Finite Group Databasebeta

← #24914302

Subgroups — #24914302

Order 10624.

512 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581778C211yesmaximal, characteristic
2C2×#14581775C211yesmaximal, characteristic
3#19127630C211yesmaximal, characteristic, complemented
4#19127629C211yesmaximal, characteristic, complemented
5#19127627C211yesmaximal, characteristic, complemented
6#19127626C211yesmaximal, characteristic, complemented
7#19127621C211yesmaximal, characteristic, complemented
8#19127621C211yesmaximal, characteristic, complemented
9#19127584C211yesmaximal, characteristic, complemented
10#19127583C211yesmaximal, characteristic, complemented
11#19127577C211yesmaximal, characteristic
12#19127574C211yesmaximal, characteristic, complemented
13#19127560C211yesmaximal, characteristic, complemented
14#19127558C211yesmaximal, characteristic, complemented
15#385×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×C22⋊D83C2211yescharacteristic
18C22×C4×D83C2211yescharacteristic
19#14581778—24no—
20#14581775—24no—
21C2×C4⋊Dic83C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×(C2×D83)⋊C4C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×C4×Dic83C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C23×C4×C83C2211yescharacteristic
47C2×C4⋊C4×C83C2211yescharacteristic
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51#95×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C2×Dic166—22no—
55C2×Dic166—41no—
56C2×Dic166—41no—
57C2×C22⋊D83—41no—
58C2×C22⋊D83—41no—
59C2×C22⋊D83D414yes—
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C2×C4×D83—22no—
68C2×C4×D83—22no—
69C2×C4×D83—41no—
70C4⋊Dic83—22no—
71Dic83⋊C4—22no—
72Dic83⋊C4—22no—
73Dic83⋊C4—22no—
74Dic83⋊C4—41no—
75Dic83⋊C4—41no—
76C22⋊Dic83—22no—
77C22⋊Dic83—22no—
78C22⋊Dic83—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—41no—
84C4×Dic83—22no—
85C24×C83C2311yescharacteristic
86C22×C4×C83—22no—
87C22×C4×C83—41no—
88C22×C4×C83—41no—
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C4⋊C4×C83—41no—
96C4⋊C4×C83—41no—
97C22⋊C4×C83—22no—
98C22⋊C4×C83—22no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83C2×D412yes—
106C23×C83C2411yescharacteristic
107C22×D83—21no—
108C22×D83—21no—
109C22×D83—21no—
110C22×D83—21no—
111C22×D83—21no—
112C22×D83—21no—
113C22×D83C2×D412yes—
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—44no—
132C2×C4×C83#4112yes—
133C22⋊D83—44no—
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—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—21no—
151C2×Dic83—21no—
152C2×Dic83—21no—
153C2×Dic83C2×D412yes—
154C2×Dic83#4112yes—
155C4×D83—44no—
156Dic166—44no—
157C22×C83—22no—
158C22×C83—22no—
159C22×C83—22no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83C22×D411yescharacteristic
165C22×C83C2×#4111yescharacteristic
166C22×C83C2×#4111yescharacteristic
167C22×C83H(5,𝔽2)11yescharacteristic
168C22×C83H(5,𝔽2)11yescharacteristic
169C22×C832−1+411yescharacteristic
170C22×C832−1+411yescharacteristic
171C2×D83—22no—
172C2×D83—22no—
173C2×D83—22no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—41no—
177C2×D83—41no—
178C4×C83—22no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—41no—
183C4×C83—41no—
184Dic83—22no—
185Dic83—22no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189Dic83—41no—
190Dic83—41no—
191C2×C83—42no—
192C2×C83#53411yescharacteristic
193C2×C83#54211yescharacteristic
194C2×C83#54211yescharacteristic
195C2×C83#54711yescharacteristic
196C2×C83#54711yescharacteristic
197C2×C83#55111yescharacteristic
198C2×C83#56311yescharacteristic
199D83—42no—
200#2717—831nomaximal, Sylow
201C83#271711yescharacteristic, complemented, Sylow
202C2×#124—831no—
203C2×#123—831no—
204#397—831no—
205#397—831no—
206#397—831no—
207#397—831no—
208#395—831no—
209#395—831no—
210#387—831no—
211#387—831no—
212#386—831no—
213#385—831no—
214#385D8311yescharacteristic, complemented
215#380—831no—
216#380—831no—
217C22×Q8—831no—
218C22×D4—831no—
219C23×C4—831no—
220C23×C4C2×D8311yescharacteristic
221#124—1664no—
222#123—1664no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4—831no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4—831no—
227C2×C4⋊C4—831no—
228C2×C4⋊C4—831no—
229C2×C4⋊C4C2×D8311yescharacteristic
230C2×C4⋊C4C2×D8311yescharacteristic
231C2×C22⋊C4—831no—
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⋊C4C2×D8311yescharacteristic
240C2×C22⋊C4C2×D8311yescharacteristic
241C2×C42—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
254C24—831no—
255C24C22×D8311yescharacteristic
256C2×Q8—3321no—
257C2×Q8—3321no—
258C2×Q8—1662no—
259C2×D4—3321no—
260C2×D4—3321no—
261C2×D4—834no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—3321no—
266C22×C4—831no—
267C22×C4—831no—
268C22×C4—831no—
269C22×C4—831no—
270C22×C4—831no—
271C22×C4—831no—
272C22×C4—831no—
273C22×C4—1662no—
274C22×C4—1662no—
275C22×C4C22×D8311yescharacteristic
276C22×C4C22×D8311yescharacteristic
277C22×C4C22×D8311yescharacteristic
278C22×C4C22×D8311yescharacteristic
279C22×C4C22×D8311yescharacteristic
280C22×C4C22×D8311yescharacteristic
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—3321no—
284C4⋊C4—3321no—
285C4⋊C4—1662no—
286C4⋊C4—1662no—
287C4⋊C4—1662no—
288C4⋊C4—1662no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—3321no—
292C22⋊C4—1662no—
293C22⋊C4—1662no—
294C22⋊C4—1662no—
295C22⋊C4—1662no—
296C22⋊C4—1662no—
297C22⋊C4—1662no—
298C22⋊C4—1662no—
299C42—1662no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—832no—
307C23—1661no—
308C23—1661no—
309C23—1661no—
310C23—1661no—
311C23—1661no—
312C23—1661no—
313C23D4×D8312yes—
314C23C23×D8311yescharacteristic
315Q8—3324no—
316D4—3324no—
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—44no—
335C2×C4—3324no—
336C2×C4—832no—
337C2×C4—832no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
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#1305059012yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—1662no—
370C22—1662no—
371C22—1662no—
372C22C2×D4×D8311yescharacteristic
373C22#41×D8311yescharacteristic
374C22C2×#1305059011yescharacteristic
375C22#1458183711yescharacteristic
376C22#1458183711yescharacteristic
377C22#1458183911yescharacteristic
378C22#1458184011yescharacteristic
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—3321no—
386C4—3321no—
387C4—3321no—
388C4—3321no—
389C4—3321no—
390C4—1662no—
391C4—1662no—
392C2—42no—
393C2—3322no—
394C2#1912826111yescharacteristic
395C2#1912831611yescharacteristic
396C2#1912832111yescharacteristic
397C2#1912834911yescharacteristic
398C2#1912835211yescharacteristic
399C2#1912838011yescharacteristic
400C2#1912845211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#385×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#385D83
hypocenterC83#2717
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #19127630C2semidirect
4. #19127629C2semidirect
5. #19127627C2semidirect
6. #19127626C2semidirect
7. #19127621C2semidirect
8. #19127621C2semidirect
9. #19127584C2semidirect
10. #19127583C2semidirect
12. #19127574C2semidirect
13. #19127560C2semidirect
14. #19127558C2semidirect
15. #385×C83C2semidirect
201. C83#2717semidirect
214. #385D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817782
C2×#145817752
#191276302
#191276292
#191276272
#191276262
#191276212
#191275842
#191275832
#191275772
#191275742
#191275602
#191275582
#385×C832
#271783