Finite Group Databasebeta

← #34273755

Subgroups — #34273755

Order 20224.

694 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774518C211yesmaximal, characteristic, complemented
2#23772887C211yesmaximal, characteristic, complemented
3#23772511C211yesmaximal, characteristic, complemented
4#23762842C212yesmaximal, complemented
5#23762835C212yesmaximal, complemented
6#23762792C212yesmaximal, complemented
7#23762787C212yesmaximal, complemented
8#23762174C211yesmaximal, characteristic, complemented
9#23762077C211yesmaximal, characteristic, complemented
10C2×#19098310C211yesmaximal, characteristic, complemented
11C79×#3035C211yesmaximal, characteristic
12C2×#14579191—21no—
13C2×#14579161C2211yescharacteristic
14#19098966C2212yes—
15#19098909C2212yes—
16#19098874—22no—
17#19098843—21no—
18#19098843—21no—
19#19098773—21no—
20#19098773C2212yes—
21#19098768—21no—
22#19098768C2212yes—
23Q8×C79⋊C8C2212yes—
24D4×C79⋊C8—22no—
25D4×C79⋊C8C2212yes—
26#19098616—21no—
27#19098616C2211yescharacteristic
28#19098587C2211yescharacteristic
29C2×C4⋊(C79⋊C8)—21no—
30C2×C4⋊(C79⋊C8)C2211yescharacteristic
31#19098571—21no—
32#19098315C2212yes—
33#19098315C2212yes—
34#19098310C2212yes—
35#19098310C2212yes—
36#19098309C2212yes—
37#19098307C2212yes—
38C2×#135×C79—21no—
39#532×C79C2211yescharacteristic
40C4×#41×C79C2211yescharacteristic
41#464×C79—21no—
42#463×C79—21no—
43#462×C79C2211yescharacteristic
44#461×C79C2211yescharacteristic
45#460×C79C2211yescharacteristic
46#459×C79C2211yescharacteristic
47C2×C4⋊C8×C79C2211yescharacteristic
48#416×C79—21no—
49C22×Dic158—21no—
50#14579191—44no—
51C2×#11929614—22no—
52C2×#11929612—22no—
53C2×#11929608—21no—
54C22×C79⋊C8—21no—
55C22×C79⋊C8—22no—
56#14579161—21no—
57#14579161—21no—
58#14579161—21no—
59#14579161C2312yes—
60C2×C4⋊Dic79—21no—
61C2×C4⋊Dic79—21no—
62#14579094—22no—
63#14579094—22no—
64C4⋊(C79⋊C8)—21no—
65C4⋊(C79⋊C8)—21no—
66C4⋊(C79⋊C8)—22no—
67C4⋊(C79⋊C8)D411yescharacteristic
68C4⋊(C79⋊C8)D411yescharacteristic
69C4⋊(C79⋊C8)C2312yes—
70C4⋊(C79⋊C8)C2312yes—
71#14579084—22no—
72#14579084—22no—
73#14579082—22no—
74#14579082—22no—
75C22⋊(C79⋊C8)—22no—
76C22⋊(C79⋊C8)D412yes—
77C4×C79⋊C8—22no—
78C4×C79⋊C8C2312yes—
79C2×#41×C79—21no—
80C2×#41×C79D411yescharacteristic
81#135×C79—44no—
82C2×Q16×C79—21no—
83C2×SD16×C79—21no—
84C2×SD16×C79—21no—
85C2×D8×C79—21no—
86C22×C8×C79—21no—
87C4⋊Q8×C79C2311yescharacteristic
88#127×C79C2311yescharacteristic
89#124×C79—21no—
90#121×C79—21no—
91#121×C79—21no—
92C4×Q8×C79D411yescharacteristic
93C4×Q8×C79C2311yescharacteristic
94C4×D4×C79—21no—
95C4×D4×C79—21no—
96C4×D4×C79—21no—
97C4×D4×C79—21no—
98C4×D4×C79D411yescharacteristic
99C4×D4×C79C2311yescharacteristic
100#117×C79—21no—
101#117×C79D411yescharacteristic
102C2×C42×C79—21no—
103C2×C42×C79C2311yescharacteristic
104C4⋊C8×C79—21no—
105C4⋊C8×C79C2311yescharacteristic
106C4⋊C8×C79C2311yescharacteristic
107#103×C79—21no—
108#103×C79—21no—
109#102×C79—21no—
110#102×C79—21no—
111C2×Dic158—22no—
112C2×Dic158—41no—
113C2×Dic158—42no—
114C2×Dic158—42no—
115C22×Dic79—41no—
116#11929614—44no—
117#11929612—44no—
118#11929608—42no—
119C2×C79⋊C8—22no—
120C2×C79⋊C8—22no—
121C2×C79⋊C8—81no—
122C2×C79⋊C8—82no—
123C2×C79⋊C8C2×D412yes—
124C2×C79⋊C8C2×D412yes—
125C4⋊Dic79—22no—
126C4⋊Dic79—22no—
127C4⋊Dic79—41no—
128C4⋊Dic79—41no—
129#41×C79—22no—
130#41×C79—42no—
131#41×C79—42no—
132#41×C79D814yes—
133C2×Q8×C79—21no—
134C2×Q8×C79C2×D411yescharacteristic
135C2×D4×C79—21no—
136C2×D4×C79—21no—
137C2×D4×C79—21no—
138C2×D4×C79—21no—
139C2×D4×C79—41no—
140C2×D4×C79C2×D411yescharacteristic
141C22×C4×C79—21no—
142C22×C4×C79—21no—
143C22×C4×C79—21no—
144C22×C4×C79—41no—
145C22×C4×C79C2×D411yescharacteristic
146C22×C4×C79C2×D411yescharacteristic
147C22×C4×C79#4111yescharacteristic
148Q16×C79—42no—
149SD16×C79—42no—
150SD16×C79—42no—
151D8×C79—42no—
152C2×C8×C79—21no—
153C2×C8×C79—21no—
154C2×C8×C79—42no—
155C4⋊C4×C79—21no—
156C4⋊C4×C79—21no—
157C4⋊C4×C79—21no—
158C4⋊C4×C79—21no—
159C4⋊C4×C79C2×D411yescharacteristic
160C4⋊C4×C79C2×D411yescharacteristic
161C22⋊C4×C79—21no—
162C22⋊C4×C79—21no—
163C22⋊C4×C79—21no—
164C22⋊C4×C79—41no—
165C22⋊C4×C79—41no—
166C42×C79—21no—
167C42×C79—21no—
168C42×C79—21no—
169C42×C79C2×D411yescharacteristic
170C42×C79C2×D411yescharacteristic
171C42×C79#4111yescharacteristic
172C42×C79C2411yescharacteristic
173C23×C79—21no—
174C23×C79—41no—
175C23×C79#12111yescharacteristic
176Q8×C79—21no—
177Q8×C79—41no—
178Q8×C79—41no—
179Q8×C79C2×D812yes—
180D4×C79—24no—
181D4×C79—21no—
182D4×C79—41no—
183D4×C79—41no—
184D4×C79—41no—
185D4×C79—41no—
186D4×C79—81no—
187D4×C79—82no—
188D4×C79C2×D812yes—
189C2×C4×C79—21no—
190C2×C4×C79—21no—
191C2×C4×C79—21no—
192C2×C4×C79—24no—
193C2×C4×C79—21no—
194C2×C4×C79—21no—
195C2×C4×C79—41no—
196C2×C4×C79—41no—
197C2×C4×C79—41no—
198C2×C4×C79—41no—
199C2×C4×C79—81no—
200C2×C4×C79—82no—
201C2×C4×C79#12111yescharacteristic
202C2×C4×C79#12111yescharacteristic
203C2×C4×C79#12111yescharacteristic
204C2×C4×C79C2×D812yes—
205C2×C4×C79#13512yes—
206C2×C4×C79C22×D411yescharacteristic
207C2×C4×C79C22×D411yescharacteristic
208C2×C4×C79C2×#4111yescharacteristic
209C2×Dic79—42no—
210C2×Dic79—82no—
211Dic158—42no—
212Dic158—42no—
213Dic158—82no—
214C8×C79—41no—
215C8×C79—41no—
216C79⋊C8—22no—
217C79⋊C8—22no—
218C79⋊C8—42no—
219C79⋊C8—42no—
220C22×C79—21no—
221C22×C79—21no—
222C22×C79—22no—
223C22×C79—41no—
224C22×C79—41no—
225C22×C79—81no—
226C22×C79—81no—
227C22×C79—81no—
228C22×C79#46612yes—
229C22×C79C2×#12111yescharacteristic
230C4×C79—21no—
231C4×C79—22no—
232C4×C79—21no—
233C4×C79—21no—
234C4×C79—41no—
235C4×C79—41no—
236C4×C79—81no—
237C4×C79#46612yes—
238C4×C79C22×D811yescharacteristic
239C4×C79C2×#13511yescharacteristic
240C4×C79#57711yescharacteristic
241C4×C79#57811yescharacteristic
242Dic79—82no—
243#19416—791nomaximal, Sylow
244C2×C79—22no—
245C2×C79—42no—
246C2×C79—161no—
247C2×C79#304611yescharacteristic
248C2×C79C2×#46611yescharacteristic
249C2×C79#306911yescharacteristic
250#3226—791no—
251#3039—791no—
252#3035D7911yescharacteristic
253#2988—791no—
254#1682—792no—
255#1681—792no—
256#1674—792no—
257#1673—792no—
258#1571—791no—
259#1547—791no—
260C2×#332—791no—
261C79#1941611yescharacteristic, complemented, Sylow
262C2×#137—1581no—
263C2×#135—21no—
264#532C2×D7911yescharacteristic
265C2×C4⋊Q8—791no—
266C4×#41C2×D7911yescharacteristic
267#500—792no—
268#486—792no—
269#480—1582no—
270#474—1581no—
271#474—1581no—
272#464—21no—
273#463—21no—
274#462—792no—
275#462—1581no—
276#462C2×D7911yescharacteristic
277#461—792no—
278#461—1581no—
279#461C2×D7911yescharacteristic
280#460C2×D7911yescharacteristic
281#459C2×D7911yescharacteristic
282C8×Q8—792no—
283C8×D4—792no—
284C8×D4—1582no—
285#431—791no—
286#431—1581no—
287#423—791no—
288C2×C4⋊C8—791no—
289C2×C4⋊C8—1581no—
290C2×C4⋊C8C2×D7911yescharacteristic
291#419—1581no—
292#416—21no—
293#335—792no—
294#335—792no—
295#332—792no—
296#332—792no—
297#332—792no—
298#331—792no—
299C2×#41—21no—
300C2×#41C22⋊D7911yescharacteristic
301C22×Q8—1581no—
302#137—3164no—
303#135—44no—
304C2×Q16—21no—
305C2×Q16—1582no—
306C2×SD16—21no—
307C2×SD16—21no—
308C2×SD16—1582no—
309C2×D8—21no—
310C2×M16—1581no—
311C22×C8—21no—
312C22×C8—1582no—
313C22×C8—1581no—
314C4⋊Q8—792no—
315C4⋊Q8—1581no—
316C4⋊Q8—1581no—
317C4⋊Q8—1581no—
318C4⋊Q8C22×D7911yescharacteristic
319#127C22×D7911yescharacteristic
320#124—21no—
321#121—21no—
322#121—21no—
323C4×Q8C22⋊D7911yescharacteristic
324C4×Q8C22×D7911yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4C22⋊D7911yescharacteristic
330C4×D4C22×D7911yescharacteristic
331#117—21no—
332#117C22⋊D7911yescharacteristic
333C2×C4⋊C4—1581no—
334C2×C4⋊C4—1581no—
335C2×C42—21no—
336C2×C42C22×D7911yescharacteristic
337#107—1582no—
338#107—1582no—
339C4⋊C8—21no—
340C4⋊C8—791no—
341C4⋊C8—791no—
342C4⋊C8—792no—
343C4⋊C8—792no—
344C4⋊C8—1582no—
345C4⋊C8—1581no—
346C4⋊C8—1581no—
347C4⋊C8C22×D7911yescharacteristic
348C4⋊C8C22×D7911yescharacteristic
349#103—21no—
350#103—21no—
351#103—1582no—
352#103—1582no—
353#102—21no—
354#102—21no—
355#102—1582no—
356#102—1582no—
357C22⋊C8—792no—
358C22⋊C8—1582no—
359C4×C8—792no—
360C4×C8—1582no—
361#41—22no—
362#41—42no—
363#41—42no—
364#41#1192960914yes—
365C2×Q8—21no—
366C2×Q8—3161no—
367C2×Q8—3162no—
368C2×Q8—3162no—
369C2×Q8—1582no—
370C2×Q8C2×C22⋊D7911yescharacteristic
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
376C2×D4C2×C22⋊D7911yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—3161no—
382C22×C4D4×D7911yescharacteristic
383C22×C4C2×C22⋊D7911yescharacteristic
384C22×C4#1192962311yescharacteristic
385Q16—42no—
386Q16—3164no—
387SD16—42no—
388SD16—42no—
389SD16—3164no—
390D8—42no—
391M16—3162no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—42no—
395C2×C8—792no—
396C2×C8—792no—
397C2×C8—6321no—
398C2×C8—6322no—
399C2×C8—1582no—
400C2×C8—1582no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—3161no—
406C4⋊C4—3161no—
407C4⋊C4—1582no—
408C4⋊C4—1582no—
409C4⋊C4C2×C22⋊D7911yescharacteristic
410C4⋊C4C2×C22⋊D7911yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42D4×D7911yescharacteristic
420C42C2×C22⋊D7911yescharacteristic
421C42#1192962311yescharacteristic
422C42C23×D7911yescharacteristic
423C23—21no—
424C23—41no—
425C23#1457913311yescharacteristic
426Q8—21no—
427Q8—41no—
428Q8—41no—
429Q8—3162no—
430Q8—3162no—
431Q8—6322no—
432Q8C2×#1192960912yes—
433D4—24no—
434D4—21no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—81no—
440D4—82no—
441D4C2×#1192960912yes—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—3162no—
455C2×C4—6322no—
456C2×C4#1457913311yescharacteristic
457C2×C4#1457913311yescharacteristic
458C2×C4#1457913311yescharacteristic
459C2×C4C2×#1192960912yes—
460C2×C4#1457918112yes—
461C2×C4C2×D4×D7911yescharacteristic
462C2×C4C22×C22⋊D7911yescharacteristic
463C2×C4C2×#1192962311yescharacteristic
464C8—41no—
465C8—41no—
466C8—3162no—
467C8—3162no—
468C8—1582no—
469C8—1582no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#1909879712yes—
479C22C2×#1457913311yescharacteristic
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—6322no—
488C4#1909879712yes—
489C4C22×#1192960911yescharacteristic
490C4C2×#1457918111yescharacteristic
491C4#1909937011yescharacteristic
492C4#1909937711yescharacteristic
493C2—22no—
494C2—42no—
495C2—161no—
496C2#2377294811yescharacteristic
497C2C2×#1909879711yescharacteristic
498C2#2377313911yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
Fitting, prime radicalC79×#3035C2
socleC22×C79C2×#121
hypercenter#3035D79
hypocenterC79#19416
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #23774518C2semidirect
2. #23772887C2semidirect
3. #23772511C2semidirect
4. #23762842C2semidirect
5. #23762835C2semidirect
6. #23762792C2semidirect
7. #23762787C2semidirect
8. #23762174C2semidirect
9. #23762077C2semidirect
10. C2×#19098310C2semidirect
261. C79#19416semidirect

Maximal subgroups

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

subgroupindex
#237745182
#237728872
#237725112
#237628422
#237628352
#237627922
#237627872
#237621742
#237620772
C2×#190983102
C79×#30352
#1941679