Finite Group Databasebeta

← #34445324

Subgroups — #34445324

Order 20224.

536 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
1D4×Dic316C211yesmaximal, characteristic
2#23765647C211yesmaximal, characteristic, complemented
3#23765608C211yesmaximal, characteristic, complemented
4#23765597C211yesmaximal, characteristic, complemented
5#23765561C211yesmaximal, characteristic, complemented
6#23765506C211yesmaximal, characteristic, complemented
7#23765489C211yesmaximal, characteristic, complemented
8#23765450C211yesmaximal, characteristic, complemented
9#23765333C211yesmaximal, characteristic, complemented
10#23765333C211yesmaximal, characteristic, complemented
11#23765269C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic
13#23765119C211yesmaximal, characteristic, complemented
14C2×#19098366C211yesmaximal, characteristic, complemented
15C79×#3336C211yesmaximal, characteristic
16C22×Dic316C2211yescharacteristic
17C22×Dic316C2211yescharacteristic
18D4×Dic158—21no—
19C2×#14579107C2211yescharacteristic
20C2×#14579107C2211yescharacteristic
21C2×#14579107C2211yescharacteristic
22C2×#14579101C2211yescharacteristic
23C22×C79⋊C16C2211yescharacteristic
24#19098943C2211yescharacteristic
25#19098804C2211yescharacteristic
26#19098804C2211yescharacteristic
27#19098774—21no—
28#19098721—21no—
29#19098721—21no—
30C4×Dic316C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098366C2211yescharacteristic
37#19098366C2211yescharacteristic
38#19098366C2211yescharacteristic
39#19098366C2211yescharacteristic
40#19098366C2211yescharacteristic
41#19098366C2211yescharacteristic
42#19098366C2211yescharacteristic
43#19098366C2211yescharacteristic
44#19098366C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47#19098339C2211yescharacteristic
48#552×C79—21no—
49#499×C79C2211yescharacteristic
50#485×C79—21no—
51#483×C79—21no—
52#479×C79—21no—
53#467×C79C2211yescharacteristic
54#467×C79C2211yescharacteristic
55C4×Q16×C79C2211yescharacteristic
56C8×D4×C79C2211yescharacteristic
57#427×C79C2211yescharacteristic
58C2×#107×C79C2211yescharacteristic
59C22×Dic158—21no—
60C22×Dic158—21no—
61C2×Dic316—22no—
62C2×Dic316—22no—
63C2×Dic316—21no—
64C2×Dic316—21no—
65C2×Dic316—21no—
66C2×Dic316—41no—
67C2×Dic316—41no—
68C2×Dic316C2311yescharacteristic
69C2×Dic316C2311yescharacteristic
70C2×Dic316C2311yescharacteristic
71#14579161—21no—
72#14579160—41no—
73#14579140—21no—
74#14579140—21no—
75#14579138—41no—
76#14579138—41no—
77C4×Dic158—21no—
78D4×Dic79—41no—
79#14579107—22no—
80#14579107—41no—
81#14579107—41no—
82#14579101—41no—
83C2×C79⋊C16—41no—
84C2×C79⋊C16C2311yescharacteristic
85C2×C79⋊C16C2311yescharacteristic
86C2×C79⋊C16C2311yescharacteristic
87C2×C79⋊C16C2311yescharacteristic
88#14579095C2311yescharacteristic
89#14579086—21no—
90#14579086—21no—
91#14579086—21no—
92C2×Q16×C79C2311yescharacteristic
93C22×C8×C79C2311yescharacteristic
94C22×C8×C79C2311yescharacteristic
95#126×C79—41no—
96#125×C79—21no—
97#125×C79—41no—
98#123×C79—41no—
99C22⋊Q8×C79—21no—
100C22⋊Q8×C79—21no—
101C4×Q8×C79—21no—
102C4×D4×C79D411yescharacteristic
103#117×C79—21no—
104#117×C79—41no—
105C2×C4⋊C4×C79—21no—
106#107×C79—21no—
107#107×C79C2311yescharacteristic
108#107×C79C2311yescharacteristic
109#107×C79C2311yescharacteristic
110#106×C79—21no—
111#106×C79—21no—
112C4⋊C8×C79D411yescharacteristic
113#103×C79—21no—
114#103×C79—21no—
115#103×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119C2×Dic158—21no—
120C2×Dic158—21no—
121C2×Dic158—21no—
122C2×Dic158—42no—
123C2×Dic158—42no—
124C2×Dic158—41no—
125C2×Dic158—41no—
126C2×Dic158—81no—
127C2×Dic158—81no—
128C22×Dic79—41no—
129C22×Dic79—41no—
130Dic316—21no—
131Dic316—21no—
132Dic316—21no—
133Dic316—21no—
134Dic316—21no—
135Dic316—42no—
136Dic316—42no—
137Dic316—81no—
138Dic316#4112yes—
139C4⋊Dic79—21no—
140C4⋊Dic79—41no—
141C4⋊Dic79—41no—
142Dic79⋊C4—41no—
143Dic79⋊C4—41no—
144Dic79⋊C4—41no—
145C22⋊Dic79—41no—
146C22⋊Dic79—41no—
147C4×Dic79—41no—
148C79⋊C16—21no—
149C79⋊C16—21no—
150C79⋊C16—21no—
151C79⋊C16—21no—
152C2×Q8×C79—21no—
153C2×D4×C79D811yescharacteristic
154C22×C4×C79—41no—
155C22×C4×C79C2×D411yescharacteristic
156C22×C4×C79C2×D411yescharacteristic
157Q16×C79—21no—
158Q16×C79—21no—
159C2×C8×C79—21no—
160C2×C8×C79—21no—
161C2×C8×C79—21no—
162C2×C8×C79—21no—
163C2×C8×C79C2×D411yescharacteristic
164C2×C8×C79C2×D411yescharacteristic
165C2×C8×C79C2×D411yescharacteristic
166C2×C8×C79C2411yescharacteristic
167C4⋊C4×C79—21no—
168C4⋊C4×C79—21no—
169C4⋊C4×C79—21no—
170C4⋊C4×C79—41no—
171C4⋊C4×C79—41no—
172C4⋊C4×C79—41no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79—41no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79D811yescharacteristic
177C22⋊C4×C79—41no—
178C22⋊C4×C79—41no—
179C22⋊C4×C79—41no—
180C22⋊C4×C79D811yescharacteristic
181C22⋊C4×C79D811yescharacteristic
182C42×C79—41no—
183C42×C79—41no—
184C42×C79C2×D411yescharacteristic
185C23×C79C2×D811yescharacteristic
186C23×C79C2×D811yescharacteristic
187Q8×C79—41no—
188Q8×C79—41no—
189D4×C79—22no—
190C2×C4×C79—21no—
191C2×C4×C79—21no—
192C2×C4×C79—21no—
193C2×C4×C79—21no—
194C2×C4×C79—41no—
195C2×C4×C79—41no—
196C2×C4×C79—41no—
197C2×C4×C79—41no—
198C2×C4×C79—82no—
199C2×C4×C79C2×D811yescharacteristic
200C2×C4×C79C2×D811yescharacteristic
201C2×C4×C79C2×D811yescharacteristic
202C2×C4×C79C2×D811yescharacteristic
203C2×C4×C79C22×D411yescharacteristic
204C2×Dic79—41no—
205C2×Dic79—41no—
206C2×Dic79—41no—
207C2×Dic79—41no—
208C2×Dic79—82no—
209C2×Dic79—82no—
210Dic158—22no—
211Dic158—41no—
212Dic158—41no—
213Dic158—41no—
214Dic158—41no—
215Dic158—41no—
216Dic158—82no—
217Dic158—82no—
218C8×C79—21no—
219C8×C79—21no—
220C8×C79—21no—
221C8×C79C2×#4111yescharacteristic
222C8×C79H(5,𝔽2)11yescharacteristic
223C22×C79—21no—
224C22×C79—21no—
225C22×C79—21no—
226C22×C79—21no—
227C22×C79—21no—
228C22×C79#51012yes—
229C22×C79C22×D811yescharacteristic
230C4×C79—21no—
231C4×C79—21no—
232C4×C79—21no—
233C4×C79—21no—
234C4×C79—81no—
235C4×C79—81no—
236C4×C79—81no—
237C4×C79—81no—
238C4×C79#54611yescharacteristic
239C4×C79#57711yescharacteristic
240Dic79—42no—
241Dic79—81no—
242Dic79—81no—
243Dic79—81no—
244#33879—791nomaximal, Sylow
245C2×C79—22no—
246C2×C79—41no—
247C2×C79#329411yescharacteristic
248C2×C79C2×#51011yescharacteristic
249C2×C79#341711yescharacteristic
250#3336D7911yescharacteristic
251D4×Q16—791no—
252#2245—791no—
253#2237—791no—
254#2235—791no—
255#2229—791no—
256#2219—791no—
257#2216—791no—
258#2209—791no—
259#2188—791no—
260#2188—791no—
261#2176—791no—
262#2168—791no—
263#2141—791no—
264C2×#358—791no—
265C79#3387911yescharacteristic, complemented, Sylow
266C22×Q16—791no—
267C22×Q16—791no—
268#552—21no—
269D4×Q8—1581no—
270C2×Q32—791no—
271C2×Q32—791no—
272C2×Q32—791no—
273C2×M32—791no—
274C22×C16—791no—
275#499C2×D7911yescharacteristic
276#494—791no—
277#485—21no—
278#483—21no—
279#479—21no—
280#467—791no—
281#467—791no—
282#467C2×D7911yescharacteristic
283#467C2×D7911yescharacteristic
284#462—1581no—
285#451—1581no—
286#451—1581no—
287C4×Q16—791no—
288C4×Q16C2×D7911yescharacteristic
289C8×D4C2×D7911yescharacteristic
290#427C2×D7911yescharacteristic
291C2×#107C2×D7911yescharacteristic
292#367—791no—
293#367—791no—
294#366—791no—
295C4⋊C16—791no—
296#358—791no—
297#358—791no—
298#358—791no—
299#358—791no—
300#358—791no—
301#358—791no—
302#358—791no—
303#358—791no—
304#358—791no—
305#358—791no—
306C22⋊C16—791no—
307C22⋊C16—791no—
308#346—791no—
309C22×Q8—1581no—
310C22×Q8—1581no—
311C2×Q16—3161no—
312C2×Q16—3161no—
313C2×Q16—791no—
314C2×Q16—791no—
315C2×Q16—791no—
316C2×Q16—1582no—
317C2×Q16—1582no—
318C2×Q16—1581no—
319C2×Q16—1581no—
320C2×Q16—1581no—
321C2×Q16C22×D7911yescharacteristic
322C22×C8C22×D7911yescharacteristic
323C22×C8C22×D7911yescharacteristic
324C4⋊Q8—3161no—
325C4⋊Q8—1581no—
326#126—41no—
327#125—21no—
328#125—41no—
329#123—41no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—3161no—
333C22⋊Q8—3161no—
334C22⋊Q8—1581no—
335C22⋊Q8—1581no—
336C4×Q8—21no—
337C4×Q8—1581no—
338C4×D4—3161no—
339C4×D4C22⋊D7911yescharacteristic
340#117—21no—
341#117—41no—
342C2×C4⋊C4—21no—
343Q32—3161no—
344Q32—3161no—
345Q32—1582no—
346M32—3161no—
347C2×C16—3161no—
348C2×C16—791no—
349C2×C16—791no—
350C2×C16—791no—
351C2×C16—791no—
352#107—21no—
353#107—791no—
354#107C22×D7911yescharacteristic
355#107C22×D7911yescharacteristic
356#107C22×D7911yescharacteristic
357#106—21no—
358#106—21no—
359C4⋊C8C22⋊D7911yescharacteristic
360#103—21no—
361#103—21no—
362#103—21no—
363#103—1581no—
364#103—1581no—
365#103—1581no—
366C22⋊C8C22⋊D7911yescharacteristic
367C22⋊C8C22⋊D7911yescharacteristic
368C4×C8C22×D7911yescharacteristic
369C2×Q8—21no—
370C2×Q8—3162no—
371C2×Q8—3162no—
372C2×Q8—3161no—
373C2×Q8—3161no—
374C2×Q8—6321no—
375C2×Q8—6321no—
376C2×Q8—1581no—
377C2×Q8—1581no—
378C2×Q8—1581no—
379C2×D4#1192960911yescharacteristic
380C22×C4—41no—
381C22×C4—3161no—
382C22×C4—3161no—
383C22×C4C2×C22⋊D7911yescharacteristic
384C22×C4C2×C22⋊D7911yescharacteristic
385Q16—21no—
386Q16—21no—
387Q16—3162no—
388Q16—3162no—
389Q16—792no—
390Q16—6321no—
391Q16—1581no—
392Q16—1581no—
393Q16—1581no—
394Q16—1581no—
395Q16—1581no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8C2×C22⋊D7911yescharacteristic
401C2×C8C2×C22⋊D7911yescharacteristic
402C2×C8C2×C22⋊D7911yescharacteristic
403C2×C8C23×D7911yescharacteristic
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—3161no—
414C4⋊C4—3161no—
415C4⋊C4—3161no—
416C4⋊C4—3161no—
417C4⋊C4—3161no—
418C4⋊C4—1581no—
419C4⋊C4#1192960911yescharacteristic
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—3161no—
424C22⋊C4—3161no—
425C22⋊C4#1192960911yescharacteristic
426C22⋊C4#1192960911yescharacteristic
427C42—41no—
428C42—41no—
429C42—3161no—
430C42C2×C22⋊D7911yescharacteristic
431C16—1581no—
432C16—1581no—
433C16—1581no—
434C16—1581no—
435C23C2×#1192960911yescharacteristic
436C23C2×#1192960911yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—3161no—
440Q8—3161no—
441Q8—3161no—
442Q8—3161no—
443Q8—3161no—
444Q8—6322no—
445Q8—6322no—
446Q8—1582no—
447D4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—3161no—
458C2×C4—3161no—
459C2×C4—3161no—
460C2×C4—3161no—
461C2×C4—6322no—
462C2×C4—6322no—
463C2×C4C2×#1192960911yescharacteristic
464C2×C4C2×#1192960911yescharacteristic
465C2×C4C2×#1192960911yescharacteristic
466C2×C4C2×#1192960911yescharacteristic
467C2×C4C22×C22⋊D7911yescharacteristic
468C8—21no—
469C8—21no—
470C8—21no—
471C8#41×D7911yescharacteristic
472C8#1457920711yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22#1909900312yes—
479C22C22×#1192960911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—3162no—
489C4—6321no—
490C4—6321no—
491C4—6321no—
492C4#1909920111yescharacteristic
493C4#1909937111yescharacteristic
494C2—22no—
495C2—41no—
496C2#2377520711yescharacteristic
497C2C2×#1909900311yescharacteristic
498C2#2377655111yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3336C2
socleC22×C79C22×D8
prime radicalC2×D4×C79D8
hypercenter#3336D79
hypocenterC79#33879
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #23765647C2semidirect
3. #23765608C2semidirect
4. #23765597C2semidirect
5. #23765561C2semidirect
6. #23765506C2semidirect
7. #23765489C2semidirect
8. #23765450C2semidirect
9. #23765333C2semidirect
10. #23765333C2semidirect
11. #23765269C2semidirect
13. #23765119C2semidirect
14. C2×#19098366C2semidirect
265. C79#33879semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3162
#237656472
#237656082
#237655972
#237655612
#237655062
#237654892
#237654502
#237653332
#237652692
#237652262
#237651192
C2×#190983662
C79×#33362
#3387979