385C2×C4⋊C4—794no—
386C2×C4⋊C4—794no—
387C2×C4⋊C4—798no—
388C2×C4⋊C4—798no—
389C2×C4⋊C4—15832no—
390C2×C4⋊C4—1584no—
391C2×C4⋊C4—1584no—
392C2×C4⋊C4—1588no—
393C2×C4⋊C4—1584no—
394C2×C4⋊C4—1584no—
395C2×C4⋊C4—1588no—
396C2×C4⋊C4C22×D7914yes—
397C2×C22⋊C4—28no—
398C2×C22⋊C4—44no—
399C2×C22⋊C4—3164no—
400C2×C22⋊C4—798no—
401C2×C22⋊C4—798no—
402C2×C22⋊C4—798no—
403C2×C22⋊C4—798no—
404C2×C22⋊C4—1588no—
405C2×C22⋊C4—1588no—
406C2×C22⋊C4—1588no—
407C2×C22⋊C4C22×D7918yes—
408C2×C22⋊C4C22×D7918yes—
409#95—794no—
410#95—798no—
411#95—798no—
412#95C22×D7914yes—
413C24—24no—
414C24—24no—
415C24—24no—
416C24—24no—
417C24—24no—
418C24—24no—
419C24—24no—
420C24—794no—
421C24—792no—
422C24—794no—
423C24—794no—
424C24—1584no—
425C24—1584no—
426C24C23×D7914yes—
427C24C23×D7911yescharacteristic
428C2×D4—28no—
429C2×D4—48no—
430C2×D4—416no—
431C2×D4—31632no—
432C2×D4—15816no—
433C2×D4—15816no—
434C2×D4—15816no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—3168no—
444C22×C4—31616no—
445C22×C4—3168no—
446C22×C4—31616no—
447C22×C4—7916no—
448C22×C4—792no—
449C22×C4—792no—
450C22×C4—794no—
451C22×C4—794no—
452C22×C4—7916no—
453C22×C4—794no—
454C22×C4—792no—
455C22×C4—794no—
456C22×C4—1588no—
457C22×C4—1588no—
458C22×C4—15832no—
459C22×C4—1588no—
460C22×C4—1584no—
461C22×C4—1584no—
462C22×C4—1584no—
463C22×C4—1584no—
464C22×C4—1584no—
465C22×C4—1584no—
466C22×C4—1582no—
467C22×C4—1584no—
468C22×C4—1582no—
469C22×C4—1584no—
470C22×C4—1582no—
471C22×C4D4×D7914yes—
472C22×C4D4×D7914yes—
473C22×C4C23×D7912yes—
474C22×C4C23×D7914yes—
475C22×C4C23×D7912yes—
476C22×C4C23×D7912yes—
477C4⋊C4—28no—
478C4⋊C4—3168no—
479C4⋊C4—1588no—
480C4⋊C4—1588no—
481C4⋊C4—15816no—
482C4⋊C4—1588no—
483C4⋊C4—15816no—
484C4⋊C4—1588no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—3168no—
488C22⋊C4—15816no—
489C22⋊C4—15816no—
490C22⋊C4—15816no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—7916no—
506C23—794no—
507C23—794no—
508C23—794no—
509C23—1584no—
510C23—1584no—
511C23—15816no—
512C23—15816no—
513C23—1588no—
514C23—1588no—
515C23C2×D4×D7912yes—
516C23C2×D4×D7912yes—
517C23#41×D7912yes—
518C23#1457920614yes—
519C23#1457920814yes—
520C23C24×D7911yescharacteristic
521D4—416no—
522D4—31632no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—31616no—
536C2×C4—31616no—
537C2×C4—3168no—
538C2×C4—3168no—
539C2×C4—3164no—
540C2×C4—794no—
541C2×C4—794no—
542C2×C4—15832no—
543C2×C4—1584no—
544C2×C4—1584no—
545C2×C4—1584no—
546C2×C4—1584no—
547C2×C4—1584no—
548C2×C4—1582no—
549C2×C4—1584no—
550C2×C4—1584no—
551C2×C4—1584no—
552C2×C4—1588no—
553C2×C4—1588no—
554C2×C4—1582no—
555C2×C4—1582no—
556C2×C4C2×D4×D7914yes—
557C2×C4C2×D4×D7914yes—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—816no—
566C22—798no—
567C22—15832no—
568C22—1584no—
569C22—1584no—
570C22—1588no—
571C22—1588no—
572C22#121×D7914yes—
573C22#1909912314yes—
574C22#1909913314yes—
575C22#1909914218yes—
576C22#1909915618yes—
577C22C22×D4×D7911yescharacteristic
578C22C22×D4×D7911yescharacteristic
579C22C2×#41×D7911yescharacteristic
580C22C2×#1457920612yes—
581C22C2×#1457920812yes—
582C4—24no—
583C4—24no—
584C4—42no—
585C4—44no—
586C4—3164no—
587C4—3162no—
588C4—3164no—
589C4—1584no—
590C4—1584no—
591C2—84no—
592C2—1588no—
593C2#2376914118yesdirect product factor
594C2C2×#121×D7911yescharacteristic
595C2C2×#1909912311yescharacteristic
596C2C2×#1909913311yescharacteristic
597C2C2×#1909914212yes—
598C2C2×#1909915612yes—
subgroupG/H
centerC24C23×D79
commutator, cosocleC23×C79C25
FrattiniC23C24×D79
FittingC2×#394×C79C2
socleC24×C79C24
prime radicalC22×#14579132C2
hypercenterC2×#394D79
hypocenterC79C2×#2618
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C22×#14579143C2semidirect
3. C22×C4⋊C4×D79C2semidirect
4. #23769141C2direct (N × H)
5. C2×#19098495C2semidirect
6. C2×#19098493C2semidirect
7. C2×#19098486C2semidirect
8. C2×#19098486C2semidirect
9. C2×#19098477C2semidirect
10. C2×#19098439C2semidirect
11. C2×#19098422C2semidirect
12. C2×#394×C79C2semidirect
18. C2×#14579143C22semidirect
29. C2×C4⋊C4×D79C22semidirect
34. #19098495C22semidirect
35. #19098493C22semidirect
36. #19098486C22semidirect
37. #19098486C22semidirect
38. #19098477C22semidirect
39. #19098439C22semidirect
40. #19098422C22semidirect
48. #394×C79C22semidirect
66. C22×C4×D79D4semidirect
150. C2×C4×D79C2×D4semidirect
293. C2×C79#2618semidirect
309. C2×#394D79semidirect
312. C79C2×#2618semidirect
343. #394C2×D79semidirect
593. C2#23769141direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791432
C22×#145791322
C22×C4⋊C4×D792
#237691412
C2×#190984952
C2×#190984932
C2×#190984862
C2×#190984772
C2×#190984392
C2×#190984222
C2×#394×C792
C2×#261879