Finite Group Databasebeta

← #34443070

Subgroups — #34443070

Order 20224.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772981C211yesmaximal, characteristic
2C2×#19098601C211yesmaximal, characteristic, complemented
3#23765609C212yesmaximal, complemented
4#23765603C211yesmaximal, characteristic, complemented
5#23765598C211yesmaximal, characteristic, complemented
6#23765592C212yesmaximal, complemented
7#23765586C211yesmaximal, characteristic, complemented
8#23765581C211yesmaximal, characteristic, complemented
9#23765184C212yesmaximal, complemented
10#23765108C212yesmaximal, complemented
11C2×C22⋊C16×C79C211yesmaximal, characteristic, complemented
12C2×#14579183C2211yescharacteristic
13#19099074—21no—
14C2×#14579120—21no—
15#19098804C2211yescharacteristic
16#19098804C2211yescharacteristic
17#19098799C2211yescharacteristic
18#19098799C2211yescharacteristic
19#19098792—21no—
20#19098792—21no—
21#19098601—21no—
22#19098601—21no—
23#19098601C2211yescharacteristic
24#19098601C2212yes—
25#19098601C2212yes—
26C2×#14579095C2211yescharacteristic
27C2×#14579095C2211yescharacteristic
28C2×#14579093—21no—
29#19098560—21no—
30#19098391C2212yescomplemented
31#19098391C2212yescomplemented
32#19098389C2212yescomplemented
33#19098389C2212yescomplemented
34#19098368C2212yescomplemented
35#19098368C2212yescomplemented
36#19098365C2212yescomplemented
37#19098365C2212yescomplemented
38C23×C8×C79C2211yescharacteristic
39C22×C16×C79C2212yes—
40C22⋊C16×C79C2211yescharacteristic
41C22⋊C16×C79C2211yescharacteristic
42C22⋊C16×C79C2212yes—
43C2×#11929624—21no—
44#14579183—24no—
45C2×Dic316C2311yescharacteristic
46C2×#11929613—21no—
47C2×D632C2311yescharacteristic
48#14579144—41no—
49#14579140—21no—
50#14579140—21no—
51#14579135—21no—
52#14579135—21no—
53#14579130—41no—
54C4×C22⋊D79—41no—
55C4×C22⋊D79—41no—
56#14579120—22no—
57#14579120—22no—
58#14579120—21no—
59#14579120—41no—
60#14579120—41no—
61C2×C4⋊Dic79—21no—
62C2×C4⋊Dic79—21no—
63C2×C22⋊Dic79—41no—
64C2×C4×Dic79—41no—
65#14579095—21no—
66#14579095—21no—
67#14579095C2312yes—
68#14579095C2312yes—
69#14579095C2311yescharacteristic
70#14579095C2311yescharacteristic
71#14579093—21no—
72#14579093—22no—
73#14579093—22no—
74#14579086—21no—
75#14579086—21no—
76#14579083—21no—
77#14579083—21no—
78C23×C4×C79D411yescharacteristic
79C22×C8×C79—24no—
80C22×C8×C79D411yescharacteristic
81C22×C8×C79D411yescharacteristic
82C22×C8×C79D411yescharacteristic
83C22×C8×C79C2311yescharacteristic
84C22×C8×C79C2311yescharacteristic
85C22×C8×C79C2311yescharacteristic
86C2×C16×C79—24no—
87C2×C16×C79C2312yes—
88C2×C16×C79C2312yes—
89#11929624—42no—
90#11929624—42no—
91C2×Dic158—21no—
92C2×D316—21no—
93C2×C22⋊D79—41no—
94C2×C22⋊D79—41no—
95C2×C22⋊D79—41no—
96C22×Dic79—41no—
97C22×Dic79—41no—
98C2×C4×D79—41no—
99Dic316—22no—
100#11929613—42no—
101D632—22no—
102C4⋊Dic79—21no—
103C4⋊Dic79—21no—
104C4⋊Dic79—22no—
105C4⋊Dic79—22no—
106C4⋊Dic79—41no—
107C4⋊Dic79—41no—
108Dic79⋊C4—41no—
109Dic79⋊C4—41no—
110C22⋊Dic79—42no—
111C22⋊Dic79—42no—
112C22⋊Dic79—41no—
113C22⋊Dic79—41no—
114C22⋊Dic79—41no—
115(C2×D79)⋊C4—41no—
116(C2×D79)⋊C4—41no—
117C4×Dic79—42no—
118C4×Dic79—42no—
119C4×Dic79—41no—
120C24×C79D811yescharacteristic
121C22×C4×C79—22no—
122C22×C4×C79—22no—
123C22×C4×C79D811yescharacteristic
124C22×C4×C79D811yescharacteristic
125C22×C4×C79D811yescharacteristic
126C22×C4×C79C2×D411yescharacteristic
127C22×C4×C79C2×D411yescharacteristic
128C22×C4×C79C2×D411yescharacteristic
129C2×C8×C79—22no—
130C2×C8×C79—22no—
131C2×C8×C79—22no—
132C2×C8×C79—22no—
133C2×C8×C79—22no—
134C2×C8×C79C2×D411yescharacteristic
135C2×C8×C79C2×D411yescharacteristic
136C2×C8×C79C2×D411yescharacteristic
137C2×C8×C79#4114yes—
138C2×C8×C79C2411yescharacteristic
139C16×C79—22no—
140C16×C79—22no—
141C23×C79—22no—
142C23×C79—22no—
143C23×C79—41no—
144C23×C79C2×D811yescharacteristic
145C23×C79C2×D811yescharacteristic
146C23×C79C2×D811yescharacteristic
147C22×D79—41no—
148C2×C4×C79—22no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—22no—
154C2×C4×C79—22no—
155C2×C4×C79#12312yes—
156C2×C4×C79C2×D811yescharacteristic
157C2×C4×C79C2×D811yescharacteristic
158C2×C4×C79C2×D811yescharacteristic
159C2×C4×C79#13712yes—
160C2×C4×C79C22×D411yescharacteristic
161C22⋊D79—82no—
162C22⋊D79—82no—
163C22⋊D79—82no—
164C2×Dic79—42no—
165C2×Dic79—42no—
166C2×Dic79—41no—
167C2×Dic79—41no—
168C2×Dic79—41no—
169C2×Dic79—82no—
170C2×Dic79—82no—
171D316—41no—
172D316—41no—
173C4×D79—82no—
174Dic158—41no—
175Dic158—41no—
176C8×C79—21no—
177C8×C79—21no—
178C8×C79#12312yes—
179C8×C79C2×#4112yes—
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—44no—
188C22×C79#48012yes—
189C22×C79#50812yes—
190C22×C79C22×D811yescharacteristic
191C2×D79—81no—
192C2×D79—81no—
193C2×D79—81no—
194C4×C79—21no—
195C4×C79—21no—
196C4×C79#48012yes—
197C4×C79C2×#12311yescharacteristic
198C4×C79C2×#13711yescharacteristic
199Dic79—81no—
200Dic79—81no—
201Dic79—82no—
202Dic79—81no—
203Dic79—82no—
204#33645—791nomaximal, Sylow
205C2×C79—22no—
206C2×C79—22no—
207C2×C79—41no—
208C2×C79C2×#48011yescharacteristic
209C2×C79C2×#50811yescharacteristic
210C2×C79#341411yescharacteristic
211D79—161no—
212#3050—791no—
213C2×#427—791no—
214#2237—792no—
215#2236—791no—
216#2235—791no—
217#2234—792no—
218#2233—791no—
219#2232—791no—
220#2158—792no—
221#2138—792no—
222C2×C22⋊C16D7911yescharacteristic, complemented
223C79#3364511yescharacteristic, complemented, Sylow
224C2×#135—791no—
225C23×C8C2×D7911yescharacteristic
226#525—1581no—
227C2×#117—1581no—
228C22×C16C2×D7912yes—
229#467—791no—
230#467—791no—
231#466—791no—
232#466—791no—
233#465—1581no—
234#465—1581no—
235#427—792no—
236#427—792no—
237#427—791no—
238#427—1581no—
239#427—1581no—
240C2×#107—791no—
241C2×#107—791no—
242C2×#106—1581no—
243#416—1581no—
244#367—792no—
245#367—792no—
246#366—792no—
247#366—792no—
248#358—792no—
249#358—792no—
250#357—792no—
251#357—792no—
252C22⋊C16C2×D7911yescharacteristic
253C22⋊C16C2×D7911yescharacteristic
254C22⋊C16C2×D7912yes—
255C2×#41—1581no—
256C23×C4D31611yescharacteristic
257#135—1584no—
258C2×Q16—791no—
259C2×SD16—1581no—
260C2×D8—791no—
261C22×C8—24no—
262C22×C8D31611yescharacteristic
263C22×C8D31611yescharacteristic
264C22×C8D31611yescharacteristic
265C22×C8C22×D7911yescharacteristic
266C22×C8C22×D7911yescharacteristic
267C22×C8C22×D7911yescharacteristic
268#123—3161no—
269C22⋊Q8—1581no—
270C22⋊Q8—1581no—
271#121—1581no—
272#121—1581no—
273#120—3161no—
274C4×D4—3161no—
275C4×D4—3161no—
276#117—3161no—
277#117—3161no—
278#117—1582no—
279#117—1582no—
280#117—1581no—
281C2×C4⋊C4—1581no—
282C2×C4⋊C4—1581no—
283C2×C22⋊C4—3161no—
284C2×C42—3161no—
285C2×C16—24no—
286C2×C16C22×D7912yes—
287C2×C16C22×D7912yes—
288#107—792no—
289#107—792no—
290#107—791no—
291#107—791no—
292#107—1581no—
293#107—1581no—
294#106—1582no—
295#106—1582no—
296#106—1581no—
297#103—1581no—
298#103—1581no—
299#102—1581no—
300#102—1581no—
301C24D63211yescharacteristic
302#41—3162no—
303#41—3162no—
304C2×Q8—1581no—
305C2×D4—3161no—
306C2×D4—3161no—
307C2×D4—3161no—
308C2×D4—1581no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—3161no—
312C22×C4—3161no—
313C22×C4—3161no—
314C22×C4D63211yescharacteristic
315C22×C4D63211yescharacteristic
316C22×C4D63211yescharacteristic
317C22×C4C2×D31611yescharacteristic
318C22×C4C2×D31611yescharacteristic
319C22×C4C2×D31611yescharacteristic
320Q16—1582no—
321SD16—3162no—
322D8—1582no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×D31611yescharacteristic
329C2×C8C2×D31611yescharacteristic
330C2×C8C2×D31611yescharacteristic
331C2×C8#1192962314yes—
332C2×C8C23×D7911yescharacteristic
333C4⋊C4—3161no—
334C4⋊C4—3161no—
335C4⋊C4—3161no—
336C4⋊C4—3161no—
337C4⋊C4—1581no—
338C4⋊C4—1581no—
339C4⋊C4—1582no—
340C4⋊C4—1582no—
341C22⋊C4—3162no—
342C22⋊C4—3162no—
343C22⋊C4—3161no—
344C22⋊C4—3161no—
345C22⋊C4—3161no—
346C22⋊C4—3161no—
347C22⋊C4—3161no—
348C42—3162no—
349C42—3162no—
350C42—3161no—
351C16—22no—
352C16—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—3161no—
357C23C2×D63211yescharacteristic
358C23C2×D63211yescharacteristic
359C23C2×D63211yescharacteristic
360Q8—3161no—
361Q8—3161no—
362D4—3161no—
363D4—3161no—
364D4—6322no—
365D4—6322no—
366D4—6322no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—3162no—
375C2×C4—3162no—
376C2×C4—3161no—
377C2×C4—3161no—
378C2×C4—3161no—
379C2×C4—6322no—
380C2×C4—6322no—
381C2×C4—6322no—
382C2×C4#1457914512yes—
383C2×C4C2×D63211yescharacteristic
384C2×C4C2×D63211yescharacteristic
385C2×C4C2×D63211yescharacteristic
386C2×C4#1457919412yes—
387C2×C4C22×D31611yescharacteristic
388C8—21no—
389C8—21no—
390C8#1457914512yes—
391C8C2×#1192962312yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—6321no—
401C22—6321no—
402C22—6321no—
403C22#1909887712yes—
404C22#1909899512yes—
405C22C22×D63211yescharacteristic
406C4—21no—
407C4—21no—
408C4—6322no—
409C4—6321no—
410C4—6322no—
411C4—6321no—
412C4—6321no—
413C4#1909887712yes—
414C4C2×#1457914511yescharacteristic
415C4C2×#1457919411yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—12641no—
420C2C2×#1909887711yescharacteristic
421C2C2×#1909899511yescharacteristic
422C2#2377653611yescharacteristic
subgroupG/H
centerC2×C4C2×D632
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC2×C22⋊C16×C79C2
socleC22×C79C22×D8
prime radical#23772981C2
hypercenterC2×C22⋊C16D79
hypocenterC79#33645
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#19098601C2semidirect
3. #23765609C2semidirect
4. #23765603C2semidirect
5. #23765598C2semidirect
6. #23765592C2semidirect
7. #23765586C2semidirect
8. #23765581C2semidirect
9. #23765184C2semidirect
10. #23765108C2semidirect
11. C2×C22⋊C16×C79C2semidirect
30. #19098391C22semidirect
31. #19098391C22semidirect
32. #19098389C22semidirect
33. #19098389C22semidirect
34. #19098368C22semidirect
35. #19098368C22semidirect
36. #19098365C22semidirect
37. #19098365C22semidirect
222. C2×C22⋊C16D79semidirect
223. C79#33645semidirect

Maximal subgroups

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

subgroupindex
#237729812
C2×#190986012
#237656092
#237656032
#237655982
#237655922
#237655862
#237655812
#237651842
#237651082
C2×C22⋊C16×C792
#3364579