Finite Group Databasebeta

← #34441657

Subgroups — #34441657

Order 20224.

674 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098994C211yesmaximal, characteristic
2#23772275C211yesmaximal, characteristic, complemented
3#23772270C211yesmaximal, characteristic, complemented
4#23765274C211yesmaximal, characteristic, complemented
5#23765273C211yesmaximal, characteristic, complemented
6#23765269C211yesmaximal, characteristic, complemented
7#23765268C211yesmaximal, characteristic, complemented
8C4×#14579106C211yesmaximal, characteristic, complemented
9SD32×Dic79C211yesmaximal, characteristic, complemented
10#23765251C211yesmaximal, characteristic, complemented
11#367×D79C211yesmaximal, characteristic, complemented
12#23765111C211yesmaximal, characteristic, complemented
13#23765109C211yesmaximal, characteristic, complemented
14#23764995C211yesmaximal, characteristic, complemented
15C4×SD32×C79C211yesmaximal, characteristic, complemented
16C2×#14579184C2211yescharacteristic
17C2×#14579180C2211yescharacteristic
18#19099045—21no—
19#19099042—21no—
20#19098994C418yescomplemented
21C2×#14579107C2211yescharacteristic
22C2×#14579106C2211yescharacteristic
23C2×#14579102C2211yescharacteristic
24C2×#14579100C2211yescharacteristic
25C2×C16×D79C2211yescharacteristic
26#19098666—21no—
27#19098663—21no—
28#19098661—21no—
29#19098660—21no—
30C4×Dic316C2211yescharacteristic
31Q16×Dic79C2211yescharacteristic
32#19098657C2211yescharacteristic
33C4×D632C2211yescharacteristic
34D8×Dic79C2211yescharacteristic
35#19098645C2211yescharacteristic
36#107×D79C2211yescharacteristic
37#107×D79C2211yescharacteristic
38#19098563—21no—
39#19098561—21no—
40#19098517C2211yescharacteristic
41#19098391C2211yescharacteristic
42#19098390C2211yescharacteristic
43#19098390C2211yescharacteristic
44#19098368C2211yescharacteristic
45#19098367C2211yescharacteristic
46#19098366C2211yescharacteristic
47#19098365C2211yescharacteristic
48#19098364C2211yescharacteristic
49#19098363C2211yescharacteristic
50#19098339C2211yescharacteristic
51#19098338C2211yescharacteristic
52C16×Dic79C2211yescharacteristic
53C2×SD32×C79C2211yescharacteristic
54C4×Q16×C79C2211yescharacteristic
55C4×D8×C79C2211yescharacteristic
56#367×C79C2211yescharacteristic
57#358×C79C2211yescharacteristic
58#357×C79C2211yescharacteristic
59C4×C16×C79C2211yescharacteristic
60C2×#11929625—21no—
61C2×#11929623—21no—
62#14579184—22no—
63#14579184C2×C414yescomplemented
64#14579180—22no—
65#14579180C2×C414yescomplemented
66C2×Dic316C2311yescharacteristic
67C2×#11929612—21no—
68C2×#11929611—21no—
69C2×D632C2311yescharacteristic
70C2×#11929607D411yescharacteristic
71C2×C8×D79C2311yescharacteristic
72Q8×Dic79—21no—
73C4×Dic158—21no—
74#14579126—21no—
75C4×D316—21no—
76C4×D316—41no—
77D4×Dic79—21no—
78#14579123—21no—
79#14579123—41no—
80C4×C22⋊D79—41no—
81(C22⋊D79)⋊C4—41no—
82#14579118—41no—
83#14579118D411yescharacteristic
84#14579117—41no—
85C2×Dic79⋊C4—41no—
86C4⋊C4×D79—21no—
87C4⋊C4×D79—21no—
88C4⋊C4×D79—41no—
89#14579107C2×C414yescomplemented
90#14579106C2×C414yescomplemented
91#14579102C2×C414yescomplemented
92#14579100—22no—
93C2×C79⋊C16C2311yescharacteristic
94C2×C79⋊C16C2311yescharacteristic
95C16×D79C2×C414yescomplemented
96#14579095C2311yescharacteristic
97#14579095C2311yescharacteristic
98#14579094—21no—
99#14579094—21no—
100#14579086—21no—
101#14579085—21no—
102#14579084—21no—
103#14579083—21no—
104#14579082—21no—
105#14579081—21no—
106#14579072D411yescharacteristic
107#14579071D411yescharacteristic
108C8×Dic79C2311yescharacteristic
109C2×Q16×C79C2311yescharacteristic
110C2×D8×C79C2311yescharacteristic
111C4×Q8×C79—21no—
112C4×D4×C79—21no—
113SD32×C79C2×C414yescomplemented
114C2×C16×C79C2311yescharacteristic
115C2×C16×C79C2311yescharacteristic
116#107×C79C2311yescharacteristic
117#107×C79C2311yescharacteristic
118#103×C79—21no—
119#102×C79—21no—
120C4×C8×C79C2311yescharacteristic
121#11929625—24no—
122#11929625—42no—
123#11929623—24no—
124#11929623—42no—
125C2×Dic158—21no—
126C2×D316—21no—
127C2×D316—41no—
128C2×C22⋊D79—41no—
129C22×Dic79—41no—
130C22×Dic79—41no—
131C2×C4×D79—41no—
132C2×C4×D79—41no—
133C2×C4×D79—41no—
134C2×C4×D79—41no—
135C2×C4×D79—41no—
136C2×C4×D79C2×D411yescharacteristic
137Dic316—21no—
138Dic316C22×C412yes—
139#11929612—24no—
140#11929611—24no—
141D632—21no—
142D632C22×C412yes—
143#11929607—22no—
144C2×C79⋊C8C2×D411yescharacteristic
145C2×C79⋊C8C2×D411yescharacteristic
146C8×D79C22×C412yes—
147C8×D79#4112yes—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—41no—
151Dic79⋊C4—41no—
152Dic79⋊C4—41no—
153Dic79⋊C4—41no—
154Dic79⋊C4—41no—
155Dic79⋊C4D811yescharacteristic
156Dic79⋊C4D811yescharacteristic
157C22⋊Dic79—41no—
158(C2×D79)⋊C4—41no—
159(C2×D79)⋊C4—41no—
160(C2×D79)⋊C4D811yescharacteristic
161(C2×D79)⋊C4D811yescharacteristic
162C4×Dic79—41no—
163C4×Dic79—41no—
164C4×Dic79—41no—
165C4×Dic79C2×D411yescharacteristic
166C79⋊C16—21no—
167C79⋊C16C22×C412yes—
168C2×Q8×C79—21no—
169C2×D4×C79—21no—
170C22×C4×C79—41no—
171Q16×C79—21no—
172Q16×C79C22×C412yes—
173D8×C79—21no—
174D8×C79C22×C412yes—
175C2×C8×C79C2×D411yescharacteristic
176C2×C8×C79C2411yescharacteristic
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C42×C79—41no—
182C42×C79C2×D411yescharacteristic
183C16×C79—21no—
184C16×C79C22×C412yes—
185C23×C79—41no—
186C22×D79—41no—
187C22×D79C2×D811yescharacteristic
188Q8×C79—22no—
189Q8×C79—41no—
190D4×C79—22no—
191D4×C79—41no—
192C2×C4×C79—41no—
193C2×C4×C79—41no—
194C2×C4×C79—41no—
195C2×C4×C79—161no—
196C2×C4×C79C2×D811yescharacteristic
197C2×C4×C79C2×D811yescharacteristic
198C2×C4×C79C22×D411yescharacteristic
199C22⋊D79—44no—
200C2×Dic79—41no—
201C2×Dic79—44no—
202C2×Dic79—41no—
203C2×Dic79—41no—
204C2×Dic79—161no—
205C2×Dic79C2×D811yescharacteristic
206C2×Dic79C2×D811yescharacteristic
207C2×Dic79C2×D811yescharacteristic
208D316—22no—
209D316—44no—
210D316—41no—
211C4×D79—44no—
212C4×D79—161no—
213C4×D79—161no—
214C4×D79—161no—
215C4×D79—161no—
216C4×D79C4×D412yes—
217C4×D79#13512yes—
218Dic158—22no—
219Dic158—41no—
220C8×C79—21no—
221C8×C79C23×C411yescharacteristic
222C8×C79C2×#4111yescharacteristic
223C79⋊C8—21no—
224C79⋊C8C4×D412yes—
225C22×C79—42no—
226C22×C79—81no—
227C22×C79—81no—
228C22×C79C22×D811yescharacteristic
229C2×D79—21no—
230C2×D79—21no—
231C2×D79—42no—
232C2×D79—81no—
233C2×D79—81no—
234C2×D79C4×D812yes—
235C4×C79—21no—
236C4×C79—21no—
237C4×C79—42no—
238C4×C79—81no—
239C4×C79—81no—
240C4×C79C2×C4×D411yescharacteristic
241C4×C79C2×#13511yescharacteristic
242Dic79—21no—
243Dic79—21no—
244Dic79—42no—
245Dic79—81no—
246Dic79—81no—
247Dic79C4×D812yes—
248#33460—791nomaximal, Sylow
249C2×C79—82no—
250C2×C79C2×C4×D811yescharacteristic
251C2×C79#341611yescharacteristic
252C2×C79#341611yescharacteristic
253D79—22no—
254D79—82no—
255C2×#508—791no—
256#2951—791no—
257#2951—791no—
258#2177—791no—
259#2177—791no—
260#2176—791no—
261#2176—791no—
262C4×SD32—791no—
263C4×SD32—791no—
264C4×SD32—791no—
265C4×SD32D7911yescharacteristic, complemented
266C2×#367—791no—
267#2139—791no—
268#2139—791no—
269#2108—791no—
270C79#3346011yescharacteristic, complemented, Sylow
271C2×#135—791no—
272C2×#135—791no—
273#520—1581no—
274#520—1581no—
275#508—798no—
276C2×Q32—791no—
277C2×SD32—791no—
278C2×SD32C2×D7911yescharacteristic
279C2×D16—791no—
280C2×M32—791no—
281C22×C16—791no—
282#440—1581no—
283#440—1581no—
284#440—1581no—
285#440—1581no—
286C4×Q16—791no—
287C4×Q16—791no—
288C4×Q16—791no—
289C4×Q16C2×D7911yescharacteristic
290C4×D8—791no—
291C4×D8—791no—
292C4×D8—791no—
293C4×D8C2×D7911yescharacteristic
294C2×#107—791no—
295C2×#107—791no—
296#417—1581no—
297#417—1581no—
298#405—791no—
299#367—791no—
300#367—791no—
301#367—791no—
302#367C2×D7911yescharacteristic
303#358—791no—
304#358—791no—
305#358—791no—
306#358C2×D7911yescharacteristic
307#357—791no—
308#357—791no—
309#357—791no—
310#357C2×D7911yescharacteristic
311#346—791no—
312#346—791no—
313C4×C16—791no—
314C4×C16C2×D7911yescharacteristic
315C2×#41—1581no—
316C2×#41—1581no—
317#135—794no—
318#135—794no—
319#135—1582no—
320#135—1582no—
321C2×Q16—791no—
322C2×Q16C22×D7911yescharacteristic
323C2×SD16—1581no—
324C2×SD16—1581no—
325C2×D8—791no—
326C2×D8C22×D7911yescharacteristic
327C2×M16—791no—
328C22×C8—791no—
329C4×Q8—21no—
330C4×Q8—1581no—
331C4×Q8—1581no—
332C4×Q8—1581no—
333C4×D4—21no—
334C4×D4—3161no—
335C4×D4—3161no—
336C4×D4—3161no—
337C4×D4—3161no—
338C4×D4—1581no—
339C4×D4—1581no—
340C4×D4—1581no—
341#117—3161no—
342#117—3161no—
343#117—791no—
344C2×C4⋊C4—3161no—
345C2×C4⋊C4—3161no—
346C2×C4⋊C4—1581no—
347C2×C4⋊C4—1581no—
348Q32—794no—
349SD32—794no—
350SD32C4×D7914yescomplemented
351D16—794no—
352M32—1582no—
353C2×C16—791no—
354C2×C16—794no—
355C2×C16—791no—
356C2×C16C22×D7911yescharacteristic
357C2×C16C22×D7911yescharacteristic
358#107—791no—
359#107—791no—
360#107—1581no—
361#107—1581no—
362#107C22×D7911yescharacteristic
363#107C22×D7911yescharacteristic
364#103—21no—
365#103—1581no—
366#103—1581no—
367#103—1581no—
368#102—21no—
369#102—1581no—
370#102—1581no—
371#102—1581no—
372#97—791no—
373#97—791no—
374C4×C8—791no—
375C4×C8C22×D7911yescharacteristic
376#41—3162no—
377#41—3162no—
378#41—1584no—
379#41—1584no—
380C2×Q8—21no—
381C2×Q8—1581no—
382C2×D4—21no—
383C2×D4—3161no—
384C2×D4—3161no—
385C2×D4—1581no—
386C22×C4—41no—
387C22×C4—3161no—
388C22×C4—3161no—
389C22×C4—3161no—
390C22×C4—3161no—
391C22×C4—3161no—
392C22×C4—3161no—
393C22×C4—3161no—
394C22×C4—791no—
395Q16—21no—
396Q16—792no—
397Q16—1581no—
398Q16C2×C4×D7912yes—
399SD16—1584no—
400SD16—1584no—
401D8—21no—
402D8—792no—
403D8—1581no—
404D8C2×C4×D7912yes—
405M16—1582no—
406C2×C8—791no—
407C2×C8—792no—
408C2×C8—792no—
409C2×C8—791no—
410C2×C8D4×D7911yescharacteristic
411C2×C8C23×D7911yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—41no—
415C4⋊C4—3161no—
416C4⋊C4—3161no—
417C4⋊C4—3161no—
418C4⋊C4—3161no—
419C4⋊C4—3161no—
420C4⋊C4—791no—
421C4⋊C4—791no—
422C4⋊C4—1581no—
423C4⋊C4—1581no—
424C22⋊C4—41no—
425C22⋊C4—3161no—
426C22⋊C4—3161no—
427C22⋊C4—3161no—
428C22⋊C4—791no—
429C22⋊C4—791no—
430C42—41no—
431C42—3161no—
432C42—3161no—
433C42—3161no—
434C42—791no—
435C42D4×D7911yescharacteristic
436C16—21no—
437C16—792no—
438C16—1581no—
439C16C2×C4×D7912yes—
440C23—41no—
441C23—3161no—
442C23—791no—
443Q8—22no—
444Q8—41no—
445Q8—3161no—
446Q8—1582no—
447D4—22no—
448D4—41no—
449D4—3164no—
450D4—3164no—
451D4—3161no—
452D4—1582no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—161no—
457C2×C4—3164no—
458C2×C4—3161no—
459C2×C4—3164no—
460C2×C4—3161no—
461C2×C4—3161no—
462C2×C4—791no—
463C2×C4—792no—
464C2×C4—792no—
465C2×C4—791no—
466C2×C4—791no—
467C2×C4—12641no—
468C2×C4—12641no—
469C2×C4—12641no—
470C2×C4—12641no—
471C2×C4—12641no—
472C2×C4D8×D7911yescharacteristic
473C2×C4D8×D7911yescharacteristic
474C2×C4C2×D4×D7911yescharacteristic
475C8—21no—
476C8—792no—
477C8—1581no—
478C8C22×C4×D7911yescharacteristic
479C8#41×D7911yescharacteristic
480C22—42no—
481C22—81no—
482C22—81no—
483C22—3162no—
484C22—792no—
485C22—6321no—
486C22—6321no—
487C22—1581no—
488C22—1581no—
489C22C2×D8×D7911yescharacteristic
490C4—21no—
491C4—21no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—3162no—
496C4—792no—
497C4—6321no—
498C4—6321no—
499C4—1581no—
500C4—1581no—
501C4C4×D4×D7911yescharacteristic
502C4#135×D7911yescharacteristic
503C2—82no—
504C2—6322no—
505C2—1582no—
506C2C4×D8×D7911yescharacteristic
507C2#2377654211yescharacteristic
508C2#2377654311yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC4×SD32×C79C2
socleC22×C79C22×D8
prime radicalC2×#19098994C2
cosocleC2×C8×C79C24
hypercenterC4×SD32D79
hypocenterC79#33460
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
2. #23772275C2semidirect
3. #23772270C2semidirect
4. #23765274C2semidirect
5. #23765273C2semidirect
6. #23765269C2semidirect
7. #23765268C2semidirect
8. C4×#14579106C2semidirect
9. SD32×Dic79C2semidirect
10. #23765251C2semidirect
11. #367×D79C2semidirect
12. #23765111C2semidirect
13. #23765109C2semidirect
14. #23764995C2semidirect
15. C4×SD32×C79C2semidirect
20. #19098994C4semidirect
63. #14579184C2×C4semidirect
65. #14579180C2×C4semidirect
89. #14579107C2×C4semidirect
90. #14579106C2×C4semidirect
91. #14579102C2×C4semidirect
95. C16×D79C2×C4semidirect
113. SD32×C79C2×C4semidirect
265. C4×SD32D79semidirect
270. C79#33460semidirect
350. SD32C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989942
#237722752
#237722702
#237652742
#237652732
#237652692
#237652682
C4×#145791062
SD32×Dic792
#237652512
#367×D792
#237651112
#237651092
#237649952
C4×SD32×C792
#3346079