Finite Group Databasebeta

← #34184569

Subgroups — #34184569

Order 20224.

512 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773431C211yesmaximal, characteristic
2#23764228C211yesmaximal, characteristic
3#23761894C211yesmaximal, characteristic
4#23761829C211yesmaximal, characteristic, complemented
5#23761303C211yesmaximal, characteristic, complemented
6#23761233C211yesmaximal, characteristic, complemented
7C79×#2295C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098881—21no—
10#19098848C412yescomplemented
11#19098763C412yescomplemented
12#19098702—21no—
13C4×#11929611—22no—
14#19098616C2211yescharacteristic
15#19098616C2211yescharacteristic
16#19098587C2211yescharacteristic
17C2×#14579089—21no—
18C2×#14579089—21no—
19#19098296—22no—
20Q8⋊(C79⋊C8)—22no—
21C2×C4×Q8×C79C2211yescharacteristic
22#391×C79—21no—
23#389×C79—21no—
24#387×C79—22no—
25C4×C4⋊C4×C79—21no—
26C4×C4⋊C4×C79C2211yescharacteristic
27#376×C79—21no—
28C43×C79C2211yescharacteristic
29C2×#11929624—21no—
30C2×#11929611—22no—
31C2×#11929608—21no—
32#14579161C2×C411yescharacteristic
33#14579158C2×C411yescharacteristic
34#14579149—21no—
35#14579135—21no—
36#14579135—21no—
37C4×D316—21no—
38C4×D316—21no—
39#14579092—22no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)C2×C412yes—
44#14579089—42no—
45#14579089—42no—
46#14579085—22no—
47#14579081—22no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C22×Q8×C79D411yescharacteristic
53C4×Q8×C79—21no—
54C4×Q8×C79—21no—
55C4×Q8×C79C2×C412yes—
56C4×Q8×C79D412yes—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79—22no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79D411yescharacteristic
61C2×C42×C79—21no—
62C2×C42×C79—21no—
63C2×C42×C79—21no—
64C2×C42×C79—21no—
65C2×C42×C79D411yescharacteristic
66C2×C42×C79D411yescharacteristic
67C2×C42×C79C2311yescharacteristic
68#95×C79—22no—
69#95×C79—22no—
70#95×C79—22no—
71#95×C79D412yes—
72#11929624—41no—
73#11929624—41no—
74#11929624—41no—
75#11929624—41no—
76C2×Dic158—21no—
77C2×D316—21no—
78C2×D316—21no—
79C2×D316—21no—
80C2×C22⋊D79—41no—
81C2×C4×D79—41no—
82#11929611—44no—
83#11929608—41no—
84#11929608—41no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C4⋊Dic79—21no—
89C4⋊Dic79—21no—
90(C2×D79)⋊C4—41no—
91(C2×D79)⋊C4—41no—
92C2×Q8×C79—21no—
93C2×Q8×C79—42no—
94C2×Q8×C79C22⋊C412yes—
95C22×C4×C79—21no—
96C22×C4×C79—21no—
97C22×C4×C79—21no—
98C22×C4×C79—21no—
99C22×C4×C79—22no—
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C4⋊C4×C79—22no—
104C4⋊C4×C79—22no—
105C4⋊C4×C79—21no—
106C4⋊C4×C79—42no—
107C4⋊C4×C79—44no—
108C4⋊C4×C79C22⋊C412yes—
109C42×C79—22no—
110C42×C79—22no—
111C42×C79—21no—
112C42×C79—21no—
113C42×C79—21no—
114C42×C79—42no—
115C42×C79—42no—
116C42×C79—41no—
117C42×C79—41no—
118C42×C79SD1612yescomplemented
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79#4111yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79#12011yescharacteristic
124C22×D79—41no—
125Q8×C79—22no—
126Q8×C79—22no—
127Q8×C79—42no—
128C2×C4×C79—22no—
129C2×C4×C79—22no—
130C2×C4×C79—22no—
131C2×C4×C79—22no—
132C2×C4×C79—22no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—42no—
138C2×C4×C79—42no—
139C2×C4×C79—41no—
140C2×C4×C79—42no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79C2×C22⋊C411yescharacteristic
144C2×C4×C79C4×D411yescharacteristic
145C2×C4×C79C4×D411yescharacteristic
146C2×C4×C79#12111yescharacteristic
147C2×C4×C79#12111yescharacteristic
148C2×C4×C79#12311yescharacteristic
149C2×C4×C79C2×SD1611yescharacteristic
150C2×C4×C79#13511yescharacteristic
151C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C79#13711yescharacteristic
153C22⋊D79—81no—
154C22⋊D79—81no—
155C2×Dic79—41no—
156D316—42no—
157D316—41no—
158D316—41no—
159D316—81no—
160C4×D79—81no—
161C4×D79—81no—
162Dic158—41no—
163Dic158—41no—
164C79⋊C8—42no—
165C79⋊C8—81no—
166C79⋊C8—81no—
167C22×C79—21no—
168C22×C79—21no—
169C22×C79#38611yescharacteristic
170C22×C79#44811yescharacteristic
171C22×C79#44911yescharacteristic
172C2×D79—81no—
173C2×D79—81no—
174C2×D79—81no—
175C4×C79—21no—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—42no—
179C4×C79—42no—
180C4×C79—42no—
181C4×C79—42no—
182C4×C79—82no—
183C4×C79#42112yes—
184C4×C79C4×SD1611yescharacteristic
185C4×C79#44011yescharacteristic
186C4×C79#46011yescharacteristic
187C4×C79#46411yescharacteristic
188Dic79—81no—
189#9238—791nomaximal, Sylow
190C2×C79—21no—
191C2×C79—21no—
192C2×C79#188011yescharacteristic
193C2×C79#189711yescharacteristic
194C2×C79#190611yescharacteristic
195D79—161no—
196#3103—791no—
197#2295D7911yescharacteristic, complemented
198#1962—791no—
199#1502—791no—
200#1487—791no—
201#1303—791no—
202#1276—791no—
203C79#923811yescharacteristic, complemented, Sylow
204#532—791no—
205C2×C4×Q8C2×D7911yescharacteristic
206#481—1581no—
207#475—792no—
208#460—792no—
209#448—1581no—
210C4×SD16—1582no—
211#431—791no—
212#431—791no—
213#423—791no—
214C2×#104—1581no—
215C2×#104—1581no—
216#391—21no—
217#389—21no—
218#387—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2×D7911yescharacteristic
221#376—21no—
222C43C2×D7911yescharacteristic
223#327—1582no—
224Q8⋊C8—1582no—
225C2×#41—1581no—
226C22×Q8C22⋊D7911yescharacteristic
227C2×SD16—1582no—
228C2×M16—1581no—
229C4⋊Q8—791no—
230#127—791no—
231#124—1581no—
232#121—1581no—
233#121—1581no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8C4×D7912yes—
237C4×Q8D31612yes—
238C4×D4—1581no—
239C4×D4—1581no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C22⋊D7911yescharacteristic
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C22⋊D7911yescharacteristic
249C2×C42C22⋊D7911yescharacteristic
250C2×C42C22×D7911yescharacteristic
251#106—1582no—
252C4⋊C8—792no—
253C4⋊C8—1581no—
254C4⋊C8—1581no—
255C4⋊C8—1581no—
256#104—3162no—
257#104—3162no—
258#103—1582no—
259#102—1582no—
260C22⋊C8—1581no—
261C22⋊C8—1581no—
262C4×C8—1581no—
263C4×C8—1581no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95C22⋊D7912yes—
268#41—3161no—
269#41—3161no—
270#41—3161no—
271#41—3161no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×Q8—1581no—
275C2×Q8(C2×D79)⋊C412yes—
276C2×D4—3161no—
277C2×D4—1581no—
278C2×D4—1581no—
279C2×D4—1581no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—3161no—
286C22×C4C2×C22⋊D7911yescharacteristic
287C22×C4C2×C22⋊D7911yescharacteristic
288C22×C4C2×C22⋊D7911yescharacteristic
289SD16—3164no—
290M16—3161no—
291M16—3161no—
292C2×C8—3161no—
293C2×C8—3161no—
294C2×C8—1582no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—1581no—
301C4⋊C4—1581no—
302C4⋊C4(C2×D79)⋊C412yes—
303C22⋊C4—3161no—
304C22⋊C4—3161no—
305C42—22no—
306C42—22no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—42no—
311C42—42no—
312C42—41no—
313C42—41no—
314C42#1192961112yescomplemented
315C42C2×C4×D7911yescharacteristic
316C42C2×D31611yescharacteristic
317C42#1192962411yescharacteristic
318C42#1192962411yescharacteristic
319C23—3161no—
320C23#1457913011yescharacteristic
321Q8—22no—
322Q8—22no—
323Q8—42no—
324Q8—3161no—
325Q8—3161no—
326D4—3161no—
327D4—3161no—
328D4—3162no—
329D4—6321no—
330D4—6321no—
331D4—6321no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—3161no—
348C2×C4—6321no—
349C2×C4—6321no—
350C2×C4C2×(C2×D79)⋊C411yescharacteristic
351C2×C4C4×C22⋊D7911yescharacteristic
352C2×C4C4×C22⋊D7911yescharacteristic
353C2×C4#1457913511yescharacteristic
354C2×C4#1457913511yescharacteristic
355C2×C4#1457914411yescharacteristic
356C2×C4C2×#1192961111yescharacteristic
357C2×C4#1457918111yescharacteristic
358C2×C4#1457918611yescharacteristic
359C2×C4#1457919211yescharacteristic
360C8—3162no—
361C8—6321no—
362C8—6321no—
363C22—21no—
364C22—21no—
365C22—6321no—
366C22—6321no—
367C22—6321no—
368C22#1909844211yescharacteristic
369C22#1909870211yescharacteristic
370C22#1909870811yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—82no—
379C4—6321no—
380C4#1909858112yes—
381C4C4×#1192961111yescharacteristic
382C4#1909866111yescharacteristic
383C4#1909876311yescharacteristic
384C4#1909878511yescharacteristic
385C2—21no—
386C2—21no—
387C2—12641no—
388C2#2376375511yescharacteristic
389C2#2376384811yescharacteristic
390C2#2376390111yescharacteristic
subgroupG/H
centerC2×C4#14579135
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2295C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#2295D79
hypocenterC79#9238
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2295

Complements

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

NH ≅ G/Ntype
4. #23761829C2semidirect
5. #23761303C2semidirect
6. #23761233C2semidirect
7. C79×#2295C2semidirect
10. #19098848C4semidirect
11. #19098763C4semidirect
118. C42×C79SD16semidirect
197. #2295D79semidirect
203. C79#9238semidirect
314. C42#11929611semidirect

Maximal subgroups

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

subgroupindex
#237734312
#237642282
#237618942
#237618292
#237613032
#237612332
C79×#22952
#923879