Finite Group Databasebeta

← #34179266

Subgroups — #34179266

Order 20224.

522 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772090C211yesmaximal, characteristic
2#23763179C211yesmaximal, characteristic, complemented
3#23763174C211yesmaximal, characteristic, complemented
4#23761892C211yesmaximal, characteristic
5#23761892C211yesmaximal, characteristic
6#23761285C211yesmaximal, characteristic, complemented
7C79×#1298C211yesmaximal, characteristic, complemented
8C2×#14579169C411yescharacteristic, complemented
9C23×C79⋊C8C2211yescharacteristic
10#19099074C2211yescharacteristic
11#19098631—21no—
12#19098631—21no—
13(C22⋊D79)⋊C8—21no—
14(C22⋊D79)⋊C8—21no—
15#19098591C411yescharacteristic, complemented
16#19098531C411yescharacteristic, complemented
17#19098521C411yescharacteristic, complemented
18C2×#14579071—21no—
19C2×C8×Dic79—21no—
20C4×C22⋊Dic79C2211yescharacteristic
21C4×C22⋊Dic79C2211yescharacteristic
22#19098317—21no—
23#19098317—21no—
24#19098301C412yescomplemented
25#19098301C412yescomplemented
26#19098298C412yescomplemented
27#19098298C412yescomplemented
28#407×C79C2211yescharacteristic
29C2×C22⋊C8×C79C2211yescharacteristic
30C2×C22⋊C8×C79C2211yescharacteristic
31C2×#11929624C2×C411yescharacteristic
32#14579169—42no—
33C2×#11929607—21no—
34C22×C79⋊C8—42no—
35C22×C79⋊C8C2×C411yescharacteristic
36C22×C79⋊C8C2×C411yescharacteristic
37C22×C79⋊C8D411yescharacteristic
38C22×C79⋊C8D411yescharacteristic
39C22×C79⋊C8D411yescharacteristic
40C22×C79⋊C8D411yescharacteristic
41C2×C8×D79—21no—
42#14579144—21no—
43#14579140C2×C411yescharacteristic
44#14579140C2×C411yescharacteristic
45#14579135C2×C411yescharacteristic
46#14579135C2×C411yescharacteristic
47#14579130—21no—
48C4×C22⋊D79—21no—
49C4×C22⋊D79—21no—
50#14579120C2×C411yescharacteristic
51C2×C22⋊Dic79—21no—
52C2×C22⋊Dic79—21no—
53C2×C4×Dic79—21no—
54C2×C4×Dic79—21no—
55Dic79⋊C8—21no—
56Dic79⋊C8—21no—
57(C2×D79)⋊C8—21no—
58(C2×D79)⋊C8—21no—
59#14579071—21no—
60#14579071—42no—
61C8×Dic79—21no—
62C8×Dic79—42no—
63C79⋊#95C2×C412yes—
64C79⋊#95C2×C412yes—
65C23×C4×C79C2311yescharacteristic
66C2×M16×C79C2×C411yescharacteristic
67C2×M16×C79C2×C411yescharacteristic
68C22×C8×C79—21no—
69C22×C8×C79—21no—
70C22⋊C8×C79—21no—
71C22⋊C8×C79—21no—
72C22⋊C8×C79—21no—
73C22⋊C8×C79—21no—
74C22⋊C8×C79C2×C412yes—
75C22⋊C8×C79C2×C412yes—
76#11929624—41no—
77#11929624—41no—
78C2×Dic158C4211yescharacteristic
79C2×D316C4211yescharacteristic
80C2×C22⋊D79—21no—
81C2×C22⋊D79—21no—
82C2×C22⋊D79—21no—
83C22×Dic79—22no—
84C22×Dic79—22no—
85C2×C4×D79—21no—
86#11929607—42no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—22no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8C22⋊C411yescharacteristic
94C2×C79⋊C8C22⋊C411yescharacteristic
95C2×C79⋊C8C22⋊C411yescharacteristic
96C2×C79⋊C8C22⋊C411yescharacteristic
97C8×D79—42no—
98C4⋊Dic79C4211yescharacteristic
99C4⋊Dic79C4211yescharacteristic
100Dic79⋊C4—21no—
101Dic79⋊C4—21no—
102C22⋊Dic79—24no—
103C22⋊Dic79—21no—
104C22⋊Dic79—21no—
105C22⋊Dic79—21no—
106C22⋊Dic79—42no—
107(C2×D79)⋊C4—21no—
108(C2×D79)⋊C4—21no—
109C4×Dic79—21no—
110C4×Dic79—41no—
111C4×Dic79—41no—
112C4×Dic79—41no—
113C4×Dic79—41no—
114C24×C79#4111yescharacteristic
115C22×C4×C79—41no—
116C22×C4×C79—41no—
117C22×C4×C79C22×C411yescharacteristic
118C22×C4×C79C22×C411yescharacteristic
119C22×C4×C79C22×C411yescharacteristic
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79#4111yescharacteristic
123M16×C79—41no—
124M16×C79—41no—
125C2×C8×C79—21no—
126C2×C8×C79—21no—
127C2×C8×C79—22no—
128C2×C8×C79—22no—
129C2×C8×C79—41no—
130C2×C8×C79—41no—
131C2×C8×C79—41no—
132C2×C8×C79—41no—
133C23×C79—22no—
134C23×C79—22no—
135C23×C79—41no—
136C23×C79#11711yescharacteristic
137C23×C79C4×D411yescharacteristic
138C23×C79C4×D411yescharacteristic
139C22×D79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79C2×C4211yescharacteristic
150C2×C4×C79C2×C22⋊C411yescharacteristic
151C2×C4×C79C4×D411yescharacteristic
152C2×C4×C79C4×D411yescharacteristic
153C22⋊D79—42no—
154C22⋊D79—42no—
155C22⋊D79—42no—
156C2×Dic79—24no—
157C2×Dic79—21no—
158C2×Dic79—21no—
159C2×Dic79—21no—
160C2×Dic79—42no—
161C2×Dic79—42no—
162C2×Dic79—42no—
163C2×Dic79—42no—
164C2×Dic79—42no—
165D316—41no—
166C4×D79—41no—
167C4×D79—41no—
168Dic158—41no—
169C8×C79—42no—
170C8×C79—42no—
171C8×C79—41no—
172C8×C79—41no—
173C79⋊C8—21no—
174C79⋊C8—21no—
175C79⋊C8—21no—
176C79⋊C8#9912yes—
177C22×C79—21no—
178C22×C79—21no—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—44no—
185C22×C79C4×C22⋊C411yescharacteristic
186C22×C79#44312yes—
187C22×C79#44412yes—
188C2×D79—41no—
189C2×D79—41no—
190C2×D79—41no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79C2×#9911yescharacteristic
195C4×C79#41011yescharacteristic
196Dic79—44no—
197Dic79—41no—
198Dic79—41no—
199Dic79—41no—
200Dic79—82no—
201#8240—791nomaximal, Sylow
202C2×C79—22no—
203C2×C79—22no—
204C2×C79—41no—
205C2×C79C4×#9911yescharacteristic
206C2×C79#176411yescharacteristic
207C2×C79#176511yescharacteristic
208D79—81no—
209#2929—791no—
210#1752—791no—
211C8×C22⋊C4—791no—
212#1501—791no—
213#1501—791no—
214#1298D7911yescharacteristic, complemented
215#1297—791no—
216C79#824011yescharacteristic, complemented, Sylow
217C2×#131—791no—
218C23×C8—791no—
219#525—791no—
220#435—1581no—
221#435—1581no—
222C8×D4—1581no—
223C8×D4—1581no—
224#424—791no—
225#408—791no—
226#407C2×D7911yescharacteristic
227C2×C22⋊C8C2×D7911yescharacteristic
228C2×C22⋊C8C2×D7911yescharacteristic
229#405—791no—
230C2×#97—1581no—
231C2×C4×C8—1581no—
232C4×C22⋊C4—791no—
233C4×C22⋊C4—791no—
234#336—1581no—
235#336—1581no—
236#328—792no—
237#328—792no—
238#327—792no—
239#327—792no—
240C2×#41—791no—
241C23×C4C22×D7911yescharacteristic
242#131—3162no—
243C2×M16—1581no—
244C2×M16C4×D7911yescharacteristic
245C2×M16C4×D7911yescharacteristic
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—3162no—
249C22×C8—791no—
250C22×C8—791no—
251C22×C8—791no—
252C22×C8—791no—
253C22×C8—791no—
254C22×C8—791no—
255C22×C8—1581no—
256#123—1581no—
257C22⋊Q8—791no—
258C22⋊Q8—791no—
259#121—791no—
260#121—791no—
261#120—1581no—
262C4×D4—1581no—
263C4×D4—1581no—
264#117—791no—
265C2×C22⋊C4—1581no—
266C2×C22⋊C4—1581no—
267C2×C42—1581no—
268C2×C42—1581no—
269C4⋊C8—1581no—
270C4⋊C8—1581no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—1581no—
276C22⋊C8—1581no—
277C22⋊C8C4×D7912yes—
278C22⋊C8C4×D7912yes—
279#97—3162no—
280#97—1581no—
281C4×C8—3162no—
282C4×C8—1581no—
283#95—792no—
284#95—792no—
285C24#1192962511yescharacteristic
286#41—3161no—
287#41—3161no—
288C2×Q8—791no—
289C2×D4—791no—
290C2×D4—1581no—
291C2×D4—1581no—
292C2×D4—1581no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—1582no—
296C22×C4—1582no—
297C22×C4—1581no—
298C22×C4C2×C4×D7911yescharacteristic
299C22×C4C2×C4×D7911yescharacteristic
300C22×C4C2×C4×D7911yescharacteristic
301C22×C4D4×D7911yescharacteristic
302C22×C4D4×D7911yescharacteristic
303C22×C4#1192962311yescharacteristic
304M16—41no—
305M16—41no—
306M16—3162no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—3162no—
316C2×C8—3161no—
317C2×C8—3161no—
318C2×C8—3161no—
319C2×C8—791no—
320C2×C8—791no—
321C2×C8—791no—
322C2×C8—791no—
323C2×C8—1582no—
324C2×C8—1582no—
325C2×C8—1582no—
326C4⋊C4—791no—
327C4⋊C4—791no—
328C4⋊C4—1581no—
329C4⋊C4—1581no—
330C22⋊C4—3162no—
331C22⋊C4—1584no—
332C22⋊C4—1581no—
333C22⋊C4—1581no—
334C22⋊C4—1581no—
335C22⋊C4—1581no—
336C22⋊C4—1581no—
337C42—3161no—
338C42—3161no—
339C42—3161no—
340C42—3161no—
341C42—1581no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—1581no—
346C23#1457911911yescharacteristic
347C23#1457912311yescharacteristic
348C23#1457912311yescharacteristic
349Q8—3161no—
350D4—3161no—
351D4—3162no—
352D4—3162no—
353D4—3162no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—3162no—
364C2×C4—3162no—
365C2×C4—3161no—
366C2×C4—3161no—
367C2×C4—3162no—
368C2×C4—3162no—
369C2×C4—3162no—
370C2×C4—1584no—
371C2×C4—1581no—
372C2×C4—1581no—
373C2×C4—1581no—
374C2×C4C42×D7911yescharacteristic
375C2×C4C22⋊C4×D7911yescharacteristic
376C2×C4(C22⋊D79)⋊C411yescharacteristic
377C2×C4(C22⋊D79)⋊C411yescharacteristic
378C8—42no—
379C8—42no—
380C8—41no—
381C8—41no—
382C8—792no—
383C8—1581no—
384C8—1581no—
385C8—1581no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22—3161no—
395C22—3161no—
396C22—3161no—
397C22#1909841011yescharacteristic
398C22#1909867912yes—
399C22#1909868412yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—3164no—
404C4—3161no—
405C4—3161no—
406C4—3161no—
407C4—6322no—
408C4#99×D7911yescharacteristic
409C4#1909853511yescharacteristic
410C2—22no—
411C2—22no—
412C2—41no—
413C2—6321no—
414C2#2376318711yescharacteristic
415C2#2376323611yescharacteristic
416C2#2376324311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D79
commutatorC2×C4×C79C2×C42
FrattiniC23×C4C22×D79
FittingC79×#1298C2
socleC22×C79C4×C22⋊C4
prime radical#19099074C22
cosocleC23×C4×C79C23
hypercenter#1298D79
hypocenterC79#8240
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. #23763179C2semidirect
3. #23763174C2semidirect
6. #23761285C2semidirect
7. C79×#1298C2semidirect
8. C2×#14579169C4semidirect
15. #19098591C4semidirect
16. #19098531C4semidirect
17. #19098521C4semidirect
24. #19098301C4semidirect
25. #19098301C4semidirect
26. #19098298C4semidirect
27. #19098298C4semidirect
214. #1298D79semidirect
216. C79#8240semidirect

Maximal subgroups

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

subgroupindex
#237720902
#237631792
#237631742
#237618922
#237612852
C79×#12982
#824079