Finite Group Databasebeta

← #34445375

Subgroups — #34445375

Order 20224.

542 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775769C211yesmaximal, characteristic, complemented
2#23765645C211yesmaximal, characteristic, complemented
3#23765605C211yesmaximal, characteristic, complemented
4#23765600C211yesmaximal, characteristic, complemented
5#23765557C211yesmaximal, characteristic, complemented
6#23765506C211yesmaximal, characteristic, complemented
7#23765489C211yesmaximal, characteristic, complemented
8#23765447C211yesmaximal, characteristic, complemented
9#23765340C211yesmaximal, characteristic, complemented
10#23765340C211yesmaximal, characteristic, complemented
11#23765264C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765112C211yesmaximal, characteristic, complemented
14#23765107C211yesmaximal, characteristic, complemented
15C79×#3283C211yesmaximal, characteristic
16#19099253—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579107C2211yescharacteristic
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#19098388C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098367C2211yescharacteristic
39#19098367C2211yescharacteristic
40#19098367C2211yescharacteristic
41#19098366C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098364C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47#19098339C2211yescharacteristic
48C2×#135×C79C2211yescharacteristic
49C2×#135×C79C2211yescharacteristic
50#547×C79—21no—
51#494×C79C2211yescharacteristic
52#467×C79C2211yescharacteristic
53#467×C79C2211yescharacteristic
54#461×C79—21no—
55#452×C79—21no—
56#452×C79—21no—
57C4×D8×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—41no—
71#14579107—41no—
72#14579105—22no—
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×C79—21no—
95C2×Q16×C79C2311yescharacteristic
96C2×Q16×C79C2311yescharacteristic
97C2×SD16×C79—21no—
98C2×SD16×C79—21no—
99C2×D8×C79C2311yescharacteristic
100C22×C8×C79C2311yescharacteristic
101C22×C8×C79C2311yescharacteristic
102C4⋊Q8×C79—21no—
103#123×C79—41no—
104#123×C79—41no—
105C22⋊Q8×C79—21no—
106C22⋊Q8×C79—21no—
107#121×C79—41no—
108C4×D4×C79—21no—
109C4×D4×C79D411yescharacteristic
110C2×C4⋊C4×C79—41no—
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
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×Q8×C79—21no—
147C2×D4×C79—21no—
148C2×D4×C79—41no—
149C2×D4×C79—41no—
150C2×D4×C79D811yescharacteristic
151C22×C4×C79—41no—
152C22×C4×C79—41no—
153C22×C4×C79—41no—
154C22×C4×C79C2×D411yescharacteristic
155C22×C4×C79C2×D411yescharacteristic
156Q16×C79—21no—
157Q16×C79—21no—
158Q16×C79—21no—
159Q16×C79—21no—
160Q16×C79—81no—
161SD16×C79—42no—
162SD16×C79—42no—
163D8×C79—21no—
164D8×C79#4112yes—
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—41no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79—41no—
178C4⋊C4×C79D811yescharacteristic
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79—41no—
182C22⋊C4×C79D811yescharacteristic
183C22⋊C4×C79D811yescharacteristic
184C42×C79C2×D411yescharacteristic
185C23×C79—41no—
186C23×C79C2×D811yescharacteristic
187C23×C79C2×D811yescharacteristic
188Q8×C79—41no—
189Q8×C79—41no—
190Q8×C79—41no—
191Q8×C79—41no—
192D4×C79—22no—
193D4×C79—41no—
194D4×C79—41no—
195D4×C79—82no—
196D4×C79—82no—
197C2×C4×C79—21no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—21no—
201C2×C4×C79—41no—
202C2×C4×C79—41no—
203C2×C4×C79—41no—
204C2×C4×C79—82no—
205C2×C4×C79—82no—
206C2×C4×C79—82no—
207C2×C4×C79C2×D811yescharacteristic
208C2×C4×C79C2×D811yescharacteristic
209C2×C4×C79C2×D811yescharacteristic
210C2×C4×C79C2×D811yescharacteristic
211C2×C4×C79C22×D411yescharacteristic
212C2×Dic79—41no—
213C2×Dic79—41no—
214C2×Dic79—41no—
215C2×Dic79—41no—
216C2×Dic79—82no—
217Dic158—41no—
218Dic158—41no—
219C8×C79—21no—
220C8×C79—21no—
221C8×C79—21no—
222C8×C79C2×#4111yescharacteristic
223C8×C79H(5,𝔽2)11yescharacteristic
224C22×C79—21no—
225C22×C79—21no—
226C22×C79—21no—
227C22×C79—21no—
228C22×C79—21no—
229C22×C79—21no—
230C22×C79—42no—
231C22×C79—81no—
232C22×C79—81no—
233C22×C79C22×D811yescharacteristic
234C4×C79—21no—
235C4×C79—21no—
236C4×C79—21no—
237C4×C79—21no—
238C4×C79—81no—
239C4×C79—81no—
240C4×C79—81no—
241C4×C79#54611yescharacteristic
242C4×C79#57711yescharacteristic
243Dic79—81no—
244Dic79—81no—
245Dic79—81no—
246Dic79—81no—
247#33882—791nomaximal, Sylow
248C2×C79—41no—
249C2×C79—41no—
250C2×C79—82no—
251C2×C79#329411yescharacteristic
252C2×C79#341411yescharacteristic
253C2×C79#341711yescharacteristic
254#3336—791no—
255#3283D7911yescharacteristic
256#2245—791no—
257#2237—791no—
258#2236—791no—
259#2228—791no—
260#2219—791no—
261#2216—791no—
262#2209—791no—
263#2189—791no—
264#2189—791no—
265#2175—791no—
266#2168—791no—
267#2139—791no—
268#2138—791no—
269C79#3388211yescharacteristic, complemented, Sylow
270C2×#135C2×D7911yescharacteristic
271C2×#135C2×D7911yescharacteristic
272#552—1581no—
273#547—21no—
274C2×Q32—791no—
275C2×Q32—791no—
276C2×SD32—791no—
277C2×M32—791no—
278C22×C16—791no—
279#499—791no—
280#494C2×D7911yescharacteristic
281#485—1581no—
282#483—1581no—
283#479—1581no—
284#467—791no—
285#467—791no—
286#467C2×D7911yescharacteristic
287#467C2×D7911yescharacteristic
288#461—21no—
289#452—21no—
290#452—21no—
291C4×Q16—791no—
292C4×D8C2×D7911yescharacteristic
293C8×D4C2×D7911yescharacteristic
294#427—791no—
295C2×#107—791no—
296#367—791no—
297#366—791no—
298#366—791no—
299C4⋊C16—791no—
300#358—791no—
301#358—791no—
302#358—791no—
303#358—791no—
304#358—791no—
305#358—791no—
306#358—791no—
307#357—791no—
308#357—791no—
309#357—791no—
310C22⋊C16—791no—
311C22⋊C16—791no—
312#346—791no—
313C2×#41—21no—
314C2×#41—21no—
315#135—22no—
316#135—22no—
317#135—41no—
318#135—41no—
319C2×Q16—21no—
320C2×Q16—791no—
321C2×Q16C22×D7911yescharacteristic
322C2×Q16C22×D7911yescharacteristic
323C2×SD16—21no—
324C2×SD16—21no—
325C2×D8C22×D7911yescharacteristic
326C22×C8C22×D7911yescharacteristic
327C22×C8C22×D7911yescharacteristic
328C4⋊Q8—21no—
329#126—3161no—
330#125—3161no—
331#125—1581no—
332#123—41no—
333#123—41no—
334#123—3161no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—1581no—
338C22⋊Q8—1581no—
339#121—41no—
340C4×Q8—1581no—
341C4×D4—21no—
342C4×D4C22⋊D7911yescharacteristic
343#117—3161no—
344#117—1581no—
345C2×C4⋊C4—41no—
346C2×C4⋊C4—1581no—
347Q32—3161no—
348Q32—3161no—
349SD32—1582no—
350M32—3161no—
351C2×C16—3161no—
352C2×C16—791no—
353C2×C16—791no—
354C2×C16—791no—
355C2×C16—791no—
356#107—791no—
357#107—791no—
358#107—791no—
359#107—1581no—
360#107C22×D7911yescharacteristic
361#106—1581no—
362#106—1581no—
363C4⋊C8C22⋊D7911yescharacteristic
364#103—21no—
365#103—21no—
366#103—1581no—
367#103—1581no—
368#103—1581no—
369#102—21no—
370C22⋊C8C22⋊D7911yescharacteristic
371C22⋊C8C22⋊D7911yescharacteristic
372C4×C8C22×D7911yescharacteristic
373#41—42no—
374#41—42no—
375#41—81no—
376#41—81no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—1581no—
380C2×D4—21no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4#1192960911yescharacteristic
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—3161no—
388C22×C4C2×C22⋊D7911yescharacteristic
389C22×C4C2×C22⋊D7911yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—21no—
393Q16—21no—
394Q16—81no—
395Q16—1581no—
396Q16—1581no—
397SD16—42no—
398SD16—42no—
399D8—21no—
400D8#1192962312yes—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D7911yescharacteristic
406C2×C8C2×C22⋊D7911yescharacteristic
407C2×C8C2×C22⋊D7911yescharacteristic
408C2×C8C23×D7911yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—3161no—
415C4⋊C4—3161no—
416C4⋊C4—3161no—
417C4⋊C4—3161no—
418C4⋊C4—3161no—
419C4⋊C4—3161no—
420C4⋊C4—1581no—
421C4⋊C4—1581no—
422C4⋊C4—1581no—
423C4⋊C4#1192960911yescharacteristic
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—3161no—
428C22⋊C4—3161no—
429C22⋊C4—3161no—
430C22⋊C4#1192960911yescharacteristic
431C22⋊C4#1192960911yescharacteristic
432C42—3161no—
433C42—3161no—
434C42C2×C22⋊D7911yescharacteristic
435C16—1581no—
436C16—1581no—
437C16—1581no—
438C16—1581no—
439C23—41no—
440C23C2×#1192960911yescharacteristic
441C23C2×#1192960911yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—3161no—
447Q8—3161no—
448D4—22no—
449D4—41no—
450D4—41no—
451D4—82no—
452D4—82no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—3161no—
464C2×C4—3161no—
465C2×C4—3161no—
466C2×C4—3161no—
467C2×C4—6322no—
468C2×C4C2×#1192960911yescharacteristic
469C2×C4C2×#1192960911yescharacteristic
470C2×C4C2×#1192960911yescharacteristic
471C2×C4C2×#1192960911yescharacteristic
472C2×C4C22×C22⋊D7911yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8C2×#1192962311yescharacteristic
477C8#1457920611yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—81no—
486C22—81no—
487C22C22×#1192960911yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—6321no—
496C4—6321no—
497C4—6321no—
498C4—6321no—
499C4#1909920711yescharacteristic
500C4#1909937411yescharacteristic
501C2—41no—
502C2—41no—
503C2—82no—
504C2#2377521311yescharacteristic
505C2#2377653411yescharacteristic
506C2#2377655111yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
Fitting, prime radicalC79×#3283C2
socleC22×C79C22×D8
hypercenter#3283D79
hypocenterC79#33882
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #23775769C2semidirect
2. #23765645C2semidirect
3. #23765605C2semidirect
4. #23765600C2semidirect
5. #23765557C2semidirect
6. #23765506C2semidirect
7. #23765489C2semidirect
8. #23765447C2semidirect
9. #23765340C2semidirect
10. #23765340C2semidirect
11. #23765264C2semidirect
12. #23765226C2semidirect
13. #23765112C2semidirect
14. #23765107C2semidirect
269. C79#33882semidirect

Maximal subgroups

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

subgroupindex
#237757692
#237656452
#237656052
#237656002
#237655572
#237655062
#237654892
#237654472
#237653402
#237652642
#237652262
#237651122
#237651072
C79×#32832
#3388279