Finite Group Databasebeta

← #34329563

Subgroups — #34329563

Order 20224.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098850C211yesmaximal, characteristic, complemented
2C2×#19098620C211yesmaximal, characteristic, complemented
3C4×#14579135C211yesmaximal, characteristic
4#23764836C212yesmaximal, complemented
5#23764582C212yesmaximal
6#23764349C211yesmaximal, characteristic, complemented
7#23764156C211yesmaximal, characteristic, complemented
8#23764135C212yesmaximal, complemented
9#23764096C212yesmaximal, complemented
10#23763397C211yesmaximal, characteristic, complemented
11C79×#2857C211yesmaximal, characteristic, complemented
12C2×#14579135C2211yescharacteristic
13C2×C4×D316C2211yescharacteristic
14C2×C4×C22⋊D79—21no—
15#19098850—24no—
16#19098620—24no—
17C2×#14579094C2212yes—
18C2×#14579092C2212yes—
19C2×C4⋊(C79⋊C8)C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579081C2212yes—
22C2×#14579081C2212yes—
23C2×C22⋊(C79⋊C8)C2212yes—
24C2×#14579072C2212yes—
25#19098465C2211yescharacteristic
26#19098461C2211yescharacteristic
27#19098454—21no—
28#19098442—21no—
29#19098423C2211yescharacteristic
30C4×C4⋊Dic79C2211yescharacteristic
31C4×(C2×D79)⋊C4—21no—
32#19098325C2212yes—
33#19098325C2212yes—
34#19098325C2212yes—
35#19098325C2212yes—
36C2×C4⋊Q8×C79C2211yescharacteristic
37C2×C22⋊Q8×C79—21no—
38C22×C42×C79C2211yescharacteristic
39#391×C79—21no—
40#389×C79—21no—
41#389×C79C2212yes—
42#384×C79C2211yescharacteristic
43#380×C79—21no—
44#380×C79C2212yes—
45C22×D316C2311yescharacteristic
46C22×C22⋊D79—21no—
47C22×C4×D79—21no—
48C22×C79⋊C8C2312yes—
49C22×C79⋊C8C2312yes—
50#14579135—22no—
51#14579135—22no—
52C4×D316—22no—
53C4×D316—22no—
54C4×C22⋊D79—44no—
55C2×C4⋊Dic79C2311yescharacteristic
56C2×C4⋊Dic79C2311yescharacteristic
57C2×C4⋊Dic79C2311yescharacteristic
58C2×Dic79⋊C4—21no—
59C2×C22⋊Dic79—21no—
60C2×(C2×D79)⋊C4—21no—
61C2×(C2×D79)⋊C4—21no—
62C2×(C2×D79)⋊C4—21no—
63C2×(C2×D79)⋊C4—21no—
64C2×C4×Dic79—21no—
65#14579094—24no—
66#14579092—24no—
67C4⋊(C79⋊C8)—22no—
68C4⋊(C79⋊C8)D414yescomplemented
69#14579081—24no—
70#14579081—24no—
71C22⋊(C79⋊C8)—24no—
72#14579072—24no—
73C79⋊#95—21no—
74C79⋊#95—21no—
75C79⋊#95—21no—
76C22×Q8×C79—21no—
77C23×C4×C79D411yescharacteristic
78C23×C4×C79D411yescharacteristic
79C23×C4×C79C2311yescharacteristic
80C4⋊Q8×C79—22no—
81C4⋊Q8×C79—22no—
82C22⋊Q8×C79—44no—
83C2×C4⋊C4×C79—21no—
84C2×C4⋊C4×C79—22no—
85C2×C4⋊C4×C79—21no—
86C2×C4⋊C4×C79—21no—
87C2×C4⋊C4×C79C2312yes—
88C2×C4⋊C4×C79C2312yes—
89C2×C22⋊C4×C79—22no—
90C2×C42×C79—22no—
91C2×C42×C79—22no—
92C2×C42×C79D411yescharacteristic
93C2×C42×C79D411yescharacteristic
94C2×C42×C79C2311yescharacteristic
95C2×C42×C79C2311yescharacteristic
96#95×C79—22no—
97#95×C79—22no—
98C23×D79—21no—
99C2×D316—21no—
100C2×D316—21no—
101C2×D316—21no—
102C2×D316—21no—
103C2×D316—21no—
104C2×D316#4112yes—
105C2×C22⋊D79—24no—
106C2×C22⋊D79—42no—
107C2×C22⋊D79—42no—
108C22×Dic79—21no—
109C22×Dic79—21no—
110C22×Dic79—21no—
111C2×C4×D79—42no—
112C2×C4×D79—42no—
113C2×C4×D79—42no—
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—21no—
122C4⋊Dic79—21no—
123C4⋊Dic79—21no—
124C4⋊Dic79—21no—
125C4⋊Dic79#4112yes—
126Dic79⋊C4—42no—
127C22⋊Dic79—42no—
128(C2×D79)⋊C4—24no—
129(C2×D79)⋊C4—42no—
130(C2×D79)⋊C4—42no—
131(C2×D79)⋊C4—42no—
132C4×Dic79—42no—
133C24×C79C2×D411yescharacteristic
134C2×Q8×C79—42no—
135C2×Q8×C79—42no—
136C2×Q8×C79—42no—
137C22×C4×C79—22no—
138C22×C4×C79—22no—
139C22×C4×C79—22no—
140C22×C4×C79—22no—
141C22×C4×C79—22no—
142C22×C4×C79—21no—
143C22×C4×C79—21no—
144C22×C4×C79—21no—
145C22×C4×C79—21no—
146C22×C4×C79—21no—
147C22×C4×C79—21no—
148C22×C4×C79—22no—
149C22×C4×C79—22no—
150C22×C4×C79—41no—
151C22×C4×C79C2×D411yescharacteristic
152C22×C4×C79C2×D411yescharacteristic
153C22×C4×C79C2×D411yescharacteristic
154C22×C4×C79C2×D411yescharacteristic
155C22×C4×C79C2×D411yescharacteristic
156C22×C4×C79C2411yescharacteristic
157C4⋊C4×C79—22no—
158C4⋊C4×C79—22no—
159C4⋊C4×C79—22no—
160C4⋊C4×C79—22no—
161C4⋊C4×C79—42no—
162C4⋊C4×C79—44no—
163C4⋊C4×C79—42no—
164C4⋊C4×C79—42no—
165C22⋊C4×C79—44no—
166C42×C79—22no—
167C42×C79—22no—
168C42×C79—21no—
169C42×C79—21no—
170C42×C79C2×D412yes—
171C42×C79#4112yes—
172C23×C79—21no—
173C23×C79—21no—
174C23×C79—21no—
175C23×C79—21no—
176C23×C79—21no—
177C23×C79—21no—
178C23×C79AGL(1,ℤ/8)12yes—
179C23×C79C22×D411yescharacteristic
180C22×D79—22no—
181C22×D79—41no—
182C22×D79—41no—
183C22×D79—41no—
184C22×D79—41no—
185C22×D79—41no—
186C22×D79—41no—
187Q8×C79—84no—
188C2×C4×C79—21no—
189C2×C4×C79—21no—
190C2×C4×C79—21no—
191C2×C4×C79—21no—
192C2×C4×C79—21no—
193C2×C4×C79—21no—
194C2×C4×C79—21no—
195C2×C4×C79—21no—
196C2×C4×C79—21no—
197C2×C4×C79—21no—
198C2×C4×C79—21no—
199C2×C4×C79—44no—
200C2×C4×C79—41no—
201C2×C4×C79—41no—
202C2×C4×C79—41no—
203C2×C4×C79—41no—
204C2×C4×C79—44no—
205C2×C4×C79—42no—
206C2×C4×C79—42no—
207C2×C4×C79—42no—
208C2×C4×C79—42no—
209C2×C4×C79—42no—
210C2×C4×C79—42no—
211C2×C4×C79#12112yes—
212C2×C4×C79#12112yes—
213C2×C4×C79AGL(1,ℤ/8)12yes—
214C2×C4×C79#13712yes—
215C2×C4×C79C22×D411yescharacteristic
216C2×C4×C79C2×#4111yescharacteristic
217C2×C4×C79C2×#4111yescharacteristic
218C2×C4×C79C2×#4111yescharacteristic
219C2×C4×C792−1+411yescharacteristic
220C2×C4×C792−1+411yescharacteristic
221C22⋊D79—48no—
222C2×Dic79—22no—
223C2×Dic79—41no—
224C2×Dic79—41no—
225C2×Dic79—41no—
226C2×Dic79—41no—
227C2×Dic79—41no—
228C2×Dic79—41no—
229C2×Dic79—41no—
230C2×Dic79—41no—
231D316—22no—
232D316—22no—
233D316—41no—
234D316—41no—
235C4×D79—84no—
236C79⋊C8—24no—
237C79⋊C8—42no—
238C22×C79—22no—
239C22×C79—22no—
240C22×C79—22no—
241C22×C79—41no—
242C22×C79—41no—
243C22×C79—41no—
244C22×C79—41no—
245C22×C79C2×#12111yescharacteristic
246C22×C79#52511yescharacteristic
247C22×C79#53611yescharacteristic
248C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C79C2×#13711yescharacteristic
251C22×C79#57511yescharacteristic
252C2×D79—42no—
253C2×D79—42no—
254C2×D79—42no—
255C2×D79—81no—
256C2×D79—81no—
257C2×D79—81no—
258C2×D79—81no—
259C4×C79—22no—
260C4×C79—41no—
261C4×C79—41no—
262C4×C79—41no—
263C4×C79—82no—
264C4×C79—82no—
265C4×C79#46814yes—
266C4×C79#54711yescharacteristic
267C4×C79#54811yescharacteristic
268C4×C79#55211yescharacteristic
269C4×C79#55611yescharacteristic
270Dic79—42no—
271Dic79—81no—
272Dic79—81no—
273#24708—791nomaximal, Sylow
274C2×C79—42no—
275C2×C79#241511yescharacteristic
276C2×C79C2×#46811yescharacteristic
277C2×C79#305511yescharacteristic
278C2×C79#310911yescharacteristic
279C2×C79#311611yescharacteristic
280C2×C79#323511yescharacteristic
281C2×C79#323811yescharacteristic
282D79—82no—
283C2×#476—791no—
284C2×#432—791no—
285#2857D7911yescharacteristic, complemented
286C4×#121—791no—
287#2077—792no—
288#2033—792no—
289#1987—791no—
290#1947—791no—
291#1943—792no—
292#1936—792no—
293#1802—791no—
294C79#2470811yescharacteristic, complemented, Sylow
295C2×C4⋊Q8C2×D7911yescharacteristic
296C2×C22⋊Q8—21no—
297C2×#121—791no—
298C2×C4×D4—791no—
299C2×C4×D4—1581no—
300C22×C42C2×D7911yescharacteristic
301#476—1584no—
302#432—1584no—
303C2×#107—792no—
304C2×#106—792no—
305C2×C4⋊C8—791no—
306C2×C4⋊C8—791no—
307C2×#102—792no—
308C2×#102—792no—
309C2×C22⋊C8—792no—
310C2×#97—792no—
311#391—21no—
312#390—791no—
313#389—21no—
314#389—791no—
315#389C2×D7912yes—
316#388—1581no—
317#386—1581no—
318#384C2×D7911yescharacteristic
319#380—21no—
320#380—791no—
321#380C2×D7912yes—
322C4×C4⋊C4—791no—
323C4×C22⋊C4—1581no—
324#340—792no—
325#340—792no—
326#340—792no—
327#340—792no—
328C22×Q8—21no—
329C22×D4—791no—
330C22×D4—1581no—
331C23×C4—1581no—
332C23×C4C22⋊D7911yescharacteristic
333C23×C4C22⋊D7911yescharacteristic
334C23×C4C22×D7911yescharacteristic
335C22×C8—792no—
336C22×C8—792no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C22⋊Q8—44no—
340#121—1582no—
341#121—1582no—
342C4×D4—3164no—
343C4×D4—1582no—
344C4×D4—1582no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—791no—
350C2×C4⋊C4—791no—
351C2×C4⋊C4—791no—
352C2×C4⋊C4—1581no—
353C2×C4⋊C4C22×D7912yes—
354C2×C4⋊C4C22×D7912yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—1581no—
357C2×C22⋊C4—1581no—
358C2×C22⋊C4—1581no—
359C2×C22⋊C4—1581no—
360C2×C22⋊C4—1581no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—1581no—
364C2×C42C22⋊D7911yescharacteristic
365C2×C42C22⋊D7911yescharacteristic
366C2×C42C22×D7911yescharacteristic
367C2×C42C22×D7911yescharacteristic
368#107—1584no—
369#106—1584no—
370C4⋊C8—794no—
371C4⋊C8—1582no—
372#102—1584no—
373#102—1584no—
374C22⋊C8—1584no—
375#97—1584no—
376#95—22no—
377#95—22no—
378#95—1581no—
379#95—1581no—
380#95—1581no—
381C24—1581no—
382C24C2×C22⋊D7911yescharacteristic
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×D4—3162no—
387C2×D4—3162no—
388C2×D4—792no—
389C2×D4—1584no—
390C2×D4—1581no—
391C2×D4—1581no—
392C2×D4—1581no—
393C2×D4—1581no—
394C2×D4—1581no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—3162no—
410C22×C4—3162no—
411C22×C4—3162no—
412C22×C4—1581no—
413C22×C4—1581no—
414C22×C4—1581no—
415C22×C4C2×C22⋊D7911yescharacteristic
416C22×C4C2×C22⋊D7911yescharacteristic
417C22×C4C2×C22⋊D7911yescharacteristic
418C22×C4C2×C22⋊D7911yescharacteristic
419C22×C4C2×C22⋊D7911yescharacteristic
420C22×C4C23×D7911yescharacteristic
421C2×C8—794no—
422C2×C8—1582no—
423C2×C8—1582no—
424C2×C8—1582no—
425C2×C8—1582no—
426C2×C8—1582no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—3162no—
436C4⋊C4—792no—
437C4⋊C4—1581no—
438C4⋊C4—1581no—
439C4⋊C4—1581no—
440C4⋊C4—1581no—
441C4⋊C4—1581no—
442C22⋊C4—44no—
443C22⋊C4—3162no—
444C22⋊C4—3162no—
445C22⋊C4—3162no—
446C22⋊C4—3162no—
447C22⋊C4—1584no—
448C42—21no—
449C42—22no—
450C42—22no—
451C42—21no—
452C42—3162no—
453C42D4×D7912yes—
454C42#1192962312yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—3161no—
462C23—3161no—
463C23—3161no—
464C23—3161no—
465C23—3161no—
466C23—3161no—
467C23—1582no—
468C23#1457918712yes—
469C23C22×C22⋊D7911yescharacteristic
470Q8—84no—
471D4—3168no—
472D4—3161no—
473D4—3161no—
474D4—1582no—
475D4—1582no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—3161no—
500C2×C4—3161no—
501C2×C4—3161no—
502C2×C4—3161no—
503C2×C4—3161no—
504C2×C4—3161no—
505C2×C4—3161no—
506C2×C4—3161no—
507C2×C4—6324no—
508C2×C4—1582no—
509C2×C4#1457913312yes—
510C2×C4#1457913312yes—
511C2×C4#1457918612yes—
512C2×C4#1457919212yes—
513C2×C4C2×D4×D7911yescharacteristic
514C2×C4#41×D7911yescharacteristic
515C2×C4#41×D7911yescharacteristic
516C2×C4C2×#1192962311yescharacteristic
517C2×C4#1457920911yescharacteristic
518C2×C4#1457920911yescharacteristic
519C8—3162no—
520C8—1584no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—3162no—
529C22—3162no—
530C22—3162no—
531C22—6321no—
532C22—6321no—
533C22—6321no—
534C22—6321no—
535C22C2×#1457913311yescharacteristic
536C22#1909907011yescharacteristic
537C22#1909913711yescharacteristic
538C22C2×#1457918611yescharacteristic
539C22C2×#1457918711yescharacteristic
540C22C2×#1457919211yescharacteristic
541C22#1909936111yescharacteristic
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—82no—
548C4—3162no—
549C4—6321no—
550C4—6321no—
551C4#1909881114yes—
552C4#1909921511yescharacteristic
553C4#1909922311yescharacteristic
554C4#1909924711yescharacteristic
555C4#1909928411yescharacteristic
556C2—42no—
557C2—6322no—
558C2#2376683011yescharacteristic
559C2C2×#1909881111yescharacteristic
560C2#2377301511yescharacteristic
561C2#2377348711yescharacteristic
562C2#2377355611yescharacteristic
563C2#2377459811yescharacteristic
564C2#2377462111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2857C2
socleC23×C79C22×D4
prime radicalC2×#14579135C22
hypercenter#2857D79
hypocenterC79#24708
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
1. C2×#19098850C2semidirect
2. C2×#19098620C2semidirect
4. #23764836C2semidirect
6. #23764349C2semidirect
7. #23764156C2semidirect
8. #23764135C2semidirect
9. #23764096C2semidirect
10. #23763397C2semidirect
11. C79×#2857C2semidirect
68. C4⋊(C79⋊C8)D4semidirect
285. #2857D79semidirect
294. C79#24708semidirect

Maximal subgroups

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

subgroupindex
C2×#190988502
C2×#190986202
C4×#145791352
#237648362
#237645822
#237643492
#237641562
#237641352
#237640962
#237633972
C79×#28572
#2470879