Finite Group Databasebeta

← #34445264

Subgroups — #34445264

Order 20224.

580 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775143C211yesmaximal, characteristic, complemented
2#23765639C211yesmaximal, characteristic, complemented
3#23765602C211yesmaximal, characteristic, complemented
4#23765580C211yesmaximal, characteristic, complemented
5#23765549C211yesmaximal, characteristic, complemented
6#23765511C211yesmaximal, characteristic, complemented
7#23765484C211yesmaximal, characteristic, complemented
8#23765455C211yesmaximal, characteristic, complemented
9#23765321C211yesmaximal, characteristic, complemented
10#23765316C211yesmaximal, characteristic, complemented
11#23765264C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765110C211yesmaximal, characteristic, complemented
14C2×#19098366C211yesmaximal, characteristic, complemented
15C79×#3334C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C22×Dic316C2211yescharacteristic
18#19099222—21no—
19C2×#14579105C2211yescharacteristic
20C2×#14579105C2211yescharacteristic
21C2×#14579102C2211yescharacteristic
22C2×#14579101C2211yescharacteristic
23C22×C79⋊C16C2211yescharacteristic
24#19098947C2211yescharacteristic
25#19098804C2211yescharacteristic
26#19098799C2211yescharacteristic
27#19098786—21no—
28#19098727—21no—
29#19098704—21no—
30C4×Dic316C2211yescharacteristic
31#19098388C2211yescharacteristic
32#19098388C2211yescharacteristic
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#547×C79—21no—
49#500×C79C2211yescharacteristic
50#480×C79—21no—
51#480×C79—21no—
52#474×C79—21no—
53#466×C79C2211yescharacteristic
54#466×C79C2211yescharacteristic
55C4×D8×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#14579105—22no—
80#14579105—41no—
81#14579102—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×#41×C79—41no—
93C2×D8×C79C2311yescharacteristic
94C22×C8×C79C2311yescharacteristic
95C22×C8×C79C2311yescharacteristic
96C4⋊Q8×C79—21no—
97#123×C79—41no—
98#123×C79—41no—
99C22⋊Q8×C79—41no—
100#121×C79—21no—
101#121×C79—21no—
102C4×D4×C79—21no—
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#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—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#4112yescomplemented
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—
152#41×C79—84no—
153C2×Q8×C79—41no—
154C2×D4×C79—21no—
155C2×D4×C79—41no—
156C2×D4×C79—41no—
157C2×D4×C79D811yescharacteristic
158C22×C4×C79—41no—
159C22×C4×C79—41no—
160C22×C4×C79—41no—
161C22×C4×C79C2×D411yescharacteristic
162C22×C4×C79C2×D411yescharacteristic
163D8×C79—21no—
164D8×C79—21no—
165C2×C8×C79—21no—
166C2×C8×C79—21no—
167C2×C8×C79—21no—
168C2×C8×C79—21no—
169C2×C8×C79C2×D411yescharacteristic
170C2×C8×C79C2×D411yescharacteristic
171C2×C8×C79C2×D411yescharacteristic
172C2×C8×C79C2411yescharacteristic
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79—41no—
178C4⋊C4×C79—41no—
179C4⋊C4×C79D811yescharacteristic
180C22⋊C4×C79—41no—
181C22⋊C4×C79—41no—
182C22⋊C4×C79—41no—
183C22⋊C4×C79D811yescharacteristic
184C22⋊C4×C79D811yescharacteristic
185C42×C79C2×D411yescharacteristic
186C23×C79—41no—
187C23×C79C2×D811yescharacteristic
188C23×C79C2×D811yescharacteristic
189C22×D79—41no—
190Q8×C79—82no—
191D4×C79—22no—
192D4×C79—41no—
193D4×C79—41no—
194D4×C79—82no—
195D4×C79—82no—
196C2×C4×C79—21no—
197C2×C4×C79—21no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—41no—
201C2×C4×C79—41no—
202C2×C4×C79—41no—
203C2×C4×C79—82no—
204C2×C4×C79—82no—
205C2×C4×C79—82no—
206C2×C4×C79C2×D811yescharacteristic
207C2×C4×C79C2×D811yescharacteristic
208C2×C4×C79C2×D811yescharacteristic
209C2×C4×C79C2×D811yescharacteristic
210C2×C4×C79C22×D411yescharacteristic
211C22⋊D79—82no—
212C22⋊D79—82no—
213C2×Dic79—41no—
214C2×Dic79—41no—
215C2×Dic79—41no—
216C2×Dic79—82no—
217D316—41no—
218D316—41no—
219C4×D79—82no—
220Dic158—22no—
221Dic158—41no—
222Dic158—41no—
223Dic158—41no—
224Dic158—82no—
225C8×C79—21no—
226C8×C79—21no—
227C8×C79—21no—
228C8×C79C2×#4111yescharacteristic
229C8×C79H(5,𝔽2)11yescharacteristic
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—21no—
234C22×C79—21no—
235C22×C79—81no—
236C22×C79—81no—
237C22×C79—81no—
238C22×C79#50912yes—
239C22×C79C22×D811yescharacteristic
240C2×D79—81no—
241C2×D79—81no—
242C2×D79—81no—
243C4×C79—21no—
244C4×C79—21no—
245C4×C79—21no—
246C4×C79—21no—
247C4×C79—81no—
248C4×C79—81no—
249C4×C79—81no—
250C4×C79#54611yescharacteristic
251C4×C79#57711yescharacteristic
252Dic79—42no—
253Dic79—81no—
254Dic79—81no—
255#33875—791nomaximal, Sylow
256C2×C79—22no—
257C2×C79—41no—
258C2×C79—161no—
259C2×C79#329411yescharacteristic
260C2×C79C2×#50911yescharacteristic
261C2×C79#341711yescharacteristic
262D79—161no—
263#3334D7911yescharacteristic, complemented
264#3285—791no—
265#2243—791no—
266#2236—791no—
267#2232—791no—
268#2227—791no—
269#2220—791no—
270#2215—791no—
271#2210—791no—
272#2186—791no—
273#2185—791no—
274#2175—791no—
275#2168—791no—
276#2139—791no—
277C2×#358—791no—
278C79#3387511yescharacteristic, complemented, Sylow
279C2×#135—791no—
280C22×Q16—791no—
281#548—1581no—
282#547—21no—
283C2×SD32—791no—
284C2×SD32—791no—
285C2×D16—791no—
286C2×M32—791no—
287C22×C16—791no—
288#500C2×D7911yescharacteristic
289#495—791no—
290#480—21no—
291#480—21no—
292#474—21no—
293#467—791no—
294#466—791no—
295#466C2×D7911yescharacteristic
296#466C2×D7911yescharacteristic
297#464—1581no—
298#452—1581no—
299#448—1581no—
300C4×Q16—791no—
301C4×D8C2×D7911yescharacteristic
302C8×D4C2×D7911yescharacteristic
303C2×#107C2×D7911yescharacteristic
304C2×#107C2×D7911yescharacteristic
305#366—791no—
306#366—791no—
307#366—791no—
308C4⋊C16—791no—
309#358—791no—
310#358—791no—
311#358—791no—
312#358—791no—
313#358—791no—
314#358—791no—
315#357—791no—
316#357—791no—
317#357—791no—
318#357—791no—
319C22⋊C16—791no—
320C22⋊C16—791no—
321#346—791no—
322C2×#41—41no—
323C2×#41—1581no—
324C22×Q8—1581no—
325#135—3161no—
326#135—1582no—
327C2×Q16—3161no—
328C2×Q16—791no—
329C2×Q16—791no—
330C2×Q16—1582no—
331C2×Q16—1581no—
332C2×SD16—1581no—
333C2×SD16—1581no—
334C2×D8—791no—
335C2×D8C22×D7911yescharacteristic
336C22×C8C22×D7911yescharacteristic
337C22×C8C22×D7911yescharacteristic
338C4⋊Q8—21no—
339#124—3161no—
340#124—1581no—
341#123—41no—
342#123—41no—
343C22⋊Q8—41no—
344C22⋊Q8—3161no—
345C22⋊Q8—1581no—
346#121—21no—
347#121—21no—
348#121—3161no—
349#121—1581no—
350C4×Q8—1581no—
351C4×D4—21no—
352C4×D4—3161no—
353C4×D4C22⋊D7911yescharacteristic
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356SD32—3161no—
357SD32—1582no—
358D16—3161no—
359M32—3161no—
360C2×C16—3161no—
361C2×C16—791no—
362C2×C16—791no—
363C2×C16—791no—
364C2×C16—791no—
365#107—21no—
366#107—21no—
367#107—21no—
368#107—791no—
369#107C22×D7911yescharacteristic
370#107C22×D7911yescharacteristic
371#107C22×D7911yescharacteristic
372C4⋊C8C22⋊D7911yescharacteristic
373#103—1581no—
374#103—1581no—
375#102—21no—
376#102—21no—
377#102—21no—
378#102—1581no—
379C22⋊C8C22⋊D7911yescharacteristic
380C22⋊C8C22⋊D7911yescharacteristic
381C4×C8C22×D7911yescharacteristic
382#41—84no—
383#41—3162no—
384#41—6321no—
385C2×Q8—41no—
386C2×Q8—3162no—
387C2×Q8—3161no—
388C2×Q8—6321no—
389C2×Q8—1581no—
390C2×Q8—1581no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—3161no—
395C2×D4—3161no—
396C2×D4—1581no—
397C2×D4#1192960911yescharacteristic
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—3161no—
402C22×C4—3161no—
403C22×C4C2×C22⋊D7911yescharacteristic
404C22×C4C2×C22⋊D7911yescharacteristic
405Q16—3162no—
406Q16—792no—
407Q16—1581no—
408Q16—1581no—
409Q16—1581no—
410SD16—3162no—
411SD16—6321no—
412D8—21no—
413D8—21no—
414D8—1581no—
415D8—1581no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8C2×C22⋊D7911yescharacteristic
421C2×C8C2×C22⋊D7911yescharacteristic
422C2×C8C2×C22⋊D7911yescharacteristic
423C2×C8C23×D7911yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—3161no—
431C4⋊C4—3161no—
432C4⋊C4—1581no—
433C4⋊C4#1192960911yescharacteristic
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—3161no—
438C22⋊C4—3161no—
439C22⋊C4—3161no—
440C22⋊C4—3161no—
441C22⋊C4#1192960911yescharacteristic
442C22⋊C4#1192960911yescharacteristic
443C42—3161no—
444C42C2×C22⋊D7911yescharacteristic
445C16—1581no—
446C16—1581no—
447C16—1581no—
448C16—1581no—
449C23—41no—
450C23—3161no—
451C23C2×#1192960911yescharacteristic
452C23C2×#1192960911yescharacteristic
453Q8—82no—
454Q8—3161no—
455Q8—3161no—
456Q8—3161no—
457Q8—6322no—
458Q8—1582no—
459D4—22no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—3161no—
465D4—3161no—
466D4—6322no—
467D4—6322no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—3161no—
479C2×C4—3161no—
480C2×C4—3161no—
481C2×C4—6322no—
482C2×C4—6322no—
483C2×C4C2×#1192960911yescharacteristic
484C2×C4C2×#1192960911yescharacteristic
485C2×C4C2×#1192960911yescharacteristic
486C2×C4C2×#1192960911yescharacteristic
487C2×C4C22×C22⋊D7911yescharacteristic
488C8—21no—
489C8—21no—
490C8—21no—
491C8#41×D7911yescharacteristic
492C8#1457920711yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—6321no—
502C22—6321no—
503C22—6321no—
504C22#1909899912yes—
505C22C22×#1192960911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—3162no—
514C4—6321no—
515C4—6321no—
516C4#1909920111yescharacteristic
517C4#1909937111yescharacteristic
518C2—22no—
519C2—41no—
520C2—161no—
521C2—12641no—
522C2#2377520711yescharacteristic
523C2C2×#1909899911yescharacteristic
524C2#2377655111yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3334C2
socleC22×C79C22×D8
hypercenter#3334D79
hypocenterC79#33875
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #23775143C2semidirect
2. #23765639C2semidirect
3. #23765602C2semidirect
4. #23765580C2semidirect
5. #23765549C2semidirect
6. #23765511C2semidirect
7. #23765484C2semidirect
8. #23765455C2semidirect
9. #23765321C2semidirect
10. #23765316C2semidirect
11. #23765264C2semidirect
12. #23765226C2semidirect
13. #23765110C2semidirect
14. C2×#19098366C2semidirect
15. C79×#3334C2semidirect
135. Dic316#41semidirect
263. #3334D79semidirect
278. C79#33875semidirect

Maximal subgroups

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

subgroupindex
#237751432
#237656392
#237656022
#237655802
#237655492
#237655112
#237654842
#237654552
#237653212
#237653162
#237652642
#237652262
#237651102
C2×#190983662
C79×#33342
#3387579