Finite Group Databasebeta

← #34185010

Subgroups — #34185010

Order 20224.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774325C211yesmaximal, characteristic
2#23764232C211yesmaximal, characteristic
3#23762081C211yesmaximal, characteristic, complemented
4#23761959C211yesmaximal, characteristic
5#23761258C211yesmaximal, characteristic, complemented
6#23761229C211yesmaximal, characteristic, complemented
7C79×#2331C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098937C412yescomplemented
10#19098913C412yescomplemented
11#19098765—22no—
12#19098702—22no—
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098587C2211yescharacteristic
16C2×#14579089—21no—
17C2×#14579089—21no—
18#19098312—22no—
19#19098296—22no—
20C2×C4⋊Q8×C79C2211yescharacteristic
21#391×C79—22no—
22#389×C79—21no—
23#387×C79—22no—
24#382×C79C2211yescharacteristic
25C4×C4⋊C4×C79—21no—
26#376×C79C2211yescharacteristic
27C2×#11929624—21no—
28C2×#11929611—24no—
29C2×#11929608—21no—
30#14579161C2×C411yescharacteristic
31#14579158C2×C411yescharacteristic
32#14579149—21no—
33#14579135—21no—
34#14579135—21no—
35C4×D316—21no—
36C4×D316—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41#14579089—42no—
42#14579089—42no—
43#14579085—24no—
44C22⋊(C79⋊C8)—21no—
45C22⋊(C79⋊C8)—21no—
46C4×C79⋊C8—21no—
47C4×C79⋊C8C2×C412yes—
48C22×Q8×C79D412yes—
49C4⋊Q8×C79—21no—
50C4⋊Q8×C79—21no—
51C4⋊Q8×C79C2×C412yes—
52C4⋊Q8×C79D412yes—
53C2×C4⋊C4×C79—21no—
54C2×C4⋊C4×C79—21no—
55C2×C4⋊C4×C79—21no—
56C2×C4⋊C4×C79—22no—
57C2×C42×C79—22no—
58C2×C42×C79D411yescharacteristic
59C2×C42×C79D411yescharacteristic
60C2×C42×C79C2311yescharacteristic
61#95×C79—22no—
62#95×C79—22no—
63#95×C79—21no—
64#95×C79—21no—
65#95×C79—21no—
66#95×C79D412yes—
67#11929624—41no—
68#11929624—41no—
69#11929624—41no—
70#11929624—41no—
71C2×Dic158—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316—21no—
75C2×C22⋊D79—41no—
76C2×C4×D79—41no—
77#11929611—84no—
78#11929608—41no—
79#11929608—41no—
80C2×C79⋊C8—22no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—41no—
83C4⋊Dic79—21no—
84C4⋊Dic79—21no—
85(C2×D79)⋊C4—41no—
86(C2×D79)⋊C4—41no—
87C2×Q8×C79—22no—
88C2×Q8×C79—82no—
89C2×Q8×C79C22⋊C414yes—
90C22×C4×C79—21no—
91C22×C4×C79—22no—
92C22×C4×C79—21no—
93C22×C4×C79—22no—
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C4⋊C4×C79—22no—
98C4⋊C4×C79—22no—
99C4⋊C4×C79—22no—
100C4⋊C4×C79—22no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—82no—
104C42×C79—21no—
105C42×C79—21no—
106C42×C79—21no—
107C42×C79—82no—
108C42×C79SD1612yescomplemented
109C42×C79C22×C411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C42×C79#4111yescharacteristic
112C42×C79#4111yescharacteristic
113C23×C79#12011yescharacteristic
114C22×D79—41no—
115Q8×C79—44no—
116Q8×C79—82no—
117C2×C4×C79—24no—
118C2×C4×C79—22no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—42no—
123C2×C4×C79—42no—
124C2×C4×C79—41no—
125C2×C4×C79—42no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—42no—
129C2×C4×C79—41no—
130C2×C4×C79—42no—
131C2×C4×C79C2×C22⋊C411yescharacteristic
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79C4×D411yescharacteristic
134C2×C4×C79#12111yescharacteristic
135C2×C4×C79#12111yescharacteristic
136C2×C4×C79#12311yescharacteristic
137C2×C4×C79C2×SD1611yescharacteristic
138C2×C4×C79#13511yescharacteristic
139C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C79#13711yescharacteristic
141C22⋊D79—81no—
142C22⋊D79—81no—
143C2×Dic79—41no—
144D316—42no—
145D316—41no—
146D316—41no—
147D316—81no—
148C4×D79—81no—
149C4×D79—81no—
150Dic158—41no—
151Dic158—41no—
152C79⋊C8—42no—
153C79⋊C8—81no—
154C79⋊C8—81no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79#38611yescharacteristic
158C22×C79#44811yescharacteristic
159C22×C79#44911yescharacteristic
160C2×D79—81no—
161C2×D79—81no—
162C2×D79—81no—
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—21no—
166C4×C79—21no—
167C4×C79—42no—
168C4×C79—44no—
169C4×C79—82no—
170C4×C79—81no—
171C4×C79C4×SD1611yescharacteristic
172C4×C79#44011yescharacteristic
173C4×C79#46011yescharacteristic
174C4×C79#46411yescharacteristic
175Dic79—81no—
176#9301—791nomaximal, Sylow
177C2×C79—21no—
178C2×C79—21no—
179C2×C79#188011yescharacteristic
180C2×C79#190511yescharacteristic
181C2×C79#191111yescharacteristic
182D79—161no—
183#3206—791no—
184#2331D7911yescharacteristic, complemented
185#1963—791no—
186#1548—791no—
187#1517—791no—
188#1286—791no—
189#1274—791no—
190C79#930111yescharacteristic, complemented, Sylow
191#532—791no—
192C2×C4⋊Q8C2×D7911yescharacteristic
193#492—792no—
194#487—792no—
195#461—1582no—
196#448—1582no—
197#431—791no—
198#431—791no—
199#423—791no—
200C2×#104—1581no—
201C2×#104—1581no—
202#391—22no—
203#389—21no—
204#387—22no—
205#382C2×D7911yescharacteristic
206C4×C4⋊C4—21no—
207#376C2×D7911yescharacteristic
208#333—1582no—
209#327—1582no—
210C2×#41—1581no—
211C22×Q8C22⋊D7912yes—
212C2×SD16—1584no—
213C2×M16—1581no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—791no—
217C4⋊Q8C4×D7912yes—
218C4⋊Q8D31612yes—
219#127—791no—
220#124—1581no—
221#121—1581no—
222#121—1581no—
223C4×D4—1581no—
224C4×D4—1581no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C42—22no—
230C2×C42C22⋊D7911yescharacteristic
231C2×C42C22⋊D7911yescharacteristic
232C2×C42C22×D7911yescharacteristic
233C4⋊C8—1581no—
234C4⋊C8—1581no—
235C4⋊C8—1581no—
236C4⋊C8—1581no—
237#104—3162no—
238#104—3162no—
239#103—1584no—
240C22⋊C8—1581no—
241C22⋊C8—1581no—
242C4×C8—792no—
243C4×C8—1581no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95C22⋊D7912yes—
250#41—3161no—
251#41—3161no—
252#41—3161no—
253#41—3161no—
254C2×Q8—22no—
255C2×Q8—82no—
256C2×Q8—1581no—
257C2×Q8(C2×D79)⋊C414yes—
258C2×D4—3161no—
259C2×D4—1581no—
260C2×D4—1581no—
261C2×D4—1581no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—3161no—
267C22×C4C2×C22⋊D7911yescharacteristic
268C22×C4C2×C22⋊D7911yescharacteristic
269C22×C4C2×C22⋊D7911yescharacteristic
270SD16—6324no—
271M16—3161no—
272M16—3161no—
273C2×C8—3161no—
274C2×C8—3161no—
275C2×C8—1582no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—82no—
283C4⋊C4—1581no—
284C4⋊C4—1581no—
285C22⋊C4—3161no—
286C22⋊C4—3161no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—82no—
291C42#1192961112yescomplemented
292C42C2×C4×D7911yescharacteristic
293C42C2×D31611yescharacteristic
294C42#1192962411yescharacteristic
295C42#1192962411yescharacteristic
296C23—3161no—
297C23#1457913011yescharacteristic
298Q8—44no—
299Q8—82no—
300Q8—3161no—
301Q8—3161no—
302D4—3162no—
303D4—3161no—
304D4—3161no—
305D4—6321no—
306D4—6321no—
307D4—6321no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—3161no—
323C2×C4—6321no—
324C2×C4—6321no—
325C2×C4C2×(C2×D79)⋊C411yescharacteristic
326C2×C4C4×C22⋊D7911yescharacteristic
327C2×C4C4×C22⋊D7911yescharacteristic
328C2×C4#1457913511yescharacteristic
329C2×C4#1457913511yescharacteristic
330C2×C4#1457914411yescharacteristic
331C2×C4C2×#1192961111yescharacteristic
332C2×C4#1457918111yescharacteristic
333C2×C4#1457918611yescharacteristic
334C2×C4#1457919211yescharacteristic
335C8—3162no—
336C8—6321no—
337C8—6321no—
338C22—21no—
339C22—21no—
340C22—6321no—
341C22—6321no—
342C22—6321no—
343C22#1909844211yescharacteristic
344C22#1909870211yescharacteristic
345C22#1909870811yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—44no—
352C4—82no—
353C4—81no—
354C4—6321no—
355C4C4×#1192961111yescharacteristic
356C4#1909866111yescharacteristic
357C4#1909876311yescharacteristic
358C4#1909878511yescharacteristic
359C2—21no—
360C2—21no—
361C2—12641no—
362C2#2376375511yescharacteristic
363C2#2376389411yescharacteristic
364C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2331C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#2331D79
hypocenterC79#9301
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #23762081C2semidirect
5. #23761258C2semidirect
6. #23761229C2semidirect
7. C79×#2331C2semidirect
9. #19098937C4semidirect
10. #19098913C4semidirect
108. C42×C79SD16semidirect
184. #2331D79semidirect
190. C79#9301semidirect
291. C42#11929611semidirect

Maximal subgroups

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

subgroupindex
#237743252
#237642322
#237620812
#237619592
#237612582
#237612292
C79×#23312
#930179