Finite Group Databasebeta

← C2×#23762414 #34260511

Subgroups — C2×#23762414 #34260511

Order 20224.

1156 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098891C211yesmaximal, characteristic, complemented
2C2×#19098874C211yesmaximal, characteristic, complemented
3C2×#19098488C211yesmaximal, characteristic, complemented
4#23762414C218yesmaximal, direct product factor
5C2×#19098300C211yesmaximal, characteristic, complemented
6C2×#19098297C211yesmaximal, characteristic, complemented
7C2×#19098287C211yesmaximal, characteristic, complemented
8C2×#538×C79C211yesmaximal, characteristic
9C22×C4⋊Dic79C2211yescharacteristic
10C22×C22⋊Dic79—21no—
11#19098891—22no—
12#19098891C2214yescomplemented
13#19098874—22no—
14#19098874C2214yescomplemented
15C2×#14579094—21no—
16C2×#14579092—21no—
17C2×#14579085—21no—
18C2×#14579082—21no—
19C2×C22⋊(C79⋊C8)C2211yescharacteristic
20C2×C22⋊(C79⋊C8)C2211yescharacteristic
21#19098488—22no—
22#19098488C2214yescomplemented
23C2×C79⋊#95—21no—
24C2×C79⋊#95C2211yescharacteristic
25#19098300C2214yescomplemented
26#19098297C2214yescomplemented
27#19098287C2214yescomplemented
28#538×C79—22no—
29#538×C79—22no—
30#538×C79—22no—
31#538×C79C2214yes—
32C2×#126×C79—21no—
33C2×#124×C79—21no—
34C2×#123×C79—21no—
35C2×C22⋊Q8×C79C2211yescharacteristic
36C2×#121×C79—21no—
37C2×#121×C79C2211yescharacteristic
38C2×#120×C79—21no—
39C2×C4×D4×C79—21no—
40C22×C22⋊C4×C79C2211yescharacteristic
41C23×Dic79—21no—
42C23×Dic79—21no—
43C22×C79⋊C8—21no—
44C22×C79⋊C8—21no—
45C2×C4⋊Dic79—21no—
46C2×C4⋊Dic79—21no—
47C2×C4⋊Dic79—21no—
48C2×C4⋊Dic79—21no—
49C2×C4⋊Dic79—21no—
50C2×C4⋊Dic79C2312yes—
51C2×C22⋊Dic79—24no—
52C2×C22⋊Dic79—44no—
53#14579094—24no—
54#14579092—24no—
55#14579085—24no—
56#14579082—24no—
57C22⋊(C79⋊C8)—21no—
58C22⋊(C79⋊C8)—21no—
59C22⋊(C79⋊C8)C2312yes—
60C22⋊(C79⋊C8)C2312yes—
61C79⋊#95—24no—
62C79⋊#95—21no—
63C79⋊#95C2312yes—
64C25×C79D411yescharacteristic
65C22×Q8×C79D411yescharacteristic
66C22×D4×C79—21no—
67C22×D4×C79—21no—
68C22×D4×C79—21no—
69C22×D4×C79D411yescharacteristic
70C23×C4×C79—21no—
71C23×C4×C79D411yescharacteristic
72C23×C4×C79C2311yescharacteristic
73#126×C79—24no—
74#126×C79—24no—
75#124×C79—24no—
76#124×C79—24no—
77#123×C79—24no—
78#123×C79—24no—
79C22⋊Q8×C79—21no—
80C22⋊Q8×C79—21no—
81C22⋊Q8×C79D412yes—
82C22⋊Q8×C79C2312yes—
83#121×C79—24no—
84#121×C79—24no—
85#121×C79—21no—
86#121×C79—21no—
87#121×C79D412yes—
88#121×C79C2312yes—
89#120×C79—24no—
90#120×C79—24no—
91C4×D4×C79—24no—
92C4×D4×C79—24no—
93C2×C4⋊C4×C79—21no—
94C2×C4⋊C4×C79—21no—
95C2×C4⋊C4×C79D411yescharacteristic
96C2×C4⋊C4×C79D411yescharacteristic
97C2×C22⋊C4×C79—21no—
98C2×C22⋊C4×C79—21no—
99C2×C22⋊C4×C79—21no—
100C2×C22⋊C4×C79—21no—
101C2×C22⋊C4×C79—21no—
102C2×C22⋊C4×C79—21no—
103C2×C22⋊C4×C79—21no—
104C2×C22⋊C4×C79—21no—
105C2×C22⋊C4×C79—21no—
106C2×C22⋊C4×C79D412yes—
107C2×C22⋊C4×C79C2312yes—
108C2×C42×C79—21no—
109C22×Dic79—22no—
110C22×Dic79—22no—
111C22×Dic79—42no—
112C22×Dic79—42no—
113C22×Dic79—41no—
114C22×Dic79—41no—
115C22×Dic79—42no—
116C22×Dic79—42no—
117C22×Dic79—42no—
118C2×C79⋊C8—22no—
119C2×C79⋊C8—22no—
120C2×C79⋊C8—42no—
121C2×C79⋊C8—42no—
122C4⋊Dic79—22no—
123C4⋊Dic79—22no—
124C4⋊Dic79—41no—
125C4⋊Dic79—41no—
126C22⋊Dic79—48no—
127C24×C79—21no—
128C24×C79—21no—
129C24×C79—22no—
130C24×C79—21no—
131C24×C79—22no—
132C24×C79—42no—
133C24×C79—42no—
134C24×C79C2×D411yescharacteristic
135C24×C79C2×D412yes—
136C2×Q8×C79—21no—
137C2×Q8×C79—42no—
138C2×Q8×C79C2×D412yes—
139C2×D4×C79—22no—
140C2×D4×C79—22no—
141C2×D4×C79—24no—
142C2×D4×C79—24no—
143C2×D4×C79—21no—
144C2×D4×C79—42no—
145C2×D4×C79—42no—
146C2×D4×C79—42no—
147C2×D4×C79—82no—
148C2×D4×C79—82no—
149C2×D4×C79C2×D412yes—
150C22×C4×C79—21no—
151C22×C4×C79—21no—
152C22×C4×C79—21no—
153C22×C4×C79—21no—
154C22×C4×C79—24no—
155C22×C4×C79—21no—
156C22×C4×C79—42no—
157C22×C4×C79—42no—
158C22×C4×C79—82no—
159C22×C4×C79C2×D411yescharacteristic
160C22×C4×C79C2×D411yescharacteristic
161C22×C4×C79C2×D412yes—
162C22×C4×C79C2×D411yescharacteristic
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79C2411yescharacteristic
166C4⋊C4×C79—24no—
167C4⋊C4×C79—22no—
168C4⋊C4×C79—21no—
169C4⋊C4×C79—21no—
170C4⋊C4×C79—22no—
171C4⋊C4×C79C2×D412yes—
172C4⋊C4×C79C2×D412yes—
173C22⋊C4×C79—24no—
174C22⋊C4×C79—24no—
175C22⋊C4×C79—24no—
176C22⋊C4×C79—22no—
177C22⋊C4×C79—22no—
178C22⋊C4×C79—22no—
179C22⋊C4×C79—22no—
180C22⋊C4×C79—22no—
181C22⋊C4×C79—22no—
182C22⋊C4×C79—22no—
183C22⋊C4×C79—22no—
184C42×C79—24no—
185C23×C79—22no—
186C23×C79—22no—
187C23×C79—22no—
188C23×C79—22no—
189C23×C79—22no—
190C23×C79—22no—
191C23×C79—42no—
192C23×C79—42no—
193C23×C79—48no—
194C23×C79—42no—
195C23×C79—42no—
196C23×C79—42no—
197C23×C79—42no—
198C23×C79—42no—
199C23×C79—42no—
200C23×C79—42no—
201C23×C79—81no—
202C23×C79—81no—
203C23×C79—84no—
204C23×C79#12011yescharacteristic
205C23×C79#12011yescharacteristic
206C23×C79#13512yes—
207C23×C79AGL(1,ℤ/8)12yes—
208C23×C79C22×D411yescharacteristic
209Q8×C79—42no—
210Q8×C79—81no—
211D4×C79—42no—
212D4×C79—42no—
213D4×C79—42no—
214D4×C79—44no—
215D4×C79—81no—
216D4×C79—84no—
217D4×C79—84no—
218C2×C4×C79—22no—
219C2×C4×C79—22no—
220C2×C4×C79—22no—
221C2×C4×C79—22no—
222C2×C4×C79—22no—
223C2×C4×C79—22no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—42no—
229C2×C4×C79—42no—
230C2×C4×C79—42no—
231C2×C4×C79—42no—
232C2×C4×C79—42no—
233C2×C4×C79—81no—
234C2×C4×C79—84no—
235C2×C4×C79#12011yescharacteristic
236C2×C4×C79#12011yescharacteristic
237C2×C4×C79C22×D411yescharacteristic
238C2×C4×C79C22×D411yescharacteristic
239C2×Dic79—42no—
240C2×Dic79—42no—
241C2×Dic79—42no—
242C2×Dic79—42no—
243C2×Dic79—42no—
244C2×Dic79—42no—
245C2×Dic79—84no—
246C2×Dic79—82no—
247C2×Dic79—82no—
248C79⋊C8—42no—
249C79⋊C8—42no—
250C22×C79—24no—
251C22×C79—24no—
252C22×C79—21no—
253C22×C79—21no—
254C22×C79—21no—
255C22×C79—21no—
256C22×C79—44no—
257C22×C79—42no—
258C22×C79—42no—
259C22×C79—42no—
260C22×C79—42no—
261C22×C79—42no—
262C22×C79—42no—
263C22×C79—82no—
264C22×C79—84no—
265C22×C79—84no—
266C22×C79—84no—
267C22×C79#44912yes—
268C22×C79#44912yes—
269C22×C79#45512yes—
270C22×C79#45812yes—
271C22×C79C2×#12011yescharacteristic
272C22×C79C2×#13511yescharacteristic
273C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
274C4×C79—21no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79—21no—
278C4×C79—42no—
279C4×C79—42no—
280C4×C79—42no—
281C4×C79—42no—
282Dic79—82no—
283Dic79—82no—
284C2×#1612—791nomaximal, Sylow
285C2×C79—21no—
286C2×C79—21no—
287C2×C79—22no—
288C2×C79—82no—
289C2×C79—82no—
290C2×C79#161214yescomplemented
291C2×C79C2×#44911yescharacteristic
292C2×C79C2×#45511yescharacteristic
293C2×C79C2×#45811yescharacteristic
294C2×#538D7911yescharacteristic
295C2×#483—791no—
296C2×#480—791no—
297C2×#397—791no—
298#1612—798no—
299C2×#328—791no—
300C2×#327—791no—
301C2×#323—791no—
302C79C2×#161211yescharacteristic, complemented, Sylow
303#538—22no—
304#538—22no—
305#538—22no—
306#538C2×D7914yes—
307C2×#126—21no—
308C2×#124—21no—
309C2×#123—21no—
310C2×C22⋊Q8C2×D7911yescharacteristic
311C2×#121—21no—
312C2×#121C2×D7911yescharacteristic
313C2×#120—21no—
314C2×C4×D4—21no—
315C22×C4⋊C4—791no—
316C22×C22⋊C4—1581no—
317C22×C22⋊C4C2×D7911yescharacteristic
318#483—794no—
319#483—1582no—
320#480—794no—
321#480—1582no—
322C2×#107—1581no—
323C2×#106—1581no—
324C2×#103—1581no—
325C2×#102—1581no—
326C2×C22⋊C8—791no—
327C2×C22⋊C8—791no—
328#397—794no—
329#397—1582no—
330C2×#95—791no—
331C2×#95—1581no—
332#328—794no—
333#327—794no—
334#323—794no—
335C25C22⋊D7911yescharacteristic
336C22×Q8C22⋊D7911yescharacteristic
337C22×D4—21no—
338C22×D4—21no—
339C22×D4—21no—
340C22×D4C22⋊D7911yescharacteristic
341C23×C4—21no—
342C23×C4—1581no—
343C23×C4—1581no—
344C23×C4C22⋊D7911yescharacteristic
345C23×C4C22×D7911yescharacteristic
346C22×C8—1581no—
347C22×C8—1581no—
348#126—24no—
349#126—24no—
350#124—24no—
351#124—24no—
352#123—24no—
353#123—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8C22⋊D7912yes—
357C22⋊Q8C22×D7912yes—
358#121—24no—
359#121—24no—
360#121—21no—
361#121—21no—
362#121C22⋊D7912yes—
363#121C22×D7912yes—
364#120—24no—
365#120—24no—
366C4×D4—24no—
367C4×D4—24no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—792no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C4⋊C4—1581no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4C22⋊D7911yescharacteristic
377C2×C4⋊C4C22⋊D7911yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—3164no—
388C2×C22⋊C4—1584no—
389C2×C22⋊C4C22⋊D7912yes—
390C2×C22⋊C4C22×D7912yes—
391C2×C42—21no—
392#107—1584no—
393#106—1584no—
394#103—1584no—
395#102—1584no—
396C22⋊C8—792no—
397C22⋊C8—792no—
398C22⋊C8—1581no—
399C22⋊C8—1581no—
400#95—792no—
401#95—1581no—
402#95—1584no—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—22no—
408C24—42no—
409C24—42no—
410C24C2×C22⋊D7911yescharacteristic
411C24C2×C22⋊D7912yes—
412C2×Q8—21no—
413C2×Q8—42no—
414C2×Q8C2×C22⋊D7912yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×C22⋊D7912yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4—3162no—
436C22×C4—3162no—
437C22×C4—3161no—
438C22×C4—3161no—
439C22×C4—3162no—
440C22×C4—3162no—
441C22×C4—3162no—
442C22×C4—1582no—
443C22×C4—1582no—
444C22×C4C2×C22⋊D7911yescharacteristic
445C22×C4C2×C22⋊D7911yescharacteristic
446C22×C4C2×C22⋊D7912yes—
447C22×C4C2×C22⋊D7911yescharacteristic
448C22×C4C2×C22⋊D7911yescharacteristic
449C22×C4C2×C22⋊D7911yescharacteristic
450C22×C4C23×D7911yescharacteristic
451C2×C8—3162no—
452C2×C8—3162no—
453C2×C8—1582no—
454C2×C8—1582no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—3161no—
461C4⋊C4—3161no—
462C4⋊C4—1582no—
463C4⋊C4—1582no—
464C4⋊C4C2×C22⋊D7912yes—
465C4⋊C4C2×C22⋊D7912yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—3168no—
478C42—24no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—48no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23—84no—
498C23#1457913011yescharacteristic
499C23#1457913011yescharacteristic
500C23#1457918112yes—
501C23#1457918612yes—
502C23C22×C22⋊D7911yescharacteristic
503Q8—42no—
504Q8—81no—
505D4—42no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—81no—
510D4—84no—
511D4—84no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—84no—
529C2×C4—3162no—
530C2×C4—3162no—
531C2×C4—3162no—
532C2×C4—3162no—
533C2×C4—3162no—
534C2×C4—3162no—
535C2×C4—6324no—
536C2×C4—6322no—
537C2×C4—6322no—
538C2×C4#1457913011yescharacteristic
539C2×C4#1457913011yescharacteristic
540C2×C4C22×C22⋊D7911yescharacteristic
541C2×C4C22×C22⋊D7911yescharacteristic
542C8—3162no—
543C8—3162no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—24no—
549C22—24no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—82no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22#1909870812yes—
562C22#1909870812yes—
563C22#1909874112yes—
564C22#1909875312yes—
565C22C2×#1457913011yescharacteristic
566C22C2×#1457918111yescharacteristic
567C22C2×#1457918611yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—6322no—
577C4—6322no—
578C2—21no—
579C2—22no—
580C2—21no—
581C2—82no—
582C2—82no—
583C2#2376241414yesdirect product factor
584C2C2×#1909870811yescharacteristic
585C2C2×#1909874111yescharacteristic
586C2C2×#1909875311yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
Fitting, prime radicalC2×#538×C79C2
socleC23×C79#120
hypercenterC2×#538D79
hypocenterC79C2×#1612
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. C2×#19098891C2semidirect
2. C2×#19098874C2semidirect
3. C2×#19098488C2semidirect
4. #23762414C2direct (N × H)
5. C2×#19098300C2semidirect
6. C2×#19098297C2semidirect
7. C2×#19098287C2semidirect
12. #19098891C22semidirect
14. #19098874C22semidirect
22. #19098488C22semidirect
25. #19098300C22semidirect
26. #19098297C22semidirect
27. #19098287C22semidirect
290. C2×C79#1612semidirect
302. C79C2×#1612semidirect
583. C2#23762414direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190988912
C2×#190988742
C2×#190984882
#237624142
C2×#190983002
C2×#190982972
C2×#190982872
C2×#538×C792
C2×#161279