Finite Group Databasebeta

← #34327013

Subgroups — #34327013

Order 20224.

1034 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098702C211yesmaximal, characteristic
2C2×#19098620C211yesmaximal, characteristic, complemented
3#23771731C211yesmaximal, characteristic
4#23764795C212yesmaximal, complemented
5#23764715C212yesmaximal, complemented
6#23764284C212yesmaximal, complemented
7#23764093C212yesmaximal, complemented
8#23763717C211yesmaximal, characteristic
9#23763588C211yesmaximal, characteristic, complemented
10#23763335C211yesmaximal, characteristic, complemented
11C2×#391×C79C211yesmaximal, characteristic, complemented
12C22×#11929611C2212yes—
13C2×#14579144—21no—
14C2×#14579135C2211yescharacteristic
15#19098702—24no—
16#19098620—24no—
17C2×#14579094C2212yes—
18C2×C4⋊(C79⋊C8)C2212yes—
19C2×#14579085C2212yes—
20C2×#14579085C2212yes—
21C2×#14579085C2212yes—
22C2×#14579081C2212yes—
23C2×C22⋊(C79⋊C8)C2211yescharacteristic
24C2×C22⋊(C79⋊C8)C2211yescharacteristic
25C2×#14579072C2212yes—
26#19098465C2211yescharacteristic
27#19098465C2211yescharacteristic
28#19098461C2211yescharacteristic
29#19098461C2211yescharacteristic
30#19098454—21no—
31#19098454—21no—
32#19098442—21no—
33#19098423C2211yescharacteristic
34#19098325C2212yes—
35#19098325C2212yes—
36C23×Q8×C79C2211yescharacteristic
37C22×C4⋊C4×C79C2211yescharacteristic
38C22×C42×C79C2211yescharacteristic
39#391×C79—21no—
40#391×C79—21no—
41#391×C79C2212yes—
42#391×C79C2212yes—
43C2×#95×C79—21no—
44C2×#95×C79—21no—
45C22×D316C2311yescharacteristic
46C22×C22⋊D79—21no—
47C2×#11929611—24no—
48C2×#11929611—24no—
49C2×#11929611—42no—
50C22×C79⋊C8C2312yes—
51C22×C79⋊C8C2312yes—
52#14579144—44no—
53#14579135—22no—
54#14579135—22no—
55C2×C4⋊Dic79C2311yescharacteristic
56C2×C4⋊Dic79C2311yescharacteristic
57C2×C4⋊Dic79C2311yescharacteristic
58C2×Dic79⋊C4—21no—
59C2×Dic79⋊C4—21no—
60C2×C22⋊Dic79—21no—
61C2×(C2×D79)⋊C4—21no—
62C2×(C2×D79)⋊C4—21no—
63C2×(C2×D79)⋊C4—21no—
64C2×(C2×D79)⋊C4—21no—
65C2×(C2×D79)⋊C4—21no—
66#14579094—42no—
67C4⋊(C79⋊C8)—24no—
68#14579085—24no—
69#14579085—24no—
70#14579085—42no—
71#14579081—42no—
72C22⋊(C79⋊C8)—22no—
73C22⋊(C79⋊C8)D414yes—
74#14579072—24no—
75C79⋊#95—21no—
76C79⋊#95—21no—
77C79⋊#95—21no—
78C22×Q8×C79—24no—
79C22×Q8×C79—21no—
80C22×Q8×C79—22no—
81C22×Q8×C79—22no—
82C22×Q8×C79—42no—
83C22×Q8×C79C2312yes—
84C23×C4×C79—21no—
85C23×C4×C79—21no—
86C23×C4×C79D411yescharacteristic
87C23×C4×C79D411yescharacteristic
88C23×C4×C79C2311yescharacteristic
89C2×C4⋊C4×C79—21no—
90C2×C4⋊C4×C79—42no—
91C2×C4⋊C4×C79C2312yes—
92C2×C42×C79—22no—
93C2×C42×C79—22no—
94C2×C42×C79D411yescharacteristic
95C2×C42×C79D411yescharacteristic
96C2×C42×C79C2311yescharacteristic
97C2×C42×C79C2311yescharacteristic
98#95×C79—22no—
99#95×C79—22no—
100#95×C79—22no—
101#95×C79—22no—
102C23×D79—21no—
103C2×D316—21no—
104C2×D316—21no—
105C2×D316—22no—
106C2×D316—22no—
107C2×C22⋊D79—42no—
108C2×C22⋊D79—42no—
109C2×C22⋊D79—42no—
110C22×Dic79—21no—
111C22×Dic79—21no—
112C22×Dic79—21no—
113#11929611—48no—
114C2×C79⋊C8—22no—
115C2×C79⋊C8—22no—
116C2×C79⋊C8—22no—
117C2×C79⋊C8—22no—
118C2×C79⋊C8—22no—
119C2×C79⋊C8C2×D414yes—
120C4⋊Dic79—21no—
121C4⋊Dic79—22no—
122C4⋊Dic79—21no—
123C4⋊Dic79—22no—
124Dic79⋊C4—42no—
125Dic79⋊C4—42no—
126C22⋊Dic79—42no—
127(C2×D79)⋊C4—42no—
128(C2×D79)⋊C4—42no—
129(C2×D79)⋊C4—42no—
130(C2×D79)⋊C4—42no—
131(C2×D79)⋊C4—42no—
132C24×C79C2×D411yescharacteristic
133C2×Q8×C79—24no—
134C2×Q8×C79—22no—
135C2×Q8×C79—22no—
136C2×Q8×C79—22no—
137C2×Q8×C79—22no—
138C2×Q8×C79—22no—
139C2×Q8×C79—48no—
140C2×Q8×C79—48no—
141C2×Q8×C79—42no—
142C2×Q8×C79—42no—
143C2×Q8×C79#4114yes—
144C22×C4×C79—22no—
145C22×C4×C79—22no—
146C22×C4×C79—24no—
147C22×C4×C79—22no—
148C22×C4×C79—22no—
149C22×C4×C79—22no—
150C22×C4×C79—22no—
151C22×C4×C79—22no—
152C22×C4×C79—22no—
153C22×C4×C79—22no—
154C22×C4×C79—41no—
155C22×C4×C79—42no—
156C22×C4×C79—42no—
157C22×C4×C79—42no—
158C22×C4×C79—42no—
159C22×C4×C79—42no—
160C22×C4×C79C2×D411yescharacteristic
161C22×C4×C79C2×D411yescharacteristic
162C22×C4×C79C2×D411yescharacteristic
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D412yes—
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79#4112yes—
167C22×C4×C79C2411yescharacteristic
168C4⋊C4×C79—22no—
169C4⋊C4×C79—22no—
170C4⋊C4×C79—42no—
171C42×C79—22no—
172C42×C79—22no—
173C42×C79—21no—
174C42×C79—21no—
175C42×C79—21no—
176C42×C79—21no—
177C23×C79—21no—
178C23×C79—21no—
179C23×C79—21no—
180C23×C79—21no—
181C23×C79#12112yes—
182C23×C79AGL(1,ℤ/8)12yes—
183C23×C79#13712yes—
184C23×C79C22×D411yescharacteristic
185C22×D79—41no—
186C22×D79—41no—
187C22×D79—41no—
188C22×D79—41no—
189C22×D79—41no—
190C22×D79—41no—
191C22×D79—41no—
192Q8×C79—24no—
193Q8×C79—24no—
194Q8×C79—48no—
195Q8×C79—42no—
196Q8×C79—42no—
197C2×C4×C79—21no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—21no—
201C2×C4×C79—24no—
202C2×C4×C79—24no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—24no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—44no—
214C2×C4×C79—48no—
215C2×C4×C79—42no—
216C2×C4×C79—42no—
217C2×C4×C79—44no—
218C2×C4×C79—42no—
219C2×C4×C79—42no—
220C2×C4×C79—42no—
221C2×C4×C79—84no—
222C2×C4×C79#12112yes—
223C2×C4×C79#13712yes—
224C2×C4×C79C22×D411yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C2×C4×C79C2×#4112yes—
227C2×C4×C79H(5,𝔽2)12yes—
228C22⋊D79—84no—
229C2×Dic79—41no—
230C2×Dic79—41no—
231C2×Dic79—41no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238D316—42no—
239D316—42no—
240C79⋊C8—24no—
241C79⋊C8—42no—
242C22×C79—22no—
243C22×C79—22no—
244C22×C79—22no—
245C22×C79—22no—
246C22×C79—22no—
247C22×C79—22no—
248C22×C79#46814yes—
249C22×C79C2×#12111yescharacteristic
250C22×C79#52511yescharacteristic
251C22×C79#53511yescharacteristic
252C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C79C2×#13711yescharacteristic
254C22×C79C2×#13711yescharacteristic
255C22×C79#57511yescharacteristic
256C2×D79—81no—
257C2×D79—81no—
258C2×D79—81no—
259C2×D79—81no—
260C2×D79—81no—
261C2×D79—81no—
262C2×D79—81no—
263C4×C79—22no—
264C4×C79—22no—
265C4×C79—44no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79—82no—
270C4×C79#54612yes—
271C4×C79#55112yes—
272Dic79—81no—
273Dic79—81no—
274Dic79—81no—
275#24509—791nomaximal, Sylow
276C2×C79—24no—
277C2×C79#241111yescharacteristic
278C2×C79C2×#46811yescharacteristic
279C2×C79#305611yescharacteristic
280C2×C79#311211yescharacteristic
281C2×C79#311711yescharacteristic
282C2×C79#321311yescharacteristic
283C2×C79#321811yescharacteristic
284D79—161no—
285C2×#448—791no—
286C2×#432—791no—
287#2854—791no—
288C2×#391D7911yescharacteristic, complemented
289#2068—792no—
290#2055—792no—
291#1973—792no—
292#1935—792no—
293#1874—791no—
294#1852—791no—
295#1787—791no—
296C79#2450911yescharacteristic, complemented, Sylow
297C23×Q8C2×D7911yescharacteristic
298C22×SD16—792no—
299C2×#123—1581no—
300C2×#121—791no—
301C22×C4⋊C4C2×D7911yescharacteristic
302C22×C42C2×D7911yescharacteristic
303#448—1584no—
304#432—1584no—
305C2×#107—792no—
306C2×C4⋊C8—792no—
307C2×#103—792no—
308C2×#103—792no—
309C2×#103—792no—
310C2×#102—792no—
311C2×C22⋊C8—791no—
312C2×C22⋊C8—791no—
313C2×#97—792no—
314#391—21no—
315#391—21no—
316#391C2×D7912yes—
317#391C2×D7912yes—
318#390—791no—
319#390—791no—
320#389—791no—
321#389—791no—
322#388—1581no—
323#388—1581no—
324#386—1581no—
325#380—791no—
326C2×#95—21no—
327C2×#95—21no—
328#340—792no—
329#340—792no—
330C22×Q8—24no—
331C22×Q8—21no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—42no—
335C22×Q8C22×D7912yes—
336C22×D4—791no—
337C22×D4—1581no—
338C23×C4—21no—
339C23×C4—21no—
340C23×C4C22⋊D7911yescharacteristic
341C23×C4C22⋊D7911yescharacteristic
342C23×C4C22×D7911yescharacteristic
343C2×SD16—3162no—
344C2×SD16—1584no—
345C2×SD16—1584no—
346C22×C8—792no—
347C22×C8—792no—
348#123—3164no—
349#121—1582no—
350#121—1582no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—791no—
354C2×C4⋊C4—791no—
355C2×C4⋊C4—791no—
356C2×C4⋊C4—1581no—
357C2×C4⋊C4—1581no—
358C2×C4⋊C4C22×D7912yes—
359C2×C22⋊C4—1581no—
360C2×C22⋊C4—1581no—
361C2×C22⋊C4—1581no—
362C2×C22⋊C4—1581no—
363C2×C22⋊C4—1581no—
364C2×C22⋊C4—1581no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42C22⋊D7911yescharacteristic
368C2×C42C22⋊D7911yescharacteristic
369C2×C42C22×D7911yescharacteristic
370C2×C42C22×D7911yescharacteristic
371#107—3162no—
372C4⋊C8—1584no—
373#103—3162no—
374#103—1584no—
375#103—1584no—
376#102—3162no—
377C22⋊C8—794no—
378C22⋊C8—1582no—
379#97—1584no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—1581no—
385#95—1581no—
386#95—1581no—
387C24—1581no—
388C24C2×C22⋊D7911yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8#1192962414yes—
400C2×D4—3162no—
401C2×D4—3162no—
402C2×D4—3162no—
403C2×D4—1581no—
404C2×D4—1582no—
405C2×D4—1582no—
406C2×D4—1581no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—1581no—
424C22×C4—1581no—
425C22×C4—1581no—
426C22×C4D4×D7912yes—
427C22×C4C2×C22⋊D7911yescharacteristic
428C22×C4C2×C22⋊D7911yescharacteristic
429C22×C4C2×C22⋊D7911yescharacteristic
430C22×C4C2×C22⋊D7911yescharacteristic
431C22×C4C2×C22⋊D7911yescharacteristic
432C22×C4#1192962312yes—
433C22×C4C23×D7911yescharacteristic
434SD16—3168no—
435C2×C8—794no—
436C2×C8—1582no—
437C2×C8—1582no—
438C2×C8—1582no—
439C2×C8—1582no—
440C2×C8—1582no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—3162no—
445C4⋊C4—3162no—
446C4⋊C4—1581no—
447C4⋊C4—1582no—
448C4⋊C4—1582no—
449C4⋊C4—1581no—
450C22⋊C4—3162no—
451C22⋊C4—3162no—
452C22⋊C4—3162no—
453C22⋊C4—3162no—
454C22⋊C4—3162no—
455C22⋊C4—3162no—
456C42—21no—
457C42—21no—
458C42—22no—
459C42—22no—
460C42—21no—
461C42—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—3161no—
467C23—3161no—
468C23—3161no—
469C23—3161no—
470C23—3161no—
471C23—3161no—
472C23—3161no—
473C23#1457913312yes—
474C23#1457918612yes—
475C23#1457919212yes—
476C23C22×C22⋊D7911yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482D4—3162no—
483D4—3162no—
484D4—6324no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—84no—
510C2×C4—3161no—
511C2×C4—3161no—
512C2×C4—3161no—
513C2×C4—3161no—
514C2×C4—3161no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4#1457913312yes—
520C2×C4#1457919212yes—
521C2×C4C2×D4×D7911yescharacteristic
522C2×C4C2×#1192962311yescharacteristic
523C2×C4C2×#1192962412yes—
524C2×C4#1457920612yes—
525C8—3162no—
526C8—1584no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—6321no—
534C22—6321no—
535C22—6321no—
536C22—6321no—
537C22—6321no—
538C22—6321no—
539C22—6321no—
540C22#1909881114yes—
541C22C2×#1457913311yescharacteristic
542C22#1909907411yescharacteristic
543C22#1909913011yescharacteristic
544C22C2×#1457918611yescharacteristic
545C22C2×#1457919211yescharacteristic
546C22C2×#1457919211yescharacteristic
547C22#1909936111yescharacteristic
548C4—22no—
549C4—22no—
550C4—44no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—82no—
555C4—6321no—
556C4—6321no—
557C4—6321no—
558C4#1909920312yes—
559C4#1909923912yes—
560C2—24no—
561C2—12641no—
562C2#2376678511yescharacteristic
563C2C2×#1909881111yescharacteristic
564C2#2377302611yescharacteristic
565C2#2377352211yescharacteristic
566C2#2377357111yescharacteristic
567C2#2377438711yescharacteristic
568C2#2377442711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#391×C79C2
socleC23×C79C22×D4
prime radicalC2×#14579135C22
hypercenterC2×#391D79
hypocenterC79#24509
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×#19098620C2semidirect
4. #23764795C2semidirect
5. #23764715C2semidirect
6. #23764284C2semidirect
7. #23764093C2semidirect
9. #23763588C2semidirect
10. #23763335C2semidirect
11. C2×#391×C79C2semidirect
288. C2×#391D79semidirect
296. C79#24509semidirect

Maximal subgroups

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

subgroupindex
C2×#190987022
C2×#190986202
#237717312
#237647952
#237647152
#237642842
#237640932
#237637172
#237635882
#237633352
C2×#391×C792
#2450979