Finite Group Databasebeta

← #34445249

Subgroups — #34445249

Order 20224.

554 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775122C211yesmaximal, characteristic, complemented
2#23765647C211yesmaximal, characteristic, complemented
3#23765608C211yesmaximal, characteristic, complemented
4#23765597C211yesmaximal, characteristic, complemented
5#23765554C211yesmaximal, characteristic, complemented
6#23765511C211yesmaximal, characteristic, complemented
7#23765494C211yesmaximal, characteristic, complemented
8#23765445C211yesmaximal, characteristic, complemented
9#23765321C211yesmaximal, characteristic, complemented
10#23765321C211yesmaximal, characteristic, complemented
11#23765274C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765110C211yesmaximal, characteristic, complemented
14#23765105C211yesmaximal, characteristic, complemented
15C79×#3339C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C2×#14579183C2211yescharacteristic
18#19099218—21no—
19C2×#14579105C2211yescharacteristic
20C2×#14579105C2211yescharacteristic
21C2×#14579102C2211yescharacteristic
22C2×#14579101C2211yescharacteristic
23C22×C79⋊C16C2211yescharacteristic
24#19098943C2211yescharacteristic
25#19098804C2211yescharacteristic
26#19098804C2211yescharacteristic
27#19098769—21no—
28#19098727—21no—
29#19098727—21no—
30C4×D632C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098366C2211yescharacteristic
36#19098366C2211yescharacteristic
37#19098366C2211yescharacteristic
38#19098366C2211yescharacteristic
39#19098366C2211yescharacteristic
40#19098366C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098363C2211yescharacteristic
43#19098363C2211yescharacteristic
44#19098363C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47#19098339C2211yescharacteristic
48#553×C79—21no—
49#499×C79C2211yescharacteristic
50#482×C79—21no—
51#481×C79—21no—
52#478×C79—21no—
53#466×C79C2211yescharacteristic
54#466×C79C2211yescharacteristic
55C4×D8×C79C2211yescharacteristic
56C8×D4×C79C2211yescharacteristic
57#427×C79C2211yescharacteristic
58C2×#107×C79C2211yescharacteristic
59C2×#11929624—21no—
60C2×#11929624—21no—
61#14579183—22no—
62#14579183—22no—
63#14579183—41no—
64#14579183—41no—
65C2×Dic316—21no—
66C2×Dic316C2311yescharacteristic
67C2×Dic316C2311yescharacteristic
68C2×#11929613—21no—
69C2×#11929613—21no—
70C2×D632C2311yescharacteristic
71#14579161—21no—
72#14579143—41no—
73#14579143—41no—
74#14579140—21no—
75#14579140—21no—
76#14579132—41no—
77C4×D316—21no—
78C4⋊C4×D79—41no—
79#14579105—41no—
80#14579105—41no—
81#14579102—22no—
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×D8×C79C2311yescharacteristic
93C22×C8×C79C2311yescharacteristic
94C22×C8×C79C2311yescharacteristic
95#126×C79—41no—
96#125×C79—21no—
97#123×C79—41no—
98#121×C79—21no—
99#121×C79—21no—
100#121×C79—41no—
101C4×D4×C79—21no—
102C4×D4×C79—41no—
103C4×D4×C79D411yescharacteristic
104#117×C79—21no—
105C2×C4⋊C4×C79—21no—
106#107×C79—21no—
107#107×C79C2311yescharacteristic
108#107×C79C2311yescharacteristic
109#107×C79C2311yescharacteristic
110#106×C79—21no—
111#106×C79—21no—
112C4⋊C8×C79D411yescharacteristic
113#102×C79—21no—
114#102×C79—21no—
115#102×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119#11929624—42no—
120#11929624—42no—
121#11929624—81no—
122#11929624—81no—
123C2×Dic158—21no—
124C2×Dic158—21no—
125C2×D316—21no—
126C2×C22⋊D79—41no—
127C2×C22⋊D79—41no—
128C2×C4×D79—41no—
129C2×C4×D79—41no—
130C2×C4×D79—41no—
131Dic316—21no—
132Dic316—21no—
133Dic316—21no—
134Dic316—21no—
135Dic316—81no—
136#11929613—42no—
137#11929613—42no—
138D632—21no—
139D632#4112yes—
140C4⋊Dic79—21no—
141C4⋊Dic79—41no—
142C4⋊Dic79—41no—
143Dic79⋊C4—41no—
144Dic79⋊C4—41no—
145C22⋊Dic79—41no—
146C22⋊Dic79—41no—
147(C2×D79)⋊C4—41no—
148C79⋊C16—21no—
149C79⋊C16—21no—
150C79⋊C16—21no—
151C79⋊C16—21no—
152C2×D4×C79—21no—
153C2×D4×C79—41no—
154C2×D4×C79—41no—
155C2×D4×C79D811yescharacteristic
156C22×C4×C79—41no—
157C22×C4×C79—41no—
158C22×C4×C79C2×D411yescharacteristic
159C22×C4×C79C2×D411yescharacteristic
160D8×C79—21no—
161D8×C79—21no—
162C2×C8×C79—21no—
163C2×C8×C79—21no—
164C2×C8×C79—21no—
165C2×C8×C79—21no—
166C2×C8×C79C2×D411yescharacteristic
167C2×C8×C79C2×D411yescharacteristic
168C2×C8×C79C2×D411yescharacteristic
169C2×C8×C79C2411yescharacteristic
170C4⋊C4×C79—21no—
171C4⋊C4×C79—21no—
172C4⋊C4×C79—21no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79—41no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79D811yescharacteristic
177C22⋊C4×C79—41no—
178C22⋊C4×C79—41no—
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79D811yescharacteristic
182C22⋊C4×C79D811yescharacteristic
183C42×C79—41no—
184C42×C79C2×D411yescharacteristic
185C23×C79—41no—
186C23×C79C2×D811yescharacteristic
187C23×C79C2×D811yescharacteristic
188C22×D79—41no—
189D4×C79—41no—
190D4×C79—41no—
191D4×C79—41no—
192D4×C79—82no—
193D4×C79—161no—
194C2×C4×C79—21no—
195C2×C4×C79—21no—
196C2×C4×C79—21no—
197C2×C4×C79—21no—
198C2×C4×C79—41no—
199C2×C4×C79—41no—
200C2×C4×C79—41no—
201C2×C4×C79—161no—
202C2×C4×C79—161no—
203C2×C4×C79C2×D811yescharacteristic
204C2×C4×C79C2×D811yescharacteristic
205C2×C4×C79C2×D811yescharacteristic
206C2×C4×C79C2×D811yescharacteristic
207C2×C4×C79C22×D411yescharacteristic
208C22⋊D79—82no—
209C22⋊D79—82no—
210C2×Dic79—41no—
211C2×Dic79—41no—
212C2×Dic79—41no—
213D316—22no—
214D316—41no—
215C4×D79—82no—
216C4×D79—82no—
217C4×D79—82no—
218Dic158—41no—
219Dic158—41no—
220Dic158—41no—
221Dic158—41no—
222C8×C79—21no—
223C8×C79—21no—
224C8×C79—21no—
225C8×C79C2×#4111yescharacteristic
226C8×C79H(5,𝔽2)11yescharacteristic
227C22×C79—21no—
228C22×C79—21no—
229C22×C79—21no—
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—81no—
234C22×C79—81no—
235C22×C79—81no—
236C22×C79C22×D811yescharacteristic
237C2×D79—42no—
238C2×D79—81no—
239C2×D79—81no—
240C4×C79—21no—
241C4×C79—21no—
242C4×C79—21no—
243C4×C79—21no—
244C4×C79—81no—
245C4×C79—81no—
246C4×C79—81no—
247C4×C79#54611yescharacteristic
248C4×C79#57711yescharacteristic
249Dic79—81no—
250Dic79—81no—
251Dic79—81no—
252#33874—791nomaximal, Sylow
253C2×C79—41no—
254C2×C79—41no—
255C2×C79—161no—
256C2×C79#329411yescharacteristic
257C2×C79#341411yescharacteristic
258C2×C79#341611yescharacteristic
259D79—82no—
260#3339D7911yescharacteristic, complemented
261#3283—791no—
262#2245—791no—
263#2237—791no—
264#2235—791no—
265#2228—791no—
266#2220—791no—
267#2217—791no—
268#2208—791no—
269#2186—791no—
270#2186—791no—
271#2177—791no—
272#2168—791no—
273#2139—791no—
274#2138—791no—
275C79#3387411yescharacteristic, complemented, Sylow
276C2×#135—791no—
277C2×#135—791no—
278#553—21no—
279#547—1581no—
280C2×SD32—791no—
281C2×SD32—791no—
282C2×D16—791no—
283C2×M32—791no—
284C22×C16—791no—
285#499C2×D7911yescharacteristic
286#494—791no—
287#482—21no—
288#481—21no—
289#478—21no—
290#467—791no—
291#467—791no—
292#466C2×D7911yescharacteristic
293#466C2×D7911yescharacteristic
294#461—1581no—
295#452—1581no—
296#452—1581no—
297C4×D8—791no—
298C4×D8C2×D7911yescharacteristic
299C8×D4C2×D7911yescharacteristic
300#427C2×D7911yescharacteristic
301C2×#107C2×D7911yescharacteristic
302#367—791no—
303#367—791no—
304#366—791no—
305C4⋊C16—791no—
306#358—791no—
307#358—791no—
308#358—791no—
309#358—791no—
310#358—791no—
311#358—791no—
312#357—791no—
313#357—791no—
314#357—791no—
315#357—791no—
316C22⋊C16—791no—
317C22⋊C16—791no—
318#346—791no—
319C2×#41—1581no—
320C2×#41—1581no—
321#135—3161no—
322#135—3161no—
323#135—1582no—
324#135—1582no—
325C2×Q16—791no—
326C2×Q16—791no—
327C2×Q16—1581no—
328C2×SD16—1581no—
329C2×SD16—1581no—
330C2×D8—791no—
331C2×D8C22×D7911yescharacteristic
332C22×C8C22×D7911yescharacteristic
333C22×C8C22×D7911yescharacteristic
334C4⋊Q8—1581no—
335#126—41no—
336#125—21no—
337#123—41no—
338#123—3161no—
339#123—3161no—
340C22⋊Q8—1581no—
341C22⋊Q8—1581no—
342#121—21no—
343#121—21no—
344#121—41no—
345#121—3161no—
346C4×D4—21no—
347C4×D4—41no—
348C4×D4—1581no—
349C4×D4C22⋊D7911yescharacteristic
350#117—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—3161no—
353SD32—3161no—
354SD32—3161no—
355D16—1582no—
356M32—3161no—
357C2×C16—3161no—
358C2×C16—791no—
359C2×C16—791no—
360C2×C16—791no—
361C2×C16—791no—
362#107—21no—
363#107—791no—
364#107C22×D7911yescharacteristic
365#107C22×D7911yescharacteristic
366#107C22×D7911yescharacteristic
367#106—21no—
368#106—21no—
369C4⋊C8C22⋊D7911yescharacteristic
370#103—1581no—
371#103—1581no—
372#102—21no—
373#102—21no—
374#102—21no—
375#102—1581no—
376C22⋊C8C22⋊D7911yescharacteristic
377C22⋊C8C22⋊D7911yescharacteristic
378C4×C8C22×D7911yescharacteristic
379#41—3162no—
380#41—3162no—
381#41—6321no—
382#41—6321no—
383C2×Q8—1581no—
384C2×Q8—1581no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—3161no—
389C2×D4—3161no—
390C2×D4—1581no—
391C2×D4#1192960911yescharacteristic
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—3161no—
395C22×C4—3161no—
396C22×C4—3161no—
397C22×C4C2×C22⋊D7911yescharacteristic
398C22×C4C2×C22⋊D7911yescharacteristic
399Q16—6321no—
400Q16—1581no—
401Q16—1581no—
402Q16—1581no—
403Q16—1581no—
404SD16—3162no—
405SD16—3162no—
406D8—21no—
407D8—21no—
408D8—792no—
409D8—1581no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8C2×C22⋊D7911yescharacteristic
415C2×C8C2×C22⋊D7911yescharacteristic
416C2×C8C2×C22⋊D7911yescharacteristic
417C2×C8C23×D7911yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—3161no—
425C4⋊C4—3161no—
426C4⋊C4—3161no—
427C4⋊C4—3161no—
428C4⋊C4—1581no—
429C4⋊C4#1192960911yescharacteristic
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—3161no—
435C22⋊C4—3161no—
436C22⋊C4—3161no—
437C22⋊C4#1192960911yescharacteristic
438C22⋊C4#1192960911yescharacteristic
439C42—41no—
440C42C2×C22⋊D7911yescharacteristic
441C16—1581no—
442C16—1581no—
443C16—1581no—
444C16—1581no—
445C23—41no—
446C23—3161no—
447C23C2×#1192960911yescharacteristic
448C23C2×#1192960911yescharacteristic
449Q8—3161no—
450Q8—3161no—
451Q8—3161no—
452Q8—3161no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—82no—
457D4—161no—
458D4—3161no—
459D4—6322no—
460D4—6322no—
461D4—1582no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—161no—
470C2×C4—161no—
471C2×C4—3161no—
472C2×C4—3161no—
473C2×C4—3161no—
474C2×C4—6322no—
475C2×C4—6322no—
476C2×C4—6322no—
477C2×C4C2×#1192960911yescharacteristic
478C2×C4C2×#1192960911yescharacteristic
479C2×C4C2×#1192960911yescharacteristic
480C2×C4C2×#1192960911yescharacteristic
481C2×C4C22×C22⋊D7911yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8#41×D7911yescharacteristic
486C8#1457920711yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—3162no—
497C22—6321no—
498C22—6321no—
499C22C22×#1192960911yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—6321no—
508C4—6321no—
509C4—6321no—
510C4#1909920111yescharacteristic
511C4#1909937111yescharacteristic
512C2—41no—
513C2—41no—
514C2—161no—
515C2—6322no—
516C2#2377520711yescharacteristic
517C2#2377653411yescharacteristic
518C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3339C2
socleC22×C79C22×D8
hypercenter#3339D79
hypocenterC79#33874
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. #23775122C2semidirect
2. #23765647C2semidirect
3. #23765608C2semidirect
4. #23765597C2semidirect
5. #23765554C2semidirect
6. #23765511C2semidirect
7. #23765494C2semidirect
8. #23765445C2semidirect
9. #23765321C2semidirect
10. #23765321C2semidirect
11. #23765274C2semidirect
12. #23765226C2semidirect
13. #23765110C2semidirect
14. #23765105C2semidirect
15. C79×#3339C2semidirect
260. #3339D79semidirect
275. C79#33874semidirect

Maximal subgroups

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

subgroupindex
#237751222
#237656472
#237656082
#237655972
#237655542
#237655112
#237654942
#237654452
#237653212
#237652742
#237652262
#237651102
#237651052
C79×#33392
#3387479