Finite Group Databasebeta

← #34445839

Subgroups — #34445839

Order 20224.

476 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775803C211yesmaximal, characteristic
2#23765676C211yesmaximal, characteristic, complemented
3#23765605C211yesmaximal, characteristic, complemented
4#23765595C211yesmaximal, characteristic, complemented
5#23765591C211yesmaximal, characteristic, complemented
6#23765585C211yesmaximal, characteristic, complemented
7#23765564C211yesmaximal, characteristic, complemented
8#23765554C211yesmaximal, characteristic, complemented
9#23765515C211yesmaximal, characteristic, complemented
10#23765515C211yesmaximal, characteristic, complemented
11C4×#14579104C211yesmaximal, characteristic, complemented
12#23765232C211yesmaximal, characteristic, complemented
13#23765189C211yesmaximal, characteristic, complemented
14#23765186C211yesmaximal, characteristic, complemented
15C79×#3336C211yesmaximal, characteristic, complemented
16#19099259—21no—
17C2×#14579104C2211yescharacteristic
18C2×#14579101C2211yescharacteristic
19C2×#14579101C2211yescharacteristic
20#19098965C2211yescharacteristic
21#19098888—21no—
22#19098882—21no—
23#19098864—21no—
24#19098799C2211yescharacteristic
25#19098799C2211yescharacteristic
26C4×D632C2211yescharacteristic
27#19098601C2211yescharacteristic
28C2×#14579095C2211yescharacteristic
29#19098390C2211yescharacteristic
30#19098390C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098390C2211yescharacteristic
34#19098388C2211yescharacteristic
35#19098388C2211yescharacteristic
36#19098388C2211yescharacteristic
37#19098388C2211yescharacteristic
38C4⋊(C79⋊C16)C2211yescharacteristic
39#19098367C2211yescharacteristic
40#19098367C2211yescharacteristic
41#19098367C2211yescharacteristic
42#19098363C2211yescharacteristic
43#19098363C2211yescharacteristic
44#19098363C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47C4×C79⋊C16C2211yescharacteristic
48#552×C79—21no—
49#499×C79C2211yescharacteristic
50#485×C79—21no—
51#483×C79—21no—
52#479×C79—21no—
53#467×C79C2211yescharacteristic
54#467×C79C2211yescharacteristic
55C4×Q16×C79C2211yescharacteristic
56C8×D4×C79C2211yescharacteristic
57#427×C79C2211yescharacteristic
58C2×#107×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#14579101—22no—
72#14579101—22no—
73C2×C79⋊C16C2311yescharacteristic
74C2×C79⋊C16C2311yescharacteristic
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77#14579095—21no—
78#14579095C2311yescharacteristic
79#14579095C2311yescharacteristic
80#14579095C2311yescharacteristic
81#14579093—21no—
82#14579093—21no—
83#14579083—21no—
84#14579083—21no—
85#14579083—21no—
86C2×Q16×C79C2311yescharacteristic
87C22×C8×C79C2311yescharacteristic
88C22×C8×C79C2311yescharacteristic
89#126×C79—41no—
90#125×C79—21no—
91#125×C79—41no—
92#123×C79—41no—
93C22⋊Q8×C79—21no—
94C22⋊Q8×C79—21no—
95C4×Q8×C79—21no—
96C4×D4×C79D411yescharacteristic
97#117×C79—21no—
98#117×C79—41no—
99C2×C4⋊C4×C79—21no—
100#107×C79—21no—
101#107×C79C2311yescharacteristic
102#107×C79C2311yescharacteristic
103#107×C79C2311yescharacteristic
104#106×C79—21no—
105#106×C79—21no—
106C4⋊C8×C79D411yescharacteristic
107#103×C79—21no—
108#103×C79—21no—
109#103×C79—21no—
110C22⋊C8×C79D411yescharacteristic
111C22⋊C8×C79D411yescharacteristic
112C4×C8×C79C2311yescharacteristic
113C2×D316—21no—
114C2×C22⋊D79—41no—
115C2×C22⋊D79—41no—
116C22×Dic79—41no—
117C2×C4×D79—41no—
118D632—21no—
119D632—21no—
120C4⋊Dic79—21no—
121C4⋊Dic79—21no—
122C4⋊Dic79—21no—
123C4⋊Dic79—41no—
124Dic79⋊C4—41no—
125Dic79⋊C4—41no—
126C22⋊Dic79—41no—
127(C2×D79)⋊C4—41no—
128(C2×D79)⋊C4—41no—
129(C2×D79)⋊C4—41no—
130C4×Dic79—41no—
131C79⋊C16—21no—
132C79⋊C16—21no—
133C79⋊C16—21no—
134C79⋊C16#4112yes—
135C2×Q8×C79—21no—
136C2×D4×C79D811yescharacteristic
137C22×C4×C79—41no—
138C22×C4×C79C2×D411yescharacteristic
139C22×C4×C79C2×D411yescharacteristic
140Q16×C79—21no—
141Q16×C79—21no—
142C2×C8×C79—21no—
143C2×C8×C79—21no—
144C2×C8×C79—21no—
145C2×C8×C79—21no—
146C2×C8×C79C2×D411yescharacteristic
147C2×C8×C79C2×D411yescharacteristic
148C2×C8×C79C2×D411yescharacteristic
149C2×C8×C79C2411yescharacteristic
150C4⋊C4×C79—21no—
151C4⋊C4×C79—21no—
152C4⋊C4×C79—21no—
153C4⋊C4×C79—41no—
154C4⋊C4×C79—41no—
155C4⋊C4×C79—41no—
156C4⋊C4×C79—41no—
157C4⋊C4×C79—41no—
158C4⋊C4×C79—41no—
159C4⋊C4×C79D811yescharacteristic
160C22⋊C4×C79—41no—
161C22⋊C4×C79—41no—
162C22⋊C4×C79—41no—
163C22⋊C4×C79D811yescharacteristic
164C22⋊C4×C79D811yescharacteristic
165C42×C79—41no—
166C42×C79—41no—
167C42×C79C2×D411yescharacteristic
168C23×C79C2×D811yescharacteristic
169C23×C79C2×D811yescharacteristic
170C22×D79—41no—
171Q8×C79—41no—
172Q8×C79—41no—
173D4×C79—41no—
174C2×C4×C79—21no—
175C2×C4×C79—21no—
176C2×C4×C79—21no—
177C2×C4×C79—21no—
178C2×C4×C79—41no—
179C2×C4×C79—41no—
180C2×C4×C79—41no—
181C2×C4×C79—41no—
182C2×C4×C79—82no—
183C2×C4×C79C2×D811yescharacteristic
184C2×C4×C79C2×D811yescharacteristic
185C2×C4×C79C2×D811yescharacteristic
186C2×C4×C79C2×D811yescharacteristic
187C2×C4×C79C22×D411yescharacteristic
188C22⋊D79—82no—
189C22⋊D79—161no—
190C2×Dic79—41no—
191C2×Dic79—41no—
192C2×Dic79—41no—
193C2×Dic79—161no—
194D316—41no—
195D316—41no—
196C4×D79—161no—
197C8×C79—21no—
198C8×C79—21no—
199C8×C79—21no—
200C8×C79C2×#4111yescharacteristic
201C8×C792−1+411yescharacteristic
202C22×C79—21no—
203C22×C79—21no—
204C22×C79—21no—
205C22×C79—21no—
206C22×C79—21no—
207C22×C79—21no—
208C22×C79C22×D811yescharacteristic
209C2×D79—81no—
210C2×D79—81no—
211C2×D79—81no—
212C4×C79—21no—
213C4×C79—21no—
214C4×C79—21no—
215C4×C79—21no—
216C4×C79—81no—
217C4×C79—81no—
218C4×C79—81no—
219C4×C79—81no—
220C4×C79#54711yescharacteristic
221C4×C79#57811yescharacteristic
222Dic79—81no—
223Dic79—81no—
224Dic79—81no—
225#33918—791nomaximal, Sylow
226C2×C79—41no—
227C2×C79—41no—
228C2×C79#333411yescharacteristic
229C2×C79#341411yescharacteristic
230C2×C79#341611yescharacteristic
231D79—161no—
232#3339—791no—
233#3336D7911yescharacteristic, complemented
234#2253—791no—
235#2237—791no—
236#2235—791no—
237#2234—791no—
238#2233—791no—
239#2229—791no—
240#2228—791no—
241#2220—791no—
242#2220—791no—
243C4×SD32—791no—
244#2169—791no—
245#2159—791no—
246#2159—791no—
247C79#3391811yescharacteristic, complemented, Sylow
248#553—1581no—
249#552—21no—
250C2×SD32—791no—
251C2×M32—791no—
252C2×M32—791no—
253#499—791no—
254#499C2×D7911yescharacteristic
255#485—21no—
256#483—21no—
257#482—1581no—
258#481—1581no—
259#479—21no—
260#478—1581no—
261#467C2×D7911yescharacteristic
262#467C2×D7911yescharacteristic
263#466—791no—
264#466—791no—
265C4×Q16C2×D7911yescharacteristic
266C4×D8—791no—
267C8×D4C2×D7911yescharacteristic
268#427—791no—
269#427C2×D7911yescharacteristic
270C2×#107—791no—
271C2×#107C2×D7911yescharacteristic
272#367—791no—
273#367—791no—
274#367—791no—
275#367—791no—
276#367—791no—
277#366—791no—
278#366—791no—
279#366—791no—
280#366—791no—
281C4⋊C16—791no—
282#358—791no—
283#358—791no—
284#358—791no—
285#357—791no—
286#357—791no—
287#357—791no—
288C22⋊C16—791no—
289C22⋊C16—791no—
290C4×C16—791no—
291C2×Q16C22×D7911yescharacteristic
292C2×D8—791no—
293C22×C8C22×D7911yescharacteristic
294C22×C8C22×D7911yescharacteristic
295#126—41no—
296#126—3161no—
297#125—21no—
298#125—41no—
299#125—1581no—
300#123—41no—
301#123—3161no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—3161no—
305#121—1581no—
306#121—1581no—
307C4×Q8—21no—
308C4×D4—3161no—
309C4×D4—1581no—
310C4×D4C22⋊D7911yescharacteristic
311#117—21no—
312#117—41no—
313#117—1581no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—1581no—
316SD32—1582no—
317M32—1582no—
318M32—1582no—
319C2×C16—791no—
320C2×C16—791no—
321C2×C16—791no—
322C2×C16—791no—
323#107—21no—
324#107—791no—
325#107—791no—
326#107—791no—
327#107—1581no—
328#107C22×D7911yescharacteristic
329#107C22×D7911yescharacteristic
330#107C22×D7911yescharacteristic
331#106—21no—
332#106—21no—
333#106—1581no—
334#106—1581no—
335C4⋊C8C22⋊D7911yescharacteristic
336#103—21no—
337#103—21no—
338#103—21no—
339#102—1581no—
340#102—1581no—
341#102—1581no—
342C22⋊C8C22⋊D7911yescharacteristic
343C22⋊C8C22⋊D7911yescharacteristic
344C4×C8C22×D7911yescharacteristic
345C2×Q8—21no—
346C2×D4—3161no—
347C2×D4—3161no—
348C2×D4—1581no—
349C2×D4#1192960911yescharacteristic
350C22×C4—41no—
351C22×C4—3161no—
352C22×C4—3161no—
353C22×C4C2×C22⋊D7911yescharacteristic
354C22×C4C2×C22⋊D7911yescharacteristic
355Q16—21no—
356Q16—21no—
357D8—1581no—
358D8—1581no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8C2×C22⋊D7911yescharacteristic
364C2×C8C2×C22⋊D7911yescharacteristic
365C2×C8C2×C22⋊D7911yescharacteristic
366C2×C8C23×D7911yescharacteristic
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—3161no—
377C4⋊C4—3161no—
378C4⋊C4—3161no—
379C4⋊C4—1581no—
380C4⋊C4—1581no—
381C4⋊C4—1581no—
382C4⋊C4#1192960911yescharacteristic
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—3161no—
387C22⋊C4—3161no—
388C22⋊C4—3161no—
389C22⋊C4—3161no—
390C22⋊C4#1192960911yescharacteristic
391C22⋊C4#1192960911yescharacteristic
392C42—41no—
393C42—41no—
394C42—3161no—
395C42C2×C22⋊D7911yescharacteristic
396C16—792no—
397C16—1581no—
398C16—1581no—
399C16—1581no—
400C23—3161no—
401C23C2×#1192960911yescharacteristic
402C23C2×#1192960911yescharacteristic
403Q8—41no—
404Q8—41no—
405D4—41no—
406D4—3161no—
407D4—3161no—
408D4—6322no—
409D4—12641no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—82no—
419C2×C4—3161no—
420C2×C4—3161no—
421C2×C4—3161no—
422C2×C4—12641no—
423C2×C4—12641no—
424C2×C4C2×#1192960911yescharacteristic
425C2×C4C2×#1192960911yescharacteristic
426C2×C4C2×#1192960911yescharacteristic
427C2×C4C2×#1192960911yescharacteristic
428C2×C4C22×C22⋊D7911yescharacteristic
429C8—21no—
430C8—21no—
431C8—21no—
432C8#41×D7911yescharacteristic
433C8#1457920811yescharacteristic
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—6321no—
441C22—6321no—
442C22—6321no—
443C22C22×#1192960911yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—6321no—
453C4—6321no—
454C4—6321no—
455C4#1909921211yescharacteristic
456C4#1909937811yescharacteristic
457C2—41no—
458C2—41no—
459C2—12641no—
460C2#2377574311yescharacteristic
461C2#2377653411yescharacteristic
462C2#2377654411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3336C2
socleC22×C79C22×D8
prime radical#23775803C2
hypercenter#3336D79
hypocenterC79#33918
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #23765676C2semidirect
3. #23765605C2semidirect
4. #23765595C2semidirect
5. #23765591C2semidirect
6. #23765585C2semidirect
7. #23765564C2semidirect
8. #23765554C2semidirect
9. #23765515C2semidirect
10. #23765515C2semidirect
11. C4×#14579104C2semidirect
12. #23765232C2semidirect
13. #23765189C2semidirect
14. #23765186C2semidirect
15. C79×#3336C2semidirect
233. #3336D79semidirect
247. C79#33918semidirect

Maximal subgroups

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

subgroupindex
#237758032
#237656762
#237656052
#237655952
#237655912
#237655852
#237655642
#237655542
#237655152
C4×#145791042
#237652322
#237651892
#237651862
C79×#33362
#3391879