Finite Group Databasebeta

← #34444643

Subgroups — #34444643

Order 20224.

496 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#23773970C211yesmaximal, characteristic, complemented
2#23765564C211yesmaximal, characteristic, complemented
3#23765561C211yesmaximal, characteristic, complemented
4#23765557C211yesmaximal, characteristic, complemented
5#23765554C211yesmaximal, characteristic, complemented
6#23765515C211yesmaximal, characteristic, complemented
7#23765511C211yesmaximal, characteristic, complemented
8#23765506C211yesmaximal, characteristic, complemented
9#23765501C211yesmaximal, characteristic
10#23765462C211yesmaximal, characteristic, complemented
11#23765459C211yesmaximal, characteristic, complemented
12#23765455C211yesmaximal, characteristic, complemented
13#23765452C211yesmaximal, characteristic, complemented
14#23765156C211yesmaximal, characteristic, complemented
15C79×#3168C211yesmaximal, characteristic, complemented
16#19099097—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579105C2211yescharacteristic
19C2×#14579104C2211yescharacteristic
20C2×#14579102C2211yescharacteristic
21C2×#14579101C2211yescharacteristic
22C2×#14579101C2211yescharacteristic
23#19098965C2211yescharacteristic
24#19098947C2211yescharacteristic
25#19098918—21no—
26#19098804C2211yescharacteristic
27#19098799C2211yescharacteristic
28#19098792—21no—
29C4×Dic316C2211yescharacteristic
30C4×#11929613—21no—
31C4×D632C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098390C2211yescharacteristic
34#19098388C2211yescharacteristic
35#19098388C2211yescharacteristic
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#529×C79—21no—
49#499×C79C2211yescharacteristic
50#495×C79C2211yescharacteristic
51#488×C79—21no—
52#467×C79C2211yescharacteristic
53#466×C79C2211yescharacteristic
54#465×C79—21no—
55C4×Q16×C79C2211yescharacteristic
56C4×SD16×C79—21no—
57C4×D8×C79C2211yescharacteristic
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—41no—
73#14579105—41no—
74#14579104—41no—
75#14579102—41no—
76#14579101—41no—
77#14579101—41no—
78C2×C79⋊C16C2311yescharacteristic
79C2×C79⋊C16C2311yescharacteristic
80C2×C79⋊C16C2311yescharacteristic
81C2×C79⋊C16C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579095C2311yescharacteristic
84#14579093—21no—
85#14579086—21no—
86#14579086—21no—
87#14579083—21no—
88#14579083—21no—
89C2×Q16×C79C2311yescharacteristic
90C2×SD16×C79—21no—
91C2×D8×C79C2311yescharacteristic
92C22×C8×C79D411yescharacteristic
93C22×C8×C79C2311yescharacteristic
94#126×C79—41no—
95#125×C79—21no—
96#124×C79—21no—
97#123×C79—41no—
98C22⋊Q8×C79—21no—
99#121×C79—21no—
100C4×Q8×C79—21no—
101C4×D4×C79—21no—
102C4×D4×C79—41no—
103#117×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—
141C2×Q8×C79—21no—
142C2×D4×C79—21no—
143C2×D4×C79—41no—
144C22×C4×C79—41no—
145C22×C4×C79D811yescharacteristic
146C22×C4×C79D811yescharacteristic
147C22×C4×C79C2×D411yescharacteristic
148Q16×C79—21no—
149Q16×C79—21no—
150SD16×C79—42no—
151D8×C79—21no—
152D8×C79—21no—
153C2×C8×C79—21no—
154C2×C8×C79—21no—
155C2×C8×C79—41no—
156C2×C8×C79C2×D411yescharacteristic
157C2×C8×C79C2×D411yescharacteristic
158C2×C8×C79C2×D411yescharacteristic
159C2×C8×C79C2411yescharacteristic
160C4⋊C4×C79—21no—
161C4⋊C4×C79—21no—
162C4⋊C4×C79—41no—
163C4⋊C4×C79—41no—
164C4⋊C4×C79—41no—
165C4⋊C4×C79—41no—
166C22⋊C4×C79—41no—
167C22⋊C4×C79—41no—
168C22⋊C4×C79—41no—
169C22⋊C4×C79—41no—
170C22⋊C4×C79—41no—
171C42×C79—41no—
172C42×C79D811yescharacteristic
173C42×C79D811yescharacteristic
174C42×C79C2×D411yescharacteristic
175C42×C79C2×D411yescharacteristic
176C23×C79—41no—
177C23×C79C2×D811yescharacteristic
178C22×D79—41no—
179Q8×C79—41no—
180Q8×C79—41no—
181D4×C79—41no—
182D4×C79—41no—
183D4×C79—82no—
184C2×C4×C79—21no—
185C2×C4×C79—21no—
186C2×C4×C79—41no—
187C2×C4×C79—41no—
188C2×C4×C79—41no—
189C2×C4×C79—41no—
190C2×C4×C79—41no—
191C2×C4×C79—82no—
192C2×C4×C79C2×D811yescharacteristic
193C2×C4×C79C2×D811yescharacteristic
194C2×C4×C79C2×D811yescharacteristic
195C2×C4×C79C2×D811yescharacteristic
196C2×C4×C79C2×D811yescharacteristic
197C2×C4×C79C22×D411yescharacteristic
198C22⋊D79—82no—
199C2×Dic79—41no—
200C2×Dic79—41no—
201C2×Dic79—41no—
202D316—41no—
203D316—41no—
204C4×D79—82no—
205Dic158—41no—
206Dic158—41no—
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#33822—791nomaximal, Sylow
236C2×C79—41no—
237C2×C79—161no—
238C2×C79#324511yescharacteristic
239C2×C79#341411yescharacteristic
240C2×C79#341411yescharacteristic
241D79—161no—
242#3168—791no—
243#3168D7911yescharacteristic, complemented
244#2229—791no—
245#2229—791no—
246#2228—791no—
247#2228—791no—
248#2220—791no—
249#2220—791no—
250#2219—791no—
251#2218—791no—
252#2211—791no—
253#2211—791no—
254#2210—791no—
255#2210—791no—
256#2150—791no—
257C79#3382211yescharacteristic, complemented, Sylow
258#529—21no—
259#529—1581no—
260C2×Q32—791no—
261C2×SD32—791no—
262C2×SD32—791no—
263C2×D16—791no—
264C2×M32—791no—
265C2×M32—791no—
266#499—791no—
267#499C2×D7911yescharacteristic
268#495—791no—
269#495C2×D7911yescharacteristic
270#488—21no—
271#488—1581no—
272#467—791no—
273#467C2×D7911yescharacteristic
274#466—791no—
275#466C2×D7911yescharacteristic
276#465—21no—
277#465—1581no—
278C4×Q16—791no—
279C4×Q16C2×D7911yescharacteristic
280C4×SD16—21no—
281C4×SD16—1581no—
282C4×D8—791no—
283C4×D8C2×D7911yescharacteristic
284C2×C4×C8C2×D7911yescharacteristic
285#367—791no—
286#367—791no—
287#366—791no—
288#366—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
307C22×C8C22⋊D7911yescharacteristic
308C22×C8C22×D7911yescharacteristic
309#126—41no—
310#126—3161no—
311#125—21no—
312#125—1581no—
313#124—21no—
314#124—1581no—
315#123—41no—
316#123—3161no—
317C22⋊Q8—21no—
318C22⋊Q8—1581no—
319#121—21no—
320#121—1581no—
321C4×Q8—21no—
322C4×Q8—1581no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—3161no—
326C4×D4—1581no—
327#117—41no—
328#117—3161no—
329C2×C42C22⋊D7911yescharacteristic
330Q32—3161no—
331SD32—3161no—
332SD32—3161no—
333D16—3161no—
334M32—3161no—
335M32—3161no—
336C2×C16—791no—
337C2×C16—791no—
338C2×C16—791no—
339C2×C16—791no—
340#107—791no—
341#107—791no—
342#107C22×D7911yescharacteristic
343#107C22×D7911yescharacteristic
344#106—21no—
345#106—1581no—
346#103—21no—
347#103—21no—
348#103—1581no—
349#103—1581no—
350#102—21no—
351#102—21no—
352#102—1581no—
353#102—1581no—
354C4×C8C22⋊D7911yescharacteristic
355C4×C8C22⋊D7911yescharacteristic
356C4×C8C22×D7911yescharacteristic
357C4×C8C22×D7911yescharacteristic
358C2×Q8—21no—
359C2×Q8—1581no—
360C2×D4—21no—
361C2×D4—41no—
362C2×D4—3161no—
363C2×D4—1581no—
364C22×C4—41no—
365C22×C4—3161no—
366C22×C4#1192960911yescharacteristic
367C22×C4#1192960911yescharacteristic
368C22×C4C2×C22⋊D7911yescharacteristic
369Q16—21no—
370Q16—21no—
371Q16—1581no—
372Q16—1581no—
373SD16—42no—
374SD16—3162no—
375D8—21no—
376D8—21no—
377D8—1581no—
378D8—1581no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—41no—
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—41no—
392C4⋊C4—3161no—
393C4⋊C4—3161no—
394C4⋊C4—3161no—
395C4⋊C4—3161no—
396C4⋊C4—1581no—
397C4⋊C4—1581no—
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—
408C42—41no—
409C42—3161no—
410C42#1192960911yescharacteristic
411C42#1192960911yescharacteristic
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—3161no—
424Q8—3161no—
425D4—41no—
426D4—41no—
427D4—82no—
428D4—3161no—
429D4—3161no—
430D4—6322no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—82no—
439C2×C4—3161no—
440C2×C4—3161no—
441C2×C4—3161no—
442C2×C4—6322no—
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#2377653411yescharacteristic
482C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3168C2
socleC22×C79C22×D8
prime radical#23765501C2
hypercenter#3168D79
hypocenterC79#33822
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #23773970C2semidirect
2. #23765564C2semidirect
3. #23765561C2semidirect
4. #23765557C2semidirect
5. #23765554C2semidirect
6. #23765515C2semidirect
7. #23765511C2semidirect
8. #23765506C2semidirect
10. #23765462C2semidirect
11. #23765459C2semidirect
12. #23765455C2semidirect
13. #23765452C2semidirect
14. #23765156C2semidirect
15. C79×#3168C2semidirect
243. #3168D79semidirect
257. C79#33822semidirect

Maximal subgroups

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

subgroupindex
#237739702
#237655642
#237655612
#237655572
#237655542
#237655152
#237655112
#237655062
#237655012
#237654622
#237654592
#237654552
#237654522
#237651562
C79×#31682
#3382279