Finite Group Databasebeta

← #35321794

Subgroups — #35321794

Order 21248.

510 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581732C211yesmaximal, characteristic, complemented
2#24917044C211yesmaximal, characteristic
3C2×#19127510C212yesmaximal, complemented
4#24905570C212yesmaximal, complemented
5C83×#1807C211yesmaximal, characteristic, complemented
6C2×#14581814C2211yescharacteristic
7#19128212—21no—
8C2×#14581732—24no—
9C2×#14581732—21no—
10C2×#14581732C2212yes—
11C22×C83⋊C16—21no—
12#19127942C411yescharacteristic, complemented
13#19127942C411yescharacteristic, complemented
14#19127937C411yescharacteristic, complemented
15#19127937C411yescharacteristic, complemented
16#19127930—21no—
17#19127930—21no—
18#19127739C2211yescharacteristic
19#19127698—21no—
20#19127510—24no—
21C23×C8×C83C2211yescharacteristic
22C2×#106×C83C2212yes—
23#380×C83—21no—
24#340×C83—21no—
25C2×#13050590—21no—
26#14581814—22no—
27#14581814—22no—
28C2×Dic332C2×C411yescharacteristic
29C2×#13050579—21no—
30C2×D664C2×C411yescharacteristic
31#14581775—41no—
32#14581771—21no—
33#14581771—21no—
34#14581766—21no—
35#14581766—21no—
36#14581761—41no—
37C4×C22⋊D83—41no—
38C4×C22⋊D83—41no—
39#14581751—21no—
40#14581732—24no—
41#14581732—22no—
42#14581732D414yescomplemented
43C2×C83⋊C16—24no—
44C2×C83⋊C16—22no—
45#14581726C2×C411yescharacteristic
46#14581726C2×C411yescharacteristic
47#14581724—21no—
48#14581717—21no—
49#14581717—21no—
50#14581714—21no—
51#14581714—21no—
52C23×C4×C83D411yescharacteristic
53C22×C8×C83—22no—
54C22×C8×C83—22no—
55C22×C8×C83C2×C411yescharacteristic
56C22×C8×C83C2×C411yescharacteristic
57C22×C8×C83D411yescharacteristic
58C22×C8×C83D411yescharacteristic
59C22×C8×C83D411yescharacteristic
60C22×C8×C83C2311yescharacteristic
61C2×C4⋊C4×C83—22no—
62C2×C22⋊C4×C83—41no—
63#106×C83—22no—
64#106×C83—22no—
65#95×C83—41no—
66#13050590—42no—
67#13050590—42no—
68C2×Dic166—21no—
69C2×D332—21no—
70C2×C22⋊D83—41no—
71C2×C22⋊D83—41no—
72C2×C22⋊D83—41no—
73C2×C4×D83—41no—
74Dic332—22no—
75#13050579—41no—
76#13050579—41no—
77D664—22no—
78C4⋊Dic83—21no—
79C4⋊Dic83—21no—
80Dic83⋊C4—41no—
81Dic83⋊C4—41no—
82C22⋊Dic83—41no—
83C22⋊Dic83—41no—
84C22⋊Dic83—41no—
85(C2×D83)⋊C4—41no—
86(C2×D83)⋊C4—41no—
87C4×Dic83—41no—
88C83⋊C16—24no—
89C24×C83D811yescharacteristic
90C22×C4×C83—22no—
91C22×C4×C83—22no—
92C22×C4×C83—42no—
93C22×C4×C83C22⋊C411yescharacteristic
94C22×C4×C83C22⋊C411yescharacteristic
95C22×C4×C83D811yescharacteristic
96C22×C4×C83SD1611yescharacteristic
97C22×C4×C83SD1611yescharacteristic
98C22×C4×C83C2×D411yescharacteristic
99C2×C8×C83—22no—
100C2×C8×C83—22no—
101C2×C8×C83—22no—
102C2×C8×C83—21no—
103C2×C8×C83—21no—
104C2×C8×C83—21no—
105C2×C8×C83—21no—
106C2×C8×C83C22⋊C411yescharacteristic
107C2×C8×C83C22⋊C411yescharacteristic
108C2×C8×C83C22×C411yescharacteristic
109C2×C8×C83C2×D412yes—
110C2×C8×C83C2×D411yescharacteristic
111C2×C8×C83#4112yes—
112C4⋊C4×C83—42no—
113C4⋊C4×C83—42no—
114C22⋊C4×C83—82no—
115C23×C83—22no—
116C23×C83—22no—
117C23×C83—41no—
118C23×C83#10211yescharacteristic
119C23×C83#10211yescharacteristic
120C23×C83C2×D811yescharacteristic
121C22×D83—41no—
122C2×C4×C83—22no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—22no—
128C2×C4×C83—22no—
129C2×C4×C83—82no—
130C2×C4×C83—82no—
131C2×C4×C83—82no—
132C2×C4×C83#10211yescharacteristic
133C2×C4×C83#10211yescharacteristic
134C2×C4×C83C2×C22⋊C411yescharacteristic
135C2×C4×C83#12112yes—
136C2×C4×C83C2×SD1611yescharacteristic
137C2×C4×C83#13512yes—
138C22⋊D83—82no—
139C22⋊D83—82no—
140C22⋊D83—82no—
141C2×Dic83—41no—
142C2×Dic83—41no—
143C2×Dic83—41no—
144D332—41no—
145D332—41no—
146C4×D83—82no—
147Dic166—41no—
148Dic166—41no—
149C8×C83—21no—
150C8×C83—21no—
151C8×C83C4×D412yes—
152C8×C83#12011yescharacteristic
153C8×C83#12311yescharacteristic
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—44no—
162C22×C83#36012yescomplemented
163C22×C83C2×#10211yescharacteristic
164C22×C83#46612yes—
165C2×D83—81no—
166C2×D83—81no—
167C2×D83—81no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—162no—
171C4×C83#38611yescharacteristic
172C4×C83#41611yescharacteristic
173C4×C83#46512yes—
174Dic83—81no—
175Dic83—81no—
176Dic83—81no—
177#13065—831nomaximal, Sylow
178C2×C83—22no—
179C2×C83—22no—
180C2×C83—41no—
181C2×C83#189311yescharacteristic
182C2×C83C2×#36011yescharacteristic
183C2×C83#214511yescharacteristic
184D83—161no—
185C22×M32—831no—
186#3050—831no—
187C2×#360—832no—
188#1807D8311yescharacteristic, complemented
189#1386—832no—
190C83#1306511yescharacteristic, complemented, Sylow
191C2×#135—831no—
192C23×C8C2×D8311yescharacteristic
193#525—1661no—
194C2×M32—832no—
195C2×M32—1664no—
196C2×M32—1661no—
197C22×C16—1661no—
198#467—831no—
199#467—831no—
200#466—831no—
201#466—831no—
202#465—1661no—
203#465—1661no—
204#427—831no—
205C2×#106C2×D8312yes—
206#416—1661no—
207#380—21no—
208#360—1664no—
209#340—21no—
210C2×#41—1661no—
211C23×C4C22⋊D8311yescharacteristic
212#135—1662no—
213#135—1662no—
214C2×Q16—831no—
215C2×SD16—1661no—
216C2×D8—831no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C4×D8311yescharacteristic
220C22×C8C4×D8311yescharacteristic
221C22×C8D33211yescharacteristic
222C22×C8D33211yescharacteristic
223C22×C8C22⋊D8311yescharacteristic
224C22×C8C22×D8311yescharacteristic
225#123—3321no—
226C22⋊Q8—1661no—
227C22⋊Q8—1661no—
228#121—1661no—
229#121—1661no—
230#120—3321no—
231C4×D4—3321no—
232C4×D4—3321no—
233#117—1661no—
234C2×C4⋊C4—22no—
235C2×C22⋊C4—41no—
236M32—834no—
237M32—1664no—
238M32—1662no—
239C2×C16—1664no—
240C2×C16—1662no—
241#107—831no—
242#107—831no—
243#106—22no—
244#106—22no—
245#106—1661no—
246#103—1661no—
247#103—1661no—
248#102—1661no—
249#102—1661no—
250#95—41no—
251C24#1305057511yescharacteristic
252#41—3322no—
253#41—3322no—
254C2×Q8—1661no—
255C2×D4—3321no—
256C2×D4—3321no—
257C2×D4—3321no—
258C2×D4—1661no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—3321no—
263C22×C4(C2×D83)⋊C411yescharacteristic
264C22×C4(C2×D83)⋊C411yescharacteristic
265C22×C4#1305057511yescharacteristic
266C22×C4#1305057711yescharacteristic
267C22×C4#1305057711yescharacteristic
268C22×C4C2×C22⋊D8311yescharacteristic
269Q16—1662no—
270SD16—3321no—
271SD16—3321no—
272D8—1662no—
273C2×C8—22no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8(C2×D83)⋊C411yescharacteristic
281C2×C8(C2×D83)⋊C411yescharacteristic
282C2×C8C2×C4×D8311yescharacteristic
283C2×C8D4×D8312yes—
284C2×C8C2×D33211yescharacteristic
285C2×C8#1305058912yes—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—3321no—
289C4⋊C4—3321no—
290C4⋊C4—1661no—
291C4⋊C4—1661no—
292C22⋊C4—82no—
293C22⋊C4—3321no—
294C22⋊C4—3321no—
295C22⋊C4—3321no—
296C22⋊C4—3321no—
297C22⋊C4—3321no—
298C42—3321no—
299C16—1664no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—3321no—
304C23#1458171211yescharacteristic
305C23#1458171211yescharacteristic
306C23C2×#1305057511yescharacteristic
307Q8—3321no—
308Q8—3321no—
309D4—3321no—
310D4—3321no—
311D4—6642no—
312D4—6642no—
313D4—6642no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—3321no—
325C2×C4—3321no—
326C2×C4—3321no—
327C2×C4—6642no—
328C2×C4#1458171211yescharacteristic
329C2×C4#1458171211yescharacteristic
330C2×C4C2×(C2×D83)⋊C411yescharacteristic
331C2×C4#1458176412yes—
332C2×C4C2×#1305057711yescharacteristic
333C2×C4#1458181212yes—
334C8—21no—
335C8—21no—
336C8(C22⋊D83)⋊C412yes—
337C8#1458176211yescharacteristic
338C8#1458177611yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—6641no—
348C22—6641no—
349C22—6641no—
350C22#1912751012yescomplemented
351C22C2×#1458171211yescharacteristic
352C22#1912793512yes—
353C4—21no—
354C4—21no—
355C4—162no—
356C4—6641no—
357C4—6641no—
358C4—6641no—
359C4#1912757811yescharacteristic
360C4#1912769511yescharacteristic
361C4#1912793212yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—13281no—
366C2#2490788911yescharacteristic
367C2C2×#1912751011yescharacteristic
368C2#2490919811yescharacteristic
subgroupG/H
centerC22C2×#14581712
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1807C2
socleC22×C83C2×#102
prime radical#24917044C2
cosocleC22×C8×C83C23
hypercenter#1807D83
hypocenterC83#13065
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #1807

Complements

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

NH ≅ G/Ntype
1. C22×#14581732C2semidirect
3. C2×#19127510C2semidirect
4. #24905570C2semidirect
5. C83×#1807C2semidirect
12. #19127942C4semidirect
13. #19127942C4semidirect
14. #19127937C4semidirect
15. #19127937C4semidirect
42. #14581732D4semidirect
162. C22×C83#360semidirect
188. #1807D83semidirect
190. C83#13065semidirect
350. C22#19127510semidirect

Maximal subgroups

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

subgroupindex
C22×#145817322
#249170442
C2×#191275102
#249055702
C83×#18072
#1306583