Finite Group Databasebeta

← #34187807

Subgroups — #34187807

Order 20224.

552 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#23764386C211yesmaximal, characteristic, complemented
2#23764237C211yesmaximal, characteristic
3#23763429C211yesmaximal, characteristic, complemented
4#23763176C211yesmaximal, characteristic, complemented
5#23761972C211yesmaximal, characteristic, complemented
6#23761968C211yesmaximal, characteristic
7C2×#465×C79C211yesmaximal, characteristic
8C2×#14579095—21no—
9C2×#14579076—22no—
10C2×C22⋊(C79⋊C8)C2211yescharacteristic
11C2×#14579071—21no—
12#19098423C2211yescharacteristic
13#19098423C2211yescharacteristic
14C4×C22⋊Dic79C2211yescharacteristic
15#19098330—22no—
16#19098326—21no—
17#19098317—21no—
18#19098300—22no—
19#19098297—22no—
20C22×SD16×C79C411yescharacteristic, complemented
21C23×C8×C79C2211yescharacteristic
22C2×C22⋊Q8×C79C2211yescharacteristic
23C2×#121×C79C2211yescharacteristic
24#465×C79—21no—
25#465×C79—21no—
26#465×C79—21no—
27#465×C79—21no—
28C2×#106×C79C411yescharacteristic, complemented
29C2×#103×C79C411yescharacteristic, complemented
30C2×#102×C79C411yescharacteristic, complemented
31C22×C79⋊C8—21no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—21no—
36C2×C22⋊Dic79D412yes—
37C2×C4×Dic79—21no—
38#14579095—42no—
39#14579076—44no—
40C22⋊(C79⋊C8)—21no—
41C22⋊(C79⋊C8)D412yes—
42#14579071—42no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C22×Q8×C79C2×C411yescharacteristic
47C22×D4×C79—21no—
48C22×D4×C79C2×C411yescharacteristic
49C23×C4×C79C2311yescharacteristic
50C2×SD16×C79—21no—
51C2×SD16×C79—21no—
52C2×SD16×C79—42no—
53C22×C8×C79—42no—
54C22×C8×C79C2×C411yescharacteristic
55C22×C8×C79C2×C411yescharacteristic
56C22×C8×C79D411yescharacteristic
57C22×C8×C79D411yescharacteristic
58C22×C8×C79D411yescharacteristic
59C22×C8×C79D411yescharacteristic
60C22⋊Q8×C79—21no—
61C22⋊Q8×C79—21no—
62C22⋊Q8×C79—21no—
63C22⋊Q8×C79—21no—
64#121×C79—21no—
65#121×C79—21no—
66#121×C79—21no—
67#121×C79—21no—
68C2×C4⋊C4×C79—21no—
69C2×C4⋊C4×C79C2×C411yescharacteristic
70C2×C4⋊C4×C79C2×C411yescharacteristic
71C2×C22⋊C4×C79—21no—
72C2×C22⋊C4×C79—21no—
73#106×C79—21no—
74#106×C79—21no—
75#103×C79—21no—
76#103×C79—21no—
77#102×C79—21no—
78#102×C79—21no—
79C22×Dic79—22no—
80C22×Dic79—41no—
81C22×Dic79—41no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C4⋊Dic79—41no—
86C4⋊Dic79—41no—
87C4⋊Dic79—41no—
88C4⋊Dic79—41no—
89C22⋊Dic79—22no—
90C22⋊Dic79—22no—
91C22⋊Dic79—42no—
92C22⋊Dic79—42no—
93C4×Dic79—41no—
94C4×Dic79—41no—
95C24×C79—21no—
96C24×C79C2×D411yescharacteristic
97C2×Q8×C79—21no—
98C2×Q8×C79—21no—
99C2×Q8×C79—41no—
100C2×Q8×C79—41no—
101C2×D4×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—41no—
104C2×D4×C79—41no—
105C2×D4×C79—41no—
106C2×D4×C79—41no—
107C2×D4×C79—42no—
108C2×D4×C79—42no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—41no—
113C22×C4×C79—41no—
114C22×C4×C79C22×C411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79C2×D411yescharacteristic
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79#4111yescharacteristic
119C22×C4×C79#4111yescharacteristic
120SD16×C79—42no—
121SD16×C79—42no—
122C2×C8×C79—22no—
123C2×C8×C79—22no—
124C2×C8×C79—22no—
125C2×C8×C79—41no—
126C2×C8×C79—41no—
127C2×C8×C79—41no—
128C2×C8×C79C22⋊C411yescharacteristic
129C2×C8×C79C22⋊C411yescharacteristic
130C2×C8×C79C22⋊C411yescharacteristic
131C2×C8×C79C22⋊C411yescharacteristic
132C4⋊C4×C79—21no—
133C4⋊C4×C79—21no—
134C4⋊C4×C79—21no—
135C4⋊C4×C79—21no—
136C4⋊C4×C79—41no—
137C4⋊C4×C79—41no—
138C22⋊C4×C79—41no—
139C22⋊C4×C79—41no—
140C22⋊C4×C79—41no—
141C22⋊C4×C79—41no—
142C23×C79—41no—
143C23×C79—41no—
144C23×C79—41no—
145C23×C79—42no—
146C23×C79—81no—
147C23×C79—81no—
148C23×C79C4×D411yescharacteristic
149C23×C79#12111yescharacteristic
150C23×C79#12711yescharacteristic
151C23×C79AGL(1,ℤ/8)12yes—
152C23×C79#13712yes—
153Q8×C79—41no—
154Q8×C79—41no—
155Q8×C79—81no—
156D4×C79—41no—
157D4×C79—41no—
158D4×C79—81no—
159D4×C79—82no—
160D4×C79—81no—
161D4×C79—81no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
165C2×C4×C79—21no—
166C2×C4×C79—21no—
167C2×C4×C79—21no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—41no—
174C2×C4×C79—42no—
175C2×C4×C79—81no—
176C2×C4×C79—81no—
177C2×C4×C79C2×C22⋊C411yescharacteristic
178C2×C4×C79C4×D411yescharacteristic
179C2×C4×C79#12111yescharacteristic
180C2×C4×C79#12411yescharacteristic
181C2×Dic79—42no—
182C2×Dic79—42no—
183C2×Dic79—42no—
184C2×Dic79—81no—
185C2×Dic79—81no—
186C2×Dic79—81no—
187C2×Dic79—81no—
188C2×Dic79—81no—
189C2×Dic79—81no—
190C8×C79—21no—
191C8×C79—21no—
192C8×C79—21no—
193C8×C79#9912yescomplemented
194C79⋊C8—42no—
195C22×C79—21no—
196C22×C79—21no—
197C22×C79—21no—
198C22×C79—21no—
199C22×C79—21no—
200C22×C79—21no—
201C22×C79—42no—
202C22×C79—42no—
203C22×C79—81no—
204C22×C79—81no—
205C22×C79—81no—
206C22×C79—162no—
207C22×C79#39011yescharacteristic
208C22×C79#44012yes—
209C22×C79#46812yes—
210C22×C79#49611yescharacteristic
211C22×C79#49711yescharacteristic
212C4×C79—21no—
213C4×C79—21no—
214C4×C79—21no—
215C4×C79—81no—
216C4×C79—81no—
217C4×C79—81no—
218C4×C79C2×#9911yescharacteristic
219C4×C79#41011yescharacteristic
220Dic79—82no—
221Dic79—161no—
222Dic79—161no—
223#9786—791nomaximal, Sylow
224C2×C79—22no—
225C2×C79—21no—
226C2×C79—22no—
227C2×C79—21no—
228C2×C79—161no—
229C2×C79#196411yescharacteristic
230C2×C79#197311yescharacteristic
231C2×C79#197611yescharacteristic
232C2×#465D7911yescharacteristic
233#1996—791no—
234#1964—791no—
235#1808—791no—
236#1752—791no—
237#1520—791no—
238#1519—791no—
239C79#978611yescharacteristic, complemented, Sylow
240C22×SD16Dic7911yescharacteristic, complemented
241C23×C8C2×D7911yescharacteristic
242C2×C22⋊Q8C2×D7911yescharacteristic
243C2×#121C2×D7911yescharacteristic
244#465—21no—
245#465—21no—
246#465—21no—
247#465—21no—
248C2×#107—1581no—
249C2×#106Dic7911yescharacteristic, complemented
250C2×#103Dic7911yescharacteristic, complemented
251C2×#102Dic7911yescharacteristic, complemented
252C2×#99—1582no—
253C2×C22⋊C8—791no—
254C2×#97—1581no—
255#380—791no—
256#380—791no—
257C4×C22⋊C4—791no—
258#342—1582no—
259#340—1581no—
260#336—1581no—
261#328—1582no—
262#327—1582no—
263C22×Q8C2×Dic7911yescharacteristic
264C22×D4—21no—
265C22×D4C2×Dic7911yescharacteristic
266C23×C4C22×D7911yescharacteristic
267C2×SD16—21no—
268C2×SD16—21no—
269C2×SD16—42no—
270C22×C8—42no—
271C22×C8—1581no—
272C22×C8C2×Dic7911yescharacteristic
273C22×C8C2×Dic7911yescharacteristic
274C22×C8C22⋊D7911yescharacteristic
275C22×C8C22⋊D7911yescharacteristic
276C22×C8C22⋊D7911yescharacteristic
277C22×C8C22⋊D7911yescharacteristic
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—21no—
283#121—21no—
284#121—21no—
285#121—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—1581no—
288C2×C4⋊C4—1581no—
289C2×C4⋊C4C2×Dic7911yescharacteristic
290C2×C4⋊C4C2×Dic7911yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—792no—
294C2×C22⋊C4—1581no—
295C2×C22⋊C4—1581no—
296C2×C42—1581no—
297#107—3162no—
298#106—21no—
299#106—21no—
300#103—21no—
301#103—21no—
302#102—21no—
303#102—21no—
304#99—3164no—
305C22⋊C8—792no—
306C22⋊C8—1581no—
307#97—3162no—
308#95—1581no—
309#95—1581no—
310#95—1581no—
311C24—21no—
312C24D4×D7911yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—3161no—
331C22×C4—3161no—
332C22×C4—1582no—
333C22×C4C22×Dic7911yescharacteristic
334C22×C4D4×D7911yescharacteristic
335C22×C4C2×C22⋊D7911yescharacteristic
336C22×C4C2×C22⋊D7911yescharacteristic
337C22×C4#1192962311yescharacteristic
338C22×C4#1192962311yescharacteristic
339SD16—42no—
340SD16—42no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—3161no—
348C2×C8—3161no—
349C2×C8—1582no—
350C2×C8C22⋊Dic7911yescharacteristic
351C2×C8C22⋊Dic7911yescharacteristic
352C2×C8C22⋊Dic7911yescharacteristic
353C2×C8C22⋊Dic7911yescharacteristic
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—3161no—
361C4⋊C4—3161no—
362C4⋊C4—3161no—
363C4⋊C4—3161no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—3162no—
369C22⋊C4—3162no—
370C22⋊C4—1582no—
371C22⋊C4—1582no—
372C42—3161no—
373C42—3161no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23D4×Dic7911yescharacteristic
381C23#1457913211yescharacteristic
382C23#1457915711yescharacteristic
383C23#1457918812yes—
384C23#1457919012yes—
385Q8—41no—
386Q8—41no—
387Q8—81no—
388D4—41no—
389D4—41no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—82no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—3162no—
410C2×C4—3162no—
411C2×C4—3162no—
412C2×C4—6321no—
413C2×C4—6321no—
414C2×C4—6321no—
415C2×C4—6321no—
416C2×C4—6321no—
417C2×C4—6321no—
418C2×C4C2×C22⋊Dic7911yescharacteristic
419C2×C4D4×Dic7911yescharacteristic
420C2×C4#1457913211yescharacteristic
421C2×C4#1457914811yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—3162no—
426C8#1457907612yescomplemented
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—162no—
439C22#1909846611yescharacteristic
440C22#1909866412yes—
441C22#1909880612yes—
442C22#1909894911yescharacteristic
443C22#1909895411yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—6322no—
451C4—12641no—
452C4—12641no—
453C4C2×#1457907611yescharacteristic
454C4#1909853711yescharacteristic
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—161no—
460C2#2376423711yescharacteristic
461C2#2376428311yescharacteristic
462C2#2376429911yescharacteristic
subgroupG/H
centerC22#19098466
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#465×C79C2
socleC22×C79#390
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenterC2×#465D79
hypocenterC79#9786
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. #23764386C2semidirect
3. #23763429C2semidirect
4. #23763176C2semidirect
5. #23761972C2semidirect
20. C22×SD16×C79C4semidirect
28. C2×#106×C79C4semidirect
29. C2×#103×C79C4semidirect
30. C2×#102×C79C4semidirect
193. C8×C79#99semidirect
239. C79#9786semidirect
240. C22×SD16Dic79semidirect
249. C2×#106Dic79semidirect
250. C2×#103Dic79semidirect
251. C2×#102Dic79semidirect
426. C8#14579076semidirect

Maximal subgroups

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

subgroupindex
#237643862
#237642372
#237634292
#237631762
#237619722
#237619682
C2×#465×C792
#978679