Finite Group Databasebeta

← #34183506

Subgroups — #34183506

Order 20224.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772144C211yesmaximal, characteristic
2#23764345C211yesmaximal, characteristic, complemented
3#23761895C211yesmaximal, characteristic
4#23761803C211yesmaximal, characteristic, complemented
5#23761240C211yesmaximal, characteristic, complemented
6#23761237C211yesmaximal, characteristic, complemented
7C79×#3167C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098641—21no—
10#19098635—22no—
11D4×C79⋊C8—21no—
12#19098616C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14#19098530—21no—
15C4×#11929608C2211yescharacteristic
16C4×C4⋊Dic79C2211yescharacteristic
17#19098322—21no—
18#19098322—21no—
19#19098297—22no—
20Q8⋊(C79⋊C8)—21no—
21Q8⋊(C79⋊C8)—21no—
22#533×C79C2211yescharacteristic
23#532×C79C2211yescharacteristic
24#498×C79C411yescharacteristic, complemented
25#492×C79C411yescharacteristic, complemented
26#487×C79C411yescharacteristic, complemented
27#486×C79C411yescharacteristic, complemented
28#465×C79—22no—
29C4×SD16×C79—22no—
30C2×C4×C8×C79C2211yescharacteristic
31#14579168—42no—
32#14579168—42no—
33C2×#11929608—22no—
34C2×#11929608D411yescharacteristic
35C2×#11929608D411yescharacteristic
36C22×C79⋊C8—22no—
37C22×C79⋊C8—21no—
38C2×C4⋊Dic79—21no—
39C2×C4×Dic79—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)D411yescharacteristic
43C4⋊(C79⋊C8)D411yescharacteristic
44C22⋊(C79⋊C8)—22no—
45C22⋊(C79⋊C8)—22no—
46C22⋊(C79⋊C8)—21no—
47#14579072—21no—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C79⋊#95—21no—
51C2×#41×C79—21no—
52C2×SD16×C79—22no—
53C22×C8×C79—21no—
54C4⋊Q8×C79C2×C411yescharacteristic
55C4⋊Q8×C79C2×C411yescharacteristic
56C4⋊Q8×C79C2×C411yescharacteristic
57#127×C79C2×C411yescharacteristic
58#125×C79—21no—
59#124×C79—21no—
60C22⋊Q8×C79—21no—
61C22⋊Q8×C79—21no—
62#121×C79—21no—
63#121×C79D412yes—
64C4×Q8×C79—21no—
65C4×Q8×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79D412yes—
68#117×C79—21no—
69C2×C42×C79C2311yescharacteristic
70#106×C79—22no—
71#103×C79—22no—
72#102×C79—22no—
73C4×C8×C79—21no—
74C4×C8×C79C2×C411yescharacteristic
75C4×C8×C79C2×C411yescharacteristic
76C22×Dic79—41no—
77#11929608—21no—
78#11929608—21no—
79#11929608—21no—
80#11929608—44no—
81#11929608SD1612yescomplemented
82C2×C79⋊C8—21no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—44no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C4⋊Dic79—22no—
91C4⋊Dic79—41no—
92C4×Dic79—41no—
93C4×Dic79—41no—
94#41×C79—41no—
95#41×C79—41no—
96#41×C79—41no—
97#41×C79—41no—
98C2×Q8×C79—21no—
99C2×Q8×C79—21no—
100C2×Q8×C79—21no—
101C2×D4×C79—22no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79C22⋊C412yes—
105C22×C4×C79—22no—
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79#4111yescharacteristic
109SD16×C79—44no—
110C2×C8×C79—21no—
111C2×C8×C79—21no—
112C2×C8×C79—42no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—41no—
119C4⋊C4×C79—41no—
120C4⋊C4×C79C22⋊C412yes—
121C22⋊C4×C79—22no—
122C22⋊C4×C79—22no—
123C22⋊C4×C79—41no—
124C42×C79—41no—
125C42×C79C22×C411yescharacteristic
126C42×C79C2×D411yescharacteristic
127C42×C79C2×D411yescharacteristic
128C42×C79#4111yescharacteristic
129C23×C79—22no—
130C23×C79#12111yescharacteristic
131Q8×C79—42no—
132Q8×C79—41no—
133Q8×C79—41no—
134Q8×C79—41no—
135Q8×C79—41no—
136D4×C79—42no—
137D4×C79—42no—
138D4×C79—42no—
139D4×C79—41no—
140D4×C79—41no—
141D4×C79—41no—
142D4×C79—41no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—22no—
146C2×C4×C79—41no—
147C2×C4×C79—42no—
148C2×C4×C79—42no—
149C2×C4×C79—41no—
150C2×C4×C79—41no—
151C2×C4×C79C2×C22⋊C411yescharacteristic
152C2×C4×C79C4×D411yescharacteristic
153C2×C4×C79C4×D411yescharacteristic
154C2×C4×C79#12011yescharacteristic
155C2×C4×C79#12111yescharacteristic
156C2×C4×C79#12311yescharacteristic
157C2×C4×C79C2×SD1611yescharacteristic
158C2×C4×C79#13511yescharacteristic
159C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C79#13711yescharacteristic
161C2×Dic79—42no—
162C2×Dic79—81no—
163C2×Dic79—81no—
164C8×C79—22no—
165C8×C79—41no—
166C79⋊C8—22no—
167C79⋊C8—41no—
168C79⋊C8—41no—
169C79⋊C8—41no—
170C79⋊C8—81no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—42no—
174C22×C79—42no—
175C22×C79—42no—
176C22×C79#38611yescharacteristic
177C22×C79#46111yescharacteristic
178C22×C79#46311yescharacteristic
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—42no—
183C4×C79—81no—
184C4×C79—81no—
185C4×C79C4×SD1611yescharacteristic
186C4×C79#44011yescharacteristic
187C4×C79#44311yescharacteristic
188C4×C79#44411yescharacteristic
189C4×C79#45011yescharacteristic
190C4×C79#45211yescharacteristic
191Dic79—82no—
192#9055—791nomaximal, Sylow
193C2×C79—21no—
194C2×C79—21no—
195C2×C79—82no—
196C2×C79#187611yescharacteristic
197C2×C79#187711yescharacteristic
198C2×C79#188411yescharacteristic
199#3167D7911yescharacteristic
200#2935—791no—
201#1986—791no—
202#1502—791no—
203#1481—791no—
204#1279—791no—
205#1278—791no—
206C79#905511yescharacteristic, complemented, Sylow
207C2×#131—1581no—
208#533C2×D7911yescharacteristic
209#532C2×D7911yescharacteristic
210#498Dic7911yescharacteristic, complemented
211#492Dic7911yescharacteristic, complemented
212#487Dic7911yescharacteristic, complemented
213#486Dic7911yescharacteristic, complemented
214#465—22no—
215C4×SD16—22no—
216#436—1581no—
217#435—1582no—
218C8×D4—1581no—
219#431—791no—
220C2×C4⋊C8—791no—
221#408—1581no—
222C4×M16—791no—
223C2×C4×C8C2×D7911yescharacteristic
224C4×C4⋊C4—791no—
225#339—1581no—
226#339—1581no—
227#327—1582no—
228Q8⋊C8—1581no—
229Q8⋊C8—1581no—
230C2×#41—21no—
231C2×SD16—22no—
232#131—3162no—
233#131—3162no—
234C2×M16—791no—
235C2×M16—791no—
236C2×M16—1582no—
237C22×C8—21no—
238C22×C8—1582no—
239C22×C8—1581no—
240C4⋊Q8C2×Dic7911yescharacteristic
241C4⋊Q8C2×Dic7911yescharacteristic
242C4⋊Q8C2×Dic7911yescharacteristic
243#127C2×Dic7911yescharacteristic
244#125—21no—
245#124—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—21no—
249#121C22⋊D7912yes—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—21no—
253C4×D4C22⋊D7912yes—
254#117—21no—
255C2×C4⋊C4—1581no—
256C2×C42—1581no—
257C2×C42C22×D7911yescharacteristic
258#106—22no—
259C4⋊C8—791no—
260C4⋊C8—791no—
261C4⋊C8—1581no—
262C4⋊C8—1581no—
263#103—22no—
264#102—22no—
265C22⋊C8—1581no—
266C22⋊C8—1582no—
267C22⋊C8—1582no—
268#97—1581no—
269C4×C8—21no—
270C4×C8—1581no—
271C4×C8—1581no—
272C4×C8C2×Dic7911yescharacteristic
273C4×C8C2×Dic7911yescharacteristic
274#95—1581no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C22⋊Dic7912yes—
286C22×C4—22no—
287C22×C4—3161no—
288C22×C4D4×D7911yescharacteristic
289C22×C4C2×C22⋊D7911yescharacteristic
290C22×C4#1192962311yescharacteristic
291SD16—44no—
292M16—3164no—
293M16—792no—
294M16—1581no—
295M16—1581no—
296M16—1581no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—42no—
300C2×C8—3161no—
301C2×C8—3161no—
302C2×C8—3164no—
303C2×C8—3161no—
304C2×C8—1581no—
305C2×C8—1581no—
306C2×C8—1581no—
307C2×C8—1581no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—3161no—
316C4⋊C4—1582no—
317C4⋊C4C22⋊Dic7912yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C42—41no—
322C42—3161no—
323C42—3161no—
324C42C22×Dic7911yescharacteristic
325C42D4×D7911yescharacteristic
326C42C2×C22⋊D7911yescharacteristic
327C42#1192962311yescharacteristic
328C23—22no—
329C23#1457913311yescharacteristic
330Q8—42no—
331Q8—41no—
332Q8—41no—
333Q8—41no—
334Q8—41no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—3162no—
351C2×C4—6321no—
352C2×C4—6321no—
353C2×C4C2×C22⋊Dic7911yescharacteristic
354C2×C4D4×Dic7911yescharacteristic
355C2×C4D4×Dic7911yescharacteristic
356C2×C4#1457912911yescharacteristic
357C2×C4#1457913311yescharacteristic
358C2×C4#1457914211yescharacteristic
359C2×C4SD16×D7911yescharacteristic
360C2×C4#1457918211yescharacteristic
361C2×C4#1457918811yescharacteristic
362C2×C4#1457919011yescharacteristic
363C8—22no—
364C8—41no—
365C8—3161no—
366C8—3161no—
367C8—3161no—
368C8—6321no—
369C8—1582no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1909844311yescharacteristic
376C22#1909876611yescharacteristic
377C22#1909878011yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—81no—
383C4—81no—
384C4—6322no—
385C4SD16×Dic7911yescharacteristic
386C4#1909866411yescharacteristic
387C4#1909868011yescharacteristic
388C4#1909868511yescharacteristic
389C4#1909871211yescharacteristic
390C4#1909872211yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2376372911yescharacteristic
395C2#2376373611yescharacteristic
396C2#2376378411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3167C2
socleC22×C79#386
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#3167D79
hypocenterC79#9055
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
2. #23764345C2semidirect
4. #23761803C2semidirect
5. #23761240C2semidirect
6. #23761237C2semidirect
24. #498×C79C4semidirect
25. #492×C79C4semidirect
26. #487×C79C4semidirect
27. #486×C79C4semidirect
81. #11929608SD16semidirect
206. C79#9055semidirect
210. #498Dic79semidirect
211. #492Dic79semidirect
212. #487Dic79semidirect
213. #486Dic79semidirect

Maximal subgroups

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

subgroupindex
#237721442
#237643452
#237618952
#237618032
#237612402
#237612372
C79×#31672
#905579