Finite Group Databasebeta

← #34258405

Subgroups — #34258405

Order 20224.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773408C211yesmaximal, characteristic
2#23772497C211yesmaximal, characteristic, complemented
3#23772424C211yesmaximal, characteristic
4#23762254C212yesmaximal, complemented
5#23762233C212yesmaximal, complemented
6#23762209C212yesmaximal, complemented
7#23762203C212yesmaximal, complemented
8#23762181C211yesmaximal, characteristic, complemented
9#23761822C211yesmaximal, characteristic, complemented
10#23761697C211yesmaximal, characteristic, complemented
11C79×#1490C211yesmaximal, characteristic
12#19099120C2211yescharacteristic
13#19098893C412yes—
14#19098853C412yes—
15#19098768C412yes—
16#19098714C412yes—
17#19098691—22no—
18#19098690C2212yescomplemented
19#19098687C2212yescomplemented
20C4×#11929612C2212yes—
21C4×#11929612C2212yes—
22D4×C79⋊C8—22no—
23D4×C79⋊C8C2212yes—
24D4×C79⋊C8C2212yes—
25#19098621—22no—
26#19098616C2211yescharacteristic
27#19098616C2211yescharacteristic
28#19098616C2211yescharacteristic
29C2×C22⋊(C79⋊C8)—22no—
30#19098315C2212yescomplemented
31#19098313C2212yescomplemented
32#19098295C2212yescomplemented
33#19098295C2212yescomplemented
34#19098285C2211yescharacteristic, complemented
35#19098285C2211yescharacteristic, complemented
36#19098284C2211yescharacteristic, complemented
37C8×C79⋊C8C2211yescharacteristic, complemented
38C2×C4×D4×C79C2211yescharacteristic
39#432×C79C2211yescharacteristic
40C4×M16×C79C2211yescharacteristic
41D4⋊C8×C79C2212yes—
42D4⋊C8×C79C2212yes—
43C2×#11929612C2×C412yes—
44C2×#11929612C2×C412yes—
45C22×C79⋊C8—22no—
46C22×C79⋊C8—22no—
47#14579161C2×C411yescharacteristic
48#14579161C2×C411yescharacteristic
49#14579153—21no—
50#14579140—21no—
51#14579140C2×C412yes—
52C4×Dic158—21no—
53C4×Dic158C2312yes—
54#14579120—21no—
55#14579092C2×C412yes—
56#14579092C2×C412yes—
57C4⋊(C79⋊C8)C2×C412yes—
58C4⋊(C79⋊C8)D412yes—
59C4⋊(C79⋊C8)C2312yes—
60Dic79⋊C8—22no—
61Dic79⋊C8—22no—
62#14579084C2×C412yes—
63#14579084C2×C412yes—
64#14579082C2×C412yes—
65#14579082C2×C412yes—
66C22⋊(C79⋊C8)—21no—
67C22⋊(C79⋊C8)—22no—
68C22⋊(C79⋊C8)—22no—
69C22⋊(C79⋊C8)C2×C412yes—
70C22⋊(C79⋊C8)D412yes—
71#14579071—22no—
72C4×C79⋊C8—21no—
73C4×C79⋊C8C2311yescharacteristic
74C4×C79⋊C8C2311yescharacteristic
75C4×C79⋊C8C2311yescharacteristic
76C4×C79⋊C8C2311yescharacteristic
77C8×Dic79—22no—
78C22×D4×C79C2×C411yescharacteristic
79C23×C4×C79—21no—
80C2×M16×C79—21no—
81C4×D4×C79—21no—
82C4×D4×C79—21no—
83C4×D4×C79C2×C412yes—
84C4×D4×C79C2312yes—
85C2×C4⋊C4×C79C2×C411yescharacteristic
86C2×C22⋊C4×C79—21no—
87C2×C42×C79C2311yescharacteristic
88C4⋊C8×C79C2312yes—
89C22⋊C8×C79—21no—
90#97×C79—21no—
91#97×C79—21no—
92C4×C8×C79C2311yescharacteristic
93C4×C8×C79C2311yescharacteristic
94C2×Dic158—21no—
95C2×Dic158C22×C412yes—
96#11929612—24no—
97#11929612—24no—
98C2×C79⋊C8—22no—
99C2×C79⋊C8—82no—
100C2×C79⋊C8—82no—
101C2×C79⋊C8C22×C412yes—
102C2×C79⋊C8C22×C412yes—
103C2×C79⋊C8C2×D412yes—
104C2×C79⋊C8C2×D412yes—
105C4⋊Dic79—21no—
106C4⋊Dic79—21no—
107C4⋊Dic79—21no—
108C4⋊Dic79C22×C412yes—
109Dic79⋊C4—22no—
110Dic79⋊C4—22no—
111C22⋊Dic79—22no—
112C4×Dic79—22no—
113C24×C79—21no—
114C2×D4×C79—24no—
115C2×D4×C79—21no—
116C2×D4×C79C22×C412yes—
117C22×C4×C79—21no—
118C22×C4×C79—22no—
119C22×C4×C79—22no—
120C22×C4×C79—82no—
121C22×C4×C79C22×C411yescharacteristic
122C22×C4×C79C2×D411yescharacteristic
123C22×C4×C79#4111yescharacteristic
124M16×C79—42no—
125C2×C8×C79—22no—
126C2×C8×C79—21no—
127C2×C8×C79—21no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79C22×C412yes—
130C22⋊C4×C79—22no—
131C22⋊C4×C79—22no—
132C42×C79C22×C411yescharacteristic
133C42×C79C2×D411yescharacteristic
134C42×C79#4111yescharacteristic
135C42×C79C2411yescharacteristic
136C23×C79—24no—
137C23×C79—22no—
138C23×C79—81no—
139C23×C79—81no—
140C23×C79C4×D411yescharacteristic
141D4×C79—24no—
142D4×C79—22no—
143D4×C79#13114yes—
144C2×C4×C79—22no—
145C2×C4×C79—22no—
146C2×C4×C79—22no—
147C2×C4×C79—41no—
148C2×C4×C79—42no—
149C2×C4×C79—42no—
150C2×C4×C79—82no—
151C2×C4×C79—81no—
152C2×C4×C79C4×D411yescharacteristic
153C2×C4×C79C4×D411yescharacteristic
154C2×C4×C79C4×D411yescharacteristic
155C2×C4×C79C23×C411yescharacteristic
156C2×C4×C79C22×D411yescharacteristic
157C2×C4×C79C2×#4111yescharacteristic
158C2×Dic79—22no—
159C2×Dic79—22no—
160Dic158—22no—
161Dic158—22no—
162Dic158—81no—
163C8×C79—22no—
164C8×C79—41no—
165C8×C79—42no—
166C79⋊C8—22no—
167C79⋊C8—22no—
168C79⋊C8—22no—
169C79⋊C8—22no—
170C79⋊C8—42no—
171C22×C79—21no—
172C22×C79—24no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—42no—
176C22×C79—42no—
177C22×C79—84no—
178C22×C79C2×C4×D411yescharacteristic
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—42no—
184C4×C79C2×#13112yes—
185C4×C79#57712yes—
186Dic79—42no—
187Dic79—42no—
188#17924—791nomaximal, Sylow
189C2×C79—41no—
190C2×C79—44no—
191C2×C79#292911yescharacteristic
192C2×C79#294911yescharacteristic
193C2×C79#295611yescharacteristic
194#3101—791no—
195#2985—791no—
196#2972—791no—
197#1585—792no—
198#1582—792no—
199SD16⋊C8—792no—
200C8×SD16—792no—
201#1572—791no—
202#1490D7911yescharacteristic
203#1485—791no—
204#1453—791no—
205C79#1792411yescharacteristic, complemented, Sylow
206#533—791no—
207C2×C4×D4C2×D7911yescharacteristic
208#483—792no—
209#476—792no—
210#461—792no—
211#450—792no—
212#446—792no—
213#446—1582no—
214C8×Q8—792no—
215C4×SD16—792no—
216C4×SD16—792no—
217C8×D4—792no—
218C8×D4—792no—
219C8×D4—1582no—
220#433—1582no—
221#432C2×D7911yescharacteristic
222#431—791no—
223#431—791no—
224#431—791no—
225C2×C22⋊C8—1582no—
226C4×M16C2×D7911yescharacteristic
227#335—792no—
228#334—792no—
229Q8⋊C8—792no—
230Q8⋊C8—792no—
231D4⋊C8C2×D7912yes—
232D4⋊C8C2×D7912yes—
233#322—791no—
234#322—791no—
235#322—791no—
236C82—791no—
237C22×D4C4×D7911yescharacteristic
238C23×C4—21no—
239C2×SD16—792no—
240C2×SD16—792no—
241C2×M16—21no—
242C22×C8—1582no—
243C22×C8—1582no—
244C4⋊Q8—791no—
245C4⋊Q8—791no—
246#125—1581no—
247C22⋊Q8—792no—
248C22⋊Q8—1581no—
249C4×Q8—792no—
250C4×Q8—1581no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4C4×D7912yes—
254C4×D4C22×D7912yes—
255#117—1581no—
256C2×C4⋊C4C4×D7911yescharacteristic
257C2×C22⋊C4—21no—
258C2×C42C22×D7911yescharacteristic
259#106—792no—
260#106—792no—
261C4⋊C8—792no—
262C4⋊C8—792no—
263C4⋊C8—792no—
264C4⋊C8—1582no—
265C4⋊C8—1582no—
266C4⋊C8C22×D7912yes—
267#103—792no—
268#103—792no—
269#102—792no—
270#102—792no—
271C22⋊C8—21no—
272C22⋊C8—792no—
273C22⋊C8—792no—
274C22⋊C8—1582no—
275C22⋊C8—1582no—
276C22⋊C8—1581no—
277#97—21no—
278#97—21no—
279#97—1582no—
280C4×C8—791no—
281C4×C8—791no—
282C4×C8—791no—
283C4×C8—791no—
284C4×C8—1582no—
285C4×C8—1581no—
286C4×C8C22×D7911yescharacteristic
287C4×C8C22×D7911yescharacteristic
288C24—21no—
289C2×Q8—792no—
290C2×Q8—1581no—
291C2×D4—24no—
292C2×D4—21no—
293C2×D4C2×C4×D7912yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—82no—
298C22×C4C2×C4×D7911yescharacteristic
299C22×C4D4×D7911yescharacteristic
300C22×C4#1192962311yescharacteristic
301SD16—1584no—
302SD16—1584no—
303M16—42no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—792no—
308C2×C8—792no—
309C2×C8—792no—
310C2×C8—792no—
311C2×C8—6322no—
312C2×C8—6322no—
313C2×C8—1582no—
314C4⋊C4—21no—
315C4⋊C4—792no—
316C4⋊C4—1582no—
317C4⋊C4—1582no—
318C4⋊C4—1581no—
319C4⋊C4—1581no—
320C4⋊C4—1581no—
321C4⋊C4C2×C4×D7912yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—1582no—
325C42—1582no—
326C42C2×C4×D7911yescharacteristic
327C42D4×D7911yescharacteristic
328C42#1192962311yescharacteristic
329C42C23×D7911yescharacteristic
330C23—24no—
331C23—22no—
332C23—81no—
333C23—81no—
334C23(C22⋊D79)⋊C411yescharacteristic
335Q8—6321no—
336Q8—1582no—
337Q8—1582no—
338D4—24no—
339D4—22no—
340D4#1457916914yes—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—1582no—
350C2×C4—1582no—
351C2×C4(C22⋊D79)⋊C411yescharacteristic
352C2×C4(C22⋊D79)⋊C411yescharacteristic
353C2×C4(C22⋊D79)⋊C411yescharacteristic
354C2×C4C22×C4×D7911yescharacteristic
355C2×C4C2×D4×D7911yescharacteristic
356C2×C4C2×#1192962311yescharacteristic
357C8—22no—
358C8—41no—
359C8—42no—
360C8—3162no—
361C8—1582no—
362C8—1582no—
363C8—1582no—
364C8—1582no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—24no—
369C22—42no—
370C22—42no—
371C22—84no—
372C22C2×(C22⋊D79)⋊C411yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—3162no—
379C4—3162no—
380C4C2×#1457916912yes—
381C4#1909937312yes—
382C2—41no—
383C2—44no—
384C2#2377209011yescharacteristic
385C2#2377226211yescharacteristic
386C2#2377230611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1490C2
socleC22×C79C2×C4×D4
prime radicalC22×D4×C79C2×C4
cosocleC42×C79C24
hypercenter#1490D79
hypocenterC79#17924
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
2. #23772497C2semidirect
4. #23762254C2semidirect
5. #23762233C2semidirect
6. #23762209C2semidirect
7. #23762203C2semidirect
8. #23762181C2semidirect
9. #23761822C2semidirect
10. #23761697C2semidirect
18. #19098690C22semidirect
19. #19098687C22semidirect
30. #19098315C22semidirect
31. #19098313C22semidirect
32. #19098295C22semidirect
33. #19098295C22semidirect
34. #19098285C22semidirect
35. #19098285C22semidirect
36. #19098284C22semidirect
37. C8×C79⋊C8C22semidirect
205. C79#17924semidirect

Maximal subgroups

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

subgroupindex
#237734082
#237724972
#237724242
#237622542
#237622332
#237622092
#237622032
#237621812
#237618222
#237616972
C79×#14902
#1792479