Finite Group Databasebeta

← #34184849

Subgroups — #34184849

Order 20224.

564 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
1#23774153C211yesmaximal, characteristic
2#23764236C211yesmaximal, characteristic
3#23761972C211yesmaximal, characteristic, complemented
4C2×#19098296C211yesmaximal, characteristic
5#23761279C211yesmaximal, characteristic, complemented
6#23761248C211yesmaximal, characteristic, complemented
7C79×#2368C211yesmaximal, characteristic, complemented
8C2×#14579135C2211yescharacteristic
9#19098891C412yescomplemented
10#19098881—21no—
11#19098794—22no—
12#19098713C412yescomplemented
13#19098702—21no—
14C2×#14579075—21no—
15C2×#14579075—21no—
16C2×C22⋊(C79⋊C8)C2211yescharacteristic
17C2×C22⋊(C79⋊C8)C2211yescharacteristic
18#19098423C2211yescharacteristic
19#19098300—22no—
20#19098296—21no—
21#19098296—21no—
22C2×C22⋊Q8×C79C2211yescharacteristic
23#391×C79—21no—
24#387×C79—21no—
25#387×C79—21no—
26#385×C79—21no—
27C4×C4⋊C4×C79—21no—
28C4×C22⋊C4×C79—21no—
29C4×C22⋊C4×C79C2211yescharacteristic
30C2×#95×C79C2211yescharacteristic
31C22×D316C2×C411yescharacteristic
32C22×C22⋊D79—21no—
33C2×#11929611—22no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
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#14579092—22no—
45#14579085—22no—
46#14579081—22no—
47#14579075—42no—
48#14579075—42no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)—21no—
51C22⋊(C79⋊C8)—21no—
52C22⋊(C79⋊C8)C2×C412yes—
53C79⋊#95—21no—
54C22×Q8×C79D411yescharacteristic
55C23×C4×C79—21no—
56C23×C4×C79C2311yescharacteristic
57C22⋊Q8×C79—21no—
58C22⋊Q8×C79—21no—
59C22⋊Q8×C79C2×C412yes—
60C22⋊Q8×C79D412yes—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79D411yescharacteristic
65C2×C22⋊C4×C79—21no—
66C2×C22⋊C4×C79D411yescharacteristic
67C2×C22⋊C4×C79D411yescharacteristic
68C2×C42×C79—21no—
69C2×C42×C79—21no—
70C2×C42×C79—21no—
71#95×C79—21no—
72#95×C79—21no—
73#95×C79—21no—
74#95×C79—21no—
75#95×C79—21no—
76#95×C79—21no—
77#95×C79D411yescharacteristic
78#95×C79D411yescharacteristic
79C23×D79—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×D316—81no—
83C2×C22⋊D79—41no—
84C2×C22⋊D79—41no—
85C2×C22⋊D79—41no—
86C2×C22⋊D79—41no—
87C2×C22⋊D79—41no—
88C2×C22⋊D79—41no—
89C22×Dic79—21no—
90C22×Dic79—41no—
91#11929611—82no—
92C2×C79⋊C8—22no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—81no—
95C2×C79⋊C8—81no—
96C4⋊Dic79—21no—
97C4⋊Dic79—21no—
98C4⋊Dic79—81no—
99C22⋊Dic79—41no—
100C22⋊Dic79—41no—
101(C2×D79)⋊C4—41no—
102(C2×D79)⋊C4—41no—
103C24×C79C2×D411yescharacteristic
104C2×Q8×C79—21no—
105C2×Q8×C79—81no—
106C2×Q8×C79C22⋊C412yes—
107C22×C4×C79—21no—
108C22×C4×C79—21no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—21no—
113C22×C4×C79—42no—
114C22×C4×C79—41no—
115C22×C4×C79—41no—
116C22×C4×C79—81no—
117C22×C4×C79—81no—
118C22×C4×C79C22×C411yescharacteristic
119C22×C4×C79C2×D411yescharacteristic
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79#4111yescharacteristic
123C22×C4×C79#4111yescharacteristic
124C4⋊C4×C79—22no—
125C4⋊C4×C79—22no—
126C4⋊C4×C79—21no—
127C4⋊C4×C79—42no—
128C4⋊C4×C79—81no—
129C4⋊C4×C79C22⋊C412yes—
130C22⋊C4×C79—22no—
131C22⋊C4×C79—22no—
132C22⋊C4×C79—21no—
133C22⋊C4×C79—21no—
134C22⋊C4×C79—21no—
135C22⋊C4×C79SD1612yescomplemented
136C42×C79—42no—
137C42×C79—81no—
138C42×C79—81no—
139C23×C79—41no—
140C23×C79—41no—
141C23×C79C4×D411yescharacteristic
142C23×C79#12011yescharacteristic
143C23×C79#12111yescharacteristic
144C23×C79C2×SD1611yescharacteristic
145C23×C79#13511yescharacteristic
146C23×C79AGL(1,ℤ/8)11yescharacteristic
147C23×C79#13711yescharacteristic
148C22×D79—41no—
149C22×D79—41no—
150C22×D79—41no—
151C22×D79—81no—
152C22×D79—81no—
153Q8×C79—42no—
154Q8×C79—81no—
155C2×C4×C79—22no—
156C2×C4×C79—22no—
157C2×C4×C79—22no—
158C2×C4×C79—22no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
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—41no—
179C2×C4×C79—82no—
180C2×C4×C79—82no—
181C2×C4×C79C2×C22⋊C411yescharacteristic
182C2×C4×C79C4×D411yescharacteristic
183C2×C4×C79#12111yescharacteristic
184C2×C4×C79#12311yescharacteristic
185C22⋊D79—81no—
186C22⋊D79—81no—
187C22⋊D79—81no—
188C22⋊D79—81no—
189C2×Dic79—41no—
190C2×Dic79—41no—
191C2×Dic79—41no—
192C2×Dic79—81no—
193C2×Dic79—81no—
194C2×Dic79—81no—
195D316—42no—
196D316—81no—
197C79⋊C8—42no—
198C79⋊C8—81no—
199C22×C79—21no—
200C22×C79—21no—
201C22×C79—21no—
202C22×C79—21no—
203C22×C79—21no—
204C22×C79—21no—
205C22×C79—41no—
206C22×C79—41no—
207C22×C79—41no—
208C22×C79—41no—
209C22×C79#38611yescharacteristic
210C22×C79C4×SD1611yescharacteristic
211C22×C79#44011yescharacteristic
212C22×C79#44811yescharacteristic
213C22×C79#44911yescharacteristic
214C22×C79#46011yescharacteristic
215C22×C79#46411yescharacteristic
216C2×D79—81no—
217C2×D79—81no—
218C2×D79—81no—
219C2×D79—161no—
220C2×D79—161no—
221C4×C79—21no—
222C4×C79—21no—
223C4×C79—21no—
224C4×C79—21no—
225C4×C79—42no—
226C4×C79—42no—
227C4×C79—42no—
228C4×C79—42no—
229C4×C79—81no—
230C4×C79—81no—
231Dic79—81no—
232Dic79—161no—
233#9278—791nomaximal, Sylow
234C2×C79—21no—
235C2×C79—21no—
236C2×C79—21no—
237C2×C79—21no—
238C2×C79—21no—
239C2×C79—21no—
240C2×C79#188011yescharacteristic
241C2×C79#190611yescharacteristic
242C2×C79#191311yescharacteristic
243D79—161no—
244#3187—791no—
245#2368D7911yescharacteristic, complemented
246#1964—791no—
247#1520—791no—
248C2×#327—791no—
249#1294—791no—
250#1282—791no—
251C79#927811yescharacteristic, complemented, Sylow
252C2×C22⋊Q8C2×D7911yescharacteristic
253C2×#121—791no—
254#483—792no—
255#481—1581no—
256#465—1582no—
257#450—792no—
258#448—1581no—
259C2×#99—1581no—
260C2×#99—1581no—
261C2×C22⋊C8—791no—
262C2×C22⋊C8—791no—
263#391—21no—
264#387—21no—
265#387—21no—
266#385—21no—
267#380—791no—
268C4×C4⋊C4—21no—
269C4×C22⋊C4—21no—
270C4×C22⋊C4C2×D7911yescharacteristic
271C2×#95C2×D7911yescharacteristic
272#328—1582no—
273#327—1581no—
274#327—1581no—
275C22×Q8C22⋊D7911yescharacteristic
276C22×D4—791no—
277C22×D4—1581no—
278C23×C4—21no—
279C23×C4C22×D7911yescharacteristic
280C2×SD16—1582no—
281C22×C8—1581no—
282C22×C8—1581no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8C4×D7912yes—
286C22⋊Q8D31612yes—
287#121—1581no—
288#121—1581no—
289#121—1581no—
290#121—1581no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—791no—
295C2×C4⋊C4—1581no—
296C2×C4⋊C4C22⋊D7911yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—1581no—
299C2×C22⋊C4—1581no—
300C2×C22⋊C4C22⋊D7911yescharacteristic
301C2×C22⋊C4C22⋊D7911yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305#106—1582no—
306#103—1582no—
307#102—1582no—
308#99—3162no—
309#99—3162no—
310C22⋊C8—792no—
311C22⋊C8—1581no—
312C22⋊C8—1581no—
313C22⋊C8—1581no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—1581no—
321#95C22⋊D7911yescharacteristic
322#95C22⋊D7911yescharacteristic
323C24—1581no—
324C24C2×C22⋊D7911yescharacteristic
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8(C2×D79)⋊C412yes—
328C2×D4—3161no—
329C2×D4—3161no—
330C2×D4—3161no—
331C2×D4—3161no—
332C2×D4—3161no—
333C2×D4—3161no—
334C2×D4—6321no—
335C2×D4—1581no—
336C2×D4—1581no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—3161no—
349C22×C4—1581no—
350C22×C4C2×C4×D7911yescharacteristic
351C22×C4C2×C22⋊D7911yescharacteristic
352C22×C4C2×C22⋊D7911yescharacteristic
353C22×C4C2×D31611yescharacteristic
354C22×C4#1192962411yescharacteristic
355C22×C4#1192962411yescharacteristic
356SD16—6322no—
357C2×C8—3161no—
358C2×C8—6321no—
359C2×C8—6321no—
360C2×C8—1582no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—81no—
366C4⋊C4—6321no—
367C4⋊C4—1581no—
368C4⋊C4—1581no—
369C4⋊C4(C2×D79)⋊C412yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—3161no—
376C22⋊C4—3161no—
377C22⋊C4—3161no—
378C22⋊C4—3161no—
379C22⋊C4#1192961112yescomplemented
380C42—42no—
381C42—81no—
382C42—81no—
383C23—41no—
384C23—41no—
385C23—3161no—
386C23—3161no—
387C23—3161no—
388C23—6321no—
389C23—6321no—
390C23C4×C22⋊D7911yescharacteristic
391C23#1457913011yescharacteristic
392C23#1457913511yescharacteristic
393C23C2×#1192961111yescharacteristic
394C23#1457918111yescharacteristic
395C23#1457918611yescharacteristic
396C23#1457919211yescharacteristic
397Q8—42no—
398Q8—81no—
399D4—3162no—
400D4—6321no—
401D4—6321no—
402D4—6321no—
403D4—6321no—
404D4—6321no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
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—82no—
430C2×C4—82no—
431C2×C4—3161no—
432C2×C4—3161no—
433C2×C4—3161no—
434C2×C4—6321no—
435C2×C4—6321no—
436C2×C4—6321no—
437C2×C4C2×(C2×D79)⋊C411yescharacteristic
438C2×C4C4×C22⋊D7911yescharacteristic
439C2×C4#1457913511yescharacteristic
440C2×C4#1457914411yescharacteristic
441C8—3162no—
442C8—6321no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—6321no—
454C22—6321no—
455C22—6321no—
456C22—12641no—
457C22—12641no—
458C22#1909844211yescharacteristic
459C22C4×#1192961111yescharacteristic
460C22#1909866111yescharacteristic
461C22#1909870211yescharacteristic
462C22#1909870811yescharacteristic
463C22#1909876311yescharacteristic
464C22#1909878511yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—6321no—
476C4—12641no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—12641no—
484C2#2376375511yescharacteristic
485C2#2376390111yescharacteristic
486C2#2376394211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2368C2
socleC22×C79#386
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#2368D79
hypocenterC79#9278
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2368

Complements

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

NH ≅ G/Ntype
3. #23761972C2semidirect
5. #23761279C2semidirect
6. #23761248C2semidirect
7. C79×#2368C2semidirect
9. #19098891C4semidirect
12. #19098713C4semidirect
135. C22⋊C4×C79SD16semidirect
245. #2368D79semidirect
251. C79#9278semidirect
379. C22⋊C4#11929611semidirect

Maximal subgroups

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

subgroupindex
#237741532
#237642362
#237619722
C2×#190982962
#237612792
#237612482
C79×#23682
#927879