Finite Group Databasebeta

← #34184830

Subgroups — #34184830

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766452C211yesmaximal, characteristic
2#23764386C211yesmaximal, characteristic, complemented
3#23761968C211yesmaximal, characteristic
4C2×#19098300C211yesmaximal, characteristic, complemented
5#23761280C211yesmaximal, characteristic, complemented
6#23761246C211yesmaximal, characteristic, complemented
7C79×#3187C211yesmaximal, characteristic
8C2×D4×Dic79—21no—
9C2×C22⋊(C79⋊C8)C2211yescharacteristic
10#19098456—21no—
11#19098449—21no—
12#19098443—21no—
13#19098443—21no—
14#19098423C2211yescharacteristic
15C22⋊C4×Dic79—21no—
16C4×C22⋊Dic79C2211yescharacteristic
17C2×C79⋊#95C2211yescharacteristic
18#19098330—21no—
19#19098330—21no—
20#19098300—21no—
21#19098300—21no—
22#19098297—22no—
23C2×C22⋊Q8×C79C2211yescharacteristic
24C2×#121×C79C2211yescharacteristic
25#483×C79—21no—
26#481×C79C412yescomplemented
27#465×C79—22no—
28#450×C79—21no—
29#448×C79C412yescomplemented
30C2×C22⋊C8×C79C2211yescharacteristic
31C23×Dic79—21no—
32C23×Dic79—21no—
33C22×C79⋊C8—21no—
34D4×Dic79—42no—
35D4×Dic79—42no—
36C2×C4⋊Dic79—21no—
37C2×C4⋊Dic79—21no—
38C2×C22⋊Dic79—21no—
39C2×C22⋊Dic79—21no—
40C2×C22⋊Dic79—21no—
41C2×C22⋊Dic79—21no—
42C2×C22⋊Dic79D411yescharacteristic
43C2×C22⋊Dic79D411yescharacteristic
44C2×C4×Dic79—21no—
45C2×C4×Dic79—21no—
46C22⋊(C79⋊C8)—21no—
47C22⋊(C79⋊C8)—21no—
48C79⋊#95—21no—
49C79⋊#95—21no—
50C79⋊#95—21no—
51C79⋊#95—21no—
52C79⋊#95—21no—
53C79⋊#95D411yescharacteristic
54C79⋊#95D411yescharacteristic
55C22×Q8×C79C2×C411yescharacteristic, complemented
56C22×D4×C79—21no—
57C22×D4×C79D411yescharacteristic
58C23×C4×C79C2311yescharacteristic
59C2×SD16×C79—22no—
60C22×C8×C79—21no—
61C22⋊Q8×C79—21no—
62C22⋊Q8×C79—21no—
63C22⋊Q8×C79—21no—
64C22⋊Q8×C79—21no—
65#121×C79—21no—
66#121×C79—21no—
67#121×C79C2×C412yes—
68#121×C79D412yes—
69C2×C4⋊C4×C79—21no—
70C2×C4⋊C4×C79C2×C411yescharacteristic, complemented
71C2×C4⋊C4×C79D411yescharacteristic
72C2×C22⋊C4×C79—21no—
73C2×C22⋊C4×C79—21no—
74#106×C79—22no—
75#103×C79—22no—
76#102×C79—22no—
77C22⋊C8×C79—21no—
78C22⋊C8×C79C2×C412yescomplemented
79C22×Dic79—21no—
80C22×Dic79—21no—
81C22×Dic79—21no—
82C22×Dic79—21no—
83C22×Dic79—42no—
84C22×Dic79—42no—
85C22×Dic79—42no—
86C22×Dic79—41no—
87C22×Dic79—41no—
88C22×Dic79—41no—
89C22×Dic79—81no—
90C22×Dic79—81no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—81no—
93C4⋊Dic79—42no—
94C4⋊Dic79—81no—
95C22⋊Dic79—21no—
96C22⋊Dic79—21no—
97C22⋊Dic79—21no—
98C22⋊Dic79—42no—
99C22⋊Dic79—42no—
100C22⋊Dic79—41no—
101C22⋊Dic79—41no—
102C22⋊Dic79—81no—
103C22⋊Dic79SD1612yescomplemented
104C4×Dic79—42no—
105C4×Dic79—81no—
106C24×C79—21no—
107C24×C79C2×D411yescharacteristic
108C2×Q8×C79—21no—
109C2×Q8×C79—21no—
110C2×Q8×C79—81no—
111C2×D4×C79—22no—
112C2×D4×C79—22no—
113C2×D4×C79—21no—
114C2×D4×C79—41no—
115C2×D4×C79—41no—
116C2×D4×C79—41no—
117C2×D4×C79—41no—
118C2×D4×C79—81no—
119C2×D4×C79C22⋊C412yes—
120C22×C4×C79—21no—
121C22×C4×C79—21no—
122C22×C4×C79—21no—
123C22×C4×C79—81no—
124C22×C4×C79C22×C411yescharacteristic
125C22×C4×C79C2×D411yescharacteristic
126C22×C4×C79C2×D411yescharacteristic
127C22×C4×C79C2×D411yescharacteristic
128C22×C4×C79#4111yescharacteristic
129C22×C4×C79#4111yescharacteristic
130SD16×C79—82no—
131C2×C8×C79—22no—
132C2×C8×C79—81no—
133C4⋊C4×C79—21no—
134C4⋊C4×C79—21no—
135C4⋊C4×C79—21no—
136C4⋊C4×C79—41no—
137C4⋊C4×C79—41no—
138C4⋊C4×C79C22⋊C412yes—
139C22⋊C4×C79—22no—
140C22⋊C4×C79—22no—
141C22⋊C4×C79—41no—
142C22⋊C4×C79—41no—
143C23×C79—22no—
144C23×C79—41no—
145C23×C79—41no—
146C23×C79—41no—
147C23×C79—41no—
148C23×C79—41no—
149C23×C79—41no—
150C23×C79—41no—
151C23×C79—41no—
152C23×C79C4×D411yescharacteristic
153C23×C79#12011yescharacteristic
154C23×C79#12111yescharacteristic
155C23×C79C2×SD1611yescharacteristic
156C23×C79#13511yescharacteristic
157C23×C79AGL(1,ℤ/8)11yescharacteristic
158C23×C79#13711yescharacteristic
159Q8×C79—42no—
160Q8×C79—81no—
161D4×C79—42no—
162D4×C79—42no—
163D4×C79—42no—
164D4×C79—42no—
165D4×C79—42no—
166D4×C79—81no—
167C2×C4×C79—22no—
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—41no—
175C2×C4×C79—41no—
176C2×C4×C79—41no—
177C2×C4×C79—41no—
178C2×C4×C79—81no—
179C2×C4×C79—81no—
180C2×C4×C79C2×C22⋊C411yescharacteristic
181C2×C4×C79C4×D411yescharacteristic
182C2×C4×C79#12111yescharacteristic
183C2×C4×C79#12311yescharacteristic
184C2×Dic79—22no—
185C2×Dic79—22no—
186C2×Dic79—41no—
187C2×Dic79—41no—
188C2×Dic79—41no—
189C2×Dic79—41no—
190C2×Dic79—41no—
191C2×Dic79—41no—
192C2×Dic79—41no—
193C2×Dic79—41no—
194C2×Dic79—41no—
195C2×Dic79—41no—
196C2×Dic79—84no—
197C2×Dic79—82no—
198C2×Dic79—82no—
199C2×Dic79—81no—
200C2×Dic79—81no—
201C2×Dic79—81no—
202C8×C79—42no—
203C79⋊C8—81no—
204C22×C79—21no—
205C22×C79—21no—
206C22×C79—21no—
207C22×C79—21no—
208C22×C79—21no—
209C22×C79—21no—
210C22×C79—41no—
211C22×C79—41no—
212C22×C79—41no—
213C22×C79—41no—
214C22×C79—42no—
215C22×C79—42no—
216C22×C79—42no—
217C22×C79—81no—
218C22×C79—81no—
219C22×C79—81no—
220C22×C79—81no—
221C22×C79#38611yescharacteristic
222C22×C79C4×SD1611yescharacteristic
223C22×C79#44011yescharacteristic
224C22×C79#45011yescharacteristic
225C22×C79#45211yescharacteristic
226C22×C79#46111yescharacteristic
227C22×C79#46311yescharacteristic
228C4×C79—21no—
229C4×C79—21no—
230C4×C79—21no—
231C4×C79—21no—
232C4×C79—42no—
233C4×C79—81no—
234C4×C79—81no—
235Dic79—42no—
236Dic79—42no—
237Dic79—81no—
238Dic79—81no—
239Dic79—161no—
240#9275—791nomaximal, Sylow
241C2×C79—21no—
242C2×C79—21no—
243C2×C79—21no—
244C2×C79—21no—
245C2×C79—21no—
246C2×C79—21no—
247C2×C79—82no—
248C2×C79#187711yescharacteristic
249C2×C79#190611yescharacteristic
250C2×C79#191311yescharacteristic
251#3187D7911yescharacteristic
252#2364—791no—
253#1996—791no—
254#1519—791no—
255C2×#328—791no—
256#1294—791no—
257#1282—791no—
258C79#927511yescharacteristic, complemented, Sylow
259C2×C22⋊Q8C2×D7911yescharacteristic
260C2×#121C2×D7911yescharacteristic
261C2×C4×D4—1581no—
262#483—21no—
263#481Dic7912yescomplemented
264#465—22no—
265#450—21no—
266#448Dic7912yescomplemented
267C2×C22⋊C8—791no—
268C2×C22⋊C8C2×D7911yescharacteristic
269#388—1581no—
270#387—1581no—
271#386—1581no—
272#386—1581no—
273#380—791no—
274C4×C22⋊C4—791no—
275C4×C22⋊C4—1581no—
276C2×#95—791no—
277#342—1581no—
278#342—1581no—
279#328—1581no—
280#328—1581no—
281#327—1582no—
282C22×Q8C2×Dic7911yescharacteristic, complemented
283C22×D4—21no—
284C22×D4C22⋊D7911yescharacteristic
285C23×C4—1581no—
286C23×C4—1581no—
287C23×C4C22×D7911yescharacteristic
288C2×SD16—22no—
289C22×C8—21no—
290C22×C8—1581no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295#121—21no—
296#121—21no—
297#121C2×Dic7912yes—
298#121C22⋊D7912yes—
299C4×D4—3162no—
300C4×D4—3162no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—1581no—
303C2×C4⋊C4—1581no—
304C2×C4⋊C4C2×Dic7911yescharacteristic, complemented
305C2×C4⋊C4C22⋊D7911yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—791no—
309C2×C22⋊C4—791no—
310C2×C22⋊C4—1581no—
311C2×C22⋊C4—1581no—
312C2×C22⋊C4—1581no—
313C2×C22⋊C4—1581no—
314C2×C42—1581no—
315C2×C42—1581no—
316#106—22no—
317#103—22no—
318#102—22no—
319C22⋊C8—21no—
320C22⋊C8—1581no—
321C22⋊C8—1581no—
322C22⋊C8C2×Dic7912yescomplemented
323#95—791no—
324#95—791no—
325#95—1581no—
326#95—1581no—
327#95—1581no—
328#95—1581no—
329#95—1581no—
330C24—21no—
331C24D4×D7911yescharacteristic
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—81no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—81no—
343C2×D4C22⋊Dic7912yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—81no—
348C22×C4—3161no—
349C22×C4—3161no—
350C22×C4—3162no—
351C22×C4—3162no—
352C22×C4—3162no—
353C22×C4—3161no—
354C22×C4—6321no—
355C22×C4—6321no—
356C22×C4—1581no—
357C22×C4—1581no—
358C22×C4—1581no—
359C22×C4—1581no—
360C22×C4C22×Dic7911yescharacteristic
361C22×C4D4×D7911yescharacteristic
362C22×C4C2×C22⋊D7911yescharacteristic
363C22×C4C2×C22⋊D7911yescharacteristic
364C22×C4#1192962311yescharacteristic
365C22×C4#1192962311yescharacteristic
366SD16—82no—
367C2×C8—22no—
368C2×C8—81no—
369C2×C8—3161no—
370C2×C8—6321no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—3162no—
377C4⋊C4—6321no—
378C4⋊C4C22⋊Dic7912yes—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—3162no—
384C22⋊C4—3162no—
385C22⋊C4—3161no—
386C22⋊C4—3161no—
387C22⋊C4—792no—
388C22⋊C4—6321no—
389C22⋊C4—1581no—
390C22⋊C4—1581no—
391C22⋊C4—1581no—
392C42—3162no—
393C42—6321no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23D4×Dic7911yescharacteristic
404C23#1457912911yescharacteristic
405C23#1457913311yescharacteristic
406C23SD16×D7911yescharacteristic
407C23#1457918211yescharacteristic
408C23#1457918811yescharacteristic
409C23#1457919011yescharacteristic
410Q8—42no—
411Q8—81no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—81no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—3161no—
432C2×C4—3161no—
433C2×C4—3161no—
434C2×C4—3161no—
435C2×C4—3161no—
436C2×C4—3161no—
437C2×C4—3161no—
438C2×C4—3161no—
439C2×C4—3161no—
440C2×C4—3161no—
441C2×C4—6324no—
442C2×C4—6322no—
443C2×C4—6322no—
444C2×C4—6321no—
445C2×C4—6321no—
446C2×C4—6321no—
447C2×C4—1582no—
448C2×C4—1582no—
449C2×C4C2×C22⋊Dic7911yescharacteristic
450C2×C4D4×Dic7911yescharacteristic
451C2×C4#1457913311yescharacteristic
452C2×C4#1457914211yescharacteristic
453C8—42no—
454C8—6321no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#1909844311yescharacteristic
473C22SD16×Dic7911yescharacteristic
474C22#1909866411yescharacteristic
475C22#1909871211yescharacteristic
476C22#1909872211yescharacteristic
477C22#1909876611yescharacteristic
478C22#1909878011yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—3162no—
487C4—3162no—
488C4—6321no—
489C4—6321no—
490C4—12641no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—82no—
498C2#2376373611yescharacteristic
499C2#2376390311yescharacteristic
500C2#2376394311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#3187C2
socleC22×C79#386
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenter#3187D79
hypocenterC79#9275
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #23764386C2semidirect
4. C2×#19098300C2semidirect
5. #23761280C2semidirect
6. #23761246C2semidirect
26. #481×C79C4semidirect
29. #448×C79C4semidirect
55. C22×Q8×C79C2×C4semidirect
70. C2×C4⋊C4×C79C2×C4semidirect
78. C22⋊C8×C79C2×C4semidirect
103. C22⋊Dic79SD16semidirect
258. C79#9275semidirect
263. #481Dic79semidirect
266. #448Dic79semidirect
282. C22×Q8C2×Dic79semidirect
304. C2×C4⋊C4C2×Dic79semidirect
322. C22⋊C8C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237664522
#237643862
#237619682
C2×#190983002
#237612802
#237612462
C79×#31872
#927579