Finite Group Databasebeta

← #34445285

Subgroups — #34445285

Order 20224.

564 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775803C211yesmaximal, characteristic, complemented
2#23765641C211yesmaximal, characteristic, complemented
3#23765588C211yesmaximal, characteristic, complemented
4#23765578C211yesmaximal, characteristic, complemented
5#23765564C211yesmaximal, characteristic, complemented
6#23765501C211yesmaximal, characteristic, complemented
7#23765484C211yesmaximal, characteristic, complemented
8#23765432C211yesmaximal, characteristic, complemented
9#23765323C211yesmaximal, characteristic, complemented
10#23765323C211yesmaximal, characteristic, complemented
11#23765262C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765112C211yesmaximal, characteristic, complemented
14#23765107C211yesmaximal, characteristic, complemented
15C79×#3287C211yesmaximal, characteristic, complemented
16#19099259—21no—
17C2×#14579104C2211yescharacteristic
18C2×#14579102C2211yescharacteristic
19C2×#14579102C2211yescharacteristic
20C2×#14579101C2211yescharacteristic
21C22×C79⋊C16C2211yescharacteristic
22#19098965C2211yescharacteristic
23#19098888—21no—
24#19098882—21no—
25#19098864—21no—
26#19098799C2211yescharacteristic
27#19098799C2211yescharacteristic
28C4×D632C2211yescharacteristic
29#19098601C2211yescharacteristic
30C2×#14579095C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098388C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098364C2211yescharacteristic
39#19098364C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098363C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47#19098339C2211yescharacteristic
48C2×#135×C79C2211yescharacteristic
49C2×#135×C79C2211yescharacteristic
50#550×C79—21no—
51#493×C79C2211yescharacteristic
52#466×C79C2211yescharacteristic
53#466×C79C2211yescharacteristic
54#460×C79—21no—
55#449×C79—21no—
56#449×C79—21no—
57C4×Q16×C79C2211yescharacteristic
58C8×D4×C79C2211yescharacteristic
59C2×D632C2311yescharacteristic
60#14579156—41no—
61#14579153—21no—
62#14579143—41no—
63#14579135—21no—
64#14579135—21no—
65#14579133—41no—
66C4×D316—21no—
67(C22⋊D79)⋊C4—41no—
68#14579120—21no—
69C2×C4⋊Dic79—21no—
70#14579104—22no—
71#14579102—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#14579095C2311yescharacteristic
81#14579095C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579093—21no—
84#14579093—21no—
85#14579083—21no—
86#14579083—21no—
87#14579083—21no—
88C2×#41×C79—21no—
89C2×#41×C79—21no—
90#135×C79—22no—
91#135×C79—22no—
92#135×C79—41no—
93#135×C79—41no—
94C2×Q16×C79C2311yescharacteristic
95C2×SD16×C79—21no—
96C2×SD16×C79—21no—
97C2×D8×C79—21no—
98C2×D8×C79C2311yescharacteristic
99C2×D8×C79C2311yescharacteristic
100C22×C8×C79C2311yescharacteristic
101C22×C8×C79C2311yescharacteristic
102#127×C79—21no—
103#127×C79—41no—
104#121×C79—21no—
105#121×C79—21no—
106#121×C79—41no—
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#102×C79—21no—
115#102×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119C2×D316—21no—
120C2×C22⋊D79—41no—
121C2×C22⋊D79—41no—
122C22×Dic79—41no—
123C2×C4×D79—41no—
124D632—21no—
125D632—21no—
126C4⋊Dic79—21no—
127C4⋊Dic79—21no—
128C4⋊Dic79—21no—
129C4⋊Dic79—41no—
130Dic79⋊C4—41no—
131Dic79⋊C4—41no—
132C22⋊Dic79—41no—
133(C2×D79)⋊C4—41no—
134(C2×D79)⋊C4—41no—
135(C2×D79)⋊C4—41no—
136C4×Dic79—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141#41×C79—42no—
142#41×C79—42no—
143#41×C79—81no—
144#41×C79—81no—
145C2×Q8×C79—21no—
146C2×D4×C79—21no—
147C2×D4×C79—21no—
148C2×D4×C79—41no—
149C2×D4×C79—41no—
150C2×D4×C79—41no—
151C2×D4×C79—41no—
152C2×D4×C79—41no—
153C2×D4×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#4112yes—
161SD16×C79—42no—
162SD16×C79—42no—
163D8×C79—21no—
164D8×C79—21no—
165D8×C79—21no—
166D8×C79—21no—
167D8×C79—81no—
168C2×C8×C79—21no—
169C2×C8×C79—21no—
170C2×C8×C79—21no—
171C2×C8×C79—21no—
172C2×C8×C79C2×D411yescharacteristic
173C2×C8×C79C2×D411yescharacteristic
174C2×C8×C79C2×D411yescharacteristic
175C2×C8×C79C2411yescharacteristic
176C4⋊C4×C79—21no—
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×C79—41no—
186C23×C79—41no—
187C23×C79C2×D811yescharacteristic
188C23×C79C2×D811yescharacteristic
189C22×D79—41no—
190Q8×C79—22no—
191Q8×C79—41no—
192D4×C79—41no—
193D4×C79—41no—
194D4×C79—41no—
195D4×C79—41no—
196D4×C79—41no—
197D4×C79—82no—
198D4×C79—82no—
199D4×C79—82no—
200D4×C79—161no—
201D4×C79—161no—
202D4×C79—161no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
209C2×C4×C79—82no—
210C2×C4×C79—82no—
211C2×C4×C79C2×D811yescharacteristic
212C2×C4×C79C2×D811yescharacteristic
213C2×C4×C79C2×D811yescharacteristic
214C2×C4×C79C2×D811yescharacteristic
215C2×C4×C79C22×D411yescharacteristic
216C22⋊D79—82no—
217C22⋊D79—161no—
218C2×Dic79—41no—
219C2×Dic79—41no—
220C2×Dic79—41no—
221C2×Dic79—161no—
222D316—41no—
223D316—41no—
224C4×D79—161no—
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—21no—
236C22×C79—81no—
237C22×C79—81no—
238C22×C79—81no—
239C22×C79—81no—
240C22×C79—81no—
241C22×C79—81no—
242C22×C79C22×D811yescharacteristic
243C2×D79—81no—
244C2×D79—81no—
245C2×D79—81no—
246C4×C79—21no—
247C4×C79—21no—
248C4×C79—21no—
249C4×C79—21no—
250C4×C79—42no—
251C4×C79—81no—
252C4×C79#54611yescharacteristic
253C4×C79#57711yescharacteristic
254Dic79—81no—
255Dic79—81no—
256Dic79—81no—
257#33876—791nomaximal, Sylow
258C2×C79—41no—
259C2×C79—41no—
260C2×C79—161no—
261C2×C79—161no—
262C2×C79#329411yescharacteristic
263C2×C79#341411yescharacteristic
264C2×C79#341511yescharacteristic
265D79—161no—
266#3339—791no—
267#3287D7911yescharacteristic, complemented
268#2244—791no—
269#2234—791no—
270#2232—791no—
271#2229—791no—
272#2218—791no—
273#2215—791no—
274#2206—791no—
275#2186—791no—
276#2186—791no—
277#2175—791no—
278#2168—791no—
279#2139—791no—
280#2138—791no—
281C79#3387611yescharacteristic, complemented, Sylow
282C2×#135C2×D7911yescharacteristic
283C2×#135C2×D7911yescharacteristic
284#553—1581no—
285#550—21no—
286C2×SD32—791no—
287C2×D16—791no—
288C2×D16—791no—
289C2×M32—791no—
290C22×C16—791no—
291#499—791no—
292#493C2×D7911yescharacteristic
293#482—1581no—
294#481—1581no—
295#478—1581no—
296#466—791no—
297#466—791no—
298#466C2×D7911yescharacteristic
299#466C2×D7911yescharacteristic
300#460—21no—
301#449—21no—
302#449—21no—
303C4×Q16C2×D7911yescharacteristic
304C4×D8—791no—
305C8×D4C2×D7911yescharacteristic
306#427—791no—
307C2×#107—791no—
308#367—791no—
309#366—791no—
310#366—791no—
311C4⋊C16—791no—
312#358—791no—
313#358—791no—
314#358—791no—
315#357—791no—
316#357—791no—
317#357—791no—
318#357—791no—
319#357—791no—
320#357—791no—
321#357—791no—
322C22⋊C16—791no—
323C22⋊C16—791no—
324#346—791no—
325C2×#41—21no—
326C2×#41—21no—
327#135—22no—
328#135—22no—
329#135—41no—
330#135—41no—
331C2×Q16C22×D7911yescharacteristic
332C2×SD16—21no—
333C2×SD16—21no—
334C2×D8—21no—
335C2×D8—791no—
336C2×D8C22×D7911yescharacteristic
337C2×D8C22×D7911yescharacteristic
338C22×C8C22×D7911yescharacteristic
339C22×C8C22×D7911yescharacteristic
340#127—21no—
341#127—41no—
342#126—3161no—
343#125—1581no—
344#123—3161no—
345#121—21no—
346#121—21no—
347#121—41no—
348#121—41no—
349#121—3161no—
350#121—1581no—
351#121—1581no—
352C4×Q8—21no—
353C4×D4—41no—
354C4×D4—3161no—
355C4×D4—1581no—
356C4×D4C22⋊D7911yescharacteristic
357#117—1581no—
358C2×C4⋊C4—1581no—
359SD32—1582no—
360D16—3161no—
361D16—3161no—
362M32—3161no—
363C2×C16—3161no—
364C2×C16—791no—
365C2×C16—791no—
366C2×C16—791no—
367C2×C16—791no—
368#107—791no—
369#107—791no—
370#107—791no—
371#107—1581no—
372#107C22×D7911yescharacteristic
373#106—1581no—
374#106—1581no—
375C4⋊C8C22⋊D7911yescharacteristic
376#103—21no—
377#102—21no—
378#102—21no—
379#102—1581no—
380#102—1581no—
381#102—1581no—
382C22⋊C8C22⋊D7911yescharacteristic
383C22⋊C8C22⋊D7911yescharacteristic
384C4×C8C22×D7911yescharacteristic
385#41—42no—
386#41—42no—
387#41—81no—
388#41—81no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—3161no—
399C2×D4—3161no—
400C2×D4—1581no—
401C2×D4#1192960911yescharacteristic
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—3161no—
405C22×C4—3161no—
406C22×C4C2×C22⋊D7911yescharacteristic
407C22×C4C2×C22⋊D7911yescharacteristic
408Q16—21no—
409Q16#1192962312yes—
410SD16—42no—
411SD16—42no—
412D8—21no—
413D8—21no—
414D8—21no—
415D8—21no—
416D8—81no—
417D8—1581no—
418D8—1581no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8C2×C22⋊D7911yescharacteristic
424C2×C8C2×C22⋊D7911yescharacteristic
425C2×C8C2×C22⋊D7911yescharacteristic
426C2×C8C23×D7911yescharacteristic
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—3161no—
430C4⋊C4—3161no—
431C4⋊C4—3161no—
432C4⋊C4—1581no—
433C4⋊C4—1581no—
434C4⋊C4—1581no—
435C4⋊C4#1192960911yescharacteristic
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—3161no—
439C22⋊C4—3161no—
440C22⋊C4—3161no—
441C22⋊C4—3161no—
442C22⋊C4#1192960911yescharacteristic
443C22⋊C4#1192960911yescharacteristic
444C42—41no—
445C42—3161no—
446C42C2×C22⋊D7911yescharacteristic
447C16—1581no—
448C16—1581no—
449C16—1581no—
450C16—1581no—
451C23—41no—
452C23—41no—
453C23—3161no—
454C23C2×#1192960911yescharacteristic
455C23C2×#1192960911yescharacteristic
456Q8—22no—
457Q8—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4—161no—
467D4—161no—
468D4—161no—
469D4—3161no—
470D4—3161no—
471D4—6322no—
472D4—12641no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—3161no—
482C2×C4—3161no—
483C2×C4—3161no—
484C2×C4—12641no—
485C2×C4—12641no—
486C2×C4C2×#1192960911yescharacteristic
487C2×C4C2×#1192960911yescharacteristic
488C2×C4C2×#1192960911yescharacteristic
489C2×C4C2×#1192960911yescharacteristic
490C2×C4C22×C22⋊D7911yescharacteristic
491C8—21no—
492C8—21no—
493C8—21no—
494C8C2×#1192962311yescharacteristic
495C8#1457920611yescharacteristic
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—6321no—
509C22—6321no—
510C22—6321no—
511C22C22×#1192960911yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—81no—
518C4—6321no—
519C4—6321no—
520C4—6321no—
521C4#1909920711yescharacteristic
522C4#1909937411yescharacteristic
523C2—41no—
524C2—41no—
525C2—161no—
526C2—161no—
527C2—12641no—
528C2#2377521311yescharacteristic
529C2#2377653411yescharacteristic
530C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3287C2
socleC22×C79C22×D8
hypercenter#3287D79
hypocenterC79#33876
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3287

Complements

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

NH ≅ G/Ntype
1. #23775803C2semidirect
2. #23765641C2semidirect
3. #23765588C2semidirect
4. #23765578C2semidirect
5. #23765564C2semidirect
6. #23765501C2semidirect
7. #23765484C2semidirect
8. #23765432C2semidirect
9. #23765323C2semidirect
10. #23765323C2semidirect
11. #23765262C2semidirect
12. #23765226C2semidirect
13. #23765112C2semidirect
14. #23765107C2semidirect
15. C79×#3287C2semidirect
267. #3287D79semidirect
281. C79#33876semidirect

Maximal subgroups

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

subgroupindex
#237758032
#237656412
#237655882
#237655782
#237655642
#237655012
#237654842
#237654322
#237653232
#237652622
#237652262
#237651122
#237651072
C79×#32872
#3387679