Finite Group Databasebeta

← #34184204

Subgroups — #34184204

Order 20224.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772864C211yesmaximal, characteristic
2#23761959C211yesmaximal, characteristic, complemented
3#23761894C211yesmaximal, characteristic, complemented
4#23761791C211yesmaximal, characteristic, complemented
5#23761303C211yesmaximal, characteristic, complemented
6#23761258C211yesmaximal, characteristic, complemented
7C79×#2349C211yesmaximal, characteristic, complemented
8C2×#14579187—21no—
9#19099113C2211yescharacteristic
10#19098778—22no—
11#19098768C412yescomplemented
12#19098758C412yescomplemented
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098587C2211yescharacteristic
16#19098562—21no—
17#19098296—22no—
18#19098296—21no—
19#19098296—21no—
20D4⋊(C79⋊C8)—22no—
21C2×C4×D4×C79—21no—
22C2×C4×D4×C79C2211yescharacteristic
23#389×C79—21no—
24#385×C79—22no—
25#382×C79C2211yescharacteristic
26#380×C79—21no—
27C4×C4⋊C4×C79C2211yescharacteristic
28C2×#95×C79—21no—
29C2×#11929624—21no—
30#14579187—42no—
31#14579187—42no—
32C2×#11929612—22no—
33C2×#11929609—22no—
34C2×#11929608—21no—
35#14579161C2×C411yescharacteristic, complemented
36#14579158C2×C411yescharacteristic, complemented
37#14579149—21no—
38#14579135—21no—
39#14579135—21no—
40C4×D316—21no—
41C4×D316—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)C2×C412yescomplemented
46#14579084—22no—
47#14579081—22no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C22×D4×C79D411yescharacteristic
53C23×C4×C79—21no—
54C23×C4×C79—22no—
55C4×D4×C79—21no—
56C4×D4×C79—21no—
57C4×D4×C79—42no—
58C4×D4×C79—42no—
59C4×D4×C79C2×C412yes—
60C4×D4×C79D412yes—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—22no—
64C2×C4⋊C4×C79D411yescharacteristic
65C2×C22⋊C4×C79—21no—
66C2×C22⋊C4×C79—22no—
67C2×C42×C79—21no—
68C2×C42×C79C2311yescharacteristic
69#95×C79—22no—
70#95×C79—21no—
71#95×C79D411yescharacteristic
72#95×C79D411yescharacteristic
73#95×C79D412yes—
74#11929624—41no—
75#11929624—41no—
76#11929624—41no—
77#11929624—41no—
78C2×Dic158—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×C22⋊D79—41no—
83C2×C4×D79—41no—
84#11929612—44no—
85#11929609—44no—
86#11929608—41no—
87#11929608—41no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C4⋊Dic79—21no—
92C4⋊Dic79—21no—
93(C2×D79)⋊C4—41no—
94(C2×D79)⋊C4—41no—
95C24×C79—21no—
96C2×D4×C79—21no—
97C2×D4×C79—21no—
98C2×D4×C79—21no—
99C2×D4×C79C22⋊C412yes—
100C2×D4×C79D812yes—
101C2×D4×C79SD1612yes—
102C22×C4×C79—21no—
103C22×C4×C79—21no—
104C22×C4×C79—22no—
105C22×C4×C79—22no—
106C22×C4×C79—22no—
107C22×C4×C79—21no—
108C22×C4×C79—44no—
109C22×C4×C79—44no—
110C22×C4×C79—82no—
111C22×C4×C79—82no—
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C4⋊C4×C79—21no—
116C4⋊C4×C79—42no—
117C4⋊C4×C79—41no—
118C4⋊C4×C79—41no—
119C4⋊C4×C79—82no—
120C4⋊C4×C79C22⋊C412yes—
121C22⋊C4×C79—22no—
122C22⋊C4×C79—22no—
123C22⋊C4×C79—44no—
124C42×C79—41no—
125C42×C79—41no—
126C42×C79C22×C411yescharacteristic
127C42×C79C2×D411yescharacteristic
128C42×C79#4111yescharacteristic
129C42×C79#4111yescharacteristic
130C23×C79—22no—
131C23×C79—22no—
132C23×C79—22no—
133C23×C79—42no—
134C23×C79—42no—
135C23×C79#12011yescharacteristic
136C22×D79—41no—
137D4×C79—22no—
138D4×C79—22no—
139D4×C79—22no—
140D4×C79#10214yes—
141C2×C4×C79—22no—
142C2×C4×C79—22no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—42no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—42no—
151C2×C4×C79—42no—
152C2×C4×C79—41no—
153C2×C4×C79—42no—
154C2×C4×C79—82no—
155C2×C4×C79—82no—
156C2×C4×C79—88no—
157C2×C4×C79C2×C22⋊C411yescharacteristic
158C2×C4×C79C4×D411yescharacteristic
159C2×C4×C79C4×D411yescharacteristic
160C2×C4×C79#12111yescharacteristic
161C2×C4×C79#12111yescharacteristic
162C2×C4×C79#12311yescharacteristic
163C2×C4×C79C2×D811yescharacteristic
164C2×C4×C79C2×SD1611yescharacteristic
165C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C79#13711yescharacteristic
167C22⋊D79—81no—
168C22⋊D79—81no—
169C2×Dic79—41no—
170D316—41no—
171D316—41no—
172D316—81no—
173D316—81no—
174C4×D79—81no—
175C4×D79—81no—
176Dic158—41no—
177Dic158—41no—
178C79⋊C8—42no—
179C79⋊C8—81no—
180C79⋊C8—81no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—24no—
184C22×C79—44no—
185C22×C79—44no—
186C22×C79—42no—
187C22×C79—42no—
188C22×C79#38611yescharacteristic
189C22×C79#44711yescharacteristic
190C22×C79#44811yescharacteristic
191C2×D79—81no—
192C2×D79—81no—
193C2×D79—81no—
194C4×C79—21no—
195C4×C79—21no—
196C4×C79—21no—
197C4×C79—21no—
198C4×C79—42no—
199C4×C79—42no—
200C4×C79—81no—
201C4×C79—82no—
202C4×C79C2×#10211yescharacteristic
203C4×C79#41711yescharacteristic
204C4×C79#48011yescharacteristic
205C4×C79#48111yescharacteristic
206Dic79—81no—
207#9179—791nomaximal, Sylow
208C2×C79—21no—
209C2×C79—21no—
210C2×C79—44no—
211C2×C79#187411yescharacteristic
212C2×C79#190411yescharacteristic
213C2×C79#190611yescharacteristic
214D79—161no—
215#3036—791no—
216#2349D7911yescharacteristic, complemented
217#1517—791no—
218#1502—791no—
219#1478—791no—
220#1303—791no—
221#1286—791no—
222C79#917911yescharacteristic, complemented, Sylow
223C2×AGL(1,ℤ/8)—1581no—
224#532—791no—
225C2×C4×D4—21no—
226C2×C4×D4C2×D7911yescharacteristic
227#463—1582no—
228#461—792no—
229#459—792no—
230#431—791no—
231#431—791no—
232#423—791no—
233#417—1581no—
234#389—21no—
235#385—22no—
236#382C2×D7911yescharacteristic
237#380—21no—
238C4×C4⋊C4C2×D7911yescharacteristic
239C2×#95—21no—
240#327—1581no—
241#327—1581no—
242#327—1582no—
243D4⋊C8—1582no—
244C2×#41—1581no—
245C22×D4C22⋊D7911yescharacteristic
246C23×C4—21no—
247C23×C4—22no—
248AGL(1,ℤ/8)—3162no—
249AGL(1,ℤ/8)—3162no—
250C2×SD16—1582no—
251C2×D8—1582no—
252C2×M16—1581no—
253C4⋊Q8—791no—
254#127—791no—
255#124—1581no—
256#121—1581no—
257#121—1581no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—1581no—
263C4×D4—1581no—
264C4×D4C4×D7912yes—
265C4×D4D31612yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4C22⋊D7911yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C42—21no—
273C2×C42C22×D7911yescharacteristic
274C4⋊C8—792no—
275C4⋊C8—1581no—
276C4⋊C8—1581no—
277C4⋊C8—1581no—
278#103—1582no—
279#102—1582no—
280C22⋊C8—1581no—
281C22⋊C8—1581no—
282C4×C8—1581no—
283C4×C8—1581no—
284#95—22no—
285#95—21no—
286#95C22⋊D7911yescharacteristic
287#95C22⋊D7911yescharacteristic
288#95C22⋊D7912yes—
289C24—21no—
290#41—3161no—
291#41—3161no—
292#41—3161no—
293#41—3161no—
294C2×Q8—1581no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—3161no—
299C2×D4—1581no—
300C2×D4—1581no—
301C2×D4—1581no—
302C2×D4(C2×D79)⋊C412yes—
303C2×D4#1192960912yes—
304C2×D4#1192961112yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—3161no—
316C22×C4C2×C22⋊D7911yescharacteristic
317C22×C4C2×C22⋊D7911yescharacteristic
318C22×C4C2×C22⋊D7911yescharacteristic
319SD16—3164no—
320D8—3164no—
321M16—3161no—
322M16—3161no—
323C2×C8—3161no—
324C2×C8—3161no—
325C2×C8—1582no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—1581no—
332C4⋊C4—1581no—
333C4⋊C4(C2×D79)⋊C412yes—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—44no—
337C22⋊C4—3161no—
338C22⋊C4—3161no—
339C42—41no—
340C42—41no—
341C42C2×C4×D7911yescharacteristic
342C42C2×D31611yescharacteristic
343C42#1192962411yescharacteristic
344C42#1192962411yescharacteristic
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—3161no—
351C23#1457913011yescharacteristic
352Q8—3161no—
353Q8—3161no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—3161no—
358D4—3161no—
359D4—6321no—
360D4—6321no—
361D4—6321no—
362D4—6321no—
363D4#1457908114yes—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—88no—
380C2×C4—3161no—
381C2×C4—6321no—
382C2×C4—6321no—
383C2×C4C2×(C2×D79)⋊C411yescharacteristic
384C2×C4C4×C22⋊D7911yescharacteristic
385C2×C4C4×C22⋊D7911yescharacteristic
386C2×C4#1457913511yescharacteristic
387C2×C4#1457913511yescharacteristic
388C2×C4#1457914411yescharacteristic
389C2×C4C2×#1192960911yescharacteristic
390C2×C4C2×#1192961111yescharacteristic
391C2×C4#1457918711yescharacteristic
392C2×C4#1457919211yescharacteristic
393C8—3162no—
394C8—6321no—
395C8—6321no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—6321no—
404C22—6321no—
405C22—6321no—
406C22#1909844211yescharacteristic
407C22#1909869711yescharacteristic
408C22#1909870211yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—6321no—
418C4C2×#1457908111yescharacteristic
419C4#1909856211yescharacteristic
420C4#1909887611yescharacteristic
421C4#1909888111yescharacteristic
422C2—21no—
423C2—21no—
424C2—44no—
425C2—12641no—
426C2#2376371711yescharacteristic
427C2#2376388711yescharacteristic
428C2#2376390111yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2349C2
socleC22×C79#386
prime radical#23772864C2
cosocleC2×C42×C79C23
hypercenter#2349D79
hypocenterC79#9179
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #23761959C2semidirect
3. #23761894C2semidirect
4. #23761791C2semidirect
5. #23761303C2semidirect
6. #23761258C2semidirect
7. C79×#2349C2semidirect
11. #19098768C4semidirect
12. #19098758C4semidirect
35. #14579161C2×C4semidirect
36. #14579158C2×C4semidirect
45. C4⋊(C79⋊C8)C2×C4semidirect
216. #2349D79semidirect
222. C79#9179semidirect

Maximal subgroups

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

subgroupindex
#237728642
#237619592
#237618942
#237617912
#237613032
#237612582
C79×#23492
#917979