Finite Group Databasebeta

← #24915178

Subgroups — #24915178

Order 10624.

434 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19127634C211yesmaximal, characteristic, complemented
2#19127630C211yesmaximal, characteristic, complemented
3#19127629C211yesmaximal, characteristic, complemented
4#19127624C211yesmaximal, characteristic, complemented
5#19127617C211yesmaximal, characteristic
6#19127601C211yesmaximal, characteristic
7#19127591C211yesmaximal, characteristic, complemented
8#19127589C211yesmaximal, characteristic, complemented
9#19127586C211yesmaximal, characteristic, complemented
10#19127584C211yesmaximal, characteristic, complemented
11#19127583C211yesmaximal, characteristic, complemented
12#19127577C211yesmaximal, characteristic
13#19127575C211yesmaximal, characteristic, complemented
14#19127560C211yesmaximal, characteristic, complemented
15#397×C83C211yesmaximal, characteristic, complemented
16C22×C22⋊D83C2211yescharacteristic
17C22×C4×D83C2211yescharacteristic
18C2×C4⋊Dic83C2211yescharacteristic
19C2×C4⋊Dic83C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×C22⋊Dic83C2211yescharacteristic
26C2×C22⋊Dic83C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×C4×Dic83C2211yescharacteristic
34C2×C4×Dic83C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C2×C4⋊C4×C83C2211yescharacteristic
45C2×C22⋊C4×C83C2211yescharacteristic
46C2×C22⋊C4×C83C2211yescharacteristic
47C2×C22⋊C4×C83C2211yescharacteristic
48#95×C83C2211yescharacteristic
49#95×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51C23×D83C2311yescharacteristic
52C2×C22⋊D83—22no—
53C2×C22⋊D83—22no—
54C2×C22⋊D83—41no—
55C22×Dic83C2311yescharacteristic
56C22×Dic83C2311yescharacteristic
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C2×C4×D83—22no—
63C2×C4×D83—41no—
64C2×C4×D83—41no—
65C4⋊Dic83—22no—
66C4⋊Dic83—22no—
67Dic83⋊C4—22no—
68Dic83⋊C4—41no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71Dic83⋊C4—41no—
72C22⋊Dic83—22no—
73C22⋊Dic83—41no—
74C22⋊Dic83—41no—
75(C2×D83)⋊C4—22no—
76(C2×D83)⋊C4—22no—
77(C2×D83)⋊C4—41no—
78(C2×D83)⋊C4—41no—
79(C2×D83)⋊C4—41no—
80C4×Dic83—41no—
81C4×Dic83—41no—
82C24×C83C2311yescharacteristic
83C22×C4×C83C2311yescharacteristic
84C22×C4×C83C2311yescharacteristic
85C22×C4×C83C2311yescharacteristic
86C22×C4×C83C2311yescharacteristic
87C22×C4×C83C2311yescharacteristic
88C22×C4×C83C2311yescharacteristic
89C4⋊C4×C83—41no—
90C22⋊C4×C83—41no—
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×C83C2411yescharacteristic
101C22×D83—21no—
102C22×D83—21no—
103C22×D83—21no—
104C22×D83—21no—
105C22×D83—21no—
106C22×D83—21no—
107C22×D83#4112yes—
108C2×C4×C83—21no—
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×C4×C83—21no—
125C2×C4×C83—21no—
126C22⋊D83—44no—
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—21no—
146C2×Dic83#4112yes—
147C2×Dic83#4112yes—
148C4×D83—44no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83C2×#4111yescharacteristic
157C22×C83C2×#4111yescharacteristic
158C22×C83C2×#4111yescharacteristic
159C22×C83H(5,𝔽2)11yescharacteristic
160C22×C83H(5,𝔽2)11yescharacteristic
161C22×C83H(5,𝔽2)11yescharacteristic
162C22×C832−1+411yescharacteristic
163C2×D83—22no—
164C2×D83—22no—
165C2×D83—22no—
166C2×D83—41no—
167C2×D83—41no—
168C2×D83—41no—
169C2×D83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—41no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176Dic83—22no—
177Dic83—22no—
178Dic83—41no—
179Dic83—41no—
180Dic83—41no—
181Dic83—41no—
182Dic83—41no—
183C2×C83—81no—
184C2×C83#53811yescharacteristic
185C2×C83#53911yescharacteristic
186C2×C83#54011yescharacteristic
187C2×C83#55111yescharacteristic
188C2×C83#55311yescharacteristic
189C2×C83#55611yescharacteristic
190C2×C83#56211yescharacteristic
191D83—42no—
192#2786—831nomaximal, Sylow
193C83#278611yescharacteristic, complemented, Sylow
194#398—831no—
195#397—831no—
196#397—831no—
197#397D8311yescharacteristic, complemented
198#396—831no—
199#394—831no—
200#390—831no—
201#388—831no—
202#388—831no—
203#387—831no—
204#387—831no—
205#387—831no—
206#386—831no—
207#385—831no—
208#380—831no—
209C22×D4—831no—
210C23×C4—831no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—831no—
213C2×C4⋊C4—831no—
214C2×C4⋊C4—831no—
215C2×C4⋊C4—831no—
216C2×C4⋊C4—831no—
217C2×C4⋊C4—831no—
218C2×C4⋊C4C2×D8311yescharacteristic
219C2×C22⋊C4—831no—
220C2×C22⋊C4—831no—
221C2×C22⋊C4—831no—
222C2×C22⋊C4—831no—
223C2×C22⋊C4—831no—
224C2×C22⋊C4—831no—
225C2×C22⋊C4—831no—
226C2×C22⋊C4—831no—
227C2×C22⋊C4C2×D8311yescharacteristic
228C2×C22⋊C4C2×D8311yescharacteristic
229C2×C22⋊C4C2×D8311yescharacteristic
230C2×C42—831no—
231C2×C42—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#95C2×D8311yescharacteristic
242#95C2×D8311yescharacteristic
243#95C2×D8311yescharacteristic
244C24—831no—
245C24C22×D8311yescharacteristic
246C2×D4—3321no—
247C2×D4—1662no—
248C2×D4—1662no—
249C22×C4—3321no—
250C22×C4—3321no—
251C22×C4—831no—
252C22×C4—831no—
253C22×C4—831no—
254C22×C4—831no—
255C22×C4—831no—
256C22×C4—831no—
257C22×C4—831no—
258C22×C4—1662no—
259C22×C4C22×D8311yescharacteristic
260C22×C4C22×D8311yescharacteristic
261C22×C4C22×D8311yescharacteristic
262C22×C4C22×D8311yescharacteristic
263C22×C4C22×D8311yescharacteristic
264C22×C4C22×D8311yescharacteristic
265C4⋊C4—41no—
266C4⋊C4—3321no—
267C4⋊C4—3321no—
268C4⋊C4—3321no—
269C4⋊C4—3321no—
270C4⋊C4—1662no—
271C4⋊C4—1662no—
272C4⋊C4—1662no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—3321no—
277C22⋊C4—3321no—
278C22⋊C4—3321no—
279C22⋊C4—3321no—
280C22⋊C4—3321no—
281C22⋊C4—1662no—
282C22⋊C4—1662no—
283C22⋊C4—1662no—
284C42—3321no—
285C42—3321no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—832no—
294C23—1661no—
295C23—1661no—
296C23—1661no—
297C23—1661no—
298C23—1661no—
299C23—1661no—
300C23C23×D8311yescharacteristic
301D4—3324no—
302C2×C4—21no—
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—21no—
319C2×C4—21no—
320C2×C4—3324no—
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—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—3321no—
350C22—3321no—
351C22—3321no—
352C22—3321no—
353C22—1662no—
354C22—1662no—
355C22—1662no—
356C22#41×D8311yescharacteristic
357C22#41×D8311yescharacteristic
358C22#41×D8311yescharacteristic
359C22#1458183711yescharacteristic
360C22#1458183711yescharacteristic
361C22#1458183811yescharacteristic
362C22#1458183911yescharacteristic
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3321no—
370C4—3321no—
371C4—3321no—
372C4—3321no—
373C4—3321no—
374C4—1662no—
375C4—1662no—
376C2—81no—
377C2—3322no—
378C2#1912828511yescharacteristic
379C2#1912829411yescharacteristic
380C2#1912830111yescharacteristic
381C2#1912837611yescharacteristic
382C2#1912839611yescharacteristic
383C2#1912841811yescharacteristic
384C2#1912844811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#397D83
hypocenterC83#2786
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. #19127634C2semidirect
2. #19127630C2semidirect
3. #19127629C2semidirect
4. #19127624C2semidirect
7. #19127591C2semidirect
8. #19127589C2semidirect
9. #19127586C2semidirect
10. #19127584C2semidirect
11. #19127583C2semidirect
13. #19127575C2semidirect
14. #19127560C2semidirect
15. #397×C83C2semidirect
193. C83#2786semidirect
197. #397D83semidirect

Maximal subgroups

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

subgroupindex
#191276342
#191276302
#191276292
#191276242
#191276172
#191276012
#191275912
#191275892
#191275862
#191275842
#191275832
#191275772
#191275752
#191275602
#397×C832
#278683