Finite Group Databasebeta

← #34273493

Subgroups — #34273493

Order 20224.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774528C211yesmaximal, characteristic, complemented
2#23772864C211yesmaximal, characteristic
3#23772424C211yesmaximal, characteristic, complemented
4#23762811C214yesmaximal, complemented
5#23762777C214yesmaximal, complemented
6#23762170C211yesmaximal, characteristic, complemented
7#23762081C211yesmaximal, characteristic, complemented
8#23762050C211yesmaximal, characteristic, complemented
9C2×#476×C79C211yesmaximal, characteristic, complemented
10C2×#14579187—21no—
11#19099113C2211yescharacteristic
12#19098959C2212yescomplemented
13#19098911C2212yescomplemented
14#19098881—22no—
15#19098850—22no—
16#19098778—22no—
17#19098768C2212yes—
18#19098758C2212yes—
19D4×C79⋊C8—22no—
20D4×C79⋊C8C2214yes—
21#19098616C2211yescharacteristic
22#19098616C2211yescharacteristic
23#19098587C2211yescharacteristic
24#19098562—21no—
25C2×C22⋊(C79⋊C8)—22no—
26#19098313C2214yescomplemented
27#19098312C2214yescomplemented
28#19098310C2212yescomplemented
29#19098306C2212yescomplemented
30C2×C4⋊Q8×C79C2211yescharacteristic
31C2×C4×D4×C79C2211yescharacteristic
32#476×C79—22no—
33#476×C79C2214yes—
34C2×#106×C79—21no—
35C2×C4⋊C8×C79C2211yescharacteristic
36C2×#102×C79—21no—
37C2×#11929624—21no—
38#14579187—44no—
39C2×#11929612—22no—
40C2×#11929609—22no—
41C2×#11929608—21no—
42C22×C79⋊C8—24no—
43#14579161C2311yescharacteristic
44#14579158C2311yescharacteristic
45#14579149—21no—
46#14579135—21no—
47#14579135—21no—
48C4×D316—21no—
49C4×D316—21no—
50#14579092—24no—
51C4⋊(C79⋊C8)—21no—
52C4⋊(C79⋊C8)D412yes—
53C4⋊(C79⋊C8)C2312yes—
54C4⋊(C79⋊C8)C2312yes—
55#14579084—22no—
56#14579081—22no—
57#14579081—24no—
58C22⋊(C79⋊C8)—21no—
59C22⋊(C79⋊C8)—24no—
60C22⋊(C79⋊C8)D412yes—
61C4×C79⋊C8—21no—
62C4×C79⋊C8C2312yes—
63C22×Q8×C79—21no—
64C22×D4×C79D411yescharacteristic
65C23×C4×C79—21no—
66C22×C8×C79—21no—
67C4⋊Q8×C79—21no—
68C4⋊Q8×C79—21no—
69C4⋊Q8×C79—21no—
70C4⋊Q8×C79C2312yes—
71C4×D4×C79—21no—
72C4×D4×C79—21no—
73C4×D4×C79D412yes—
74C4×D4×C79C2312yes—
75C2×C4⋊C4×C79—21no—
76C2×C4⋊C4×C79—21no—
77C2×C4⋊C4×C79D411yescharacteristic
78C2×C22⋊C4×C79—21no—
79C2×C42×C79C2311yescharacteristic
80#106×C79—24no—
81C4⋊C8×C79—21no—
82C4⋊C8×C79C2312yes—
83#102×C79—24no—
84#11929624—42no—
85#11929624—42no—
86C2×Dic158—21no—
87C2×D316—21no—
88C2×D316—21no—
89C2×D316—21no—
90C2×C22⋊D79—41no—
91C2×C4×D79—41no—
92#11929612—44no—
93#11929609—44no—
94#11929608—42no—
95C2×C79⋊C8—22no—
96C2×C79⋊C8—22no—
97C2×C79⋊C8—84no—
98C2×C79⋊C8C2×D414yes—
99C4⋊Dic79—21no—
100C4⋊Dic79—21no—
101(C2×D79)⋊C4—41no—
102(C2×D79)⋊C4—41no—
103C24×C79—21no—
104C2×Q8×C79—22no—
105C2×Q8×C79—41no—
106C2×Q8×C79—82no—
107C2×D4×C79—21no—
108C2×D4×C79—22no—
109C2×D4×C79SD1614yes—
110C2×D4×C79C2×D412yes—
111C22×C4×C79—21no—
112C22×C4×C79—22no—
113C22×C4×C79—22no—
114C22×C4×C79—41no—
115C22×C4×C79—82no—
116C22×C4×C79C2×D411yescharacteristic
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79#4111yescharacteristic
119C2×C8×C79—22no—
120C2×C8×C79—42no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—22no—
123C4⋊C4×C79—22no—
124C4⋊C4×C79—41no—
125C4⋊C4×C79—41no—
126C4⋊C4×C79C2×D412yes—
127C22⋊C4×C79—22no—
128C22⋊C4×C79—22no—
129C42×C79C2×D411yescharacteristic
130C42×C79C2×D411yescharacteristic
131C42×C79#4111yescharacteristic
132C42×C79C2411yescharacteristic
133C23×C79—22no—
134C23×C79—24no—
135C23×C79—81no—
136C23×C79—81no—
137C23×C79#12111yescharacteristic
138C22×D79—41no—
139Q8×C79—82no—
140Q8×C79—82no—
141D4×C79—22no—
142D4×C79—24no—
143D4×C79C2×SD1614yes—
144C2×C4×C79—22no—
145C2×C4×C79—22no—
146C2×C4×C79—41no—
147C2×C4×C79—44no—
148C2×C4×C79—42no—
149C2×C4×C79—82no—
150C2×C4×C79—81no—
151C2×C4×C79—82no—
152C2×C4×C79#12111yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79#12111yescharacteristic
155C2×C4×C79C2×SD1612yes—
156C2×C4×C79#13512yes—
157C2×C4×C79C22×D411yescharacteristic
158C2×C4×C79C22×D411yescharacteristic
159C2×C4×C79C2×#4111yescharacteristic
160C22⋊D79—82no—
161C2×Dic79—41no—
162D316—41no—
163D316—41no—
164D316—42no—
165D316—81no—
166C4×D79—82no—
167Dic158—41no—
168Dic158—41no—
169C8×C79—42no—
170C79⋊C8—24no—
171C79⋊C8—42no—
172C79⋊C8—42no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—24no—
176C22×C79—42no—
177C22×C79—42no—
178C22×C79—84no—
179C22×C79#46512yes—
180C22×C79C2×#12111yescharacteristic
181C2×D79—81no—
182C2×D79—81no—
183C2×D79—81no—
184C4×C79—21no—
185C4×C79—21no—
186C4×C79—21no—
187C4×C79—42no—
188C4×C79—82no—
189C4×C79#46512yes—
190C4×C79C22×SD1611yescharacteristic
191C4×C79C2×#13511yescharacteristic
192C4×C79#57712yes—
193Dic79—81no—
194#19391—791nomaximal, Sylow
195C2×C79—22no—
196C2×C79—44no—
197C2×C79#304211yescharacteristic
198C2×C79C2×#46511yescharacteristic
199C2×C79#306811yescharacteristic
200D79—161no—
201#3227—791no—
202C2×#476D7911yescharacteristic, complemented
203#3036—791no—
204#2972—791no—
205#1677—794no—
206#1671—794no—
207#1570—791no—
208#1548—791no—
209#1540—791no—
210C79#1939111yescharacteristic, complemented, Sylow
211C2×AGL(1,ℤ/8)—1581no—
212#532—791no—
213C2×C4⋊Q8C2×D7911yescharacteristic
214C2×C4×D4C2×D7911yescharacteristic
215#498—792no—
216#486—792no—
217#481—1582no—
218#476—22no—
219#476—1582no—
220#476C2×D7914yes—
221#463—1582no—
222#461—792no—
223#459—792no—
224C8×D4—794no—
225C8×D4—1582no—
226#431—791no—
227#431—791no—
228C2×#106—21no—
229#423—791no—
230C2×C4⋊C8C2×D7911yescharacteristic
231#417—1581no—
232C2×#102—21no—
233C2×C22⋊C8—1582no—
234#334—794no—
235#333—794no—
236#332—792no—
237#331—792no—
238C2×#41—1581no—
239C22×Q8—21no—
240C22×D4C22⋊D7911yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—3164no—
243C2×SD16—1582no—
244C2×D8—1582no—
245C2×M16—1581no—
246C22×C8—21no—
247C22×C8—1584no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—791no—
252C4⋊Q8C22×D7912yes—
253#127—791no—
254#124—1581no—
255#121—1581no—
256#121—1581no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—1581no—
260C4×D4—1581no—
261C4×D4C22⋊D7912yes—
262C4×D4C22×D7912yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D7911yescharacteristic
266C2×C22⋊C4—21no—
267C2×C42C22×D7911yescharacteristic
268#106—24no—
269#106—1584no—
270C4⋊C8—21no—
271C4⋊C8—792no—
272C4⋊C8—792no—
273C4⋊C8—792no—
274C4⋊C8—1581no—
275C4⋊C8C22×D7912yes—
276#103—1582no—
277#102—24no—
278#102—1584no—
279#102—1582no—
280C22⋊C8—792no—
281C22⋊C8—1584no—
282C22⋊C8—1581no—
283C4×C8—792no—
284C4×C8—1581no—
285C24—21no—
286#41—3162no—
287#41—3162no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—82no—
291C2×Q8—1581no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—3161no—
295C2×D4—1581no—
296C2×D4—1581no—
297C2×D4—1581no—
298C2×D4#1192961114yes—
299C2×D4C2×C22⋊D7912yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—3161no—
306C22×C4D4×D7911yescharacteristic
307C22×C4C2×C22⋊D7911yescharacteristic
308C22×C4#1192962311yescharacteristic
309SD16—3164no—
310D8—3164no—
311M16—3162no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—794no—
315C2×C8—6324no—
316C2×C8—1582no—
317C2×C8—1582no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—1581no—
324C4⋊C4—1581no—
325C4⋊C4C2×C22⋊D7912yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—3161no—
329C22⋊C4—3161no—
330C42D4×D7911yescharacteristic
331C42C2×C22⋊D7911yescharacteristic
332C42#1192962311yescharacteristic
333C42C23×D7911yescharacteristic
334C23—22no—
335C23—24no—
336C23—81no—
337C23—81no—
338C23—3161no—
339C23#1457913311yescharacteristic
340Q8—82no—
341Q8—82no—
342Q8—3161no—
343Q8—3161no—
344D4—24no—
345D4—22no—
346D4—3161no—
347D4—3161no—
348D4—3162no—
349D4—6321no—
350D4—6322no—
351D4C2×#1192961114yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—3161no—
361C2×C4—6322no—
362C2×C4#1457913311yescharacteristic
363C2×C4#1457913311yescharacteristic
364C2×C4#1457913311yescharacteristic
365C2×C4C2×#1192961112yes—
366C2×C4#1457918112yes—
367C2×C4C2×D4×D7911yescharacteristic
368C2×C4C22×C22⋊D7911yescharacteristic
369C2×C4C2×#1192962311yescharacteristic
370C8—42no—
371C8—3162no—
372C8—3162no—
373C8—1584no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—84no—
380C22—6321no—
381C22—6321no—
382C22—6321no—
383C22#1909879412yes—
384C22C2×#1457913311yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—82no—
390C4—6321no—
391C4#1909879412yes—
392C4C22×#1192961111yescharacteristic
393C4C2×#1457918111yescharacteristic
394C4#1909936912yes—
395C2—22no—
396C2—44no—
397C2—12641no—
398C2#2377290911yescharacteristic
399C2C2×#1909879411yescharacteristic
400C2#2377312511yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC2×#476×C79C2
socleC22×C79C2×#121
prime radical#23772864C2
hypercenterC2×#476D79
hypocenterC79#19391
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #23774528C2semidirect
3. #23772424C2semidirect
4. #23762811C2semidirect
5. #23762777C2semidirect
6. #23762170C2semidirect
7. #23762081C2semidirect
8. #23762050C2semidirect
9. C2×#476×C79C2semidirect
12. #19098959C22semidirect
13. #19098911C22semidirect
26. #19098313C22semidirect
27. #19098312C22semidirect
28. #19098310C22semidirect
29. #19098306C22semidirect
202. C2×#476D79semidirect
210. C79#19391semidirect

Maximal subgroups

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

subgroupindex
#237745282
#237728642
#237724242
#237628112
#237627772
#237621702
#237620812
#237620502
C2×#476×C792
#1939179