Finite Group Databasebeta

← #34184828

Subgroups — #34184828

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#23774157C211yesmaximal, characteristic
2#23764385C211yesmaximal, characteristic, complemented
3#23761968C211yesmaximal, characteristic
4C2×#19098299C211yesmaximal, characteristic, complemented
5#23761279C211yesmaximal, characteristic, complemented
6#23761248C211yesmaximal, characteristic, complemented
7C79×#2364C211yesmaximal, characteristic
8C2×#14579140C2211yescharacteristic
9#19098893—21no—
10#19098879C412yescomplemented
11#19098790—22no—
12#19098714—21no—
13#19098701C412yescomplemented
14C2×C22⋊(C79⋊C8)C2211yescharacteristic
15C2×C22⋊(C79⋊C8)C2211yescharacteristic
16#19098423C2211yescharacteristic
17#19098329—21no—
18#19098329—21no—
19#19098299—21no—
20#19098299—21no—
21#19098297—22no—
22C2×#121×C79C2211yescharacteristic
23C2×C4×D4×C79—21no—
24#388×C79—21no—
25#387×C79—21no—
26#386×C79—21no—
27#386×C79—21no—
28C4×C22⋊C4×C79—21no—
29C4×C22⋊C4×C79C2211yescharacteristic
30C2×#95×C79C2211yescharacteristic
31C22×Dic158C2×C411yescharacteristic, complemented
32C2×#11929612—22no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8—21no—
35#14579140—21no—
36#14579140—21no—
37#14579140—21no—
38#14579140—21no—
39C2×C4⋊Dic79—21no—
40C2×C4⋊Dic79C2×C411yescharacteristic, complemented
41C2×Dic79⋊C4—21no—
42C2×C22⋊Dic79—21no—
43C2×C22⋊Dic79—21no—
44#14579092—22no—
45#14579084—22no—
46#14579082—22no—
47C22⋊(C79⋊C8)—21no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)C2×C412yescomplemented
51C79⋊#95—21no—
52C22×D4×C79—21no—
53C22×D4×C79D411yescharacteristic
54C23×C4×C79—21no—
55C23×C4×C79—21no—
56C23×C4×C79C2311yescharacteristic
57#121×C79—21no—
58#121×C79—21no—
59#121×C79C2×C412yes—
60#121×C79D412yes—
61C4×D4×C79—42no—
62C4×D4×C79—42no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79D411yescharacteristic
65C2×C22⋊C4×C79—21no—
66C2×C22⋊C4×C79—21no—
67C2×C22⋊C4×C79—21no—
68C2×C22⋊C4×C79—21no—
69C2×C22⋊C4×C79D411yescharacteristic
70C2×C22⋊C4×C79D411yescharacteristic
71C2×C42×C79—21no—
72C2×C42×C79—21no—
73#95×C79—21no—
74#95×C79—21no—
75#95×C79—21no—
76#95×C79—21no—
77#95×C79D411yescharacteristic
78#95×C79D411yescharacteristic
79C2×Dic158—21no—
80C2×Dic158—21no—
81C2×Dic158—81no—
82C22×Dic79—21no—
83C22×Dic79—21no—
84C22×Dic79—41no—
85#11929612—82no—
86C2×C79⋊C8—22no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—81no—
89C2×C79⋊C8—81no—
90C4⋊Dic79—21no—
91C4⋊Dic79—21no—
92C4⋊Dic79—81no—
93Dic79⋊C4—41no—
94Dic79⋊C4—41no—
95C22⋊Dic79—41no—
96C22⋊Dic79—41no—
97C22⋊Dic79—41no—
98C22⋊Dic79—41no—
99C24×C79—21no—
100C24×C79C2×D411yescharacteristic
101C2×D4×C79—22no—
102C2×D4×C79—22no—
103C2×D4×C79—21no—
104C2×D4×C79—41no—
105C2×D4×C79—41no—
106C2×D4×C79—41no—
107C2×D4×C79—41no—
108C2×D4×C79—81no—
109C2×D4×C79C22⋊C412yes—
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—21no—
113C22×C4×C79—21no—
114C22×C4×C79—21no—
115C22×C4×C79—42no—
116C22×C4×C79—42no—
117C22×C4×C79—42no—
118C22×C4×C79—41no—
119C22×C4×C79—41no—
120C22×C4×C79—81no—
121C22×C4×C79—81no—
122C22×C4×C79—81no—
123C22×C4×C79C22×C411yescharacteristic
124C22×C4×C79C2×D411yescharacteristic
125C22×C4×C79C2×D411yescharacteristic
126C22×C4×C79C2×D411yescharacteristic
127C22×C4×C79#4111yescharacteristic
128C22×C4×C79#4111yescharacteristic
129C4⋊C4×C79—21no—
130C4⋊C4×C79—42no—
131C4⋊C4×C79C22⋊C412yes—
132C22⋊C4×C79—21no—
133C22⋊C4×C79—22no—
134C22⋊C4×C79—22no—
135C22⋊C4×C79—21no—
136C22⋊C4×C79—21no—
137C22⋊C4×C79—42no—
138C22⋊C4×C79—42no—
139C22⋊C4×C79—81no—
140C22⋊C4×C79SD1612yescomplemented
141C42×C79—42no—
142C42×C79—81no—
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
159D4×C79—42no—
160D4×C79—42no—
161D4×C79—42no—
162D4×C79—42no—
163D4×C79—42no—
164D4×C79—81no—
165C2×C4×C79—22no—
166C2×C4×C79—22no—
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—41no—
179C2×C4×C79—41no—
180C2×C4×C79—41no—
181C2×C4×C79—41no—
182C2×C4×C79—41no—
183C2×C4×C79—41no—
184C2×C4×C79—41no—
185C2×C4×C79—41no—
186C2×C4×C79—84no—
187C2×C4×C79—82no—
188C2×C4×C79—82no—
189C2×C4×C79C2×C22⋊C411yescharacteristic
190C2×C4×C79C4×D411yescharacteristic
191C2×C4×C79#12111yescharacteristic
192C2×C4×C79#12311yescharacteristic
193C2×Dic79—41no—
194C2×Dic79—41no—
195C2×Dic79—81no—
196C2×Dic79—81no—
197C2×Dic79—81no—
198C2×Dic79—81no—
199C2×Dic79—81no—
200Dic158—42no—
201Dic158—81no—
202C79⋊C8—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—42no—
234C4×C79—42no—
235C4×C79—81no—
236C4×C79—81no—
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#3187—791no—
252#2364D7911yescharacteristic
253#1996—791no—
254#1519—791no—
255C2×#328—791no—
256#1294—791no—
257#1282—791no—
258C79#927511yescharacteristic, complemented, Sylow
259C2×C22⋊Q8—791no—
260C2×#121C2×D7911yescharacteristic
261C2×C4×D4—21no—
262#483—1581no—
263#481—792no—
264#465—1582no—
265#450—1581no—
266#448—792no—
267C2×C22⋊C8—791no—
268C2×C22⋊C8—791no—
269#388—21no—
270#387—21no—
271#386—21no—
272#386—21no—
273#380—791no—
274C4×C22⋊C4—21no—
275C4×C22⋊C4C2×D7911yescharacteristic
276C2×#95C2×D7911yescharacteristic
277#342—1581no—
278#342—1581no—
279#328—1581no—
280#328—1581no—
281#327—1582no—
282C22×Q8—791no—
283C22×D4—21no—
284C22×D4C22⋊D7911yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C22×D7911yescharacteristic
288C2×SD16—1582no—
289C22×C8—1581no—
290C22×C8—1581no—
291C22⋊Q8—1581no—
292C22⋊Q8—1581no—
293C22⋊Q8—1581no—
294C22⋊Q8—1581no—
295#121—21no—
296#121—21no—
297#121C4×D7912yes—
298#121D31612yes—
299C4×D4—42no—
300C4×D4—42no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—791no—
303C2×C4⋊C4—1581no—
304C2×C4⋊C4—1581no—
305C2×C4⋊C4C22⋊D7911yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—1581no—
311C2×C22⋊C4—1581no—
312C2×C22⋊C4C22⋊D7911yescharacteristic
313C2×C22⋊C4C22⋊D7911yescharacteristic
314C2×C42—21no—
315C2×C42—21no—
316#106—1582no—
317#103—1582no—
318#102—1582no—
319C22⋊C8—792no—
320C22⋊C8—1581no—
321C22⋊C8—1581no—
322C22⋊C8—1581no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—1581no—
328#95C22⋊D7911yescharacteristic
329#95C22⋊D7911yescharacteristic
330C24—21no—
331C24C2×C22⋊D7911yescharacteristic
332C2×Q8—6321no—
333C2×Q8—1581no—
334C2×Q8—1581no—
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×D4(C2×D79)⋊C412yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—3161no—
358C22×C4—1581no—
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—6322no—
367C2×C8—3161no—
368C2×C8—6321no—
369C2×C8—6321no—
370C2×C8—1582no—
371C4⋊C4—21no—
372C4⋊C4—42no—
373C4⋊C4—3161no—
374C4⋊C4—3161no—
375C4⋊C4—6321no—
376C4⋊C4—1581no—
377C4⋊C4—1581no—
378C4⋊C4(C2×D79)⋊C412yes—
379C22⋊C4—21no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—81no—
387C22⋊C4—3161no—
388C22⋊C4—3161no—
389C22⋊C4—3161no—
390C22⋊C4—3161no—
391C22⋊C4#1192961212yescomplemented
392C42—42no—
393C42—81no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C4×C22⋊D7911yescharacteristic
404C23#1457913011yescharacteristic
405C23#1457913511yescharacteristic
406C23C2×#1192961211yescharacteristic
407C23#1457918111yescharacteristic
408C23#1457918711yescharacteristic
409C23#1457919111yescharacteristic
410Q8—3162no—
411Q8—6321no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—81no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
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—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—3161no—
443C2×C4—3161no—
444C2×C4—6321no—
445C2×C4—6321no—
446C2×C4—6321no—
447C2×C4—6321no—
448C2×C4—6321no—
449C2×C4C2×(C2×D79)⋊C411yescharacteristic
450C2×C4C4×C22⋊D7911yescharacteristic
451C2×C4#1457913511yescharacteristic
452C2×C4#1457914411yescharacteristic
453C8—3162no—
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#1909844211yescharacteristic
473C22C4×#1192961211yescharacteristic
474C22#1909866311yescharacteristic
475C22#1909871411yescharacteristic
476C22#1909872511yescharacteristic
477C22#1909876811yescharacteristic
478C22#1909877811yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—6321no—
489C4—6321no—
490C4—12641no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—82no—
498C2#2376373411yescharacteristic
499C2#2376390111yescharacteristic
500C2#2376394211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2364C2
socleC22×C79#386
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenter#2364D79
hypocenterC79#9275
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2364

Complements

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

NH ≅ G/Ntype
2. #23764385C2semidirect
4. C2×#19098299C2semidirect
5. #23761279C2semidirect
6. #23761248C2semidirect
10. #19098879C4semidirect
13. #19098701C4semidirect
31. C22×Dic158C2×C4semidirect
40. C2×C4⋊Dic79C2×C4semidirect
50. C22⋊(C79⋊C8)C2×C4semidirect
140. C22⋊C4×C79SD16semidirect
258. C79#9275semidirect
391. C22⋊C4#11929612semidirect

Maximal subgroups

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

subgroupindex
#237741572
#237643852
#237619682
C2×#190982992
#237612792
#237612482
C79×#23642
#927579