Finite Group Databasebeta

← #34190589

Subgroups — #34190589

Order 20224.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098765C211yesmaximal, characteristic
2#23768318C211yesmaximal, characteristic
3#23764362C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
5C2×#19098290C211yesmaximal, characteristic
6#23761284C211yesmaximal, characteristic, complemented
7C2×#475×C79C211yesmaximal, characteristic, complemented
8C22×#11929611—21no—
9C2×#14579137—21no—
10C2×C4×D316C2211yescharacteristic
11#19098765—22no—
12#19098765—22no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579085—21no—
15C2×C4×C79⋊C8C2211yescharacteristic
16#19098485—22no—
17#19098465—21no—
18#19098461—21no—
19#19098433C2211yescharacteristic
20#19098420—21no—
21#19098325—22no—
22Q8⋊(C79⋊C8)—22no—
23#19098290—22no—
24C2×C4⋊Q8×C79C2211yescharacteristic
25C2×C4×Q8×C79C2211yescharacteristic
26#475×C79—22no—
27#475×C79—22no—
28C2×#106×C79—21no—
29C2×C4⋊C8×C79C2211yescharacteristic
30C2×#103×C79—21no—
31C22×D316D411yescharacteristic
32C22×C4×D79—21no—
33C2×#11929611—24no—
34C2×#11929611—42no—
35C2×#11929611—42no—
36C22×C79⋊C8—21no—
37C22×C79⋊C8—21no—
38#14579137—42no—
39#14579137—42no—
40C4×D316—21no—
41C4×D316—21no—
42C4×D316—21no—
43C4×D316—21no—
44C2×C4⋊Dic79—21no—
45C2×C4⋊Dic79—21no—
46C2×C4⋊Dic79D411yescharacteristic
47C2×C4⋊Dic79D412yes—
48C2×Dic79⋊C4—22no—
49C2×(C2×D79)⋊C4—22no—
50C2×(C2×D79)⋊C4—21no—
51C2×(C2×D79)⋊C4—22no—
52C4⋊(C79⋊C8)—21no—
53C4⋊(C79⋊C8)D412yes—
54#14579085—24no—
55C4×C79⋊C8—21no—
56C4×C79⋊C8—21no—
57C79⋊#95—22no—
58C22×Q8×C79—21no—
59C22×Q8×C79D411yescharacteristic
60C22×C8×C79—21no—
61C4⋊Q8×C79—21no—
62C4⋊Q8×C79—21no—
63C4⋊Q8×C79—21no—
64C4⋊Q8×C79D412yes—
65C4×Q8×C79—21no—
66C4×Q8×C79—21no—
67C4×Q8×C79—21no—
68C4×Q8×C79—21no—
69C2×C4⋊C4×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79—21no—
72C2×C4⋊C4×C79D411yescharacteristic
73C2×C42×C79—21no—
74C2×C42×C79C2311yescharacteristic
75#106×C79—24no—
76C4⋊C8×C79—21no—
77C4⋊C8×C79D412yes—
78#103×C79—24no—
79C23×D79—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×D316—21no—
83C2×D316—21no—
84C2×D316—21no—
85C2×D316SD1612yes—
86C22×Dic79—21no—
87C22×Dic79—22no—
88C2×C4×D79—41no—
89C2×C4×D79—41no—
90C2×C4×D79—41no—
91C2×C4×D79—41no—
92C2×C4×D79—41no—
93C2×C4×D79—41no—
94#11929611—48no—
95C2×C79⋊C8—22no—
96C2×C79⋊C8—22no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C2×C79⋊C8—41no—
101C4⋊Dic79—21no—
102C4⋊Dic79—21no—
103C4⋊Dic79—42no—
104C4⋊Dic79—41no—
105C4⋊Dic79—41no—
106C4⋊Dic79—41no—
107C4⋊Dic79—41no—
108Dic79⋊C4—44no—
109(C2×D79)⋊C4—44no—
110(C2×D79)⋊C4—41no—
111(C2×D79)⋊C4—41no—
112(C2×D79)⋊C4—82no—
113C2×Q8×C79—21no—
114C2×Q8×C79—21no—
115C2×Q8×C79—22no—
116C2×Q8×C79—21no—
117C2×Q8×C79—21no—
118C2×Q8×C79—21no—
119C2×Q8×C79—41no—
120C2×Q8×C79—41no—
121C2×Q8×C79—41no—
122C2×Q8×C79—41no—
123C2×Q8×C79—41no—
124C2×Q8×C79SD1612yescomplemented
125C22×C4×C79—21no—
126C22×C4×C79—21no—
127C22×C4×C79—41no—
128C22×C4×C79C2×D411yescharacteristic
129C22×C4×C79C2×D411yescharacteristic
130C22×C4×C79C2×D411yescharacteristic
131C2×C8×C79—22no—
132C2×C8×C79—41no—
133C2×C8×C79—41no—
134C4⋊C4×C79—22no—
135C4⋊C4×C79—21no—
136C4⋊C4×C79—22no—
137C4⋊C4×C79—21no—
138C4⋊C4×C79—41no—
139C4⋊C4×C79—41no—
140C4⋊C4×C79—41no—
141C4⋊C4×C79—41no—
142C42×C79—41no—
143C42×C79—41no—
144C42×C79C2×D411yescharacteristic
145C42×C79C2×D411yescharacteristic
146C42×C79C2×D411yescharacteristic
147C42×C79#4111yescharacteristic
148C23×C79#12011yescharacteristic
149C22×D79—22no—
150C22×D79—41no—
151C22×D79—41no—
152C22×D79—41no—
153C22×D79—41no—
154C22×D79—41no—
155C22×D79—41no—
156Q8×C79—22no—
157Q8×C79—22no—
158Q8×C79—41no—
159Q8×C79—41no—
160Q8×C79—42no—
161Q8×C79—42no—
162Q8×C79—81no—
163Q8×C79—81no—
164C2×C4×C79—21no—
165C2×C4×C79—21no—
166C2×C4×C79—22no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—42no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—81no—
174C2×C4×C79—81no—
175C2×C4×C79#12011yescharacteristic
176C2×C4×C79#12011yescharacteristic
177C2×C4×C79#12111yescharacteristic
178C2×C4×C79#12111yescharacteristic
179C2×C4×C79#12111yescharacteristic
180C2×C4×C79#12711yescharacteristic
181C2×C4×C79C2×SD1611yescharacteristic
182C2×C4×C79C2×SD1611yescharacteristic
183C2×C4×C79#13511yescharacteristic
184C2×C4×C79#13511yescharacteristic
185C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C79#13711yescharacteristic
188C2×C4×C79#13711yescharacteristic
189C2×Dic79—41no—
190C2×Dic79—42no—
191C2×Dic79—42no—
192C2×Dic79—41no—
193C2×Dic79—41no—
194C2×Dic79—42no—
195D316—22no—
196D316—22no—
197D316—41no—
198D316—41no—
199C4×D79—81no—
200C4×D79—81no—
201C4×D79—81no—
202C4×D79—81no—
203C8×C79—42no—
204C79⋊C8—42no—
205C79⋊C8—42no—
206C22×C79#39211yescharacteristic
207C22×C79#44811yescharacteristic
208C22×C79#44911yescharacteristic
209C22×C79#45011yescharacteristic
210C22×C79#45211yescharacteristic
211C22×C79#45511yescharacteristic
212C22×C79#45611yescharacteristic
213C2×D79—42no—
214C2×D79—42no—
215C2×D79—42no—
216C2×D79—81no—
217C2×D79—81no—
218C2×D79—81no—
219C2×D79—81no—
220C4×C79—21no—
221C4×C79—21no—
222C4×C79—42no—
223C4×C79—82no—
224C4×C79—81no—
225C4×C79#44811yescharacteristic
226C4×C79#44911yescharacteristic
227C4×C79#45011yescharacteristic
228C4×C79#45211yescharacteristic
229C4×C79#46511yescharacteristic
230C4×C79#46511yescharacteristic
231C4×C79#46811yescharacteristic
232C4×C79#46811yescharacteristic
233Dic79—82no—
234Dic79—81no—
235#10232—791nomaximal, Sylow
236C2×C79#164211yescharacteristic
237C2×C79#164311yescharacteristic
238C2×C79#165011yescharacteristic
239C2×C79#165111yescharacteristic
240C2×C79#200011yescharacteristic
241C2×C79#200011yescharacteristic
242C2×C79#200711yescharacteristic
243D79—82no—
244C2×#475D7911yescharacteristic, complemented
245C2×#461—791no—
246#2538—791no—
247#1990—791no—
248C2×Q8⋊C8—791no—
249C2×D4⋊C8—791no—
250#1296—791no—
251C79#1023211yescharacteristic, complemented, Sylow
252C22×SD16—1581no—
253C2×C4⋊Q8C2×D7911yescharacteristic
254C2×C22⋊Q8—1581no—
255C2×C4×Q8C2×D7911yescharacteristic
256C2×C4×D4—791no—
257#475—22no—
258#475—22no—
259#461—1582no—
260#461—1582no—
261C2×#106—21no—
262C2×C4⋊C8—791no—
263C2×C4⋊C8C2×D7911yescharacteristic
264C2×#103—21no—
265C2×#103—1581no—
266C2×C4×C8—791no—
267#396—1582no—
268#390—1581no—
269#389—1581no—
270#384—791no—
271#380—1581no—
272#340—1582no—
273Q8⋊C8—1582no—
274D4⋊C8—1582no—
275C22×Q8—21no—
276C22×Q8C22⋊D7911yescharacteristic
277C22×D4—791no—
278C23×C4—1581no—
279C2×SD16—3162no—
280C2×SD16—3162no—
281C2×SD16—1584no—
282C22×C8—21no—
283C22×C8—1581no—
284C22×C8—1581no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8C22⋊D7912yes—
289C22⋊Q8—3162no—
290C22⋊Q8—3162no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×D4—1581no—
296C4×D4—1581no—
297C4×D4—1581no—
298C4×D4—1581no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—792no—
303C2×C4⋊C4—791no—
304C2×C4⋊C4—1582no—
305C2×C4⋊C4—1581no—
306C2×C4⋊C4—1581no—
307C2×C4⋊C4C22⋊D7911yescharacteristic
308C2×C22⋊C4—1582no—
309C2×C22⋊C4—1581no—
310C2×C22⋊C4—1582no—
311C2×C42—21no—
312C2×C42C22×D7911yescharacteristic
313#106—24no—
314C4⋊C8—21no—
315C4⋊C8—792no—
316C4⋊C8—1581no—
317C4⋊C8C22⋊D7912yes—
318#103—24no—
319#103—1584no—
320C4×C8—1581no—
321C4×C8—1581no—
322#95—1582no—
323C24—1581no—
324C2×Q8—21no—
325C2×Q8—22no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8#1192961112yescomplemented
336C2×D4—792no—
337C2×D4—1581no—
338C2×D4—1581no—
339C2×D4—1581no—
340C2×D4—1581no—
341C2×D4—1581no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4—3161no—
346C22×C4—3161no—
347C22×C4—3161no—
348C22×C4—3161no—
349C22×C4—3161no—
350C22×C4—3161no—
351C22×C4—1581no—
352C22×C4—1582no—
353C22×C4D4×D7911yescharacteristic
354C22×C4D4×D7911yescharacteristic
355C22×C4C2×C22⋊D7911yescharacteristic
356SD16—3168no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—3161no—
361C2×C8—3161no—
362C2×C8—3161no—
363C2×C8—3161no—
364C2×C8—1582no—
365C2×C8—1582no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—3162no—
375C4⋊C4—3164no—
376C4⋊C4—3161no—
377C4⋊C4—3161no—
378C4⋊C4—3161no—
379C4⋊C4—3161no—
380C4⋊C4—1581no—
381C4⋊C4—1581no—
382C22⋊C4—3164no—
383C22⋊C4—3161no—
384C22⋊C4—3161no—
385C22⋊C4—6322no—
386C42—41no—
387C42—41no—
388C42D4×D7911yescharacteristic
389C42C2×C22⋊D7911yescharacteristic
390C42C2×C22⋊D7911yescharacteristic
391C42#1192962311yescharacteristic
392C23—3161no—
393C23—3161no—
394C23—3161no—
395C23—3161no—
396C23—3161no—
397C23—3161no—
398C23—1582no—
399C23#1457912911yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—42no—
405Q8—42no—
406Q8—81no—
407Q8—81no—
408D4—3161no—
409D4—3161no—
410D4—1582no—
411D4—1582no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—3162no—
424C2×C4—3162no—
425C2×C4—3161no—
426C2×C4—3161no—
427C2×C4—3161no—
428C2×C4—3162no—
429C2×C4—6321no—
430C2×C4—6321no—
431C2×C4—6321no—
432C2×C4—6321no—
433C2×C4#1457912911yescharacteristic
434C2×C4#1457913011yescharacteristic
435C2×C4#1457913211yescharacteristic
436C2×C4#1457913311yescharacteristic
437C2×C4#1457913311yescharacteristic
438C2×C4#1457915711yescharacteristic
439C2×C4SD16×D7911yescharacteristic
440C2×C4C2×#1192961111yescharacteristic
441C2×C4#1457918111yescharacteristic
442C2×C4#1457918211yescharacteristic
443C2×C4#1457918611yescharacteristic
444C2×C4#1457918811yescharacteristic
445C2×C4#1457919011yescharacteristic
446C2×C4#1457919211yescharacteristic
447C8—42no—
448C8—3162no—
449C8—3162no—
450C22—3162no—
451C22—3162no—
452C22—3162no—
453C22—6321no—
454C22—6321no—
455C22—6321no—
456C22—6321no—
457C22#1909847111yescharacteristic
458C22#1909870011yescharacteristic
459C22#1909870711yescharacteristic
460C22#1909871311yescharacteristic
461C22#1909872311yescharacteristic
462C22#1909873911yescharacteristic
463C22#1909874411yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—82no—
468C4—81no—
469C4—6322no—
470C4—6321no—
471C4#1909870211yescharacteristic
472C4#1909870811yescharacteristic
473C4#1909871211yescharacteristic
474C4#1909872211yescharacteristic
475C4#1909878911yescharacteristic
476C4#1909879411yescharacteristic
477C4#1909880611yescharacteristic
478C4#1909881111yescharacteristic
479C2—6322no—
480C2#2376259911yescharacteristic
481C2#2376260611yescharacteristic
482C2#2376265511yescharacteristic
483C2#2376266211yescharacteristic
484C2#2376440011yescharacteristic
485C2#2376440111yescharacteristic
486C2#2376444511yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#475×C79C2
socleC23×C79#120
prime radicalC22×D316D4
hypercenterC2×#475D79
hypocenterC79#10232
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #23764362C2semidirect
4. C2×Q8⋊(C79⋊C8)C2semidirect
6. #23761284C2semidirect
7. C2×#475×C79C2semidirect
124. C2×Q8×C79SD16semidirect
244. C2×#475D79semidirect
251. C79#10232semidirect
335. C2×Q8#11929611semidirect

Maximal subgroups

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

subgroupindex
C2×#190987652
#237683182
#237643622
C2×Q8⋊(C79⋊C8)2
C2×#190982902
#237612842
C2×#475×C792
#1023279