Finite Group Databasebeta

← #34183116

Subgroups — #34183116

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
1D4×#11929608C211yesmaximal, characteristic
2#23764345C211yesmaximal, characteristic, complemented
3C4×#14579082C211yesmaximal, characteristic
4#23761855C211yesmaximal, characteristic, complemented
5#23761302C211yesmaximal, characteristic, complemented
6#23761236C211yesmaximal, characteristic, complemented
7C79×#3101C211yesmaximal, characteristic
8C22×#11929608—21no—
9#19098641—21no—
10#19098635—22no—
11D4×C79⋊C8—21no—
12#19098587C2211yescharacteristic
13C2×#14579082—21no—
14#19098527—21no—
15C4×#11929608C2211yescharacteristic
16C2×C4×C79⋊C8C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098325—21no—
19#19098322—21no—
20#19098322—21no—
21Q8⋊(C79⋊C8)—22no—
22#533×C79C2211yescharacteristic
23C2×C4×D4×C79C2211yescharacteristic
24#483×C79—21no—
25#476×C79C411yescharacteristic, complemented
26#476×C79C411yescharacteristic, complemented
27#461×C79C411yescharacteristic, complemented
28#461×C79C411yescharacteristic, complemented
29#450×C79—21no—
30C4×SD16×C79—21no—
31C4×SD16×C79—21no—
32#431×C79C2211yescharacteristic
33C2×#11929608—22no—
34C2×#11929608—21no—
35C2×#11929608—42no—
36C2×#11929608—42no—
37C2×#11929608D411yescharacteristic
38C2×#11929608D411yescharacteristic
39C22×C79⋊C8—22no—
40C22×C79⋊C8—21no—
41C2×C4⋊Dic79—21no—
42C2×C4×Dic79—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)D412yes—
45#14579082—24no—
46C22⋊(C79⋊C8)—22no—
47C22⋊(C79⋊C8)—22no—
48#14579072—21no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C79⋊#95—21no—
53C22×D4×C79D411yescharacteristic
54C23×C4×C79—21no—
55C2×SD16×C79—21no—
56C2×SD16×C79—21no—
57C4⋊Q8×C79C2×C411yescharacteristic
58C4⋊Q8×C79C2×C411yescharacteristic
59#125×C79—21no—
60C22⋊Q8×C79—21no—
61C22⋊Q8×C79—21no—
62C4×Q8×C79—21no—
63C4×Q8×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79C2×C411yescharacteristic
67C4×D4×C79C2×C411yescharacteristic
68C4×D4×C79D411yescharacteristic
69C4×D4×C79D411yescharacteristic
70#117×C79—21no—
71C2×C4⋊C4×C79D411yescharacteristic
72C2×C22⋊C4×C79—21no—
73C2×C42×C79C2311yescharacteristic
74#106×C79—21no—
75#106×C79—21no—
76C4⋊C8×C79C2×C411yescharacteristic
77C4⋊C8×C79C2×C411yescharacteristic
78#103×C79—21no—
79#103×C79—21no—
80#102×C79—21no—
81#102×C79—21no—
82C22⋊C8×C79—21no—
83C4×C8×C79—21no—
84C22×Dic79—41no—
85#11929608—21no—
86#11929608—21no—
87#11929608—21no—
88#11929608—44no—
89#11929608—41no—
90#11929608—41no—
91#11929608SD1612yescomplemented
92C2×C79⋊C8—22no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8—22no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—44no—
98C2×C79⋊C8—41no—
99C4⋊Dic79—22no—
100C4⋊Dic79—41no—
101C4×Dic79—41no—
102C4×Dic79—41no—
103C24×C79—21no—
104C2×Q8×C79—21no—
105C2×Q8×C79—21no—
106C2×D4×C79—22no—
107C2×D4×C79—21no—
108C2×D4×C79—41no—
109C2×D4×C79C22⋊C411yescharacteristic
110C2×D4×C79C22⋊C411yescharacteristic
111C22×C4×C79—21no—
112C22×C4×C79—21no—
113C22×C4×C79—21no—
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—41no—
117C22×C4×C79—41no—
118C22×C4×C79—41no—
119C22×C4×C79—41no—
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79#4111yescharacteristic
123SD16×C79—42no—
124SD16×C79—42no—
125C2×C8×C79—21no—
126C2×C8×C79—21no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79—41no—
133C4⋊C4×C79—41no—
134C4⋊C4×C79C22⋊C411yescharacteristic
135C4⋊C4×C79C22⋊C411yescharacteristic
136C22⋊C4×C79—21no—
137C22⋊C4×C79—21no—
138C22⋊C4×C79—21no—
139C22⋊C4×C79—21no—
140C22⋊C4×C79—41no—
141C42×C79—41no—
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
153Q8×C79—42no—
154Q8×C79—41no—
155Q8×C79—41no—
156D4×C79—21no—
157D4×C79—21no—
158D4×C79—21no—
159D4×C79—42no—
160D4×C79#10412yes—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
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×C79—41no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—41no—
177C2×C4×C79—41no—
178C2×C4×C79C2×C22⋊C411yescharacteristic
179C2×C4×C79C4×D411yescharacteristic
180C2×C4×C79C4×D411yescharacteristic
181C2×C4×C79#12011yescharacteristic
182C2×C4×C79#12111yescharacteristic
183C2×C4×C79#12311yescharacteristic
184C2×C4×C79C2×SD1611yescharacteristic
185C2×C4×C79#13511yescharacteristic
186C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C79#13711yescharacteristic
188C2×Dic79—42no—
189C2×Dic79—81no—
190C2×Dic79—81no—
191C8×C79—41no—
192C8×C79—41no—
193C79⋊C8—22no—
194C79⋊C8—42no—
195C79⋊C8—42no—
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×C79—81no—
216C4×C79—81no—
217C4×C79C2×#10411yescharacteristic
218C4×C79#42111yescharacteristic
219C4×C79C4×SD1611yescharacteristic
220C4×C79#44011yescharacteristic
221C4×C79#45011yescharacteristic
222C4×C79#45211yescharacteristic
223Dic79—82no—
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#3101D7911yescharacteristic
233D4×M16—791no—
234#1986—791no—
235C4×#102—791no—
236#1493—791no—
237#1303—791no—
238#1278—791no—
239C79#898911yescharacteristic, complemented, Sylow
240C22×M16—1581no—
241#533C2×D7911yescharacteristic
242C2×C4×D4C2×D7911yescharacteristic
243#483—21no—
244#476Dic7911yescharacteristic, complemented
245#476Dic7911yescharacteristic, complemented
246#461Dic7911yescharacteristic, complemented
247#461Dic7911yescharacteristic, complemented
248#450—21no—
249C4×SD16—21no—
250C4×SD16—21no—
251#436—1581no—
252#435—1582no—
253C8×D4—1581no—
254#431C2×D7911yescharacteristic
255#423—791no—
256C2×#102—1581no—
257#407—1581no—
258C4×M16—791no—
259C2×C4×C8—791no—
260C4×C4⋊C4—791no—
261#340—1581no—
262#339—1581no—
263#339—1581no—
264Q8⋊C8—1582no—
265C22×D4C22⋊D7911yescharacteristic
266C23×C4—21no—
267C2×SD16—21no—
268C2×SD16—21no—
269C2×M16—3162no—
270C2×M16—3162no—
271C2×M16—791no—
272C2×M16—791no—
273C2×M16—1582no—
274C2×M16—1581no—
275C22×C8—1582no—
276C22×C8—1581no—
277C4⋊Q8C2×Dic7911yescharacteristic
278C4⋊Q8C2×Dic7911yescharacteristic
279#125—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C2×Dic7911yescharacteristic
287C4×D4C2×Dic7911yescharacteristic
288C4×D4C22⋊D7911yescharacteristic
289C4×D4C22⋊D7911yescharacteristic
290#117—21no—
291C2×C4⋊C4—1581no—
292C2×C4⋊C4C22⋊D7911yescharacteristic
293C2×C22⋊C4—21no—
294C2×C42—1581no—
295C2×C42C22×D7911yescharacteristic
296#106—21no—
297#106—21no—
298C4⋊C8—792no—
299C4⋊C8—1581no—
300C4⋊C8C2×Dic7911yescharacteristic
301C4⋊C8C2×Dic7911yescharacteristic
302#103—21no—
303#103—21no—
304#102—21no—
305#102—21no—
306#102—1584no—
307C22⋊C8—21no—
308C22⋊C8—1582no—
309C22⋊C8—1582no—
310#97—1581no—
311C4×C8—21no—
312C4×C8—1581no—
313C4×C8—1581no—
314C4×C8—1581no—
315#95—1581no—
316C24—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4C22⋊Dic7911yescharacteristic
323C2×D4C22⋊Dic7911yescharacteristic
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×C4D4×D7911yescharacteristic
335C22×C4C2×C22⋊D7911yescharacteristic
336C22×C4#1192962311yescharacteristic
337SD16—42no—
338SD16—42no—
339M16—3161no—
340M16—3161no—
341M16—3164no—
342M16—792no—
343M16—1581no—
344M16—1581no—
345M16—1581no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—3161no—
349C2×C8—3161no—
350C2×C8—3164no—
351C2×C8—1582no—
352C2×C8—1581no—
353C2×C8—1582no—
354C2×C8—1581no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—3161no—
363C4⋊C4—1582no—
364C4⋊C4C22⋊Dic7911yescharacteristic
365C4⋊C4C22⋊Dic7911yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—41no—
371C42—41no—
372C42—3161no—
373C42—3161no—
374C42C22×Dic7911yescharacteristic
375C42D4×D7911yescharacteristic
376C42C2×C22⋊D7911yescharacteristic
377C42#1192962311yescharacteristic
378C23—21no—
379C23—21no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23#1457913311yescharacteristic
385Q8—42no—
386Q8—41no—
387Q8—41no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—42no—
392D4#1457908812yes—
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—41no—
409C2×C4—41no—
410C2×C4—3162no—
411C2×C4—6321no—
412C2×C4—6321no—
413C2×C4C2×C22⋊Dic7911yescharacteristic
414C2×C4D4×Dic7911yescharacteristic
415C2×C4D4×Dic7911yescharacteristic
416C2×C4#1457912911yescharacteristic
417C2×C4#1457913311yescharacteristic
418C2×C4#1457914211yescharacteristic
419C2×C4SD16×D7911yescharacteristic
420C2×C4#1457918211yescharacteristic
421C2×C4#1457918811yescharacteristic
422C2×C4#1457919011yescharacteristic
423C8—41no—
424C8—41no—
425C8—3162no—
426C8—3162no—
427C8—3161no—
428C8—1582no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—82no—
438C22—82no—
439C22#1909844311yescharacteristic
440C22#1909876611yescharacteristic
441C22#1909878011yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—81no—
448C4—81no—
449C4—6322no—
450C4C2×#1457908811yescharacteristic
451C4#1909858011yescharacteristic
452C4SD16×Dic7911yescharacteristic
453C4#1909866411yescharacteristic
454C4#1909871211yescharacteristic
455C4#1909872211yescharacteristic
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#2376370311yescharacteristic
461C2#2376373611yescharacteristic
462C2#2376376411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3101C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenter#3101D79
hypocenterC79#8989
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
2. #23764345C2semidirect
4. #23761855C2semidirect
5. #23761302C2semidirect
6. #23761236C2semidirect
25. #476×C79C4semidirect
26. #476×C79C4semidirect
27. #461×C79C4semidirect
28. #461×C79C4semidirect
91. #11929608SD16semidirect
239. C79#8989semidirect
244. #476Dic79semidirect
245. #476Dic79semidirect
246. #461Dic79semidirect
247. #461Dic79semidirect

Maximal subgroups

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

subgroupindex
D4×#119296082
#237643452
C4×#145790822
#237618552
#237613022
#237612362
C79×#31012
#898979