Finite Group Databasebeta

← #34445270

Subgroups — #34445270

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#23775748C211yesmaximal, characteristic, complemented
2#23765636C211yesmaximal, characteristic, complemented
3#23765600C211yesmaximal, characteristic, complemented
4#23765578C211yesmaximal, characteristic, complemented
5#23765552C211yesmaximal, characteristic, complemented
6#23765515C211yesmaximal, characteristic, complemented
7#23765484C211yesmaximal, characteristic, complemented
8#23765452C211yesmaximal, characteristic, complemented
9#23765323C211yesmaximal, characteristic, complemented
10#23765317C211yesmaximal, characteristic, complemented
11#23765262C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765112C211yesmaximal, characteristic, complemented
14C2×#19098367C211yesmaximal, characteristic, complemented
15C79×#3285C211yesmaximal, characteristic, complemented
16#19099217—21no—
17C2×#14579104C2211yescharacteristic
18C2×#14579104C2211yescharacteristic
19C2×#14579102C2211yescharacteristic
20C2×#14579101C2211yescharacteristic
21C22×C79⋊C16C2211yescharacteristic
22#19098968C2211yescharacteristic
23#19098877—21no—
24#19098877—21no—
25#19098844—21no—
26#19098799C2211yescharacteristic
27#19098799C2211yescharacteristic
28C4×D632C2211yescharacteristic
29C2×#14579095C2211yescharacteristic
30C2×#14579095C2211yescharacteristic
31#19098388C2211yescharacteristic
32#19098388C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098367C2211yescharacteristic
39#19098367C2211yescharacteristic
40#19098367C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098363C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47#19098339C2211yescharacteristic
48C2×#135×C79C2211yescharacteristic
49C22×Q16×C79C2211yescharacteristic
50#548×C79—21no—
51#495×C79C2211yescharacteristic
52#467×C79C2211yescharacteristic
53#466×C79C2211yescharacteristic
54#464×C79—21no—
55#452×C79—21no—
56#448×C79—21no—
57C4×Q16×C79C2211yescharacteristic
58C8×D4×C79C2211yescharacteristic
59C2×#11929623—41no—
60C2×D632C2311yescharacteristic
61#14579161—21no—
62#14579145—41no—
63#14579145—41no—
64#14579141—41no—
65#14579135—21no—
66#14579135—21no—
67C4×D316—21no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—21no—
70#14579104—22no—
71#14579104—41no—
72#14579102—41no—
73#14579101—41no—
74C2×C79⋊C16—41no—
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77C2×C79⋊C16C2311yescharacteristic
78C2×C79⋊C16C2311yescharacteristic
79#14579095—21no—
80#14579095—21no—
81#14579095—21no—
82#14579095C2311yescharacteristic
83#14579095C2311yescharacteristic
84#14579095C2311yescharacteristic
85#14579083—21no—
86#14579083—21no—
87#14579083—21no—
88C2×#41×C79—21no—
89C22×Q8×C79—21no—
90#135×C79—22no—
91#135×C79—41no—
92C2×Q16×C79—22no—
93C2×Q16×C79—21no—
94C2×Q16×C79—41no—
95C2×Q16×C79C2311yescharacteristic
96C2×Q16×C79C2311yescharacteristic
97C2×SD16×C79—21no—
98C2×SD16×C79—21no—
99C2×D8×C79C2311yescharacteristic
100C22×C8×C79C2311yescharacteristic
101C22×C8×C79C2311yescharacteristic
102#124×C79—21no—
103#124×C79—41no—
104C22⋊Q8×C79—21no—
105C22⋊Q8×C79—41no—
106#121×C79—21no—
107#121×C79—41no—
108C4×Q8×C79—21no—
109C4×D4×C79—41no—
110C4×D4×C79D411yescharacteristic
111#107×C79C2311yescharacteristic
112C4⋊C8×C79D411yescharacteristic
113#103×C79—21no—
114#103×C79—21no—
115#102×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119#11929623—84no—
120C2×Dic158—41no—
121C2×D316—21no—
122C2×C22⋊D79—41no—
123C2×C22⋊D79—41no—
124C22×Dic79—41no—
125C22×Dic79—41no—
126C2×C4×D79—41no—
127D632—21no—
128D632—21no—
129C4⋊Dic79—21no—
130C4⋊Dic79—21no—
131C4⋊Dic79—21no—
132C4⋊Dic79—41no—
133C4⋊Dic79—41no—
134C4⋊Dic79—41no—
135(C2×D79)⋊C4—41no—
136(C2×D79)⋊C4—41no—
137(C2×D79)⋊C4—41no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C79⋊C16—21no—
142#41×C79—42no—
143#41×C79—81no—
144C2×Q8×C79—21no—
145C2×Q8×C79—21no—
146C2×Q8×C79—42no—
147C2×Q8×C79—41no—
148C2×Q8×C79—81no—
149C2×D4×C79—21no—
150C2×D4×C79—41no—
151C2×D4×C79—41no—
152C2×D4×C79D811yescharacteristic
153C22×C4×C79—41no—
154C22×C4×C79—41no—
155C22×C4×C79C2×D411yescharacteristic
156C22×C4×C79C2×D411yescharacteristic
157Q16×C79—21no—
158Q16×C79—21no—
159Q16×C79—21no—
160Q16×C79—42no—
161Q16×C79#4112yescomplemented
162SD16×C79—42no—
163SD16×C79—81no—
164D8×C79—21no—
165D8×C79—21no—
166C2×C8×C79—21no—
167C2×C8×C79—21no—
168C2×C8×C79—21no—
169C2×C8×C79—21no—
170C2×C8×C79C2×D411yescharacteristic
171C2×C8×C79C2×D411yescharacteristic
172C2×C8×C79C2×D411yescharacteristic
173C2×C8×C79C2411yescharacteristic
174C4⋊C4×C79—21no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79D811yescharacteristic
178C22⋊C4×C79—41no—
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79—41no—
182C22⋊C4×C79D811yescharacteristic
183C22⋊C4×C79D811yescharacteristic
184C42×C79—41no—
185C42×C79C2×D411yescharacteristic
186C23×C79—41no—
187C23×C79C2×D811yescharacteristic
188C23×C79C2×D811yescharacteristic
189C22×D79—41no—
190Q8×C79—22no—
191Q8×C79—41no—
192Q8×C79—41no—
193Q8×C79—41no—
194Q8×C79—82no—
195D4×C79—22no—
196D4×C79—41no—
197D4×C79—41no—
198D4×C79—82no—
199D4×C79—82no—
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—41no—
205C2×C4×C79—41no—
206C2×C4×C79—41no—
207C2×C4×C79—82no—
208C2×C4×C79—82no—
209C2×C4×C79C2×D811yescharacteristic
210C2×C4×C79C2×D811yescharacteristic
211C2×C4×C79C2×D811yescharacteristic
212C2×C4×C79C2×D811yescharacteristic
213C2×C4×C79C22×D411yescharacteristic
214C22⋊D79—82no—
215C22⋊D79—82no—
216C2×Dic79—41no—
217C2×Dic79—41no—
218C2×Dic79—41no—
219C2×Dic79—82no—
220C2×Dic79—82no—
221D316—41no—
222D316—41no—
223C4×D79—82no—
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—42no—
248C4×C79—81no—
249C4×C79—81no—
250C4×C79#54611yescharacteristic
251C4×C79#57711yescharacteristic
252Dic79—81no—
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#3334—791no—
264#3285D7911yescharacteristic, complemented
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×#135C2×D7911yescharacteristic
280C22×Q16C2×D7911yescharacteristic
281#548—21no—
282#547—1581no—
283C2×SD32—791no—
284C2×SD32—791no—
285C2×D16—791no—
286C2×M32—791no—
287C22×C16—791no—
288#500—791no—
289#495C2×D7911yescharacteristic
290#480—1581no—
291#480—1581no—
292#474—1581no—
293#467C2×D7911yescharacteristic
294#466—791no—
295#466—791no—
296#466C2×D7911yescharacteristic
297#464—21no—
298#452—21no—
299#448—21no—
300C4×Q16C2×D7911yescharacteristic
301C4×D8—791no—
302C8×D4C2×D7911yescharacteristic
303C2×#107—791no—
304C2×#107—791no—
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—21no—
323C2×#41—3161no—
324C22×Q8—21no—
325#135—22no—
326#135—41no—
327C2×Q16—22no—
328C2×Q16—21no—
329C2×Q16—41no—
330C2×Q16C22×D7911yescharacteristic
331C2×Q16C22×D7911yescharacteristic
332C2×SD16—21no—
333C2×SD16—21no—
334C2×D8—791no—
335C2×D8C22×D7911yescharacteristic
336C22×C8C22×D7911yescharacteristic
337C22×C8C22×D7911yescharacteristic
338C4⋊Q8—1581no—
339#124—21no—
340#124—41no—
341#123—3161no—
342#123—3161no—
343C22⋊Q8—21no—
344C22⋊Q8—41no—
345C22⋊Q8—3161no—
346#121—21no—
347#121—41no—
348#121—1581no—
349#121—1581no—
350C4×Q8—21no—
351C4×D4—41no—
352C4×D4—1581no—
353C4×D4C22⋊D7911yescharacteristic
354C2×C4⋊C4—1581no—
355C2×C4⋊C4—1581no—
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—791no—
366#107—791no—
367#107—791no—
368#107—1581no—
369#107—1581no—
370#107—1581no—
371#107C22×D7911yescharacteristic
372C4⋊C8C22⋊D7911yescharacteristic
373#103—21no—
374#103—21no—
375#102—21no—
376#102—1581no—
377#102—1581no—
378#102—1581no—
379C22⋊C8C22⋊D7911yescharacteristic
380C22⋊C8C22⋊D7911yescharacteristic
381C4×C8C22×D7911yescharacteristic
382#41—42no—
383#41—81no—
384#41—6324no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—42no—
388C2×Q8—41no—
389C2×Q8—81no—
390C2×Q8—3161no—
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—3161no—
401C22×C4—3161no—
402C22×C4—3161no—
403C22×C4C2×C22⋊D7911yescharacteristic
404C22×C4C2×C22⋊D7911yescharacteristic
405Q16—21no—
406Q16—21no—
407Q16—21no—
408Q16—42no—
409Q16#1192962312yescomplemented
410SD16—42no—
411SD16—81no—
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—41no—
426C4⋊C4—41no—
427C4⋊C4—3161no—
428C4⋊C4—3161no—
429C4⋊C4—3161no—
430C4⋊C4—1581no—
431C4⋊C4—1581no—
432C4⋊C4—1581no—
433C4⋊C4#1192960911yescharacteristic
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—3161no—
439C22⋊C4—3161no—
440C22⋊C4—3161no—
441C22⋊C4#1192960911yescharacteristic
442C22⋊C4#1192960911yescharacteristic
443C42—41no—
444C42C2×C22⋊D7911yescharacteristic
445C16—1581no—
446C16—1581no—
447C16—1581no—
448C16—1581no—
449C23—41no—
450C23—3161no—
451C23C2×#1192960911yescharacteristic
452C23C2×#1192960911yescharacteristic
453Q8—22no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—82no—
458Q8—6322no—
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—3161no—
478C2×C4—3161no—
479C2×C4—3161no—
480C2×C4—6322no—
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—
491C8C2×#1192962311yescharacteristic
492C8#1457920611yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—6321no—
502C22—6321no—
503C22—6321no—
504C22#1909899812yes—
505C22C22×#1192960911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—6321no—
514C4—6321no—
515C4—6321no—
516C4#1909920711yescharacteristic
517C4#1909937411yescharacteristic
518C2—22no—
519C2—41no—
520C2—161no—
521C2—12641no—
522C2#2377521311yescharacteristic
523C2C2×#1909899811yescharacteristic
524C2#2377655111yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3285C2
socleC22×C79C22×D8
hypercenter#3285D79
hypocenterC79#33875
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. #23775748C2semidirect
2. #23765636C2semidirect
3. #23765600C2semidirect
4. #23765578C2semidirect
5. #23765552C2semidirect
6. #23765515C2semidirect
7. #23765484C2semidirect
8. #23765452C2semidirect
9. #23765323C2semidirect
10. #23765317C2semidirect
11. #23765262C2semidirect
12. #23765226C2semidirect
13. #23765112C2semidirect
14. C2×#19098367C2semidirect
15. C79×#3285C2semidirect
161. Q16×C79#41semidirect
264. #3285D79semidirect
278. C79#33875semidirect
409. Q16#11929623semidirect

Maximal subgroups

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

subgroupindex
#237757482
#237656362
#237656002
#237655782
#237655522
#237655152
#237654842
#237654522
#237653232
#237653172
#237652622
#237652262
#237651122
C2×#190983672
C79×#32852
#3387579