Finite Group Databasebeta

← #34298843

Subgroups — #34298843

Order 20224.

830 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098618C211yesmaximal, characteristic
2#23765806C211yesmaximal, characteristic
3#23765805C211yesmaximal, characteristic
4#23764162C212yesmaximal, complemented
5#23764157C212yesmaximal, complemented
6#23763602C212yesmaximal
7#23763600C212yesmaximal
8#23763569C211yesmaximal, characteristic, complemented
9#23763568C211yesmaximal, characteristic, complemented
10#23763535C211yesmaximal, characteristic, complemented
11C79×#1836C211yesmaximal, characteristic, complemented
12C2×#14579119—21no—
13C2×#14579119C2211yescharacteristic
14C2×#14579119C2211yescharacteristic
15C2×C42×D79C2211yescharacteristic
16#19098618C418yescomplemented
17C2×#14579095C2212yes—
18C2×#14579094C2212yes—
19C2×#14579093C2212yes—
20C2×#14579092C2212yes—
21C2×Dic79⋊C8C2212yes—
22C2×(C2×D79)⋊C8C2212yes—
23C2×#14579072C2211yescharacteristic
24#19098433C2211yescharacteristic
25#19098432C2211yescharacteristic
26#19098432C2211yescharacteristic
27#19098429C2211yescharacteristic
28#19098428—21no—
29#19098422—21no—
30#19098422—21no—
31#19098420—21no—
32#19098420C2212yes—
33C4⋊C4×Dic79—21no—
34C4⋊C4×Dic79C2212yes—
35#19098414—21no—
36#19098414—21no—
37#19098325C2212yes—
38#19098325C2212yes—
39C2×#97×C79C2211yescharacteristic
40#384×C79C2211yescharacteristic
41#382×C79C2211yescharacteristic
42#340×C79C2212yes—
43#340×C79C2212yes—
44C22×C4×D79D411yescharacteristic
45C22×C4×D79D411yescharacteristic
46C22×C4×D79C2311yescharacteristic
47C22×C79⋊C8C2312yes—
48#14579119—42no—
49#14579119—42no—
50#14579119—44no—
51C2×C4⋊Dic79—21no—
52C2×C4⋊Dic79—21no—
53C2×C4⋊Dic79C2312yes—
54C2×C4⋊Dic79C2311yescharacteristic
55C2×C4⋊Dic79C2311yescharacteristic
56C2×Dic79⋊C4—21no—
57C2×Dic79⋊C4—21no—
58C2×Dic79⋊C4—21no—
59C2×Dic79⋊C4—21no—
60C2×(C2×D79)⋊C4—22no—
61C2×(C2×D79)⋊C4—21no—
62C2×(C2×D79)⋊C4—21no—
63C2×C4×Dic79—21no—
64C2×C4×Dic79—21no—
65C2×C4×Dic79—22no—
66C2×C4×Dic79D411yescharacteristic
67C2×C4×Dic79D411yescharacteristic
68C2×C4×Dic79C2311yescharacteristic
69C42×D79C2×C414yes—
70C42×D79Q814yes—
71#14579095—42no—
72#14579094—42no—
73#14579093—42no—
74#14579092—42no—
75Dic79⋊C8C2×C418yes—
76(C2×D79)⋊C8C2×C418yes—
77#14579072C2×C414yes—
78C79⋊#95—21no—
79C79⋊#95—21no—
80C79⋊#95—22no—
81C79⋊#95—22no—
82C22×C8×C79C2312yes—
83C2×C4⋊C4×C79—21no—
84C2×C4⋊C4×C79—21no—
85C2×C4⋊C4×C79C2312yes—
86C2×C4⋊C4×C79C2311yescharacteristic
87C2×C4⋊C4×C79C2311yescharacteristic
88C2×C42×C79C2311yescharacteristic
89#97×C79C2×C414yes—
90#95×C79—21no—
91#95×C79—21no—
92C23×D79C2×D411yescharacteristic
93C22×Dic79—21no—
94C22×Dic79—21no—
95C22×Dic79—22no—
96C22×Dic79C2×D411yescharacteristic
97C22×Dic79C2×D411yescharacteristic
98C22×Dic79C2×D411yescharacteristic
99C2×C4×D79—22no—
100C2×C4×D79—22no—
101C2×C4×D79—22no—
102C2×C4×D79—22no—
103C2×C4×D79—41no—
104C2×C4×D79—41no—
105C2×C4×D79C4⋊C414yes—
106C2×C4×D79C4⋊C414yes—
107C2×C4×D79C22×C412yes—
108C2×C4×D79C2×Q812yes—
109C2×C79⋊C8—22no—
110C2×C79⋊C8—22no—
111C2×C79⋊C8C22×C414yes—
112C4⋊Dic79—22no—
113C4⋊Dic79—22no—
114C4⋊Dic79—22no—
115C4⋊Dic79—22no—
116C4⋊Dic79—42no—
117C4⋊Dic79—42no—
118Dic79⋊C4—42no—
119Dic79⋊C4—42no—
120Dic79⋊C4—42no—
121Dic79⋊C4—42no—
122(C2×D79)⋊C4—42no—
123(C2×D79)⋊C4—42no—
124(C2×D79)⋊C4—44no—
125C4×Dic79—42no—
126C4×Dic79—42no—
127C4×Dic79—44no—
128C4×Dic79C4⋊C414yes—
129C4×Dic79C4⋊C414yes—
130C4×Dic79C22×C412yes—
131C4×Dic79C2×Q812yes—
132C22×C4×C79—21no—
133C22×C4×C79—21no—
134C22×C4×C79—22no—
135C22×C4×C79C2×D411yescharacteristic
136C22×C4×C79C2×D411yescharacteristic
137C22×C4×C79C2411yescharacteristic
138C2×C8×C79—22no—
139C2×C8×C79—22no—
140C2×C8×C79C22×C414yes—
141C4⋊C4×C79—22no—
142C4⋊C4×C79—22no—
143C4⋊C4×C79—24no—
144C4⋊C4×C79—42no—
145C4⋊C4×C79—42no—
146C42×C79C22×C412yes—
147C42×C79C2×Q812yes—
148C23×C79C22×D411yescharacteristic
149C22×D79—21no—
150C22×D79—21no—
151C22×D79—21no—
152C22×D79—21no—
153C22×D79—21no—
154C22×D79—21no—
155C22×D79C2×C4⋊C412yes—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—42no—
163C2×C4×C79—42no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—42no—
169C2×C4×C79C2×C4⋊C412yes—
170C2×C4×C79C2×C4⋊C412yes—
171C2×C4×C79C23×C411yescharacteristic
172C2×C4×C79C22×Q811yescharacteristic
173C2×C4×C79H(5,𝔽2)12yes—
174C2×C4×C792−1+412yes—
175C2×Dic79—21no—
176C2×Dic79—21no—
177C2×Dic79—21no—
178C2×Dic79—21no—
179C2×Dic79—41no—
180C2×Dic79—41no—
181C2×Dic79—42no—
182C2×Dic79—42no—
183C2×Dic79—41no—
184C2×Dic79—41no—
185C2×Dic79—41no—
186C2×Dic79—41no—
187C2×Dic79—42no—
188C2×Dic79C2×C4⋊C412yes—
189C2×Dic79C2×C4⋊C412yes—
190C2×Dic79C2×C4⋊C412yes—
191C2×Dic79AGL(1,ℤ/8)12yes—
192C2×Dic79#13712yes—
193C4×D79—24no—
194C4×D79—24no—
195C4×D79—28no—
196C4×D79—44no—
197C8×C79—24no—
198C79⋊C8—24no—
199C22×C79C22×C4⋊C411yescharacteristic
200C22×C79#53511yescharacteristic
201C22×C79#53611yescharacteristic
202C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
203C22×C79C2×#13711yescharacteristic
204C22×C79#57511yescharacteristic
205C22×C79#57511yescharacteristic
206C2×D79—22no—
207C2×D79—22no—
208C2×D79—22no—
209C2×D79—41no—
210C2×D79—41no—
211C2×D79—41no—
212C2×D79—41no—
213C4×C79—22no—
214C4×C79—22no—
215C4×C79—81no—
216C4×C79—81no—
217C4×C79—82no—
218C4×C79#52012yes—
219C4×C79#54412yes—
220Dic79—22no—
221Dic79—22no—
222Dic79—81no—
223Dic79—82no—
224Dic79—81no—
225Dic79#42814yes—
226#21988—791nomaximal, Sylow
227C2×C79#231711yescharacteristic
228C2×C79C2×#42811yescharacteristic
229C2×C79#291111yescharacteristic
230C2×C79#321411yescharacteristic
231C2×C79#321611yescharacteristic
232C2×C79#323511yescharacteristic
233C2×C79#323711yescharacteristic
234D79—42no—
235C2×#432—791no—
236#2285—791no—
237#2285—791no—
238#1948—792no—
239#1947—792no—
240#1854—792no—
241#1854—792no—
242#1847—791no—
243#1847—791no—
244#1836—791no—
245#1836D7911yescharacteristic, complemented
246C79#2198811yescharacteristic, complemented, Sylow
247C2×#117—791no—
248C2×#117—791no—
249C2×#117—1581no—
250C22×C42—791no—
251#432—798no—
252C2×#107—792no—
253C2×#107—792no—
254C2×#106—792no—
255C2×#106—792no—
256C2×C4⋊C8—792no—
257C2×C22⋊C8—792no—
258C2×#97—791no—
259C2×#97C2×D7911yescharacteristic
260#384—791no—
261#384—791no—
262#384—791no—
263#384C2×D7911yescharacteristic
264#382—791no—
265#382—1581no—
266#382C2×D7911yescharacteristic
267#380—792no—
268#380—1581no—
269#380—1581no—
270#380—1581no—
271C4×C4⋊C4—792no—
272C4×C4⋊C4—1581no—
273C4×C4⋊C4—1581no—
274C4×C4⋊C4—1581no—
275#340—792no—
276#340—792no—
277#340C2×D7912yes—
278#340C2×D7912yes—
279C23×C4—791no—
280C23×C4—791no—
281C23×C4—791no—
282C22×C8—792no—
283C22×C8C22×D7912yes—
284#117—3162no—
285#117—3162no—
286#117—3164no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—791no—
290C2×C4⋊C4—791no—
291C2×C4⋊C4—792no—
292C2×C4⋊C4—1581no—
293C2×C4⋊C4—1581no—
294C2×C4⋊C4—1581no—
295C2×C4⋊C4—1581no—
296C2×C4⋊C4—1581no—
297C2×C4⋊C4—1581no—
298C2×C4⋊C4C22×D7911yescharacteristic
299C2×C4⋊C4C22×D7911yescharacteristic
300C2×C4⋊C4C22×D7912yes—
301C2×C22⋊C4—1581no—
302C2×C22⋊C4—1581no—
303C2×C22⋊C4—1582no—
304C2×C42—791no—
305C2×C42—791no—
306C2×C42—791no—
307C2×C42—794no—
308C2×C42—794no—
309C2×C42—1581no—
310C2×C42—1581no—
311C2×C42—1582no—
312C2×C42C22×D7911yescharacteristic
313#107—3162no—
314#107—3162no—
315#106—3162no—
316#106—3162no—
317C4⋊C8—798no—
318C22⋊C8—798no—
319#97—794no—
320#97C4×D7914yes—
321#95—21no—
322#95—21no—
323#95—1582no—
324#95—1582no—
325#95—1581no—
326#95—1581no—
327C24—791no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—3161no—
332C22×C4—3161no—
333C22×C4—794no—
334C22×C4—791no—
335C22×C4—791no—
336C22×C4—791no—
337C22×C4—792no—
338C22×C4—794no—
339C22×C4—792no—
340C22×C4—1582no—
341C22×C4—1582no—
342C22×C4—1582no—
343C22×C4—1581no—
344C22×C4—1581no—
345C22×C4—1582no—
346C22×C4—1582no—
347C22×C4D4×D7911yescharacteristic
348C22×C4D4×D7911yescharacteristic
349C22×C4C23×D7911yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—794no—
353C2×C8—1582no—
354C2×C8—1582no—
355C2×C8C2×C4×D7914yes—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—3162no—
362C4⋊C4—3162no—
363C4⋊C4—3162no—
364C4⋊C4—3162no—
365C4⋊C4—3162no—
366C4⋊C4—3162no—
367C4⋊C4—1582no—
368C4⋊C4—1582no—
369C4⋊C4—1582no—
370C4⋊C4—1582no—
371C22⋊C4—3164no—
372C22⋊C4—3162no—
373C22⋊C4—3162no—
374C42—3164no—
375C42—3162no—
376C42—3162no—
377C42—792no—
378C42—794no—
379C42—794no—
380C42—792no—
381C42C2×C4×D7912yes—
382C42Q8×D7912yes—
383C23—792no—
384C23—1581no—
385C23—1581no—
386C23—1581no—
387C23—1581no—
388C23—1581no—
389C23—1581no—
390C23C2×D4×D7911yescharacteristic
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—3164no—
405C2×C4—3162no—
406C2×C4—3161no—
407C2×C4—3161no—
408C2×C4—3161no—
409C2×C4—3162no—
410C2×C4—3161no—
411C2×C4—3161no—
412C2×C4—3161no—
413C2×C4—3162no—
414C2×C4—792no—
415C2×C4—792no—
416C2×C4—792no—
417C2×C4—792no—
418C2×C4—792no—
419C2×C4—1588no—
420C2×C4—1581no—
421C2×C4—1581no—
422C2×C4—1581no—
423C2×C4—1581no—
424C2×C4—1584no—
425C2×C4—1584no—
426C2×C4C4⋊C4×D7912yes—
427C2×C4C4⋊C4×D7912yes—
428C2×C4C22×C4×D7911yescharacteristic
429C2×C4C2×Q8×D7911yescharacteristic
430C2×C4#1457920712yes—
431C2×C4#1457920812yes—
432C8—24no—
433C8—1584no—
434C22—3161no—
435C22—3161no—
436C22—3161no—
437C22—3161no—
438C22—1582no—
439C22—1582no—
440C22—1582no—
441C22C2×C4⋊C4×D7911yescharacteristic
442C22#1909913111yescharacteristic
443C22#1909913311yescharacteristic
444C22AGL(1,ℤ/8)×D7911yescharacteristic
445C22#137×D7911yescharacteristic
446C22#1909935911yescharacteristic
447C22#1909936211yescharacteristic
448C4—22no—
449C4—22no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—794no—
454C4—6322no—
455C4—6321no—
456C4—6321no—
457C4—1582no—
458C4—1582no—
459C4#1909904312yes—
460C4#1909919312yes—
461C2—3162no—
462C2#2376602511yescharacteristic
463C2#428×D7911yescharacteristic
464C2#2377198711yescharacteristic
465C2#2377439611yescharacteristic
466C2#2377441211yescharacteristic
467C2#2377459311yescharacteristic
468C2#2377460911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1836C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1836D79
hypocenterC79#21988
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
4. #23764162C2semidirect
5. #23764157C2semidirect
8. #23763569C2semidirect
9. #23763568C2semidirect
10. #23763535C2semidirect
11. C79×#1836C2semidirect
16. #19098618C4semidirect
245. #1836D79semidirect
246. C79#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#190986182
#237658062
#237658052
#237641622
#237641572
#237636022
#237636002
#237635692
#237635682
#237635352
C79×#18362
#2198879