Finite Group Databasebeta

← #24910941

Subgroups — #24910941

Order 10624.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581777C211yesmaximal, characteristic
2C2×#14581754C211yesmaximal, characteristic
3#19127639C211yesmaximal, characteristic, complemented
4#19127628C211yesmaximal, characteristic, complemented
5#19127627C211yesmaximal, characteristic, complemented
6#19127586C211yesmaximal, characteristic, complemented
7#19127582C211yesmaximal, characteristic, complemented
8#19127579C211yesmaximal, characteristic
9#19127572C211yesmaximal, characteristic, complemented
10#19127572C211yesmaximal, characteristic, complemented
11#19127563C211yesmaximal, characteristic, complemented
12#19127563C211yesmaximal, characteristic, complemented
13#19127558C211yesmaximal, characteristic, complemented
14#95×D83C211yesmaximal, characteristic, complemented
15#398×C83C211yesmaximal, characteristic, complemented
16C22×D332C2211yescharacteristic
17C22×C4×D83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21#14581777—24no—
22#14581754—24no—
23C2×C4⋊Dic83C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2211yescharacteristic
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×C4×Dic83C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C4⋊C4×C83C2211yescharacteristic
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic
50#95×C83C2211yescharacteristic
51#95×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C23×D83C2311yescharacteristic
55C2×D332—22no—
56C2×D332—22no—
57C2×D332D414yes—
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C2×C4×D83—22no—
65C2×C4×D83—22no—
66C2×C4×D83—22no—
67C2×C4×D83—22no—
68C2×C4×D83—22no—
69C2×C4×D83—22no—
70C2×C4×D83—22no—
71C2×C4×D83—22no—
72C2×C4×D83—22no—
73C2×C4×D83—41no—
74C2×C4×D83—41no—
75C2×C4×D83—41no—
76C4⋊Dic83—22no—
77Dic83⋊C4—22no—
78Dic83⋊C4—22no—
79(C2×D83)⋊C4—22no—
80(C2×D83)⋊C4—22no—
81(C2×D83)⋊C4—22no—
82(C2×D83)⋊C4—22no—
83(C2×D83)⋊C4—22no—
84(C2×D83)⋊C4—22no—
85(C2×D83)⋊C4—22no—
86(C2×D83)⋊C4—41no—
87C4×Dic83—22no—
88C22×C4×C83C2311yescharacteristic
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C4⋊C4×C83—22no—
96C4⋊C4×C83—22no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C23×C83C2411yescharacteristic
100C22×D83—21no—
101C22×D83—21no—
102C22×D83—21no—
103C22×D83—21no—
104C22×D83—21no—
105C22×D83—21no—
106C22×D83—21no—
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83C2×D412yes—
111C22×D83C2×D412yes—
112C22×D83#4112yes—
113C22×D83#4112yes—
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—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83C2×D412yes—
134C2×C4×C83#4112yes—
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—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—21no—
151C2×Dic83—21no—
152C2×Dic83#4112yes—
153D332—28no—
154C4×D83—44no—
155C4×D83—44no—
156C4×D83—44no—
157C4×D83—44no—
158C22×C83C22×D411yescharacteristic
159C22×C83C2×#4111yescharacteristic
160C22×C83C2×#4111yescharacteristic
161C22×C83C2×#4111yescharacteristic
162C22×C83C2×#4111yescharacteristic
163C22×C83H(5,𝔽2)11yescharacteristic
164C22×C832−1+411yescharacteristic
165C2×D83—22no—
166C2×D83—22no—
167C2×D83—22no—
168C2×D83—22no—
169C2×D83—22no—
170C2×D83—22no—
171C2×D83—22no—
172C2×D83—22no—
173C2×D83—22no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C2×D83—41no—
178C2×D83#12314yes—
179C4×C83—22no—
180C4×C83—22no—
181C4×C83—41no—
182C4×C83—41no—
183C4×C83—41no—
184C4×C83—41no—
185C4×C83—41no—
186Dic83—22no—
187Dic83—41no—
188Dic83—41no—
189Dic83—41no—
190Dic83—41no—
191Dic83—41no—
192C2×C83C2×#12311yescharacteristic
193C2×C83#52911yescharacteristic
194C2×C83#53911yescharacteristic
195C2×C83#54611yescharacteristic
196C2×C83#54711yescharacteristic
197C2×C83#55111yescharacteristic
198C2×C83#55211yescharacteristic
199D83—24no—
200D83—42no—
201#2419—831nomaximal, Sylow
202C83#241911yescharacteristic, complemented, Sylow
203C2×#123—831no—
204C2×C4×D4—831no—
205#400—831no—
206#398D8311yescharacteristic, complemented
207#397—831no—
208#397—831no—
209#387—831no—
210#387—831no—
211#386—831no—
212#385—831no—
213#385—831no—
214#381—831no—
215#381—831no—
216#380—831no—
217C2×#95—831no—
218C22×D4—831no—
219C23×C4—831no—
220C23×C4—831no—
221C23×C4—831no—
222C23×C4—831no—
223#123—1664no—
224C4×D4—1664no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4—831no—
227C2×C4⋊C4—831no—
228C2×C4⋊C4C2×D8311yescharacteristic
229C2×C4⋊C4C2×D8311yescharacteristic
230C2×C4⋊C4C2×D8311yescharacteristic
231C2×C4⋊C4C2×D8311yescharacteristic
232C2×C22⋊C4—831no—
233C2×C22⋊C4—831no—
234C2×C22⋊C4—831no—
235C2×C22⋊C4—831no—
236C2×C22⋊C4—831no—
237C2×C22⋊C4—831no—
238C2×C22⋊C4—831no—
239C2×C22⋊C4—831no—
240C2×C42—831no—
241#95—831no—
242#95—831no—
243#95—831no—
244#95—831no—
245#95—831no—
246#95—831no—
247#95—831no—
248#95—831no—
249#95—831no—
250#95—831no—
251#95—831no—
252#95C2×D8311yescharacteristic
253#95C2×D8311yescharacteristic
254#95C2×D8311yescharacteristic
255C24—831no—
256C24—831no—
257C2×D4—834no—
258C2×D4—1662no—
259C2×D4—1662no—
260C22×C4—3321no—
261C22×C4—3321no—
262C22×C4—3321no—
263C22×C4—831no—
264C22×C4—831no—
265C22×C4—831no—
266C22×C4—831no—
267C22×C4—831no—
268C22×C4—831no—
269C22×C4—1662no—
270C22×C4—1662no—
271C22×C4—1662no—
272C22×C4—1662no—
273C22×C4—1662no—
274C22×C4—1662no—
275C22×C4—1662no—
276C22×C4—1662no—
277C22×C4—1662no—
278C22×C4C22×D8311yescharacteristic
279C22×C4C22×D8311yescharacteristic
280C22×C4C22×D8311yescharacteristic
281C22×C4C22×D8311yescharacteristic
282C22×C4C22×D8311yescharacteristic
283C22×C4C22×D8311yescharacteristic
284C22×C4C22×D8311yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—1662no—
290C4⋊C4—1662no—
291C4⋊C4—1662no—
292C22⋊C4—3321no—
293C22⋊C4—1662no—
294C22⋊C4—1662no—
295C22⋊C4—1662no—
296C22⋊C4—1662no—
297C22⋊C4—1662no—
298C22⋊C4—1662no—
299C22⋊C4—1662no—
300C42—1662no—
301C23—832no—
302C23—832no—
303C23—832no—
304C23—832no—
305C23—1661no—
306C23—1661no—
307C23—1661no—
308C23—1661no—
309C23—1661no—
310C23—1661no—
311C23—1661no—
312C23—1661no—
313C23—1661no—
314C23—1661no—
315C23C23×D8311yescharacteristic
316D4—1668no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—3324no—
337C2×C4—3324no—
338C2×C4—3324no—
339C2×C4—3324no—
340C2×C4—832no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4—1661no—
354C2×C4—1661no—
355C2×C4—1661no—
356C2×C4—1661no—
357C2×C4—1661no—
358C2×C4D4×D8312yes—
359C2×C4#1305059112yes—
360C22—3321no—
361C22—3321no—
362C22—3321no—
363C22—3321no—
364C22—834no—
365C22—1662no—
366C22—1662no—
367C22—1662no—
368C22—1662no—
369C22—1662no—
370C22—1662no—
371C22—1662no—
372C22—1662no—
373C22—1662no—
374C22C2×D4×D8311yescharacteristic
375C22#41×D8311yescharacteristic
376C22#41×D8311yescharacteristic
377C22#41×D8311yescharacteristic
378C22C2×#1305059111yescharacteristic
379C22#1458183711yescharacteristic
380C22#1458184011yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—3321no—
389C4—3321no—
390C4—3321no—
391C4—3321no—
392C4—3321no—
393C4—1662no—
394C2—3322no—
395C2—1664no—
396C2#123×D8311yescharacteristic
397C2#1912823611yescharacteristic
398C2#1912829711yescharacteristic
399C2#1912834811yescharacteristic
400C2#1912835411yescharacteristic
401C2#1912837811yescharacteristic
402C2#1912838311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC22×D332C22
hypercenter#398D83
hypocenterC83#2419
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #19127639C2semidirect
4. #19127628C2semidirect
5. #19127627C2semidirect
6. #19127586C2semidirect
7. #19127582C2semidirect
9. #19127572C2semidirect
10. #19127572C2semidirect
11. #19127563C2semidirect
12. #19127563C2semidirect
13. #19127558C2semidirect
14. #95×D83C2semidirect
15. #398×C83C2semidirect
202. C83#2419semidirect
206. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817772
C2×#145817542
#191276392
#191276282
#191276272
#191275862
#191275822
#191275792
#191275722
#191275632
#191275582
#95×D832
#398×C832
#241983