Finite Group Databasebeta

← #35412459

Subgroups — #35412459

Order 21248.

834 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128058C211yesmaximal, characteristic
2C4×#14581780C211yesmaximal, characteristic
3#24908429C211yesmaximal, characteristic, complemented
4#24908272C211yesmaximal, characteristic
5#24908159C212yesmaximal
6#24908152C212yesmaximal, complemented
7#24907593C211yesmaximal, characteristic, complemented
8#24907296C212yesmaximal, complemented
9#24907291C212yesmaximal
10C4×#14581703C211yesmaximal, characteristic, complemented
11C4×#125×C83C211yesmaximal, characteristic, complemented
12C2×#14581780C2211yescharacteristic
13C2×C4×Dic166C2211yescharacteristic
14C2×C4×D332C2211yescharacteristic
15#19128058C418yescomplemented
16C2×#14581715C2212yes—
17C2×#14581715C2212yes—
18C2×#14581712C2212yes—
19C2×#14581712C2212yes—
20C2×#14581703C2212yes—
21C2×#14581703C2212yes—
22C2×C4×C83⋊C8C2211yescharacteristic
23C2×C4×C83⋊C8C2211yescharacteristic
24#19127607C2211yescharacteristic
25#19127603C2211yescharacteristic
26#19127592—22no—
27#19127567C2211yescharacteristic
28C4×(C2×D83)⋊C4—22no—
29#19127463C2212yes—
30#19127463C2212yes—
31#19127463C2212yes—
32#19127463C2212yes—
33C2×#125×C83C2211yescharacteristic
34#389×C83C2212yes—
35#387×C83—22no—
36#382×C83C2211yescharacteristic
37C4×C4⋊C4×C83—22no—
38C4×C4⋊C4×C83C2212yes—
39C43×C83C2211yescharacteristic
40C22×Dic166C2311yescharacteristic
41C22×D332C2311yescharacteristic
42C22×C4×D83—21no—
43C22×C83⋊C8C2312yes—
44C22×C83⋊C8C2312yes—
45#14581780—22no—
46#14581780C2×C414yes—
47C4×Dic166—24no—
48C4×D332—24no—
49C2×C4⋊Dic83C2311yescharacteristic
50C2×C4⋊Dic83C2311yescharacteristic
51C2×Dic83⋊C4—21no—
52C2×(C2×D83)⋊C4—22no—
53C2×(C2×D83)⋊C4—21no—
54C2×(C2×D83)⋊C4—22no—
55C2×C4×Dic83—21no—
56#14581715—24no—
57#14581715C2×C418yescomplemented
58#14581712—24no—
59#14581712C2×C418yes—
60#14581703—24no—
61#14581703—24no—
62C4×C83⋊C8—22no—
63C4×C83⋊C8C2×C414yescomplemented
64C83⋊#95—22no—
65C83⋊#95—22no—
66#125×C83—22no—
67#125×C83C2×C414yescomplemented
68C2×C4⋊C4×C83—22no—
69C2×C4⋊C4×C83—22no—
70C2×C4⋊C4×C83C2312yes—
71C2×C4⋊C4×C83C2312yes—
72C2×C42×C83—22no—
73C2×C42×C83D412yes—
74C2×C42×C83D412yes—
75C2×C42×C83C2311yescharacteristic
76C2×C42×C83C2311yescharacteristic
77C2×C42×C83C2311yescharacteristic
78#95×C83—22no—
79#95×C83—22no—
80#95×C83—22no—
81C23×D83—21no—
82C2×Dic166—22no—
83C2×Dic166—22no—
84C2×Dic166—21no—
85C2×Dic166C22×C412yes—
86C2×D332—22no—
87C2×D332—22no—
88C2×D332—21no—
89C2×D332C22×C412yes—
90C22×Dic83—21no—
91C22×Dic83—21no—
92C22×Dic83—21no—
93C2×C4×D83—44no—
94C2×C4×D83—42no—
95C2×C83⋊C8—22no—
96C2×C83⋊C8—22no—
97C2×C83⋊C8—22no—
98C2×C83⋊C8—22no—
99C2×C83⋊C8C22×C414yes—
100C2×C83⋊C8#4114yes—
101C4⋊Dic83—22no—
102C4⋊Dic83—22no—
103Dic83⋊C4—42no—
104(C2×D83)⋊C4—28no—
105(C2×D83)⋊C4—44no—
106(C2×D83)⋊C4—42no—
107C4×Dic83—42no—
108C22×C4×C83—22no—
109C22×C4×C83—22no—
110C22×C4×C83C2×D412yes—
111C22×C4×C83C2×D412yes—
112C22×C4×C83C2×D411yescharacteristic
113C22×C4×C83C2×D411yescharacteristic
114C22×C4×C83C2411yescharacteristic
115C4⋊C4×C83—28no—
116C4⋊C4×C83—22no—
117C4⋊C4×C83—22no—
118C4⋊C4×C83—22no—
119C4⋊C4×C83—44no—
120C4⋊C4×C83C22×C414yes—
121C42×C83—22no—
122C42×C83—42no—
123C42×C83—42no—
124C42×C83—44no—
125C42×C83C22×C412yes—
126C42×C83#4112yes—
127C42×C83#4114yes—
128C23×C83C22×D411yescharacteristic
129C22×D83—22no—
130C22×D83—41no—
131C22×D83—41no—
132C22×D83—42no—
133C22×D83—42no—
134C2×C4×C83—22no—
135C2×C4×C83—22no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—24no—
139C2×C4×C83—22no—
140C2×C4×C83—22no—
141C2×C4×C83—22no—
142C2×C4×C83—42no—
143C2×C4×C83—42no—
144C2×C4×C83—42no—
145C2×C4×C83—42no—
146C2×C4×C83—42no—
147C2×C4×C83C4×D414yes—
148C2×C4×C83#12412yes—
149C2×C4×C83#12412yes—
150C2×C4×C83C23×C411yescharacteristic
151C2×C4×C83C2×#4112yes—
152C2×C4×C83C2×#4111yescharacteristic
153C2×C4×C83C2×#4112yes—
154C2×Dic83—22no—
155C2×Dic83—41no—
156C2×Dic83—41no—
157C2×Dic83—41no—
158C2×Dic83—41no—
159C2×Dic83—42no—
160C2×Dic83—42no—
161D332—24no—
162D332—42no—
163C4×D83—84no—
164Dic166—24no—
165Dic166—42no—
166C83⋊C8—24no—
167C83⋊C8—24no—
168C22×C83C2×C4×D411yescharacteristic
169C22×C83C2×#12411yescharacteristic
170C22×C83#53211yescharacteristic
171C22×C83#57512yes—
172C22×C83#57512yes—
173C2×D83—44no—
174C2×D83—42no—
175C2×D83—81no—
176C2×D83—81no—
177C2×D83—82no—
178C4×C83—22no—
179C4×C83—22no—
180C4×C83—24no—
181C4×C83—42no—
182C4×C83—44no—
183C4×C83—82no—
184C4×C83C4×#4112yes—
185C4×C83#52912yes—
186Dic83—42no—
187Dic83—81no—
188Dic83—81no—
189#21648—831nomaximal, Sylow
190C2×C83C4×#12411yescharacteristic
191C2×C83#294312yes—
192C2×C83#314011yescharacteristic
193C2×C83#314211yescharacteristic
194C2×C83#318312yes—
195D83—82no—
196C2×#488—831no—
197C4×#125D8311yescharacteristic, complemented
198C4×#124—831no—
199#1991—831no—
200#1958—831no—
201#1936—832no—
202#1935—832no—
203#1835—831no—
204#1763—832no—
205#1762—832no—
206C4×#97—831no—
207C83#2164811yescharacteristic, complemented, Sylow
208C2×#125C2×D8311yescharacteristic
209C2×#124—831no—
210C2×C4×Q8—831no—
211C2×C4×D4—831no—
212#488—838no—
213C2×#103—832no—
214C2×#103—832no—
215C2×#102—832no—
216C2×#102—832no—
217C2×#97—832no—
218C2×#97—832no—
219C2×C4×C8—831no—
220C2×C4×C8—831no—
221#391—831no—
222#390—831no—
223#389C2×D8312yes—
224#388—1662no—
225#387—22no—
226#382—831no—
227#382C2×D8311yescharacteristic
228C4×C4⋊C4—22no—
229C4×C4⋊C4C2×D8312yes—
230C4×C22⋊C4—1662no—
231C43C2×D8311yescharacteristic
232#340—832no—
233#340—832no—
234#340—832no—
235#340—832no—
236C22×Q8—831no—
237C22×D4—831no—
238C23×C4—1661no—
239C22×C8—832no—
240C22×C8—832no—
241#125—22no—
242#125C4×D8314yescomplemented
243#124—834no—
244#124—1662no—
245C4×Q8—1664no—
246C4×D4—1664no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—831no—
250C2×C4⋊C4—831no—
251C2×C4⋊C4—1661no—
252C2×C4⋊C4C22×D8312yes—
253C2×C4⋊C4C22×D8312yes—
254C2×C22⋊C4—1661no—
255C2×C22⋊C4—1662no—
256C2×C22⋊C4—1662no—
257C2×C42—22no—
258C2×C42—1661no—
259C2×C42C22⋊D8312yes—
260C2×C42C22⋊D8312yes—
261C2×C42C22×D8311yescharacteristic
262C2×C42C22×D8311yescharacteristic
263C2×C42C22×D8311yescharacteristic
264#103—838no—
265#103—1664no—
266#102—838no—
267#102—1664no—
268#97—1664no—
269#97—1664no—
270C4×C8—834no—
271C4×C8—1662no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—1662no—
276#95—1662no—
277C24—1661no—
278C2×Q8—832no—
279C2×Q8—1662no—
280C2×Q8—1662no—
281C2×Q8—1661no—
282C2×D4—832no—
283C2×D4—1662no—
284C2×D4—1661no—
285C2×D4—1662no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—3324no—
289C22×C4—3322no—
290C22×C4—1661no—
291C22×C4—1661no—
292C22×C4—1661no—
293C22×C4C2×C22⋊D8312yes—
294C22×C4C2×C22⋊D8311yescharacteristic
295C22×C4C2×C22⋊D8311yescharacteristic
296C22×C4C2×C22⋊D8312yes—
297C22×C4C23×D8311yescharacteristic
298C2×C8—834no—
299C2×C8—834no—
300C2×C8—1662no—
301C2×C8—1662no—
302C2×C8—1662no—
303C2×C8—1662no—
304C4⋊C4—22no—
305C4⋊C4—28no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—44no—
309C4⋊C4—3322no—
310C4⋊C4—1662no—
311C4⋊C4—1662no—
312C4⋊C4C2×C4×D8314yes—
313C22⋊C4—3324no—
314C22⋊C4—3322no—
315C22⋊C4—1668no—
316C42—22no—
317C42—42no—
318C42—42no—
319C42—44no—
320C42—3322no—
321C42C2×C4×D8312yes—
322C42#1305059012yes—
323C42#1305059114yes—
324C23—3321no—
325C23—3322no—
326C23—3322no—
327C23—3321no—
328C23—1662no—
329C23C22×C22⋊D8311yescharacteristic
330Q8—3322no—
331Q8—1664no—
332D4—3322no—
333D4—1664no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—3321no—
348C2×C4—3321no—
349C2×C4—3321no—
350C2×C4—3321no—
351C2×C4—3322no—
352C2×C4—3322no—
353C2×C4—6644no—
354C2×C4—1662no—
355C2×C4C4×C22⋊D8314yes—
356C2×C4#1458178112yes—
357C2×C4#1458178112yes—
358C2×C4C22×C4×D8311yescharacteristic
359C2×C4#41×D8312yes—
360C2×C4C2×#1305059011yescharacteristic
361C2×C4C2×#1305059112yes—
362C8—1664no—
363C8—1664no—
364C22—3324no—
365C22—3322no—
366C22—6641no—
367C22—6641no—
368C22—6642no—
369C22C2×C4×C22⋊D8311yescharacteristic
370C22C2×#1458178111yescharacteristic
371C22#1912824711yescharacteristic
372C22#1912849912yes—
373C22#1912849912yes—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—42no—
379C4—82no—
380C4—3322no—
381C4—6641no—
382C4—6641no—
383C4C4×#1305059112yes—
384C4#1912823812yes—
385C2—6642no—
386C2C4×#1458178111yescharacteristic
387C2#2491626112yes—
388C2#2491785211yescharacteristic
389C2#2491786611yescharacteristic
390C2#2491818612yes—
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC4×#125×C83C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenterC4×#125D83
hypocenterC83#21648
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C43 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
3. #24908429C2semidirect
6. #24908152C2semidirect
7. #24907593C2semidirect
8. #24907296C2semidirect
10. C4×#14581703C2semidirect
11. C4×#125×C83C2semidirect
15. #19128058C4semidirect
57. #14581715C2×C4semidirect
63. C4×C83⋊C8C2×C4semidirect
67. #125×C83C2×C4semidirect
197. C4×#125D83semidirect
207. C83#21648semidirect
242. #125C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191280582
C4×#145817802
#249084292
#249082722
#249081592
#249081522
#249075932
#249072962
#249072912
C4×#145817032
C4×#125×C832
#2164883