Finite Group Databasebeta

← #24912798

Subgroups — #24912798

Order 10624.

468 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19127639C211yesmaximal, characteristic, complemented
2#19127629C211yesmaximal, characteristic, complemented
3#19127629C211yesmaximal, characteristic, complemented
4#19127595C211yesmaximal, characteristic, complemented
5#19127589C211yesmaximal, characteristic, complemented
6#19127583C211yesmaximal, characteristic, complemented
7#19127582C211yesmaximal, characteristic, complemented
8#19127581C211yesmaximal, characteristic
9#19127581C211yesmaximal, characteristic
10#19127573C211yesmaximal, characteristic, complemented
11#19127568C211yesmaximal, characteristic, complemented
12#19127559C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
15#394×C83C211yesmaximal, characteristic
16C23×Dic83C2211yescharacteristic
17C23×Dic83C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×C22⋊Dic83C2211yescharacteristic
23C2×C22⋊Dic83C2211yescharacteristic
24C2×C22⋊Dic83C2211yescharacteristic
25C2×C22⋊Dic83C2211yescharacteristic
26C2×C22⋊Dic83C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C4×Dic83C2211yescharacteristic
29C2×C4×Dic83C2211yescharacteristic
30C2×C4×Dic83C2211yescharacteristic
31C83⋊#95C2211yescharacteristic
32C83⋊#95C2211yescharacteristic
33C83⋊#95C2211yescharacteristic
34C83⋊#95C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C22×D4×C83C2211yescharacteristic
45C2×C4⋊C4×C83C2211yescharacteristic
46C2×C22⋊C4×C83C2211yescharacteristic
47C2×C22⋊C4×C83C2211yescharacteristic
48C2×C22⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51C22×Dic83—22no—
52C22×Dic83—22no—
53C22×Dic83—22no—
54C22×Dic83—41no—
55C22×Dic83—41no—
56C22×Dic83—41no—
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65Dic83⋊C4—22no—
66Dic83⋊C4—22no—
67Dic83⋊C4—41no—
68Dic83⋊C4—41no—
69C22⋊Dic83—22no—
70C22⋊Dic83—22no—
71C22⋊Dic83—22no—
72C22⋊Dic83—22no—
73C22⋊Dic83—22no—
74C22⋊Dic83—41no—
75C4×Dic83—22no—
76C4×Dic83—41no—
77C4×Dic83—41no—
78C24×C83C2311yescharacteristic
79C24×C83C2311yescharacteristic
80C2×D4×C83—22no—
81C2×D4×C83—22no—
82C2×D4×C83—41no—
83C22×C4×C83C2311yescharacteristic
84C22×C4×C83C2311yescharacteristic
85C22×C4×C83C2311yescharacteristic
86C22×C4×C83C2311yescharacteristic
87C22×C4×C83C2311yescharacteristic
88C4⋊C4×C83—41no—
89C22⋊C4×C83—22no—
90C22⋊C4×C83—22no—
91C22⋊C4×C83—41no—
92C22⋊C4×C83—41no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83#4112yes—
106C23×C83#4112yes—
107C23×C83C2411yescharacteristic
108D4×C83—44no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×Dic83—21no—
125C2×Dic83—21no—
126C2×Dic83—21no—
127C2×Dic83—21no—
128C2×Dic83—21no—
129C2×Dic83—21no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—44no—
146C2×Dic83—44no—
147C2×Dic83#4112yes—
148C2×Dic83#4112yes—
149C2×Dic83#4112yes—
150C22×C83—22no—
151C22×C83—22no—
152C22×C83—22no—
153C22×C83—22no—
154C22×C83—22no—
155C22×C83—22no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83C2×#4111yescharacteristic
165C22×C83C2×#4111yescharacteristic
166C22×C83C2×#4111yescharacteristic
167C22×C83C2×#4111yescharacteristic
168C22×C83C2×#4111yescharacteristic
169C22×C83H(5,𝔽2)11yescharacteristic
170C22×C83H(5,𝔽2)11yescharacteristic
171C4×C83—41no—
172C4×C83—41no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176Dic83—22no—
177Dic83—22no—
178Dic83—22no—
179Dic83—41no—
180Dic83—41no—
181Dic83—41no—
182Dic83—41no—
183Dic83—41no—
184C2×C83—42no—
185C2×C83—42no—
186C2×C83#52911yescharacteristic
187C2×C83#52911yescharacteristic
188C2×C83#53811yescharacteristic
189C2×C83#55111yescharacteristic
190C2×C83#55111yescharacteristic
191C2×C83#55311yescharacteristic
192C2×C83#55311yescharacteristic
193#2578—831nomaximal, Sylow
194C83#257811yescharacteristic, complemented, Sylow
195#400—831no—
196#397—831no—
197#397—831no—
198#394D8311yescharacteristic
199#388—831no—
200#388—831no—
201#387—831no—
202#387—831no—
203#386—831no—
204#386—831no—
205#385—831no—
206#383—831no—
207#380—831no—
208C4×C22⋊C4—831no—
209C4×C22⋊C4—831no—
210C22×D4C2×D8311yescharacteristic
211C23×C4—831no—
212C23×C4—831no—
213C2×C4⋊C4—831no—
214C2×C4⋊C4—831no—
215C2×C4⋊C4—831no—
216C2×C4⋊C4—831no—
217C2×C4⋊C4C2×D8311yescharacteristic
218C2×C22⋊C4—831no—
219C2×C22⋊C4—831no—
220C2×C22⋊C4—831no—
221C2×C22⋊C4—831no—
222C2×C22⋊C4—831no—
223C2×C22⋊C4—831no—
224C2×C22⋊C4C2×D8311yescharacteristic
225C2×C22⋊C4C2×D8311yescharacteristic
226C2×C22⋊C4C2×D8311yescharacteristic
227C2×C22⋊C4C2×D8311yescharacteristic
228C2×C42—831no—
229C2×C42—831no—
230C2×C42—831no—
231#95—831no—
232#95—831no—
233#95—831no—
234#95—831no—
235#95—831no—
236#95—831no—
237#95—831no—
238#95—831no—
239#95—831no—
240#95—831no—
241#95—831no—
242#95—831no—
243#95—831no—
244#95C2×D8311yescharacteristic
245C24C22×D8311yescharacteristic
246C24C22×D8311yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C22×C4—3321no—
251C22×C4—3321no—
252C22×C4—3321no—
253C22×C4—831no—
254C22×C4—831no—
255C22×C4—831no—
256C22×C4—831no—
257C22×C4—831no—
258C22×C4—831no—
259C22×C4—831no—
260C22×C4—831no—
261C22×C4—1662no—
262C22×C4—1662no—
263C22×C4—1662no—
264C22×C4C22×D8311yescharacteristic
265C22×C4C22×D8311yescharacteristic
266C22×C4C22×D8311yescharacteristic
267C22×C4C22×D8311yescharacteristic
268C22×C4C22×D8311yescharacteristic
269C4⋊C4—41no—
270C4⋊C4—3321no—
271C4⋊C4—3321no—
272C4⋊C4—1662no—
273C4⋊C4—1662no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—3321no—
279C22⋊C4—1662no—
280C22⋊C4—1662no—
281C22⋊C4—1662no—
282C22⋊C4—1662no—
283C22⋊C4—1662no—
284C42—3321no—
285C42—3321no—
286C42—1662no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1305058912yes—
300C23#1305058912yes—
301C23C23×D8311yescharacteristic
302D4—44no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—3324no—
319C2×C4—3324no—
320C2×C4—832no—
321C2×C4—832no—
322C2×C4—832no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4—1661no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22#41×D8311yescharacteristic
359C22#41×D8311yescharacteristic
360C22#41×D8311yescharacteristic
361C22C2×#1305058911yescharacteristic
362C22C2×#1305058911yescharacteristic
363C22#1458183711yescharacteristic
364C22#1458183711yescharacteristic
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—3321no—
371C4—3321no—
372C4—3321no—
373C4—3321no—
374C4—3321no—
375C4—1662no—
376C4—1662no—
377C4—1662no—
378C2—42no—
379C2—42no—
380C2#1912822811yescharacteristic
381C2#1912822811yescharacteristic
382C2#1912828311yescharacteristic
383C2#1912838111yescharacteristic
384C2#1912838211yescharacteristic
385C2#1912839411yescharacteristic
386C2#1912839411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2578
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #19127639C2semidirect
2. #19127629C2semidirect
3. #19127629C2semidirect
4. #19127595C2semidirect
5. #19127589C2semidirect
6. #19127583C2semidirect
7. #19127582C2semidirect
10. #19127573C2semidirect
11. #19127568C2semidirect
12. #19127559C2semidirect
13. C22⋊C4×Dic83C2semidirect
14. C22⋊C4×Dic83C2semidirect
194. C83#2578semidirect

Maximal subgroups

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

subgroupindex
#191276392
#191276292
#191275952
#191275892
#191275832
#191275822
#191275812
#191275732
#191275682
#191275592
C22⋊C4×Dic832
#394×C832
#257883