Finite Group Databasebeta

← #34443302

Subgroups — #34443302

Order 20224.

554 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773970C211yesmaximal, characteristic, complemented
2#23765608C211yesmaximal, characteristic, complemented
3#23765591C211yesmaximal, characteristic, complemented
4#23765564C211yesmaximal, characteristic, complemented
5#23765557C211yesmaximal, characteristic, complemented
6#23765462C211yesmaximal, characteristic, complemented
7#23765455C211yesmaximal, characteristic, complemented
8#23765340C211yesmaximal, characteristic, complemented
9#23765323C211yesmaximal, characteristic
10#23765274C211yesmaximal, characteristic, complemented
11#23765269C211yesmaximal, characteristic, complemented
12#23765264C211yesmaximal, characteristic, complemented
13#23765262C211yesmaximal, characteristic, complemented
14#23765203C211yesmaximal, characteristic, complemented
15C4×#135×C79C211yesmaximal, characteristic, complemented
16#19099097—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579105C2211yescharacteristic
19C2×#14579104C2211yescharacteristic
20C2×#14579102C2211yescharacteristic
21#19098965C2211yescharacteristic
22#19098947C2211yescharacteristic
23#19098918—21no—
24#19098804C2211yescharacteristic
25#19098799C2211yescharacteristic
26#19098792—21no—
27C4×Dic316C2211yescharacteristic
28C4×#11929613—21no—
29C4×D632C2211yescharacteristic
30#19098390C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098388C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35C4⋊(C79⋊C16)C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098366C2211yescharacteristic
39#19098366C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098363C2211yescharacteristic
43#19098363C2211yescharacteristic
44C22⋊(C79⋊C16)C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46#19098339C2211yescharacteristic
47#19098339C2211yescharacteristic
48C2×#135×C79C2211yescharacteristic
49C4×#41×C79—21no—
50C4×Q16×C79C2211yescharacteristic
51C4×Q16×C79C2211yescharacteristic
52C4×SD16×C79—21no—
53C4×SD16×C79—21no—
54C4×D8×C79C2211yescharacteristic
55C4×D8×C79C2211yescharacteristic
56#427×C79C2211yescharacteristic
57#416×C79—21no—
58C2×C4×C8×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60C2×#11929613—21no—
61C2×D632C2311yescharacteristic
62#14579155—41no—
63#14579153—21no—
64#14579149—21no—
65#14579144—41no—
66#14579140—21no—
67#14579135—21no—
68C4×Dic158—21no—
69C4×D316—21no—
70C4×C22⋊D79—41no—
71#14579118—41no—
72#14579107—22no—
73#14579105—22no—
74#14579104—22no—
75#14579102—22no—
76C2×C79⋊C16C2311yescharacteristic
77C2×C79⋊C16C2311yescharacteristic
78C2×C79⋊C16C2311yescharacteristic
79C2×C79⋊C16C2311yescharacteristic
80#14579095C2311yescharacteristic
81#14579095C2311yescharacteristic
82#14579093—21no—
83#14579086—21no—
84#14579086—21no—
85#14579083—21no—
86#14579083—21no—
87C2×#41×C79—21no—
88#135×C79—22no—
89#135×C79—22no—
90C2×Q16×C79C2311yescharacteristic
91C2×SD16×C79—21no—
92C2×D8×C79C2311yescharacteristic
93C22×C8×C79D411yescharacteristic
94C22×C8×C79C2311yescharacteristic
95C4×Q8×C79—21no—
96C4×Q8×C79—21no—
97C4×D4×C79—21no—
98C4×D4×C79—21no—
99C4×D4×C79—41no—
100C4×D4×C79—41no—
101#117×C79—21no—
102#117×C79—41no—
103C2×C42×C79—41no—
104C2×C42×C79D411yescharacteristic
105#107×C79C2311yescharacteristic
106#107×C79C2311yescharacteristic
107#106×C79—21no—
108#103×C79—21no—
109#103×C79—21no—
110#102×C79—21no—
111#102×C79—21no—
112C4×C8×C79D411yescharacteristic
113C4×C8×C79D411yescharacteristic
114C4×C8×C79C2311yescharacteristic
115C4×C8×C79C2311yescharacteristic
116C2×Dic158—21no—
117C2×D316—21no—
118C2×C22⋊D79—41no—
119C2×C4×D79—41no—
120Dic316—21no—
121Dic316—21no—
122#11929613—42no—
123D632—21no—
124D632—21no—
125C4⋊Dic79—21no—
126C4⋊Dic79—21no—
127Dic79⋊C4—41no—
128Dic79⋊C4—41no—
129Dic79⋊C4—41no—
130Dic79⋊C4—41no—
131C22⋊Dic79—41no—
132(C2×D79)⋊C4—41no—
133(C2×D79)⋊C4—41no—
134(C2×D79)⋊C4—41no—
135(C2×D79)⋊C4—41no—
136C4×Dic79—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141#41×C79—24no—
142#41×C79—42no—
143C2×Q8×C79—21no—
144C2×D4×C79—21no—
145C2×D4×C79—41no—
146C22×C4×C79—41no—
147C22×C4×C79—41no—
148C22×C4×C79—41no—
149C22×C4×C79D811yescharacteristic
150C22×C4×C79D811yescharacteristic
151C22×C4×C79C2×D411yescharacteristic
152Q16×C79—21no—
153Q16×C79#4112yes—
154SD16×C79—24no—
155D8×C79—21no—
156D8×C79#4112yes—
157C2×C8×C79—21no—
158C2×C8×C79—21no—
159C2×C8×C79—41no—
160C2×C8×C79C2×D411yescharacteristic
161C2×C8×C79C2×D411yescharacteristic
162C2×C8×C79C2×D411yescharacteristic
163C2×C8×C79C2411yescharacteristic
164C4⋊C4×C79—21no—
165C4⋊C4×C79—21no—
166C4⋊C4×C79—41no—
167C4⋊C4×C79—41no—
168C22⋊C4×C79—41no—
169C22⋊C4×C79—41no—
170C22⋊C4×C79—41no—
171C42×C79—41no—
172C42×C79—41no—
173C42×C79—41no—
174C42×C79D811yescharacteristic
175C42×C79D811yescharacteristic
176C42×C79C2×D411yescharacteristic
177C42×C79C2×D411yescharacteristic
178C23×C79—41no—
179C23×C79C2×D811yescharacteristic
180C22×D79—41no—
181Q8×C79—22no—
182Q8×C79—41no—
183D4×C79—22no—
184D4×C79—44no—
185D4×C79—41no—
186C2×C4×C79—21no—
187C2×C4×C79—21no—
188C2×C4×C79—41no—
189C2×C4×C79—41no—
190C2×C4×C79—44no—
191C2×C4×C79—41no—
192C2×C4×C79—41no—
193C2×C4×C79—41no—
194C2×C4×C79—82no—
195C2×C4×C79—82no—
196C2×C4×C79C2×D811yescharacteristic
197C2×C4×C79C2×D811yescharacteristic
198C2×C4×C79C2×D811yescharacteristic
199C2×C4×C79C2×D811yescharacteristic
200C2×C4×C79C2×D811yescharacteristic
201C2×C4×C79C22×D411yescharacteristic
202C22⋊D79—82no—
203C2×Dic79—41no—
204C2×Dic79—41no—
205C2×Dic79—41no—
206D316—41no—
207D316—41no—
208C4×D79—82no—
209Dic158—41no—
210Dic158—41no—
211C8×C79—21no—
212C8×C79—21no—
213C8×C79—21no—
214C8×C79C2×#4111yescharacteristic
215C8×C79C2×#4111yescharacteristic
216C22×C79—21no—
217C22×C79—21no—
218C22×C79—21no—
219C22×C79—42no—
220C22×C79—81no—
221C22×C79—81no—
222C22×C79C22×D811yescharacteristic
223C2×D79—81no—
224C2×D79—81no—
225C2×D79—81no—
226C4×C79—21no—
227C4×C79—21no—
228C4×C79—21no—
229C4×C79—21no—
230C4×C79—21no—
231C4×C79—42no—
232C4×C79—81no—
233C4×C79—81no—
234C4×C79#52511yescharacteristic
235C4×C79#57511yescharacteristic
236Dic79—81no—
237Dic79—81no—
238Dic79—81no—
239#33671—791nomaximal, Sylow
240C2×C79—41no—
241C2×C79—82no—
242C2×C79#310011yescharacteristic
243C2×C79#341411yescharacteristic
244C2×C79#341411yescharacteristic
245D79—161no—
246#3168—791no—
247C4×#135D7911yescharacteristic, complemented
248#2237—791no—
249#2234—791no—
250#2229—791no—
251#2228—791no—
252#2211—791no—
253#2210—791no—
254#2189—791no—
255#2186—791no—
256#2177—791no—
257#2176—791no—
258#2175—791no—
259#2175—791no—
260#2163—791no—
261C79#3367111yescharacteristic, complemented, Sylow
262C2×#135C2×D7911yescharacteristic
263#529—1581no—
264C4×#41—21no—
265C2×Q32—791no—
266C2×SD32—791no—
267C2×SD32—791no—
268C2×D16—791no—
269#499—791no—
270#495—791no—
271#488—1581no—
272#467—791no—
273#466—791no—
274#465—1581no—
275C4×Q16—791no—
276C4×Q16C2×D7911yescharacteristic
277C4×Q16C2×D7911yescharacteristic
278C4×SD16—21no—
279C4×SD16—21no—
280C4×SD16—1581no—
281C4×D8—791no—
282C4×D8C2×D7911yescharacteristic
283C4×D8C2×D7911yescharacteristic
284#427C2×D7911yescharacteristic
285#416—21no—
286C2×C4×C8C2×D7911yescharacteristic
287#367—791no—
288#367—791no—
289#366—791no—
290#366—791no—
291C4⋊C16—791no—
292C4⋊C16—791no—
293#358—791no—
294#358—791no—
295#358—791no—
296#358—791no—
297#357—791no—
298#357—791no—
299#357—791no—
300#357—791no—
301C22⋊C16—791no—
302C22⋊C16—791no—
303#346—791no—
304#346—791no—
305C2×#41—21no—
306#135—22no—
307#135—22no—
308C2×Q16—791no—
309C2×Q16C22×D7911yescharacteristic
310C2×SD16—21no—
311C2×SD16—1581no—
312C2×D8—791no—
313C2×D8C22×D7911yescharacteristic
314C22×C8C22⋊D7911yescharacteristic
315C22×C8C22×D7911yescharacteristic
316#126—3161no—
317#125—1581no—
318#124—1581no—
319#123—3161no—
320C22⋊Q8—1581no—
321#121—1581no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—1581no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—41no—
328C4×D4—41no—
329C4×D4—3161no—
330C4×D4—1581no—
331#117—21no—
332#117—41no—
333#117—3161no—
334C2×C42—41no—
335C2×C42C22⋊D7911yescharacteristic
336Q32—1582no—
337SD32—1582no—
338SD32—1582no—
339D16—1582no—
340C2×C16—791no—
341C2×C16—791no—
342C2×C16—791no—
343C2×C16—791no—
344#107—791no—
345#107—791no—
346#107C22×D7911yescharacteristic
347#107C22×D7911yescharacteristic
348#106—21no—
349#106—1581no—
350#103—21no—
351#103—21no—
352#103—1581no—
353#103—1581no—
354#102—21no—
355#102—21no—
356#102—1581no—
357#102—1581no—
358C4×C8C22⋊D7911yescharacteristic
359C4×C8C22⋊D7911yescharacteristic
360C4×C8C22×D7911yescharacteristic
361C4×C8C22×D7911yescharacteristic
362#41—24no—
363#41—42no—
364C2×Q8—21no—
365C2×Q8—1581no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—3161no—
369C2×D4—1581no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—3161no—
374C22×C4#1192960911yescharacteristic
375C22×C4#1192960911yescharacteristic
376C22×C4C2×C22⋊D7911yescharacteristic
377Q16—21no—
378Q16—1581no—
379Q16—1581no—
380Q16#1192962412yes—
381SD16—24no—
382SD16—3162no—
383D8—21no—
384D8—1581no—
385D8—1581no—
386D8#1192962412yes—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
390C2×C8C2×C22⋊D7911yescharacteristic
391C2×C8C2×C22⋊D7911yescharacteristic
392C2×C8C2×C22⋊D7911yescharacteristic
393C2×C8C23×D7911yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—3161no—
399C4⋊C4—3161no—
400C4⋊C4—3161no—
401C4⋊C4—3161no—
402C4⋊C4—1581no—
403C4⋊C4—1581no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—3161no—
408C22⋊C4—3161no—
409C22⋊C4—3161no—
410C22⋊C4—3161no—
411C22⋊C4—3161no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—3161no—
416C42#1192960911yescharacteristic
417C42#1192960911yescharacteristic
418C42C2×C22⋊D7911yescharacteristic
419C42C2×C22⋊D7911yescharacteristic
420C16—1581no—
421C16—1581no—
422C16—1581no—
423C16—1581no—
424C23—41no—
425C23—3161no—
426C23C2×#1192960911yescharacteristic
427Q8—22no—
428Q8—41no—
429Q8—3161no—
430Q8—3161no—
431D4—22no—
432D4—44no—
433D4—41no—
434D4—3161no—
435D4—3161no—
436D4—6322no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—3161no—
448C2×C4—3161no—
449C2×C4—3161no—
450C2×C4—6322no—
451C2×C4C2×#1192960911yescharacteristic
452C2×C4C2×#1192960911yescharacteristic
453C2×C4C2×#1192960911yescharacteristic
454C2×C4C2×#1192960911yescharacteristic
455C2×C4C2×#1192960911yescharacteristic
456C2×C4C22×C22⋊D7911yescharacteristic
457C8—21no—
458C8—21no—
459C8—21no—
460C8C2×#1192962411yescharacteristic
461C8C2×#1192962411yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—81no—
467C22—81no—
468C22—6321no—
469C22—6321no—
470C22—6321no—
471C22C22×#1192960911yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—81no—
479C4—81no—
480C4—6321no—
481C4—6321no—
482C4—6321no—
483C4#1909907411yescharacteristic
484C4#1909936111yescharacteristic
485C2—41no—
486C2—82no—
487C2—12641no—
488C2#2377340011yescharacteristic
489C2#2377653411yescharacteristic
490C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC4×#135×C79C2
socleC22×C79C22×D8
prime radical#23765323C2
hypercenterC4×#135D79
hypocenterC79#33671
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #23773970C2semidirect
2. #23765608C2semidirect
3. #23765591C2semidirect
4. #23765564C2semidirect
5. #23765557C2semidirect
6. #23765462C2semidirect
7. #23765455C2semidirect
8. #23765340C2semidirect
10. #23765274C2semidirect
11. #23765269C2semidirect
12. #23765264C2semidirect
13. #23765262C2semidirect
14. #23765203C2semidirect
15. C4×#135×C79C2semidirect
247. C4×#135D79semidirect
261. C79#33671semidirect

Maximal subgroups

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

subgroupindex
#237739702
#237656082
#237655912
#237655642
#237655572
#237654622
#237654552
#237653402
#237653232
#237652742
#237652692
#237652642
#237652622
#237652032
C4×#135×C792
#3367179