Finite Group Databasebeta

← #35315543

Subgroups — #35315543

Order 21248.

452 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912639C211yesmaximal, characteristic
2#24905784C211yesmaximal, characteristic
3#24905486C212yesmaximal, complemented
4#24905483C212yesmaximal, complemented
5C83×#1799C211yesmaximal, characteristic
6C2×#14581771C411yescharacteristic, complemented
7#19127754—21no—
8#19127640C411yescharacteristic, complemented
9#19127637C2211yescharacteristic
10#19127621C411yescharacteristic, complemented
11#19127613C2211yescharacteristic
12#19127585—21no—
13#19127581—21no—
14#19127574—21no—
15#19127559—21no—
16C2×C83⋊#95C411yescharacteristic, complemented
17#19127425—21no—
18#19127425C2212yes—
19C2×C4×D4×C83C2211yescharacteristic
20C2×#99×C83—21no—
21#380×C83—21no—
22#342×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C23×Dic83—21no—
25#14581771—41no—
26#14581771—41no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C22⋊Dic83—21no—
34C4⋊(C83⋊C8)—41no—
35C22⋊(C83⋊C8)—22no—
36C4×C83⋊C8—41no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C22×D4×C83D411yescharacteristic
45C23×C4×C83C2×C411yescharacteristic
46C23×C4×C83D411yescharacteristic
47C4×D4×C83—41no—
48C4×D4×C83—41no—
49C2×C4⋊C4×C83—41no—
50C2×C4⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83—22no—
52C2×C22⋊C4×C83D411yescharacteristic
53C2×C22⋊C4×C83C2311yescharacteristic
54C2×C42×C83D411yescharacteristic
55#99×C83—42no—
56#95×C83—41no—
57C2×Dic166—41no—
58C2×Dic166—81no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—42no—
64C22×Dic83—42no—
65C22×Dic83—81no—
66C2×C83⋊C8—42no—
67C4⋊Dic83—41no—
68Dic83⋊C4—42no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71Dic83⋊C4—81no—
72C22⋊Dic83—42no—
73C22⋊Dic83—42no—
74C22⋊Dic83—81no—
75C24×C83C22⋊C411yescharacteristic
76C24×C83C2×D411yescharacteristic
77C2×D4×C83—41no—
78C2×D4×C83—41no—
79C2×D4×C83SD1612yes—
80C2×D4×C83Q1612yes—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—42no—
86C22×C4×C83—81no—
87C22×C4×C83—81no—
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22×C411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C4⋊C4×C83—41no—
94C4⋊C4×C83—82no—
95C22⋊C4×C83—41no—
96C22⋊C4×C83—41no—
97C22⋊C4×C83—42no—
98C22⋊C4×C83—42no—
99C42×C83—21no—
100C42×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83#10312yes—
110C23×C83C2×C22⋊C411yescharacteristic
111C23×C83C2×SD1611yescharacteristic
112C23×C83C2×Q1611yescharacteristic
113D4×C83—42no—
114D4×C83—42no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—41no—
121C2×C4×C83—42no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—81no—
126C2×C4×C83—81no—
127C2×C4×C83—81no—
128C2×C4×C83—82no—
129C2×C4×C83—82no—
130C2×C4×C83—82no—
131C2×C4×C83#9911yescharacteristic
132C2×C4×C83#9911yescharacteristic
133C2×C4×C83#10312yes—
134C2×Dic83—22no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—41no—
142C2×Dic83—41no—
143C2×Dic83—41no—
144C2×Dic83—41no—
145C2×Dic83—41no—
146C2×Dic83—84no—
147Dic166—82no—
148C83⋊C8—82no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—22no—
153C22×C83—21no—
154C22×C83—42no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83—42no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—81no—
163C22×C83C2×#9911yescharacteristic
164C22×C83C2×#10311yescharacteristic
165C22×C83#42111yescharacteristic
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—22no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—162no—
173Dic83—42no—
174Dic83—81no—
175Dic83—81no—
176Dic83—81no—
177#11902—831nomaximal, Sylow
178C2×C83—21no—
179C2×C83—21no—
180C2×C83—41no—
181C2×C83—42no—
182C2×C83—41no—
183C2×C83#152011yescharacteristic
184C2×C83#212111yescharacteristic
185C2×C83#212611yescharacteristic
186#2564—831no—
187#1799D8311yescharacteristic
188#1460—831no—
189#1352—832no—
190#1351—832no—
191C83#1190211yescharacteristic, complemented, Sylow
192C2×C22⋊Q8—831no—
193C2×C4×D4C2×D8311yescharacteristic
194#431—1661no—
195C2×#99—21no—
196#400—831no—
197#399—831no—
198#395—831no—
199#393—831no—
200#387—1661no—
201#386—1661no—
202#385—1661no—
203#380—21no—
204#380—1661no—
205C2×#95—831no—
206#342C2×D8312yes—
207#323—832no—
208#323—1661no—
209C22×Q8—831no—
210C22×D4C22⋊D8311yescharacteristic
211C23×C4—1661no—
212C23×C4C4×D8311yescharacteristic
213C23×C4C22⋊D8311yescharacteristic
214C22⋊Q8—3321no—
215C22⋊Q8—3321no—
216C4×D4—41no—
217C4×D4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—831no—
220C2×C4⋊C4—1661no—
221C2×C4⋊C4—1661no—
222C2×C4⋊C4—1661no—
223C2×C4⋊C4C4×D8311yescharacteristic
224C2×C22⋊C4—22no—
225C2×C22⋊C4—1661no—
226C2×C22⋊C4—1661no—
227C2×C22⋊C4—1661no—
228C2×C22⋊C4D33211yescharacteristic
229C2×C22⋊C4C22×D8311yescharacteristic
230C2×C42D33211yescharacteristic
231C4⋊C8—3321no—
232#99—42no—
233C22⋊C8—1662no—
234C4×C8—3321no—
235#95—41no—
236#95—831no—
237#95—831no—
238#95—1661no—
239#95—1661no—
240#95—1661no—
241#95—1661no—
242#95—1661no—
243C24(C2×D83)⋊C411yescharacteristic
244C24C2×C22⋊D8311yescharacteristic
245C2×Q8—3321no—
246C2×Q8—6641no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4#1305057812yes—
250C2×D4#1305058012yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—3322no—
259C22×C4—3322no—
260C22×C4—6641no—
261C22×C4—1661no—
262C22×C4—1661no—
263C22×C4—1661no—
264C22×C4—1661no—
265C22×C4(C2×D83)⋊C411yescharacteristic
266C22×C4(C2×D83)⋊C411yescharacteristic
267C22×C4(C2×D83)⋊C411yescharacteristic
268C22×C4C2×C4×D8311yescharacteristic
269C22×C4C2×D33211yescharacteristic
270C2×C8—3322no—
271C4⋊C4—41no—
272C4⋊C4—82no—
273C4⋊C4—3321no—
274C4⋊C4—3322no—
275C4⋊C4—3321no—
276C4⋊C4—3321no—
277C4⋊C4—6641no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—3322no—
283C22⋊C4—3322no—
284C22⋊C4—6641no—
285C42—21no—
286C42—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23#1458171512yes—
296C23C2×(C2×D83)⋊C411yescharacteristic
297C23C2×#1305057811yescharacteristic
298C23C2×#1305058011yescharacteristic
299Q8—6642no—
300D4—42no—
301D4—42no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—3321no—
319C2×C4—3321no—
320C2×C4—3321no—
321C2×C4—3321no—
322C2×C4—3321no—
323C2×C4—3321no—
324C2×C4—3321no—
325C2×C4—3321no—
326C2×C4—3321no—
327C2×C4—3321no—
328C2×C4—3321no—
329C2×C4—6644no—
330C2×C4—1662no—
331C2×C4#1458170611yescharacteristic
332C2×C4#1458170611yescharacteristic
333C2×C4#1458171512yes—
334C8—6642no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22C2×#1458170611yescharacteristic
350C22C2×#1458171511yescharacteristic
351C22#1912771911yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—162no—
359C4—3322no—
360C4—6641no—
361C4—6641no—
362C4—6641no—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—42no—
367C2—41no—
368C2#2490603411yescharacteristic
369C2#2490910711yescharacteristic
370C2#2490912611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1799C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#1799D83
hypocenterC83#11902
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #24905486C2semidirect
4. #24905483C2semidirect
6. C2×#14581771C4semidirect
8. #19127640C4semidirect
10. #19127621C4semidirect
16. C2×C83⋊#95C4semidirect
191. C83#11902semidirect

Maximal subgroups

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

subgroupindex
#249126392
#249057842
#249054862
#249054832
C83×#17992
#1190283