Finite Group Databasebeta

← #34183114

Subgroups — #34183114

Order 20224.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773408C211yesmaximal, characteristic
2#23764344C211yesmaximal, characteristic, complemented
3C4×#14579082C211yesmaximal, characteristic
4#23761854C211yesmaximal, characteristic, complemented
5#23761301C211yesmaximal, characteristic, complemented
6#23761238C211yesmaximal, characteristic, complemented
7D4×M16×C79C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098893—21no—
10#19098853C411yescharacteristic, complemented
11#19098853C411yescharacteristic, complemented
12#19098768C411yescharacteristic, complemented
13#19098768C411yescharacteristic, complemented
14#19098714—21no—
15C4×#11929612—21no—
16C4×#11929612—21no—
17#19098616C2211yescharacteristic
18C2×#14579082—21no—
19C2×C4×C79⋊C8C2211yescharacteristic
20C4×C4⋊Dic79C2211yescharacteristic
21#19098325—21no—
22#19098324—21no—
23#19098324—21no—
24#19098293—22no—
25C22×M16×C79—21no—
26C2×C4×D4×C79C2211yescharacteristic
27#436×C79—21no—
28#435×C79—22no—
29C8×D4×C79—21no—
30#423×C79C2211yescharacteristic
31#407×C79—21no—
32C4×M16×C79C2211yescharacteristic
33C2×#11929612—21no—
34C2×#11929612—21no—
35C22×C79⋊C8—21no—
36#14579161C2×C411yescharacteristic
37#14579161C2×C411yescharacteristic
38#14579153—21no—
39#14579140—21no—
40#14579140—21no—
41C4×Dic158—21no—
42C4×Dic158—21no—
43#14579120—21no—
44C2×C4⋊Dic79—21no—
45C2×C4×Dic79—21no—
46#14579092—21no—
47#14579092—21no—
48C4⋊(C79⋊C8)C2×C411yescharacteristic
49C4⋊(C79⋊C8)C2×C411yescharacteristic
50#14579084—21no—
51#14579084—21no—
52#14579082—24no—
53#14579082—21no—
54#14579082—21no—
55C22⋊(C79⋊C8)—21no—
56C4×C79⋊C8—21no—
57C4×C79⋊C8—21no—
58C4×C79⋊C8—21no—
59C79⋊#95—21no—
60C22×D4×C79D411yescharacteristic
61C23×C4×C79—21no—
62C2×M16×C79—22no—
63C2×M16×C79—21no—
64C2×M16×C79—42no—
65C2×M16×C79—42no—
66C2×M16×C79D411yescharacteristic
67C2×M16×C79D411yescharacteristic
68C22×C8×C79—22no—
69C4×D4×C79—21no—
70C4×D4×C79—21no—
71C4×D4×C79C2×C411yescharacteristic
72C4×D4×C79C2×C411yescharacteristic
73C4×D4×C79D411yescharacteristic
74C4×D4×C79D411yescharacteristic
75C2×C4⋊C4×C79D411yescharacteristic
76C2×C22⋊C4×C79—21no—
77C2×C42×C79C2311yescharacteristic
78C4⋊C8×C79—21no—
79C4⋊C8×C79D412yes—
80C22⋊C8×C79—22no—
81C22⋊C8×C79—22no—
82#97×C79—21no—
83C4×C8×C79—21no—
84C2×Dic158—21no—
85C2×Dic158—21no—
86C22×Dic79—41no—
87#11929612—42no—
88#11929612—42no—
89C2×C79⋊C8—22no—
90C2×C79⋊C8—21no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C4⋊Dic79—22no—
95C4⋊Dic79—21no—
96C4⋊Dic79—21no—
97C4⋊Dic79—21no—
98C4⋊Dic79—21no—
99C4⋊Dic79—41no—
100Dic79⋊C4—41no—
101Dic79⋊C4—41no—
102C22⋊Dic79—41no—
103C4×Dic79—41no—
104C4×Dic79—41no—
105C4×Dic79—41no—
106C24×C79—21no—
107C2×D4×C79—22no—
108C2×D4×C79—21no—
109C2×D4×C79—41no—
110C2×D4×C79C22⋊C411yescharacteristic
111C2×D4×C79C22⋊C411yescharacteristic
112C22×C4×C79—21no—
113C22×C4×C79—21no—
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—21no—
117C22×C4×C79—41no—
118C22×C4×C79—41no—
119C22×C4×C79—41no—
120C22×C4×C79—41no—
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79C2×D411yescharacteristic
123C22×C4×C79#4111yescharacteristic
124M16×C79—21no—
125M16×C79—21no—
126M16×C79—21no—
127M16×C79—44no—
128M16×C79—41no—
129M16×C79—41no—
130M16×C79SD1612yescomplemented
131C2×C8×C79—21no—
132C2×C8×C79—21no—
133C2×C8×C79—22no—
134C2×C8×C79—44no—
135C4⋊C4×C79—21no—
136C4⋊C4×C79C22⋊C411yescharacteristic
137C4⋊C4×C79C22⋊C411yescharacteristic
138C22⋊C4×C79—21no—
139C22⋊C4×C79—21no—
140C22⋊C4×C79—21no—
141C22⋊C4×C79—21no—
142C42×C79C22×C411yescharacteristic
143C42×C79C2×D411yescharacteristic
144C42×C79C2×D411yescharacteristic
145C42×C79#4111yescharacteristic
146C23×C79—21no—
147C23×C79—21no—
148C23×C79—42no—
149C23×C79—41no—
150C23×C79—41no—
151C23×C79—81no—
152C23×C79#12111yescharacteristic
153D4×C79—21no—
154D4×C79—21no—
155D4×C79—21no—
156D4×C79—42no—
157D4×C79#10412yes—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79C2×C22⋊C411yescharacteristic
174C2×C4×C79C4×D411yescharacteristic
175C2×C4×C79C4×D411yescharacteristic
176C2×C4×C79#12011yescharacteristic
177C2×C4×C79#12111yescharacteristic
178C2×C4×C79#12311yescharacteristic
179C2×C4×C79C2×SD1611yescharacteristic
180C2×C4×C79#13511yescharacteristic
181C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C79#13711yescharacteristic
183C2×Dic79—42no—
184C2×Dic79—41no—
185C2×Dic79—41no—
186C2×Dic79—81no—
187C2×Dic79—81no—
188Dic158—42no—
189Dic158—41no—
190Dic158—41no—
191C8×C79—22no—
192C8×C79—42no—
193C8×C79—41no—
194C79⋊C8—42no—
195C79⋊C8—41no—
196C79⋊C8—41no—
197C22×C79—21no—
198C22×C79—21no—
199C22×C79—22no—
200C22×C79—41no—
201C22×C79—41no—
202C22×C79—41no—
203C22×C79—41no—
204C22×C79—41no—
205C22×C79—82no—
206C22×C79—82no—
207C22×C79#38611yescharacteristic
208C22×C79#46111yescharacteristic
209C22×C79#46311yescharacteristic
210C4×C79—21no—
211C4×C79—21no—
212C4×C79—21no—
213C4×C79—41no—
214C4×C79—41no—
215C4×C79C2×#10411yescharacteristic
216C4×C79#42111yescharacteristic
217C4×C79C4×SD1611yescharacteristic
218C4×C79#44011yescharacteristic
219C4×C79#45011yescharacteristic
220C4×C79#45211yescharacteristic
221Dic79—82no—
222Dic79—81no—
223Dic79—81no—
224#8989—791nomaximal, Sylow
225C2×C79—21no—
226C2×C79—21no—
227C2×C79—42no—
228C2×C79—81no—
229C2×C79#187111yescharacteristic
230C2×C79#187711yescharacteristic
231C2×C79#188111yescharacteristic
232#3101—791no—
233D4×M16D7911yescharacteristic
234#1986—791no—
235C4×#102—791no—
236#1493—791no—
237#1303—791no—
238#1278—791no—
239C79#898911yescharacteristic, complemented, Sylow
240C22×M16—21no—
241#533—791no—
242C2×C4×D4C2×D7911yescharacteristic
243#483—1581no—
244#476—791no—
245#476—791no—
246#461—791no—
247#461—791no—
248#450—1581no—
249C4×SD16—1581no—
250C4×SD16—1581no—
251#436—21no—
252#435—22no—
253C8×D4—21no—
254#431—791no—
255#423C2×D7911yescharacteristic
256C2×#102—1581no—
257#407—21no—
258C4×M16C2×D7911yescharacteristic
259C2×C4×C8—791no—
260C4×C4⋊C4—791no—
261#340—1581no—
262#339—1581no—
263#339—1581no—
264Q8⋊C8—1582no—
265C22×D4D31611yescharacteristic
266C23×C4—21no—
267C2×SD16—1581no—
268C2×SD16—1581no—
269C2×M16—22no—
270C2×M16—21no—
271C2×M16—42no—
272C2×M16—42no—
273C2×M16C22⋊D7911yescharacteristic
274C2×M16C22⋊D7911yescharacteristic
275C22×C8—22no—
276C22×C8—1581no—
277C4⋊Q8—791no—
278C4⋊Q8—791no—
279#125—1581no—
280C22⋊Q8—1581no—
281C22⋊Q8—1581no—
282C4×Q8—1581no—
283C4×Q8—1581no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C4×D7911yescharacteristic
287C4×D4C4×D7911yescharacteristic
288C4×D4C22⋊D7911yescharacteristic
289C4×D4C22⋊D7911yescharacteristic
290#117—1581no—
291C2×C4⋊C4—1581no—
292C2×C4⋊C4D31611yescharacteristic
293C2×C22⋊C4—21no—
294C2×C42—1581no—
295C2×C42C22×D7911yescharacteristic
296#106—1581no—
297#106—1581no—
298C4⋊C8—21no—
299C4⋊C8—791no—
300C4⋊C8—791no—
301C4⋊C8C22⋊D7912yes—
302#103—1581no—
303#103—1581no—
304#102—1584no—
305#102—1581no—
306#102—1581no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—1581no—
310#97—21no—
311C4×C8—21no—
312C4×C8—1581no—
313C4×C8—1581no—
314C4×C8—1581no—
315#95—1581no—
316C24—21no—
317C2×Q8—1581no—
318C2×Q8—1581no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4(C2×D79)⋊C411yescharacteristic
323C2×D4(C2×D79)⋊C411yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—3161no—
334C22×C4C2×C22⋊D7911yescharacteristic
335C22×C4C2×D31611yescharacteristic
336C22×C4#1192962411yescharacteristic
337SD16—3162no—
338SD16—3162no—
339M16—21no—
340M16—21no—
341M16—21no—
342M16—44no—
343M16—41no—
344M16—41no—
345M16#1192961212yescomplemented
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—44no—
350C2×C8—3161no—
351C2×C8—3161no—
352C2×C8—1582no—
353C2×C8—1581no—
354C2×C8—1581no—
355C4⋊C4—21no—
356C4⋊C4—3161no—
357C4⋊C4—3161no—
358C4⋊C4—3161no—
359C4⋊C4—1582no—
360C4⋊C4—1581no—
361C4⋊C4—1581no—
362C4⋊C4—1581no—
363C4⋊C4—1581no—
364C4⋊C4(C2×D79)⋊C411yescharacteristic
365C4⋊C4(C2×D79)⋊C411yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—3161no—
371C42—3161no—
372C42—3161no—
373C42—3161no—
374C42C2×C4×D7911yescharacteristic
375C42C2×C22⋊D7911yescharacteristic
376C42C2×C22⋊D7911yescharacteristic
377C42#1192962411yescharacteristic
378C23—21no—
379C23—21no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23#1457913511yescharacteristic
385Q8—3162no—
386Q8—3161no—
387Q8—3161no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—42no—
392D4#1457908712yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—3162no—
409C2×C4—3161no—
410C2×C4—3161no—
411C2×C4—6321no—
412C2×C4—6321no—
413C2×C4C2×(C2×D79)⋊C411yescharacteristic
414C2×C4C4×C22⋊D7911yescharacteristic
415C2×C4C4×C22⋊D7911yescharacteristic
416C2×C4#1457913011yescharacteristic
417C2×C4#1457913511yescharacteristic
418C2×C4#1457914411yescharacteristic
419C2×C4C2×#1192961211yescharacteristic
420C2×C4#1457918111yescharacteristic
421C2×C4#1457918711yescharacteristic
422C2×C4#1457919111yescharacteristic
423C8—22no—
424C8—42no—
425C8—41no—
426C8—3162no—
427C8—3161no—
428C8—3161no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—82no—
438C22—82no—
439C22#1909844211yescharacteristic
440C22#1909876811yescharacteristic
441C22#1909877811yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—6322no—
448C4—6321no—
449C4—6321no—
450C4C2×#1457908711yescharacteristic
451C4#1909857911yescharacteristic
452C4C4×#1192961211yescharacteristic
453C4#1909866311yescharacteristic
454C4#1909871411yescharacteristic
455C4#1909872511yescharacteristic
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#2376370211yescharacteristic
461C2#2376373411yescharacteristic
462C2#2376376211yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingD4×M16×C79C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenterD4×M16D79
hypocenterC79#8989
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
2. #23764344C2semidirect
4. #23761854C2semidirect
5. #23761301C2semidirect
6. #23761238C2semidirect
10. #19098853C4semidirect
11. #19098853C4semidirect
12. #19098768C4semidirect
13. #19098768C4semidirect
130. M16×C79SD16semidirect
239. C79#8989semidirect
345. M16#11929612semidirect

Maximal subgroups

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

subgroupindex
#237734082
#237643442
C4×#145790822
#237618542
#237613012
#237612382
D4×M16×C792
#898979