Finite Group Databasebeta

← #34445384

Subgroups — #34445384

Order 20224.

548 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775143C211yesmaximal, characteristic
2#23765639C211yesmaximal, characteristic, complemented
3#23765597C211yesmaximal, characteristic, complemented
4#23765585C211yesmaximal, characteristic, complemented
5#23765539C211yesmaximal, characteristic, complemented
6#23765506C211yesmaximal, characteristic, complemented
7#23765498C211yesmaximal, characteristic, complemented
8#23765462C211yesmaximal, characteristic, complemented
9#23765338C211yesmaximal, characteristic, complemented
10#23765333C211yesmaximal, characteristic, complemented
11#23765269C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765110C211yesmaximal, characteristic, complemented
14C2×#19098366C211yesmaximal, characteristic, complemented
15C79×#3338C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C22×Dic316C2211yescharacteristic
18#19099222—21no—
19C2×#14579107C2211yescharacteristic
20C2×#14579107C2211yescharacteristic
21C2×#14579104C2211yescharacteristic
22C2×#14579101C2211yescharacteristic
23C22×C79⋊C16C2211yescharacteristic
24#19098947C2211yescharacteristic
25#19098804C2211yescharacteristic
26#19098799C2211yescharacteristic
27#19098786—21no—
28#19098727—21no—
29#19098704—21no—
30C4×Dic316C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098390C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098366C2211yescharacteristic
37#19098366C2211yescharacteristic
38#19098366C2211yescharacteristic
39#19098366C2211yescharacteristic
40#19098366C2211yescharacteristic
41#19098366C2211yescharacteristic
42#19098363C2211yescharacteristic
43#19098363C2211yescharacteristic
44#19098363C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47#19098339C2211yescharacteristic
48#554×C79—21no—
49#500×C79C2211yescharacteristic
50#484×C79—21no—
51#484×C79—21no—
52#477×C79—21no—
53#467×C79C2211yescharacteristic
54#467×C79C2211yescharacteristic
55C4×Q16×C79C2211yescharacteristic
56C8×D4×C79C2211yescharacteristic
57C2×#107×C79C2211yescharacteristic
58C2×#107×C79C2211yescharacteristic
59C2×#11929624—21no—
60C22×Dic158—21no—
61#14579183—22no—
62#14579183—41no—
63C2×Dic316—22no—
64C2×Dic316—21no—
65C2×Dic316—41no—
66C2×Dic316C2311yescharacteristic
67C2×Dic316C2311yescharacteristic
68C2×#11929613—21no—
69C2×#11929613—21no—
70C2×D632C2311yescharacteristic
71#14579149—21no—
72#14579147—41no—
73#14579140—21no—
74#14579139—41no—
75#14579135—21no—
76#14579133—41no—
77C4×Dic158—21no—
78(C22⋊D79)⋊C4—41no—
79#14579107—22no—
80#14579107—41no—
81#14579104—41no—
82#14579101—41no—
83C2×C79⋊C16—41no—
84C2×C79⋊C16C2311yescharacteristic
85C2×C79⋊C16C2311yescharacteristic
86C2×C79⋊C16C2311yescharacteristic
87C2×C79⋊C16C2311yescharacteristic
88#14579095C2311yescharacteristic
89#14579086—21no—
90#14579086—21no—
91#14579083—21no—
92C2×Q16×C79C2311yescharacteristic
93C22×C8×C79C2311yescharacteristic
94C22×C8×C79C2311yescharacteristic
95C4⋊Q8×C79—21no—
96#125×C79—41no—
97C22⋊Q8×C79—21no—
98C22⋊Q8×C79—21no—
99C22⋊Q8×C79—41no—
100C22⋊Q8×C79—41no—
101C4×Q8×C79—21no—
102C4×D4×C79—41no—
103C4×D4×C79D411yescharacteristic
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79—21no—
106#107×C79—21no—
107#107×C79—21no—
108#107×C79—21no—
109#107×C79C2311yescharacteristic
110#107×C79C2311yescharacteristic
111#107×C79C2311yescharacteristic
112C4⋊C8×C79D411yescharacteristic
113#103×C79—21no—
114#103×C79—21no—
115#103×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119#11929624—42no—
120#11929624—81no—
121C2×Dic158—21no—
122C2×Dic158—21no—
123C2×Dic158—42no—
124C2×Dic158—41no—
125C2×Dic158—81no—
126C2×D316—21no—
127C2×C22⋊D79—41no—
128C2×C22⋊D79—41no—
129C22×Dic79—41no—
130C2×C4×D79—41no—
131Dic316—21no—
132Dic316—21no—
133Dic316—21no—
134Dic316—42no—
135Dic316#4112yes—
136#11929613—42no—
137#11929613—81no—
138D632—21no—
139D632—21no—
140C4⋊Dic79—21no—
141Dic79⋊C4—41no—
142Dic79⋊C4—41no—
143C22⋊Dic79—41no—
144(C2×D79)⋊C4—41no—
145(C2×D79)⋊C4—41no—
146(C2×D79)⋊C4—41no—
147C4×Dic79—41no—
148C79⋊C16—21no—
149C79⋊C16—21no—
150C79⋊C16—21no—
151C79⋊C16—21no—
152C2×Q8×C79—21no—
153C2×Q8×C79—41no—
154C2×D4×C79D811yescharacteristic
155C22×C4×C79—41no—
156C22×C4×C79—41no—
157C22×C4×C79C2×D411yescharacteristic
158C22×C4×C79C2×D411yescharacteristic
159Q16×C79—21no—
160Q16×C79—21no—
161C2×C8×C79—21no—
162C2×C8×C79—21no—
163C2×C8×C79—21no—
164C2×C8×C79—21no—
165C2×C8×C79C2×D411yescharacteristic
166C2×C8×C79C2×D411yescharacteristic
167C2×C8×C79C2×D411yescharacteristic
168C2×C8×C79C2411yescharacteristic
169C4⋊C4×C79—21no—
170C4⋊C4×C79—21no—
171C4⋊C4×C79—21no—
172C4⋊C4×C79—41no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79—41no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79—41no—
178C4⋊C4×C79D811yescharacteristic
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79D811yescharacteristic
182C22⋊C4×C79D811yescharacteristic
183C42×C79—41no—
184C42×C79C2×D411yescharacteristic
185C23×C79C2×D811yescharacteristic
186C23×C79C2×D811yescharacteristic
187C22×D79—41no—
188Q8×C79—41no—
189Q8×C79—41no—
190Q8×C79—161no—
191D4×C79—22no—
192C2×C4×C79—21no—
193C2×C4×C79—21no—
194C2×C4×C79—21no—
195C2×C4×C79—21no—
196C2×C4×C79—41no—
197C2×C4×C79—41no—
198C2×C4×C79—41no—
199C2×C4×C79—41no—
200C2×C4×C79—161no—
201C2×C4×C79—161no—
202C2×C4×C79C2×D811yescharacteristic
203C2×C4×C79C2×D811yescharacteristic
204C2×C4×C79C2×D811yescharacteristic
205C2×C4×C79C2×D811yescharacteristic
206C2×C4×C79C22×D411yescharacteristic
207C22⋊D79—82no—
208C22⋊D79—82no—
209C2×Dic79—41no—
210C2×Dic79—41no—
211C2×Dic79—41no—
212C2×Dic79—82no—
213D316—41no—
214D316—41no—
215C4×D79—82no—
216Dic158—22no—
217Dic158—41no—
218Dic158—41no—
219Dic158—41no—
220Dic158—82no—
221C8×C79—21no—
222C8×C79—21no—
223C8×C79—21no—
224C8×C79C2×#4111yescharacteristic
225C8×C79H(5,𝔽2)11yescharacteristic
226C22×C79—21no—
227C22×C79—21no—
228C22×C79—21no—
229C22×C79—21no—
230C22×C79—21no—
231C22×C79#51012yes—
232C22×C79C22×D811yescharacteristic
233C2×D79—81no—
234C2×D79—81no—
235C2×D79—81no—
236C4×C79—21no—
237C4×C79—21no—
238C4×C79—21no—
239C4×C79—21no—
240C4×C79—81no—
241C4×C79—81no—
242C4×C79—81no—
243C4×C79—81no—
244C4×C79#54611yescharacteristic
245C4×C79#57711yescharacteristic
246Dic79—42no—
247Dic79—81no—
248Dic79—81no—
249#33883—791nomaximal, Sylow
250C2×C79—22no—
251C2×C79—41no—
252C2×C79#329411yescharacteristic
253C2×C79C2×#51011yescharacteristic
254C2×C79#341611yescharacteristic
255D79—161no—
256#3338D7911yescharacteristic, complemented
257#3285—791no—
258#2243—791no—
259#2235—791no—
260#2233—791no—
261#2225—791no—
262#2219—791no—
263#2217—791no—
264#2211—791no—
265#2189—791no—
266#2188—791no—
267#2176—791no—
268#2168—791no—
269#2139—791no—
270C2×#358—791no—
271C79#3388311yescharacteristic, complemented, Sylow
272C2×#135—791no—
273C22×Q16—791no—
274#554—21no—
275#548—1581no—
276C2×Q32—791no—
277C2×Q32—791no—
278C2×SD32—791no—
279C2×M32—791no—
280C22×C16—791no—
281#500C2×D7911yescharacteristic
282#495—791no—
283#484—21no—
284#484—21no—
285#477—21no—
286#467—791no—
287#467C2×D7911yescharacteristic
288#467C2×D7911yescharacteristic
289#466—791no—
290#464—1581no—
291#452—1581no—
292#448—1581no—
293C4×Q16—791no—
294C4×Q16C2×D7911yescharacteristic
295C8×D4C2×D7911yescharacteristic
296C2×#107C2×D7911yescharacteristic
297C2×#107C2×D7911yescharacteristic
298#367—791no—
299#367—791no—
300#367—791no—
301C4⋊C16—791no—
302#358—791no—
303#358—791no—
304#358—791no—
305#358—791no—
306#358—791no—
307#358—791no—
308#358—791no—
309#357—791no—
310#357—791no—
311#357—791no—
312C22⋊C16—791no—
313C22⋊C16—791no—
314#346—791no—
315C2×#41—1581no—
316C22×Q8—1581no—
317#135—3161no—
318#135—1582no—
319C2×Q16—3161no—
320C2×Q16—791no—
321C2×Q16—791no—
322C2×Q16—1582no—
323C2×Q16—1581no—
324C2×Q16C22×D7911yescharacteristic
325C2×SD16—1581no—
326C2×SD16—1581no—
327C2×D8—791no—
328C22×C8C22×D7911yescharacteristic
329C22×C8C22×D7911yescharacteristic
330C4⋊Q8—21no—
331#125—41no—
332#124—3161no—
333#124—1581no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—41no—
337C22⋊Q8—41no—
338C22⋊Q8—3161no—
339C22⋊Q8—1581no—
340#121—3161no—
341#121—1581no—
342C4×Q8—21no—
343C4×Q8—1581no—
344C4×D4—41no—
345C4×D4—3161no—
346C4×D4C22⋊D7911yescharacteristic
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349Q32—3161no—
350Q32—1582no—
351SD32—3161no—
352M32—3161no—
353C2×C16—3161no—
354C2×C16—791no—
355C2×C16—791no—
356C2×C16—791no—
357C2×C16—791no—
358#107—21no—
359#107—21no—
360#107—21no—
361#107—791no—
362#107C22×D7911yescharacteristic
363#107C22×D7911yescharacteristic
364#107C22×D7911yescharacteristic
365C4⋊C8C22⋊D7911yescharacteristic
366#103—21no—
367#103—21no—
368#103—21no—
369#103—1581no—
370#103—1581no—
371#102—1581no—
372C22⋊C8C22⋊D7911yescharacteristic
373C22⋊C8C22⋊D7911yescharacteristic
374C4×C8C22×D7911yescharacteristic
375#41—3162no—
376#41—6321no—
377C2×Q8—21no—
378C2×Q8—41no—
379C2×Q8—3162no—
380C2×Q8—3161no—
381C2×Q8—6321no—
382C2×Q8—1581no—
383C2×Q8—1581no—
384C2×D4—3161no—
385C2×D4—3161no—
386C2×D4—1581no—
387C2×D4#1192960911yescharacteristic
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—3161no—
391C22×C4—3161no—
392C22×C4C2×C22⋊D7911yescharacteristic
393C22×C4C2×C22⋊D7911yescharacteristic
394Q16—21no—
395Q16—21no—
396Q16—3162no—
397Q16—792no—
398Q16—1581no—
399Q16—1581no—
400Q16—1581no—
401SD16—3162no—
402SD16—6321no—
403D8—1581no—
404D8—1581no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8C2×C22⋊D7911yescharacteristic
410C2×C8C2×C22⋊D7911yescharacteristic
411C2×C8C2×C22⋊D7911yescharacteristic
412C2×C8C23×D7911yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—3161no—
423C4⋊C4—3161no—
424C4⋊C4—1581no—
425C4⋊C4#1192960911yescharacteristic
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—3161no—
429C22⋊C4—3161no—
430C22⋊C4—3161no—
431C22⋊C4—3161no—
432C22⋊C4#1192960911yescharacteristic
433C22⋊C4#1192960911yescharacteristic
434C42—41no—
435C42—3161no—
436C42C2×C22⋊D7911yescharacteristic
437C16—1581no—
438C16—1581no—
439C16—1581no—
440C16—1581no—
441C23—3161no—
442C23C2×#1192960911yescharacteristic
443C23C2×#1192960911yescharacteristic
444Q8—41no—
445Q8—41no—
446Q8—161no—
447Q8—3161no—
448Q8—3161no—
449Q8—3161no—
450Q8—6322no—
451Q8—1582no—
452D4—22no—
453D4—3161no—
454D4—3161no—
455D4—6322no—
456D4—6322no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—161no—
466C2×C4—161no—
467C2×C4—3161no—
468C2×C4—3161no—
469C2×C4—3161no—
470C2×C4—6322no—
471C2×C4—6322no—
472C2×C4C2×#1192960911yescharacteristic
473C2×C4C2×#1192960911yescharacteristic
474C2×C4C2×#1192960911yescharacteristic
475C2×C4C2×#1192960911yescharacteristic
476C2×C4C22×C22⋊D7911yescharacteristic
477C8—21no—
478C8—21no—
479C8—21no—
480C8#41×D7911yescharacteristic
481C8#1457920711yescharacteristic
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—6321no—
488C22—6321no—
489C22—6321no—
490C22#1909900412yes—
491C22C22×#1192960911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—3162no—
501C4—6321no—
502C4—6321no—
503C4#1909920111yescharacteristic
504C4#1909937111yescharacteristic
505C2—22no—
506C2—41no—
507C2—12641no—
508C2#2377520711yescharacteristic
509C2C2×#1909900411yescharacteristic
510C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3338C2
socleC22×C79C22×D8
prime radical#23775143C2
hypercenter#3338D79
hypocenterC79#33883
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3338

Complements

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

NH ≅ G/Ntype
2. #23765639C2semidirect
3. #23765597C2semidirect
4. #23765585C2semidirect
5. #23765539C2semidirect
6. #23765506C2semidirect
7. #23765498C2semidirect
8. #23765462C2semidirect
9. #23765338C2semidirect
10. #23765333C2semidirect
11. #23765269C2semidirect
12. #23765226C2semidirect
13. #23765110C2semidirect
14. C2×#19098366C2semidirect
15. C79×#3338C2semidirect
256. #3338D79semidirect
271. C79#33883semidirect

Maximal subgroups

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

subgroupindex
#237751432
#237656392
#237655972
#237655852
#237655392
#237655062
#237654982
#237654622
#237653382
#237653332
#237652692
#237652262
#237651102
C2×#190983662
C79×#33382
#3388379