Finite Group Databasebeta

← #34183499

Subgroups — #34183499

Order 20224.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772349C211yesmaximal, characteristic, complemented
2#23764320C211yesmaximal, characteristic
3#23761899C211yesmaximal, characteristic, complemented
4#23761799C211yesmaximal, characteristic
5#23761240C211yesmaximal, characteristic, complemented
6#23761237C211yesmaximal, characteristic, complemented
7C79×#3167C211yesmaximal, characteristic
8#19098694—21no—
9#19098694—22no—
10Q8×C79⋊C8—21no—
11#19098616C2211yescharacteristic
12#19098590—21no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579088—21no—
15C2×#14579088—21no—
16#19098519—21no—
17C4×#11929608C2211yescharacteristic
18C4×C4⋊Dic79C2211yescharacteristic
19#19098300—22no—
20D4⋊(C79⋊C8)—21no—
21D4⋊(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
31C2×#11929608D411yescharacteristic
32C2×#11929608D411yescharacteristic
33C22×C79⋊C8—21no—
34C2×C4⋊Dic79—21no—
35C2×C4×Dic79—21no—
36C4⋊(C79⋊C8)—22no—
37C4⋊(C79⋊C8)—22no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)D411yescharacteristic
41C4⋊(C79⋊C8)D411yescharacteristic
42#14579088—42no—
43#14579088—42no—
44C22⋊(C79⋊C8)—21no—
45#14579072—22no—
46#14579072—21no—
47C4×C79⋊C8—22no—
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×C79D412yes—
62#121×C79—21no—
63#121×C79—21no—
64C4×Q8×C79—21no—
65C4×Q8×C79D412yes—
66C4×D4×C79—21no—
67C4×D4×C79—21no—
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#11929608SD1612yescomplemented
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C4⋊Dic79—22no—
89C4⋊Dic79—41no—
90C4×Dic79—41no—
91C4×Dic79—41no—
92#41×C79—41no—
93#41×C79—41no—
94#41×C79—41no—
95#41×C79—41no—
96C2×Q8×C79—21no—
97C2×Q8×C79—21no—
98C2×Q8×C79C22⋊C412yes—
99C2×D4×C79—21no—
100C2×D4×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—41no—
103C22×C4×C79—41no—
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79#4111yescharacteristic
107SD16×C79—44no—
108C2×C8×C79—21no—
109C2×C8×C79—21no—
110C2×C8×C79—42no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—22no—
115C4⋊C4×C79—22no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79C22⋊C412yes—
119C22⋊C4×C79—22no—
120C22⋊C4×C79—41no—
121C22⋊C4×C79—41no—
122C42×C79—22no—
123C42×C79C22×C411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79#4111yescharacteristic
127C23×C79—41no—
128C23×C79#12111yescharacteristic
129Q8×C79—42no—
130Q8×C79—41no—
131Q8×C79—41no—
132Q8×C79—41no—
133Q8×C79—41no—
134D4×C79—42no—
135D4×C79—41no—
136D4×C79—41no—
137D4×C79—41no—
138D4×C79—41no—
139D4×C79—81no—
140D4×C79—81no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—22no—
144C2×C4×C79—22no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79—81no—
148C2×C4×C79—81no—
149C2×C4×C79C2×C22⋊C411yescharacteristic
150C2×C4×C79C4×D411yescharacteristic
151C2×C4×C79C4×D411yescharacteristic
152C2×C4×C79#12011yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79#12311yescharacteristic
155C2×C4×C79C2×SD1611yescharacteristic
156C2×C4×C79#13511yescharacteristic
157C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C79#13711yescharacteristic
159C2×Dic79—42no—
160C2×Dic79—81no—
161C2×Dic79—81no—
162C8×C79—22no—
163C8×C79—41no—
164C79⋊C8—22no—
165C79⋊C8—41no—
166C79⋊C8—41no—
167C79⋊C8—41no—
168C79⋊C8—81no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79—81no—
172C22×C79—81no—
173C22×C79—81no—
174C22×C79#38611yescharacteristic
175C22×C79#46011yescharacteristic
176C22×C79#46411yescharacteristic
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—42no—
181C4×C79—42no—
182C4×C79—81no—
183C4×C79C4×SD1611yescharacteristic
184C4×C79#44011yescharacteristic
185C4×C79#44311yescharacteristic
186C4×C79#44411yescharacteristic
187C4×C79#44811yescharacteristic
188C4×C79#44911yescharacteristic
189Dic79—82no—
190#9054—791nomaximal, Sylow
191C2×C79—21no—
192C2×C79—21no—
193C2×C79—161no—
194C2×C79#187611yescharacteristic
195C2×C79#188011yescharacteristic
196C2×C79#188411yescharacteristic
197#3167D7911yescharacteristic
198#2962—791no—
199#1980—791no—
200#1503—791no—
201#1480—791no—
202#1279—791no—
203#1278—791no—
204C79#905411yescharacteristic, complemented, Sylow
205#533C2×D7911yescharacteristic
206#532C2×D7911yescharacteristic
207#498Dic7911yescharacteristic, complemented
208#492Dic7911yescharacteristic, complemented
209#487Dic7911yescharacteristic, complemented
210#486Dic7911yescharacteristic, complemented
211#465—22no—
212#446—1581no—
213#446—1582no—
214C8×Q8—1581no—
215C4×SD16—22no—
216#431—791no—
217#424—1581no—
218C2×C4⋊C8—791no—
219C2×#104—1581no—
220C2×#104—1581no—
221#405—1581no—
222C4×M16—791no—
223C2×C4×C8C2×D7911yescharacteristic
224C4×C4⋊C4—791no—
225#328—1582no—
226D4⋊C8—1581no—
227D4⋊C8—1581no—
228C2×#41—21no—
229C2×SD16—22no—
230C2×M16—791no—
231C2×M16—791no—
232C22×C8—21no—
233C22×C8—1581no—
234C4⋊Q8C2×Dic7911yescharacteristic
235C4⋊Q8C2×Dic7911yescharacteristic
236C4⋊Q8C2×Dic7911yescharacteristic
237#127C2×Dic7911yescharacteristic
238#125—21no—
239#124—21no—
240C22⋊Q8—21no—
241C22⋊Q8C22⋊D7912yes—
242#121—21no—
243#121—21no—
244C4×Q8—21no—
245C4×Q8C22⋊D7912yes—
246C4×D4—21no—
247C4×D4—21no—
248#117—21no—
249C2×C4⋊C4—1581no—
250C2×C42—1581no—
251C2×C42C22×D7911yescharacteristic
252#106—22no—
253C4⋊C8—791no—
254C4⋊C8—791no—
255C4⋊C8—1582no—
256C4⋊C8—1582no—
257C4⋊C8—1581no—
258C4⋊C8—1581no—
259#104—3162no—
260#104—3162no—
261#103—22no—
262#102—22no—
263C22⋊C8—1581no—
264#97—1582no—
265#97—1581no—
266C4×C8—21no—
267C4×C8—1582no—
268C4×C8—1581no—
269C4×C8—1581no—
270C4×C8C2×Dic7911yescharacteristic
271C4×C8C2×Dic7911yescharacteristic
272#95—1581no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C22⋊Dic7912yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—3161no—
286C22×C4D4×D7911yescharacteristic
287C22×C4C2×C22⋊D7911yescharacteristic
288C22×C4#1192962311yescharacteristic
289SD16—44no—
290M16—792no—
291M16—1581no—
292M16—1581no—
293M16—1581no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—42no—
297C2×C8—3161no—
298C2×C8—3161no—
299C2×C8—3161no—
300C2×C8—1581no—
301C2×C8—1581no—
302C2×C8—1581no—
303C2×C8—1581no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—3161no—
312C4⋊C4—1582no—
313C4⋊C4C22⋊Dic7912yes—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—3161no—
319C42—3161no—
320C42C22×Dic7911yescharacteristic
321C42D4×D7911yescharacteristic
322C42C2×C22⋊D7911yescharacteristic
323C42#1192962311yescharacteristic
324C23—41no—
325C23#1457913311yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—3162no—
347C2×C4—6321no—
348C2×C4—6321no—
349C2×C4C2×C22⋊Dic7911yescharacteristic
350C2×C4D4×Dic7911yescharacteristic
351C2×C4D4×Dic7911yescharacteristic
352C2×C4#1457912911yescharacteristic
353C2×C4#1457913311yescharacteristic
354C2×C4#1457914211yescharacteristic
355C2×C4SD16×D7911yescharacteristic
356C2×C4#1457918211yescharacteristic
357C2×C4#1457918811yescharacteristic
358C2×C4#1457919011yescharacteristic
359C8—22no—
360C8—41no—
361C8—3161no—
362C8—3161no—
363C8—3161no—
364C8—6321no—
365C8—1582no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1909844311yescharacteristic
372C22#1909876111yescharacteristic
373C22#1909878311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—6322no—
381C4SD16×Dic7911yescharacteristic
382C4#1909866411yescharacteristic
383C4#1909868011yescharacteristic
384C4#1909868511yescharacteristic
385C4#1909870011yescharacteristic
386C4#1909870711yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2376372911yescharacteristic
391C2#2376375711yescharacteristic
392C2#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#9054
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
1. #23772349C2semidirect
3. #23761899C2semidirect
5. #23761240C2semidirect
6. #23761237C2semidirect
24. #498×C79C4semidirect
25. #492×C79C4semidirect
26. #487×C79C4semidirect
27. #486×C79C4semidirect
80. #11929608SD16semidirect
204. C79#9054semidirect
207. #498Dic79semidirect
208. #492Dic79semidirect
209. #487Dic79semidirect
210. #486Dic79semidirect

Maximal subgroups

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

subgroupindex
#237723492
#237643202
#237618992
#237617992
#237612402
#237612372
C79×#31672
#905479