Finite Group Databasebeta

← #34187623

Subgroups — #34187623

Order 20224.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773725C211yesmaximal, characteristic
2#23763265C211yesmaximal, characteristic, complemented
3#23762146C211yesmaximal, characteristic, complemented
4#23761899C211yesmaximal, characteristic, complemented
5#23761896C211yesmaximal, characteristic
6#23761798C211yesmaximal, characteristic
7C79×#1481C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098920—21no—
10#19098920—21no—
11#19098915C411yescharacteristic, complemented
12#19098915C411yescharacteristic, complemented
13#19098911C411yescharacteristic, complemented
14#19098911C411yescharacteristic, complemented
15#19098616C2211yescharacteristic
16#19098557—21no—
17#19098557—21no—
18C2×C4×C79⋊C8C2211yescharacteristic
19C2×C8×Dic79—21no—
20C4×C4⋊Dic79C2211yescharacteristic
21#19098317—21no—
22#19098315—21no—
23#19098313—21no—
24#19098300—22no—
25#19098298—22no—
26#19098290—22no—
27#533×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29C2×C4⋊C8×C79C2211yescharacteristic
30Q8⋊C8×C79—22no—
31C2×#11929624—21no—
32C22×C79⋊C8D411yescharacteristic
33C22×C79⋊C8D411yescharacteristic
34#14579161C2×C411yescharacteristic
35#14579158C2×C411yescharacteristic
36#14579149—21no—
37#14579135—21no—
38#14579135—21no—
39C4×D316—21no—
40C4×D316—21no—
41C2×C4⋊Dic79—21no—
42C2×C4×Dic79—21no—
43C4⋊(C79⋊C8)—21no—
44#14579084—21no—
45#14579084—21no—
46#14579084—21no—
47#14579084—21no—
48#14579081—21no—
49#14579081—21no—
50#14579081—21no—
51#14579081—21no—
52C22⋊(C79⋊C8)—21no—
53C4×C79⋊C8—21no—
54C4×C79⋊C8C2×C411yescharacteristic
55C4×C79⋊C8C2×C411yescharacteristic
56C4×C79⋊C8D411yescharacteristic
57C4×C79⋊C8D411yescharacteristic
58C8×Dic79—42no—
59C79⋊#95D412yes—
60C22×C8×C79—21no—
61C4⋊Q8×C79C2×C411yescharacteristic
62C4⋊Q8×C79C2×C411yescharacteristic
63#125×C79—21no—
64C22⋊Q8×C79—21no—
65C22⋊Q8×C79—21no—
66C4×Q8×C79—21no—
67C4×Q8×C79—21no—
68#117×C79—21no—
69C2×C42×C79C2311yescharacteristic
70C4⋊C8×C79—21no—
71C4⋊C8×C79—21no—
72C4⋊C8×C79D412yes—
73C22⋊C8×C79—21no—
74C4×C8×C79—21no—
75#11929624—41no—
76#11929624—41no—
77#11929624—41no—
78#11929624—41no—
79C2×Dic158—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×D316—21no—
83C2×C22⋊D79—41no—
84C22×Dic79—22no—
85C2×C4×D79—41no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8C22⋊C411yescharacteristic
92C2×C79⋊C8C22⋊C411yescharacteristic
93C2×C79⋊C8C22⋊C411yescharacteristic
94C2×C79⋊C8C22⋊C411yescharacteristic
95C4⋊Dic79—24no—
96C4⋊Dic79—21no—
97C4⋊Dic79—21no—
98(C2×D79)⋊C4—41no—
99(C2×D79)⋊C4—41no—
100C4×Dic79—41no—
101C4×Dic79—41no—
102C2×Q8×C79—21no—
103C2×Q8×C79—21no—
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79#4111yescharacteristic
107C2×C8×C79—22no—
108C2×C8×C79—41no—
109C2×C8×C79—41no—
110C2×C8×C79—41no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C22⋊C4×C79—41no—
118C42×C79—41no—
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79C2×D411yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79#12111yescharacteristic
124C22×D79—41no—
125Q8×C79—42no—
126Q8×C79—81no—
127C2×C4×C79—22no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79C2×C22⋊C411yescharacteristic
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79C4×D411yescharacteristic
134C2×C4×C79#12111yescharacteristic
135C2×C4×C79#12411yescharacteristic
136C2×C4×C79#12711yescharacteristic
137C2×C4×C79#13511yescharacteristic
138C2×C4×C79#13511yescharacteristic
139C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C79#13711yescharacteristic
141C22⋊D79—81no—
142C22⋊D79—81no—
143C2×Dic79—24no—
144C2×Dic79—42no—
145C2×Dic79—42no—
146C2×Dic79—41no—
147D316—41no—
148D316—41no—
149D316—42no—
150D316—81no—
151C4×D79—81no—
152C4×D79—81no—
153Dic158—41no—
154Dic158—41no—
155C8×C79—42no—
156C8×C79—81no—
157C79⋊C8—21no—
158C79⋊C8—21no—
159C79⋊C8—21no—
160C79⋊C8—21no—
161C79⋊C8—81no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79#39011yescharacteristic
165C22×C79#46311yescharacteristic
166C22×C79#46411yescharacteristic
167C2×D79—81no—
168C2×D79—81no—
169C2×D79—81no—
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—81no—
174C4×C79—81no—
175C4×C79#41611yescharacteristic
176C4×C79#41711yescharacteristic
177C4×C79#44312yes—
178C4×C79#46311yescharacteristic
179C4×C79#46411yescharacteristic
180Dic79—44no—
181Dic79—81no—
182#9751—791nomaximal, Sylow
183C2×C79—21no—
184C2×C79—21no—
185C2×C79#197411yescharacteristic
186C2×C79#197611yescharacteristic
187C2×C79#198311yescharacteristic
188D79—161no—
189#3131—791no—
190C8×C4⋊C4—791no—
191#1564—791no—
192#1503—791no—
193#1502—791no—
194#1481D7911yescharacteristic, complemented
195#1480—791no—
196C79#975111yescharacteristic, complemented, Sylow
197#533C2×D7911yescharacteristic
198#532—791no—
199#488—1581no—
200#488—1581no—
201#487—791no—
202#487—791no—
203#486—791no—
204#486—791no—
205#431—791no—
206#431C2×D7911yescharacteristic
207C2×C4⋊C8C2×D7911yescharacteristic
208#416—1581no—
209#416—1581no—
210C2×C4×C8—791no—
211C2×C4×C8—1581no—
212C4×C4⋊C4—791no—
213#336—1581no—
214#335—1581no—
215#334—1581no—
216#328—1582no—
217#327—1582no—
218Q8⋊C8—22no—
219D4⋊C8—1582no—
220C2×#41—1581no—
221C22×C8—21no—
222C22×C8—791no—
223C22×C8—791no—
224C4⋊Q8—791no—
225C4⋊Q8C4×D7911yescharacteristic
226C4⋊Q8C4×D7911yescharacteristic
227#127—791no—
228#125—21no—
229#124—1581no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—1581no—
233#121—1581no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—1581no—
237C4×D4—1581no—
238#117—21no—
239C2×C4⋊C4—1581no—
240C2×C42—1581no—
241C2×C42C22×D7911yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—1581no—
245C4⋊C8C22⋊D7912yes—
246#103—1581no—
247#103—1581no—
248#103—1581no—
249#103—1581no—
250#102—1581no—
251#102—1581no—
252#102—1581no—
253#102—1581no—
254C22⋊C8—21no—
255C22⋊C8—1581no—
256C4×C8—21no—
257C4×C8—3162no—
258C4×C8—791no—
259C4×C8—791no—
260C4×C8—791no—
261C4×C8—791no—
262C4×C8—1581no—
263#95—792no—
264#41—3161no—
265#41—3161no—
266#41—3161no—
267#41—3161no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—1581no—
271C2×D4—3161no—
272C2×D4—1581no—
273C2×D4—1581no—
274C2×D4—1581no—
275C22×C4—3161no—
276C22×C4—1582no—
277C22×C4D4×D7911yescharacteristic
278C22×C4D4×D7911yescharacteristic
279C22×C4#1192962411yescharacteristic
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—3161no—
285C2×C8—791no—
286C2×C8—791no—
287C2×C8—791no—
288C2×C8—791no—
289C2×C8—1581no—
290C2×C8—1581no—
291C2×C8—1581no—
292C2×C8—1581no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—1584no—
300C4⋊C4—1581no—
301C4⋊C4—1581no—
302C22⋊C4—41no—
303C22⋊C4—3161no—
304C22⋊C4—3161no—
305C42—41no—
306C42—3161no—
307C42—3161no—
308C42C2×C4×D7911yescharacteristic
309C42D4×D7911yescharacteristic
310C42C2×C22⋊D7911yescharacteristic
311C42#1192962311yescharacteristic
312C23—3161no—
313C23#1457913411yescharacteristic
314Q8—42no—
315Q8—81no—
316Q8—3161no—
317Q8—3161no—
318D4—3162no—
319D4—3161no—
320D4—3161no—
321D4—6321no—
322D4—6321no—
323D4—6321no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—3162no—
329C2×C4—3162no—
330C2×C4—3161no—
331C2×C4—6321no—
332C2×C4—6321no—
333C2×C4—1584no—
334C2×C4C22⋊C4×D7911yescharacteristic
335C2×C4(C22⋊D79)⋊C411yescharacteristic
336C2×C4C4×C22⋊D7911yescharacteristic
337C2×C4#1457913311yescharacteristic
338C2×C4#1457914711yescharacteristic
339C2×C4#1457915711yescharacteristic
340C2×C4#1457918211yescharacteristic
341C2×C4#1457918411yescharacteristic
342C2×C4#1457918811yescharacteristic
343C2×C4#1457919311yescharacteristic
344C8—42no—
345C8—81no—
346C8—6321no—
347C8—1581no—
348C8—1581no—
349C8—1581no—
350C8—1581no—
351C22—21no—
352C22—21no—
353C22—6321no—
354C22—6321no—
355C22—6321no—
356C22#1909846311yescharacteristic
357C22#1909878111yescharacteristic
358C22#1909878811yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—81no—
364C4—3164no—
365C4—6321no—
366C4#1909855811yescharacteristic
367C4#1909856311yescharacteristic
368C4#1909867812yes—
369C4#1909878011yescharacteristic
370C4#1909878711yescharacteristic
371C2—21no—
372C2—21no—
373C2—12641no—
374C2#2376428711yescharacteristic
375C2#2376429711yescharacteristic
376C2#2376432911yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1481C2
socleC22×C79#390
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1481D79
hypocenterC79#9751
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
2. #23763265C2semidirect
3. #23762146C2semidirect
4. #23761899C2semidirect
7. C79×#1481C2semidirect
11. #19098915C4semidirect
12. #19098915C4semidirect
13. #19098911C4semidirect
14. #19098911C4semidirect
194. #1481D79semidirect
196. C79#9751semidirect

Maximal subgroups

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

subgroupindex
#237737252
#237632652
#237621462
#237618992
#237618962
#237617982
C79×#14812
#975179