Finite Group Databasebeta

← #34445405

Subgroups — #34445405

Order 20224.

552 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#23775769C211yesmaximal, characteristic, complemented
2#23765641C211yesmaximal, characteristic, complemented
3#23765588C211yesmaximal, characteristic, complemented
4#23765583C211yesmaximal, characteristic, complemented
5#23765564C211yesmaximal, characteristic, complemented
6#23765511C211yesmaximal, characteristic, complemented
7#23765494C211yesmaximal, characteristic, complemented
8#23765437C211yesmaximal, characteristic, complemented
9#23765340C211yesmaximal, characteristic, complemented
10#23765340C211yesmaximal, characteristic, complemented
11#23765269C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765112C211yesmaximal, characteristic, complemented
14#23765107C211yesmaximal, characteristic, complemented
15C79×#3287C211yesmaximal, characteristic
16#19099253—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579105C2211yescharacteristic
19C2×#14579105C2211yescharacteristic
20C2×#14579101C2211yescharacteristic
21C22×C79⋊C16C2211yescharacteristic
22#19098965C2211yescharacteristic
23#19098905—21no—
24#19098894—21no—
25#19098871—21no—
26#19098804C2211yescharacteristic
27#19098804C2211yescharacteristic
28C4×Dic316C2211yescharacteristic
29#19098601C2211yescharacteristic
30C2×#14579095C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098366C2211yescharacteristic
39#19098364C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098364C2211yescharacteristic
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×Dic316C2311yescharacteristic
60#14579154—41no—
61#14579153—21no—
62#14579151—41no—
63#14579142—41no—
64#14579140—21no—
65#14579140—21no—
66C4×Dic158—21no—
67#14579120—21no—
68#14579117—41no—
69C2×C4⋊Dic79—21no—
70#14579107—22no—
71#14579105—41no—
72#14579105—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#14579086—21no—
86#14579086—21no—
87#14579086—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×Dic158—21no—
120C22×Dic79—41no—
121Dic316—21no—
122Dic316—21no—
123C4⋊Dic79—21no—
124C4⋊Dic79—21no—
125C4⋊Dic79—21no—
126C4⋊Dic79—41no—
127Dic79⋊C4—41no—
128Dic79⋊C4—41no—
129Dic79⋊C4—41no—
130Dic79⋊C4—41no—
131Dic79⋊C4—41no—
132C22⋊Dic79—41no—
133C22⋊Dic79—41no—
134C22⋊Dic79—41no—
135C4×Dic79—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
189Q8×C79—22no—
190Q8×C79—41no—
191D4×C79—41no—
192D4×C79—41no—
193D4×C79—41no—
194D4×C79—41no—
195D4×C79—41no—
196D4×C79—82no—
197D4×C79—82no—
198D4×C79—82no—
199D4×C79—161no—
200D4×C79—161no—
201D4×C79—161no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79—82no—
209C2×C4×C79—82no—
210C2×C4×C79C2×D811yescharacteristic
211C2×C4×C79C2×D811yescharacteristic
212C2×C4×C79C2×D811yescharacteristic
213C2×C4×C79C2×D811yescharacteristic
214C2×C4×C79C22×D411yescharacteristic
215C2×Dic79—41no—
216C2×Dic79—41no—
217C2×Dic79—41no—
218C2×Dic79—41no—
219C2×Dic79—82no—
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×C79—81no—
237C22×C79—81no—
238C22×C79—81no—
239C22×C79C22×D811yescharacteristic
240C4×C79—21no—
241C4×C79—21no—
242C4×C79—21no—
243C4×C79—21no—
244C4×C79—42no—
245C4×C79—81no—
246C4×C79#54611yescharacteristic
247C4×C79#57711yescharacteristic
248Dic79—81no—
249Dic79—81no—
250Dic79—81no—
251Dic79—81no—
252#33884—791nomaximal, Sylow
253C2×C79—41no—
254C2×C79—41no—
255C2×C79—161no—
256C2×C79—161no—
257C2×C79#329411yescharacteristic
258C2×C79#341411yescharacteristic
259C2×C79#341611yescharacteristic
260#3336—791no—
261#3287D7911yescharacteristic
262#2244—791no—
263#2234—791no—
264#2233—791no—
265#2229—791no—
266#2220—791no—
267#2217—791no—
268#2207—791no—
269#2189—791no—
270#2189—791no—
271#2176—791no—
272#2168—791no—
273#2139—791no—
274#2138—791no—
275C79#3388411yescharacteristic, complemented, Sylow
276C2×#135C2×D7911yescharacteristic
277C2×#135C2×D7911yescharacteristic
278#552—1581no—
279#550—21no—
280C2×Q32—791no—
281C2×SD32—791no—
282C2×SD32—791no—
283C2×M32—791no—
284C22×C16—791no—
285#499—791no—
286#493C2×D7911yescharacteristic
287#485—1581no—
288#483—1581no—
289#479—1581no—
290#467—791no—
291#467—791no—
292#466C2×D7911yescharacteristic
293#466C2×D7911yescharacteristic
294#460—21no—
295#449—21no—
296#449—21no—
297C4×Q16—791no—
298C4×Q16C2×D7911yescharacteristic
299C8×D4C2×D7911yescharacteristic
300#427—791no—
301C2×#107—791no—
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#357—791no—
311#357—791no—
312#357—791no—
313#357—791no—
314#357—791no—
315#357—791no—
316C22⋊C16—791no—
317C22⋊C16—791no—
318#346—791no—
319C2×#41—21no—
320C2×#41—21no—
321#135—22no—
322#135—22no—
323#135—41no—
324#135—41no—
325C2×Q16—791no—
326C2×Q16C22×D7911yescharacteristic
327C2×SD16—21no—
328C2×SD16—21no—
329C2×D8—21no—
330C2×D8C22×D7911yescharacteristic
331C2×D8C22×D7911yescharacteristic
332C22×C8C22×D7911yescharacteristic
333C22×C8C22×D7911yescharacteristic
334#127—21no—
335#127—41no—
336#126—3161no—
337#125—3161no—
338#125—1581no—
339#123—3161no—
340C22⋊Q8—1581no—
341C22⋊Q8—1581no—
342#121—21no—
343#121—21no—
344#121—41no—
345#121—41no—
346C4×Q8—21no—
347C4×Q8—1581no—
348C4×D4—41no—
349C4×D4C22⋊D7911yescharacteristic
350#117—3161no—
351#117—1581no—
352C2×C4⋊C4—1581no—
353Q32—1582no—
354SD32—3161no—
355SD32—3161no—
356M32—3161no—
357C2×C16—3161no—
358C2×C16—791no—
359C2×C16—791no—
360C2×C16—791no—
361C2×C16—791no—
362#107—791no—
363#107—791no—
364#107—791no—
365#107—1581no—
366#107C22×D7911yescharacteristic
367#106—1581no—
368#106—1581no—
369C4⋊C8C22⋊D7911yescharacteristic
370#103—21no—
371#103—1581no—
372#103—1581no—
373#103—1581no—
374#102—21no—
375#102—21no—
376C22⋊C8C22⋊D7911yescharacteristic
377C22⋊C8C22⋊D7911yescharacteristic
378C4×C8C22×D7911yescharacteristic
379#41—42no—
380#41—42no—
381#41—81no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—1581no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4#1192960911yescharacteristic
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—3161no—
397C22×C4C2×C22⋊D7911yescharacteristic
398C22×C4C2×C22⋊D7911yescharacteristic
399Q16—21no—
400Q16—1581no—
401Q16—1581no—
402Q16#1192962312yes—
403SD16—42no—
404SD16—42no—
405D8—21no—
406D8—21no—
407D8—21no—
408D8—21no—
409D8—81no—
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—41no—
420C4⋊C4—3161no—
421C4⋊C4—3161no—
422C4⋊C4—3161no—
423C4⋊C4—3161no—
424C4⋊C4—3161no—
425C4⋊C4—3161no—
426C4⋊C4—1581no—
427C4⋊C4—1581no—
428C4⋊C4—1581no—
429C4⋊C4#1192960911yescharacteristic
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—3161no—
433C22⋊C4—3161no—
434C22⋊C4—3161no—
435C22⋊C4#1192960911yescharacteristic
436C22⋊C4#1192960911yescharacteristic
437C42—41no—
438C42—3161no—
439C42—3161no—
440C42C2×C22⋊D7911yescharacteristic
441C16—1581no—
442C16—1581no—
443C16—1581no—
444C16—1581no—
445C23—41no—
446C23—41no—
447C23C2×#1192960911yescharacteristic
448C23C2×#1192960911yescharacteristic
449Q8—22no—
450Q8—41no—
451Q8—3161no—
452Q8—3161no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—161no—
462D4—161no—
463D4—161no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—3161no—
473C2×C4—3161no—
474C2×C4—3161no—
475C2×C4—3161no—
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—
485C8C2×#1192962311yescharacteristic
486C8#1457920611yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22C22×#1192960911yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—81no—
506C4—6321no—
507C4—6321no—
508C4—6321no—
509C4—6321no—
510C4#1909920711yescharacteristic
511C4#1909937411yescharacteristic
512C2—41no—
513C2—41no—
514C2—161no—
515C2—161no—
516C2#2377521311yescharacteristic
517C2#2377653411yescharacteristic
518C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
Fitting, prime radicalC79×#3287C2
socleC22×C79C22×D8
hypercenter#3287D79
hypocenterC79#33884
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. #23775769C2semidirect
2. #23765641C2semidirect
3. #23765588C2semidirect
4. #23765583C2semidirect
5. #23765564C2semidirect
6. #23765511C2semidirect
7. #23765494C2semidirect
8. #23765437C2semidirect
9. #23765340C2semidirect
10. #23765340C2semidirect
11. #23765269C2semidirect
12. #23765226C2semidirect
13. #23765112C2semidirect
14. #23765107C2semidirect
275. C79#33884semidirect

Maximal subgroups

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

subgroupindex
#237757692
#237656412
#237655882
#237655832
#237655642
#237655112
#237654942
#237654372
#237653402
#237652692
#237652262
#237651122
#237651072
C79×#32872
#3388479