Finite Group Databasebeta

← #34184737

Subgroups — #34184737

Order 20224.

750 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098702C211yesmaximal, characteristic
2#23764236C211yesmaximal, characteristic
3#23763335C211yesmaximal, characteristic, complemented
4#23761967C211yesmaximal, characteristic
5#23761267C211yesmaximal, characteristic, complemented
6#23761248C211yesmaximal, characteristic, complemented
7Q8×C22⋊C4×C79C211yesmaximal, characteristic, complemented
8C22×#11929611C412yescomplemented
9C2×#14579135C2211yescharacteristic
10#19098702—21no—
11#19098702—21no—
12#19098702—22no—
13C2×#14579085—22no—
14C2×#14579085C412yescomplemented
15C2×#14579075—21no—
16C2×#14579075—21no—
17C2×C22⋊(C79⋊C8)C2211yescharacteristic
18C2×C22⋊(C79⋊C8)C2211yescharacteristic
19#19098423C2211yescharacteristic
20#19098296—22no—
21C23×Q8×C79C2211yescharacteristic
22C2×C4×Q8×C79—21no—
23#391×C79—21no—
24#391×C79—22no—
25#380×C79—21no—
26#380×C79C2211yescharacteristic
27C4×C22⋊C4×C79—21no—
28C4×C22⋊C4×C79C2211yescharacteristic
29C22×D316C2×C411yescharacteristic
30C22×C22⋊D79—21no—
31C2×#11929611—22no—
32C2×#11929611—22no—
33C2×#11929611—44no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8C2×C412yes—
36#14579135—21no—
37#14579135—21no—
38#14579135—21no—
39#14579135—21no—
40C2×C4⋊Dic79—21no—
41C2×C4⋊Dic79C2×C411yescharacteristic
42C2×C22⋊Dic79—21no—
43C2×(C2×D79)⋊C4—21no—
44#14579085—22no—
45#14579085—22no—
46#14579085—44no—
47#14579075—42no—
48#14579075—42no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)—21no—
51C22⋊(C79⋊C8)—21no—
52C22⋊(C79⋊C8)—21no—
53C79⋊#95—21no—
54C22×Q8×C79—21no—
55C22×Q8×C79—21no—
56C22×Q8×C79—21no—
57C22×Q8×C79—44no—
58C22×Q8×C79C2×C412yes—
59C22×Q8×C79D411yescharacteristic
60C22×Q8×C79D411yescharacteristic
61C22×Q8×C79D412yes—
62C23×C4×C79—21no—
63C23×C4×C79C2311yescharacteristic
64C4×Q8×C79—44no—
65C2×C4⋊C4×C79—22no—
66C2×C4⋊C4×C79—21no—
67C2×C22⋊C4×C79—21no—
68C2×C22⋊C4×C79D411yescharacteristic
69C2×C22⋊C4×C79D411yescharacteristic
70C2×C42×C79—22no—
71C2×C42×C79—21no—
72#95×C79—22no—
73#95×C79—22no—
74#95×C79—21no—
75#95×C79D412yes—
76C23×D79—21no—
77C2×D316—21no—
78C2×D316—21no—
79C2×D316—42no—
80C2×C22⋊D79—41no—
81C2×C22⋊D79—41no—
82C2×C22⋊D79—41no—
83C2×C22⋊D79—41no—
84C2×C22⋊D79—41no—
85C2×C22⋊D79—41no—
86C22×Dic79—21no—
87C22×Dic79—41no—
88#11929611—44no—
89#11929611—44no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—42no—
92C2×C79⋊C8—42no—
93C2×C79⋊C8—41no—
94C4⋊Dic79—21no—
95C4⋊Dic79—21no—
96C4⋊Dic79—42no—
97C22⋊Dic79—41no—
98C22⋊Dic79—41no—
99(C2×D79)⋊C4—41no—
100(C2×D79)⋊C4—41no—
101C24×C79C2×D411yescharacteristic
102C2×Q8×C79—24no—
103C2×Q8×C79—22no—
104C2×Q8×C79—22no—
105C2×Q8×C79—21no—
106C2×Q8×C79—21no—
107C2×Q8×C79—24no—
108C2×Q8×C79—24no—
109C2×Q8×C79—42no—
110C2×Q8×C79—42no—
111C2×Q8×C79—42no—
112C2×Q8×C79—44no—
113C2×Q8×C79—44no—
114C2×Q8×C79—44no—
115C2×Q8×C79C22⋊C412yes—
116C2×Q8×C79C22⋊C412yes—
117C22×C4×C79—21no—
118C22×C4×C79—22no—
119C22×C4×C79—21no—
120C22×C4×C79—21no—
121C22×C4×C79—22no—
122C22×C4×C79—21no—
123C22×C4×C79—22no—
124C22×C4×C79—42no—
125C22×C4×C79—44no—
126C22×C4×C79C22×C411yescharacteristic
127C22×C4×C79C2×D411yescharacteristic
128C22×C4×C79C2×D411yescharacteristic
129C22×C4×C79C2×D411yescharacteristic
130C22×C4×C79#4111yescharacteristic
131C22×C4×C79#4111yescharacteristic
132C4⋊C4×C79—44no—
133C4⋊C4×C79—42no—
134C22⋊C4×C79—21no—
135C22⋊C4×C79—21no—
136C22⋊C4×C79—21no—
137C22⋊C4×C79—41no—
138C22⋊C4×C79—41no—
139C22⋊C4×C79SD1612yescomplemented
140C42×C79—44no—
141C42×C79—42no—
142C23×C79—41no—
143C23×C79—41no—
144C23×C79C4×D411yescharacteristic
145C23×C79#12011yescharacteristic
146C23×C79#12111yescharacteristic
147C23×C79C2×SD1611yescharacteristic
148C23×C79#13511yescharacteristic
149C23×C79AGL(1,ℤ/8)11yescharacteristic
150C23×C79#13711yescharacteristic
151C22×D79—41no—
152C22×D79—41no—
153C22×D79—41no—
154C22×D79—81no—
155C22×D79—81no—
156Q8×C79—24no—
157Q8×C79—22no—
158Q8×C79—22no—
159Q8×C79—22no—
160Q8×C79—44no—
161Q8×C79—44no—
162Q8×C79—42no—
163Q8×C79#9914yescomplemented
164C2×C4×C79—24no—
165C2×C4×C79—22no—
166C2×C4×C79—22no—
167C2×C4×C79—22no—
168C2×C4×C79—22no—
169C2×C4×C79—22no—
170C2×C4×C79—22no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—41no—
174C2×C4×C79—44no—
175C2×C4×C79—44no—
176C2×C4×C79—41no—
177C2×C4×C79—41no—
178C2×C4×C79—42no—
179C2×C4×C79—41no—
180C2×C4×C79—42no—
181C2×C4×C79—41no—
182C2×C4×C79—42no—
183C2×C4×C79—41no—
184C2×C4×C79—42no—
185C2×C4×C79C2×C22⋊C411yescharacteristic
186C2×C4×C79C4×D411yescharacteristic
187C2×C4×C79#12111yescharacteristic
188C2×C4×C79#12311yescharacteristic
189C22⋊D79—81no—
190C22⋊D79—81no—
191C22⋊D79—81no—
192C22⋊D79—81no—
193C2×Dic79—41no—
194C2×Dic79—41no—
195C2×Dic79—41no—
196C2×Dic79—81no—
197C2×Dic79—81no—
198C2×Dic79—81no—
199D316—42no—
200D316—81no—
201C79⋊C8—42no—
202C79⋊C8—81no—
203C22×C79—21no—
204C22×C79—21no—
205C22×C79—21no—
206C22×C79—21no—
207C22×C79—21no—
208C22×C79—21no—
209C22×C79—41no—
210C22×C79—41no—
211C22×C79—41no—
212C22×C79—41no—
213C22×C79#38611yescharacteristic
214C22×C79C4×SD1611yescharacteristic
215C22×C79#44011yescharacteristic
216C22×C79#44811yescharacteristic
217C22×C79#44911yescharacteristic
218C22×C79#46011yescharacteristic
219C22×C79#46411yescharacteristic
220C2×D79—81no—
221C2×D79—81no—
222C2×D79—81no—
223C2×D79—161no—
224C2×D79—161no—
225C4×C79—24no—
226C4×C79—21no—
227C4×C79—21no—
228C4×C79—21no—
229C4×C79—42no—
230C4×C79—42no—
231C4×C79—81no—
232C4×C79—82no—
233C4×C79C2×#9912yes—
234Dic79—81no—
235Dic79—161no—
236#9262—791nomaximal, Sylow
237C2×C79—21no—
238C2×C79—21no—
239C2×C79—21no—
240C2×C79—21no—
241C2×C79—21no—
242C2×C79—21no—
243C2×C79#188011yescharacteristic
244C2×C79#189811yescharacteristic
245C2×C79#190511yescharacteristic
246D79—161no—
247C2×#448—791no—
248Q8×C22⋊C4D7911yescharacteristic, complemented
249#1964—791no—
250#1787—791no—
251#1519—791no—
252#1289—791no—
253#1282—791no—
254C79#926211yescharacteristic, complemented, Sylow
255C23×Q8C2×D7911yescharacteristic
256C22×SD16—792no—
257C2×#121—791no—
258C2×C4×Q8—21no—
259#448—1581no—
260#448—1581no—
261#448—1582no—
262C2×#103—792no—
263C2×#103—1582no—
264C2×#99—1581no—
265C2×#99—1581no—
266C2×C22⋊C8—791no—
267C2×C22⋊C8—791no—
268#391—21no—
269#391—22no—
270#380—21no—
271#380—791no—
272#380C2×D7911yescharacteristic
273C4×C22⋊C4—21no—
274C4×C22⋊C4C2×D7911yescharacteristic
275#327—1582no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—44no—
280C22×Q8C4×D7912yes—
281C22×Q8D31612yes—
282C22×Q8C22⋊D7911yescharacteristic
283C22×Q8C22⋊D7911yescharacteristic
284C22×D4—791no—
285C22×D4—1581no—
286C23×C4—21no—
287C23×C4C22×D7911yescharacteristic
288C2×SD16—3164no—
289C2×SD16—1582no—
290C2×SD16—1582no—
291C22×C8—792no—
292C22×C8—1581no—
293#121—1581no—
294#121—1581no—
295#121—1581no—
296#121—1581no—
297C4×Q8—44no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—791no—
301C2×C4⋊C4—1581no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—1581no—
304C2×C22⋊C4—1581no—
305C2×C22⋊C4C22⋊D7911yescharacteristic
306C2×C22⋊C4C22⋊D7911yescharacteristic
307C2×C42—22no—
308C2×C42—21no—
309#103—3164no—
310#103—1582no—
311#103—1582no—
312#99—3162no—
313#99—3162no—
314C22⋊C8—1581no—
315C22⋊C8—1581no—
316C22⋊C8—1581no—
317C22⋊C8—1581no—
318#95—22no—
319#95—22no—
320#95—21no—
321#95—1581no—
322#95C22⋊D7912yes—
323C24—1581no—
324C24C2×C22⋊D7911yescharacteristic
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8(C2×D79)⋊C412yes—
339C2×Q8(C2×D79)⋊C412yes—
340C2×D4—3162no—
341C2×D4—3161no—
342C2×D4—3161no—
343C2×D4—3161no—
344C2×D4—3161no—
345C2×D4—3161no—
346C2×D4—3161no—
347C2×D4—1581no—
348C2×D4—1581no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—3161no—
359C22×C4—1581no—
360C22×C4C2×C4×D7911yescharacteristic
361C22×C4C2×C22⋊D7911yescharacteristic
362C22×C4C2×C22⋊D7911yescharacteristic
363C22×C4C2×D31611yescharacteristic
364C22×C4#1192962411yescharacteristic
365C22×C4#1192962411yescharacteristic
366SD16—3164no—
367SD16—3164no—
368C2×C8—3162no—
369C2×C8—3162no—
370C2×C8—3161no—
371C2×C8—1582no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—3162no—
375C4⋊C4—1581no—
376C4⋊C4—1581no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—3161no—
383C22⋊C4—3161no—
384C22⋊C4—3161no—
385C22⋊C4—3161no—
386C22⋊C4#1192961112yescomplemented
387C42—44no—
388C42—42no—
389C23—41no—
390C23—41no—
391C23—3161no—
392C23—3161no—
393C23—3161no—
394C23—6321no—
395C23—6321no—
396C23C4×C22⋊D7911yescharacteristic
397C23#1457913011yescharacteristic
398C23#1457913511yescharacteristic
399C23C2×#1192961111yescharacteristic
400C23#1457918111yescharacteristic
401C23#1457918611yescharacteristic
402C23#1457919211yescharacteristic
403Q8—24no—
404Q8—22no—
405Q8—22no—
406Q8—22no—
407Q8—44no—
408Q8—44no—
409Q8—42no—
410Q8#1457907514yescomplemented
411D4—3162no—
412D4—6321no—
413D4—6321no—
414D4—6321no—
415D4—6321no—
416D4—6321no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—3161no—
439C2×C4—3161no—
440C2×C4—3161no—
441C2×C4—6321no—
442C2×C4—6321no—
443C2×C4—6321no—
444C2×C4C2×(C2×D79)⋊C411yescharacteristic
445C2×C4C4×C22⋊D7911yescharacteristic
446C2×C4#1457913511yescharacteristic
447C2×C4#1457914411yescharacteristic
448C8—3162no—
449C8—6321no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—6321no—
461C22—6321no—
462C22—6321no—
463C22—12641no—
464C22—12641no—
465C22#1909844211yescharacteristic
466C22C4×#1192961111yescharacteristic
467C22#1909866111yescharacteristic
468C22#1909870211yescharacteristic
469C22#1909870811yescharacteristic
470C22#1909876311yescharacteristic
471C22#1909878511yescharacteristic
472C4—21no—
473C4—24no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—82no—
480C4—6321no—
481C4—12641no—
482C4C2×#1457907512yes—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—12641no—
490C2#2376375511yescharacteristic
491C2#2376385311yescharacteristic
492C2#2376389411yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingQ8×C22⋊C4×C79C2
socleC22×C79#386
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenterQ8×C22⋊C4D79
hypocenterC79#9262
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #23763335C2semidirect
5. #23761267C2semidirect
6. #23761248C2semidirect
7. Q8×C22⋊C4×C79C2semidirect
8. C22×#11929611C4semidirect
14. C2×#14579085C4semidirect
139. C22⋊C4×C79SD16semidirect
163. Q8×C79#99semidirect
248. Q8×C22⋊C4D79semidirect
254. C79#9262semidirect
386. C22⋊C4#11929611semidirect
410. Q8#14579075semidirect

Maximal subgroups

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

subgroupindex
C2×#190987022
#237642362
#237633352
#237619672
#237612672
#237612482
Q8×C22⋊C4×C792
#926279