Finite Group Databasebeta

← #34445330

Subgroups — #34445330

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
1#23775769C211yesmaximal, characteristic
2#23765645C211yesmaximal, characteristic, complemented
3#23765605C211yesmaximal, characteristic, complemented
4#23765595C211yesmaximal, characteristic, complemented
5#23765564C211yesmaximal, characteristic, complemented
6#23765506C211yesmaximal, characteristic, complemented
7#23765489C211yesmaximal, characteristic, complemented
8#23765447C211yesmaximal, characteristic, complemented
9#23765334C211yesmaximal, characteristic, complemented
10#23765334C211yesmaximal, characteristic, complemented
11#23765269C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic
13#23765120C211yesmaximal, characteristic, complemented
14C2×#19098367C211yesmaximal, characteristic, complemented
15D4×Q16×C79C211yesmaximal, characteristic
16#19099253—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579107C2211yescharacteristic
19C2×#14579107C2211yescharacteristic
20C2×#14579101C2211yescharacteristic
21C22×C79⋊C16C2211yescharacteristic
22#19098965C2211yescharacteristic
23#19098905—21no—
24#19098894—21no—
25#19098871—21no—
26#19098804C2211yescharacteristic
27#19098804C2211yescharacteristic
28C4×Dic316C2211yescharacteristic
29#19098601C2211yescharacteristic
30C2×#14579095C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098367C2211yescharacteristic
39#19098367C2211yescharacteristic
40#19098367C2211yescharacteristic
41#19098367C2211yescharacteristic
42#19098367C2211yescharacteristic
43#19098367C2211yescharacteristic
44#19098366C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47#19098339C2211yescharacteristic
48C22×Q16×C79C2211yescharacteristic
49C22×Q16×C79C2211yescharacteristic
50D4×Q8×C79—21no—
51#494×C79C2211yescharacteristic
52#467×C79C2211yescharacteristic
53#467×C79C2211yescharacteristic
54#462×C79—21no—
55#451×C79—21no—
56#451×C79—21no—
57C4×Q16×C79C2211yescharacteristic
58C8×D4×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60#14579154—41no—
61#14579153—21no—
62#14579151—41no—
63#14579142—41no—
64#14579140—21no—
65#14579140—21no—
66C4×Dic158—21no—
67#14579120—21no—
68#14579117—41no—
69C2×C4⋊Dic79—21no—
70#14579107—22no—
71#14579107—41no—
72#14579107—41no—
73#14579101—41no—
74C2×C79⋊C16—41no—
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77C2×C79⋊C16C2311yescharacteristic
78C2×C79⋊C16C2311yescharacteristic
79#14579095—21no—
80#14579095C2311yescharacteristic
81#14579095C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579093—21no—
84#14579093—21no—
85#14579086—21no—
86#14579086—21no—
87#14579086—21no—
88C22×Q8×C79—21no—
89C22×Q8×C79—21no—
90C2×Q16×C79—22no—
91C2×Q16×C79—22no—
92C2×Q16×C79—21no—
93C2×Q16×C79—21no—
94C2×Q16×C79—21no—
95C2×Q16×C79—41no—
96C2×Q16×C79—41no—
97C2×Q16×C79C2311yescharacteristic
98C2×Q16×C79C2311yescharacteristic
99C2×Q16×C79C2311yescharacteristic
100C22×C8×C79C2311yescharacteristic
101C22×C8×C79C2311yescharacteristic
102C4⋊Q8×C79—21no—
103C4⋊Q8×C79—41no—
104C22⋊Q8×C79—21no—
105C22⋊Q8×C79—21no—
106C22⋊Q8×C79—41no—
107C22⋊Q8×C79—41no—
108C4×Q8×C79—21no—
109C4×D4×C79—41no—
110C4×D4×C79D411yescharacteristic
111#107×C79C2311yescharacteristic
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—
120C22×Dic79—41no—
121Dic316—21no—
122Dic316—21no—
123C4⋊Dic79—21no—
124C4⋊Dic79—21no—
125C4⋊Dic79—21no—
126C4⋊Dic79—41no—
127Dic79⋊C4—41no—
128Dic79⋊C4—41no—
129Dic79⋊C4—41no—
130Dic79⋊C4—41no—
131Dic79⋊C4—41no—
132C22⋊Dic79—41no—
133C22⋊Dic79—41no—
134C22⋊Dic79—41no—
135C4×Dic79—41no—
136C4×Dic79—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C2×Q8×C79—21no—
142C2×Q8×C79—21no—
143C2×Q8×C79—21no—
144C2×Q8×C79—42no—
145C2×Q8×C79—42no—
146C2×Q8×C79—41no—
147C2×Q8×C79—41no—
148C2×Q8×C79—81no—
149C2×Q8×C79—81no—
150C2×D4×C79D811yescharacteristic
151C22×C4×C79—41no—
152C22×C4×C79—41no—
153C22×C4×C79C2×D411yescharacteristic
154C22×C4×C79C2×D411yescharacteristic
155Q16×C79—21no—
156Q16×C79—21no—
157Q16×C79—21no—
158Q16×C79—21no—
159Q16×C79—21no—
160Q16×C79—42no—
161Q16×C79—42no—
162Q16×C79—81no—
163Q16×C79#4112yes—
164C2×C8×C79—21no—
165C2×C8×C79—21no—
166C2×C8×C79—21no—
167C2×C8×C79—21no—
168C2×C8×C79C2×D411yescharacteristic
169C2×C8×C79C2×D411yescharacteristic
170C2×C8×C79C2×D411yescharacteristic
171C2×C8×C79C2411yescharacteristic
172C4⋊C4×C79—21no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79—41no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79—41no—
178C4⋊C4×C79D811yescharacteristic
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79D811yescharacteristic
182C22⋊C4×C79D811yescharacteristic
183C42×C79—41no—
184C42×C79C2×D411yescharacteristic
185C23×C79C2×D811yescharacteristic
186C23×C79C2×D811yescharacteristic
187Q8×C79—22no—
188Q8×C79—41no—
189Q8×C79—41no—
190Q8×C79—41no—
191Q8×C79—41no—
192Q8×C79—41no—
193Q8×C79—82no—
194Q8×C79—82no—
195D4×C79—22no—
196C2×C4×C79—21no—
197C2×C4×C79—21no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—41no—
201C2×C4×C79—41no—
202C2×C4×C79—41no—
203C2×C4×C79—41no—
204C2×C4×C79—82no—
205C2×C4×C79—82no—
206C2×C4×C79C2×D811yescharacteristic
207C2×C4×C79C2×D811yescharacteristic
208C2×C4×C79C2×D811yescharacteristic
209C2×C4×C79C2×D811yescharacteristic
210C2×C4×C79C22×D411yescharacteristic
211C2×Dic79—41no—
212C2×Dic79—41no—
213C2×Dic79—41no—
214C2×Dic79—41no—
215C2×Dic79—82no—
216Dic158—41no—
217Dic158—41no—
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—42no—
235C4×C79—81no—
236C4×C79—81no—
237C4×C79—81no—
238C4×C79#54611yescharacteristic
239C4×C79#57711yescharacteristic
240Dic79—81no—
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#3336—791no—
251D4×Q16D7911yescharacteristic
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×Q16C2×D7911yescharacteristic
267C22×Q16C2×D7911yescharacteristic
268#552—1581no—
269D4×Q8—21no—
270C2×Q32—791no—
271C2×Q32—791no—
272C2×Q32—791no—
273C2×M32—791no—
274C22×C16—791no—
275#499—791no—
276#494C2×D7911yescharacteristic
277#485—1581no—
278#483—1581no—
279#479—1581no—
280#467—791no—
281#467—791no—
282#467C2×D7911yescharacteristic
283#467C2×D7911yescharacteristic
284#462—21no—
285#451—21no—
286#451—21no—
287C4×Q16—791no—
288C4×Q16C2×D7911yescharacteristic
289C8×D4C2×D7911yescharacteristic
290#427—791no—
291C2×#107—791no—
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—21no—
310C22×Q8—21no—
311C2×Q16—22no—
312C2×Q16—22no—
313C2×Q16—21no—
314C2×Q16—21no—
315C2×Q16—21no—
316C2×Q16—41no—
317C2×Q16—41no—
318C2×Q16—791no—
319C2×Q16C22×D7911yescharacteristic
320C2×Q16C22×D7911yescharacteristic
321C2×Q16C22×D7911yescharacteristic
322C22×C8C22×D7911yescharacteristic
323C22×C8C22×D7911yescharacteristic
324C4⋊Q8—21no—
325C4⋊Q8—41no—
326#126—3161no—
327#125—3161no—
328#125—1581no—
329#123—3161no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—41no—
333C22⋊Q8—41no—
334C22⋊Q8—1581no—
335C22⋊Q8—1581no—
336C4×Q8—21no—
337C4×Q8—1581no—
338C4×D4—41no—
339C4×D4C22⋊D7911yescharacteristic
340#117—3161no—
341#117—1581no—
342C2×C4⋊C4—1581no—
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—791no—
353#107—791no—
354#107—791no—
355#107—1581no—
356#107C22×D7911yescharacteristic
357#106—1581no—
358#106—1581no—
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—21no—
371C2×Q8—21no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—81no—
377C2×Q8—81no—
378C2×Q8—1581no—
379C2×D4#1192960911yescharacteristic
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—3161no—
383C22×C4C2×C22⋊D7911yescharacteristic
384C22×C4C2×C22⋊D7911yescharacteristic
385Q16—21no—
386Q16—21no—
387Q16—21no—
388Q16—21no—
389Q16—21no—
390Q16—42no—
391Q16—42no—
392Q16—81no—
393Q16—1581no—
394Q16—1581no—
395Q16#1192962312yes—
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—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—3161no—
411C4⋊C4—3161no—
412C4⋊C4—3161no—
413C4⋊C4—3161no—
414C4⋊C4—3161no—
415C4⋊C4—3161no—
416C4⋊C4—1581no—
417C4⋊C4—1581no—
418C4⋊C4—1581no—
419C4⋊C4#1192960911yescharacteristic
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—3161no—
423C22⋊C4—3161no—
424C22⋊C4—3161no—
425C22⋊C4#1192960911yescharacteristic
426C22⋊C4#1192960911yescharacteristic
427C42—41no—
428C42—3161no—
429C42—3161no—
430C42C2×C22⋊D7911yescharacteristic
431C16—1581no—
432C16—1581no—
433C16—1581no—
434C16—1581no—
435C23C2×#1192960911yescharacteristic
436C23C2×#1192960911yescharacteristic
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443Q8—82no—
444Q8—82no—
445Q8—3161no—
446Q8—3161no—
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—82no—
458C2×C4—3161no—
459C2×C4—3161no—
460C2×C4—3161no—
461C2×C4—3161no—
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—
471C8C2×#1192962311yescharacteristic
472C8#1457920611yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22#1909900412yes—
479C22C22×#1192960911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—6321no—
489C4—6321no—
490C4—6321no—
491C4—6321no—
492C4#1909920711yescharacteristic
493C4#1909937411yescharacteristic
494C2—22no—
495C2—41no—
496C2#2377521311yescharacteristic
497C2C2×#1909900411yescharacteristic
498C2#2377655111yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingD4×Q16×C79C2
socleC22×C79C22×D8
prime radicalC2×D4×C79D8
hypercenterD4×Q16D79
hypocenterC79#33879
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
2. #23765645C2semidirect
3. #23765605C2semidirect
4. #23765595C2semidirect
5. #23765564C2semidirect
6. #23765506C2semidirect
7. #23765489C2semidirect
8. #23765447C2semidirect
9. #23765334C2semidirect
10. #23765334C2semidirect
11. #23765269C2semidirect
13. #23765120C2semidirect
14. C2×#19098367C2semidirect
265. C79#33879semidirect

Maximal subgroups

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

subgroupindex
#237757692
#237656452
#237656052
#237655952
#237655642
#237655062
#237654892
#237654472
#237653342
#237652692
#237652262
#237651202
C2×#190983672
D4×Q16×C792
#3387979