Finite Group Databasebeta

← #34444712

Subgroups — #34444712

Order 20224.

488 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774039C211yesmaximal, characteristic, complemented
2#23765547C211yesmaximal, characteristic, complemented
3#23765544C211yesmaximal, characteristic, complemented
4#23765542C211yesmaximal, characteristic, complemented
5#23765539C211yesmaximal, characteristic, complemented
6#23765506C211yesmaximal, characteristic, complemented
7#23765501C211yesmaximal, characteristic
8#23765498C211yesmaximal, characteristic, complemented
9#23765494C211yesmaximal, characteristic, complemented
10#23765462C211yesmaximal, characteristic, complemented
11#23765459C211yesmaximal, characteristic, complemented
12#23765455C211yesmaximal, characteristic, complemented
13#23765452C211yesmaximal, characteristic, complemented
14#23765159C211yesmaximal, characteristic, complemented
15C79×#3175C211yesmaximal, characteristic, complemented
16#19099184—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579105C2211yescharacteristic
19C2×#14579104C2211yescharacteristic
20C2×#14579102C2211yescharacteristic
21C2×#14579101C2211yescharacteristic
22C22×C79⋊C16C2211yescharacteristic
23#19098968C2211yescharacteristic
24#19098947C2211yescharacteristic
25#19098923—21no—
26#19098809—21no—
27#19098804C2211yescharacteristic
28#19098799C2211yescharacteristic
29#19098665—21no—
30C4×Dic316C2211yescharacteristic
31C4×D632C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098390C2211yescharacteristic
34#19098390C2211yescharacteristic
35#19098390C2211yescharacteristic
36C4⋊(C79⋊C16)C2211yescharacteristic
37C4⋊(C79⋊C16)C2211yescharacteristic
38C4⋊(C79⋊C16)C2211yescharacteristic
39C4⋊(C79⋊C16)C2211yescharacteristic
40#19098367C2211yescharacteristic
41#19098367C2211yescharacteristic
42#19098366C2211yescharacteristic
43#19098366C2211yescharacteristic
44#19098364C2211yescharacteristic
45#19098364C2211yescharacteristic
46#19098363C2211yescharacteristic
47#19098363C2211yescharacteristic
48#542×C79—21no—
49#500×C79C2211yescharacteristic
50#495×C79C2211yescharacteristic
51#489×C79—21no—
52#468×C79—21no—
53#467×C79C2211yescharacteristic
54#466×C79C2211yescharacteristic
55#440×C79—21no—
56C4×Q16×C79C2211yescharacteristic
57C4×D8×C79C2211yescharacteristic
58#405×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60C2×#11929613—21no—
61C2×D632C2311yescharacteristic
62#14579161—21no—
63#14579156—41no—
64#14579149—21no—
65#14579147—41no—
66#14579143—41no—
67#14579140—21no—
68#14579139—41no—
69#14579135—21no—
70C4×Dic158—21no—
71C4×D316—21no—
72#14579107—41no—
73#14579105—41no—
74#14579104—41no—
75#14579102—41no—
76#14579101—41no—
77C2×C79⋊C16—41no—
78C2×C79⋊C16C2311yescharacteristic
79C2×C79⋊C16C2311yescharacteristic
80C2×C79⋊C16C2311yescharacteristic
81C2×C79⋊C16C2311yescharacteristic
82#14579095—21no—
83#14579095C2311yescharacteristic
84#14579095C2311yescharacteristic
85#14579086—21no—
86#14579086—21no—
87#14579083—21no—
88#14579083—21no—
89C2×Q16×C79C2311yescharacteristic
90C2×SD16×C79—21no—
91C2×D8×C79C2311yescharacteristic
92C2×M16×C79D411yescharacteristic
93C22×C8×C79C2311yescharacteristic
94C4⋊Q8×C79—21no—
95#126×C79—41no—
96#124×C79—21no—
97#124×C79—41no—
98#123×C79—41no—
99C22⋊Q8×C79—21no—
100C22⋊Q8×C79—41no—
101#121×C79—21no—
102C4×Q8×C79—21no—
103C4×D4×C79—21no—
104#117×C79D411yescharacteristic
105#107×C79—21no—
106#107×C79C2311yescharacteristic
107#107×C79C2311yescharacteristic
108#103×C79—21no—
109#103×C79—21no—
110#102×C79—21no—
111#102×C79—21no—
112#97×C79D411yescharacteristic
113#97×C79D411yescharacteristic
114C4×C8×C79C2311yescharacteristic
115C4×C8×C79C2311yescharacteristic
116C2×Dic158—21no—
117C2×Dic158—41no—
118C2×D316—21no—
119C2×C22⋊D79—41no—
120C2×C4×D79—41no—
121Dic316—21no—
122Dic316—21no—
123#11929613—42no—
124D632—21no—
125D632—21no—
126C4⋊Dic79—21no—
127C4⋊Dic79—21no—
128C4⋊Dic79—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—
141C2×Q8×C79—21no—
142C2×Q8×C79—41no—
143C2×D4×C79—21no—
144C2×D4×C79—41no—
145C22×C4×C79—41no—
146C22×C4×C79C2×D411yescharacteristic
147Q16×C79—21no—
148Q16×C79—21no—
149SD16×C79—42no—
150D8×C79—21no—
151D8×C79—21no—
152M16×C79—41no—
153C2×C8×C79—21no—
154C2×C8×C79—21no—
155C2×C8×C79C2×D411yescharacteristic
156C2×C8×C79C2×D411yescharacteristic
157C2×C8×C79C2×D411yescharacteristic
158C2×C8×C79C2411yescharacteristic
159C4⋊C4×C79—21no—
160C4⋊C4×C79—21no—
161C4⋊C4×C79—41no—
162C4⋊C4×C79—41no—
163C4⋊C4×C79—41no—
164C4⋊C4×C79D811yescharacteristic
165C4⋊C4×C79D811yescharacteristic
166C22⋊C4×C79—41no—
167C22⋊C4×C79—41no—
168C22⋊C4×C79—41no—
169C22⋊C4×C79—41no—
170C22⋊C4×C79—41no—
171C22⋊C4×C79D811yescharacteristic
172C22⋊C4×C79D811yescharacteristic
173C42×C79—41no—
174C42×C79C2×D411yescharacteristic
175C42×C79C2×D411yescharacteristic
176C23×C79—41no—
177C23×C79C2×D811yescharacteristic
178C22×D79—41no—
179Q8×C79—41no—
180Q8×C79—41no—
181Q8×C79—161no—
182D4×C79—41no—
183D4×C79—41no—
184D4×C79—161no—
185C2×C4×C79—21no—
186C2×C4×C79—21no—
187C2×C4×C79—41no—
188C2×C4×C79—41no—
189C2×C4×C79—41no—
190C2×C4×C79—161no—
191C2×C4×C79C2×D811yescharacteristic
192C2×C4×C79C2×D811yescharacteristic
193C2×C4×C79C2×D811yescharacteristic
194C2×C4×C79C2×D811yescharacteristic
195C2×C4×C79C2×D811yescharacteristic
196C2×C4×C79C22×D411yescharacteristic
197C22⋊D79—161no—
198C2×Dic79—41no—
199C2×Dic79—41no—
200C2×Dic79—41no—
201D316—41no—
202D316—41no—
203C4×D79—161no—
204Dic158—41no—
205Dic158—41no—
206Dic158—161no—
207C8×C79—21no—
208C8×C79—21no—
209C8×C79—21no—
210C8×C79H(5,𝔽2)11yescharacteristic
211C8×C792−1+411yescharacteristic
212C22×C79—21no—
213C22×C79—21no—
214C22×C79—21no—
215C22×C79—81no—
216C22×C79—81no—
217C22×C79—81no—
218C22×C79C22×D811yescharacteristic
219C2×D79—81no—
220C2×D79—81no—
221C2×D79—81no—
222C4×C79—21no—
223C4×C79—21no—
224C4×C79—21no—
225C4×C79—21no—
226C4×C79—21no—
227C4×C79—81no—
228C4×C79—81no—
229C4×C79—81no—
230C4×C79#53711yescharacteristic
231C4×C79#57511yescharacteristic
232Dic79—81no—
233Dic79—81no—
234Dic79—81no—
235#33829—791nomaximal, Sylow
236C2×C79—41no—
237C2×C79—161no—
238C2×C79#324511yescharacteristic
239C2×C79#341611yescharacteristic
240C2×C79#341611yescharacteristic
241D79—161no—
242#3175—791no—
243#3175D7911yescharacteristic, complemented
244#2226—791no—
245#2226—791no—
246#2225—791no—
247#2225—791no—
248#2219—791no—
249#2218—791no—
250#2217—791no—
251#2217—791no—
252#2211—791no—
253#2211—791no—
254#2210—791no—
255#2210—791no—
256#2151—791no—
257C79#3382911yescharacteristic, complemented, Sylow
258#542—21no—
259#542—1581no—
260C2×Q32—791no—
261C2×SD32—791no—
262C2×SD32—791no—
263C2×D16—791no—
264C2×M32—791no—
265C22×C16—791no—
266#500—791no—
267#500C2×D7911yescharacteristic
268#495—791no—
269#495C2×D7911yescharacteristic
270#489—21no—
271#489—1581no—
272#468—21no—
273#468—1581no—
274#467—791no—
275#467C2×D7911yescharacteristic
276#466—791no—
277#466C2×D7911yescharacteristic
278#440—21no—
279#440—1581no—
280C4×Q16—791no—
281C4×Q16C2×D7911yescharacteristic
282C4×D8—791no—
283C4×D8C2×D7911yescharacteristic
284#405C2×D7911yescharacteristic
285#367—791no—
286#367—791no—
287#367—791no—
288#367—791no—
289C4⋊C16—791no—
290C4⋊C16—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—
301C2×Q16—791no—
302C2×Q16C22×D7911yescharacteristic
303C2×SD16—21no—
304C2×SD16—1581no—
305C2×D8—791no—
306C2×D8C22×D7911yescharacteristic
307C2×M16C22⋊D7911yescharacteristic
308C22×C8C22×D7911yescharacteristic
309C4⋊Q8—21no—
310C4⋊Q8—1581no—
311#126—41no—
312#126—3161no—
313#124—21no—
314#124—41no—
315#124—3161no—
316#124—1581no—
317#123—41no—
318#123—3161no—
319C22⋊Q8—21no—
320C22⋊Q8—41no—
321C22⋊Q8—3161no—
322C22⋊Q8—1581no—
323#121—21no—
324#121—1581no—
325C4×Q8—21no—
326C4×Q8—1581no—
327C4×D4—21no—
328C4×D4—1581no—
329#117C22⋊D7911yescharacteristic
330Q32—3161no—
331SD32—3161no—
332SD32—3161no—
333D16—3161no—
334M32—3161no—
335C2×C16—3161no—
336C2×C16—791no—
337C2×C16—791no—
338C2×C16—791no—
339C2×C16—791no—
340#107—21no—
341#107—791no—
342#107—791no—
343#107—1581no—
344#107C22×D7911yescharacteristic
345#107C22×D7911yescharacteristic
346#103—21no—
347#103—21no—
348#103—1581no—
349#103—1581no—
350#102—21no—
351#102—21no—
352#102—1581no—
353#102—1581no—
354#97C22⋊D7911yescharacteristic
355#97C22⋊D7911yescharacteristic
356C4×C8C22×D7911yescharacteristic
357C4×C8C22×D7911yescharacteristic
358C2×Q8—21no—
359C2×Q8—41no—
360C2×Q8—3161no—
361C2×Q8—1581no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—3161no—
365C2×D4—1581no—
366C22×C4—41no—
367C22×C4—3161no—
368C22×C4C2×C22⋊D7911yescharacteristic
369Q16—21no—
370Q16—21no—
371Q16—1581no—
372Q16—1581no—
373SD16—42no—
374SD16—3162no—
375D8—21no—
376D8—21no—
377D8—1581no—
378D8—1581no—
379M16—41no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8C2×C22⋊D7911yescharacteristic
383C2×C8C2×C22⋊D7911yescharacteristic
384C2×C8C2×C22⋊D7911yescharacteristic
385C2×C8C23×D7911yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—3161no—
392C4⋊C4—3161no—
393C4⋊C4—3161no—
394C4⋊C4—1581no—
395C4⋊C4—1581no—
396C4⋊C4#1192960911yescharacteristic
397C4⋊C4#1192960911yescharacteristic
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—3161no—
404C22⋊C4—3161no—
405C22⋊C4—3161no—
406C22⋊C4—3161no—
407C22⋊C4—3161no—
408C22⋊C4#1192960911yescharacteristic
409C22⋊C4#1192960911yescharacteristic
410C42—41no—
411C42—3161no—
412C42C2×C22⋊D7911yescharacteristic
413C42C2×C22⋊D7911yescharacteristic
414C16—1581no—
415C16—1581no—
416C16—1581no—
417C16—1581no—
418C23—41no—
419C23—3161no—
420C23C2×#1192960911yescharacteristic
421Q8—41no—
422Q8—41no—
423Q8—161no—
424Q8—3161no—
425Q8—3161no—
426Q8—12641no—
427D4—41no—
428D4—41no—
429D4—161no—
430D4—3161no—
431D4—3161no—
432D4—12641no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—161no—
439C2×C4—3161no—
440C2×C4—3161no—
441C2×C4—3161no—
442C2×C4—12641no—
443C2×C4C2×#1192960911yescharacteristic
444C2×C4C2×#1192960911yescharacteristic
445C2×C4C2×#1192960911yescharacteristic
446C2×C4C2×#1192960911yescharacteristic
447C2×C4C2×#1192960911yescharacteristic
448C2×C4C22×C22⋊D7911yescharacteristic
449C8—21no—
450C8—21no—
451C8—21no—
452C8#1457920711yescharacteristic
453C8#1457920811yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—6321no—
461C22—6321no—
462C22—6321no—
463C22C22×#1192960911yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—6321no—
473C4—6321no—
474C4—6321no—
475C4#1909914111yescharacteristic
476C4#1909936111yescharacteristic
477C2—41no—
478C2—161no—
479C2—12641no—
480C2#2377470711yescharacteristic
481C2#2377654411yescharacteristic
482C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3175C2
socleC22×C79C22×D8
prime radical#23765501C2
hypercenter#3175D79
hypocenterC79#33829
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #23774039C2semidirect
2. #23765547C2semidirect
3. #23765544C2semidirect
4. #23765542C2semidirect
5. #23765539C2semidirect
6. #23765506C2semidirect
8. #23765498C2semidirect
9. #23765494C2semidirect
10. #23765462C2semidirect
11. #23765459C2semidirect
12. #23765455C2semidirect
13. #23765452C2semidirect
14. #23765159C2semidirect
15. C79×#3175C2semidirect
243. #3175D79semidirect
257. C79#33829semidirect

Maximal subgroups

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

subgroupindex
#237740392
#237655472
#237655442
#237655422
#237655392
#237655062
#237655012
#237654982
#237654942
#237654622
#237654592
#237654552
#237654522
#237651592
C79×#31752
#3382979