Finite Group Databasebeta

← #24919640

Subgroups — #24919640

Order 10624.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128403C211yesmaximal, characteristic
2#19128364C211yesmaximal, characteristic, complemented
3#19128068C211yesmaximal, characteristic, complemented
4#19128045C211yesmaximal, characteristic, complemented
5#19128034C211yesmaximal, characteristic, complemented
6#19128011C211yesmaximal, characteristic, complemented
7#19127946C211yesmaximal, characteristic, complemented
8#19127904C211yesmaximal, characteristic, complemented
9#19127839C211yesmaximal, characteristic, complemented
10#19127838C211yesmaximal, characteristic, complemented
11#19127798C211yesmaximal, characteristic, complemented
12#19127772C211yesmaximal, characteristic, complemented
13#19127708C211yesmaximal, characteristic, complemented
14C2×#14581717C211yesmaximal, characteristic, complemented
15#465×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C2×Q8×D83C2211yescharacteristic
18C2×#13050578C2211yescharacteristic
19C2×#13050578C2211yescharacteristic
20C2×#13050573C2211yescharacteristic
21#14581791—21no—
22#14581790C2211yescharacteristic
23#14581785—21no—
24#14581783C2211yescharacteristic
25#14581776—21no—
26#14581772C2211yescharacteristic
27#14581770—21no—
28#14581769—21no—
29#14581764—21no—
30#14581763C2211yescharacteristic
31#14581757C2211yescharacteristic
32D4×Dic83—21no—
33D4×Dic83C2211yescharacteristic
34C4×C22⋊D83C2211yescharacteristic
35(C22⋊D83)⋊C4—21no—
36#14581750C2211yescharacteristic
37C2×C4⋊Dic83C2211yescharacteristic
38#14581725C2211yescharacteristic
39#14581725C2211yescharacteristic
40Dic83⋊C8C2211yescharacteristic
41#14581717C2211yescharacteristic
42#14581717C2211yescharacteristic
43#14581717C2211yescharacteristic
44#14581717C2211yescharacteristic
45#14581716C2211yescharacteristic
46#14581716C2211yescharacteristic
47#14581715C2211yescharacteristic
48#14581715C2211yescharacteristic
49C22⋊(C83⋊C8)C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51#14581702C2211yescharacteristic
52C2×SD16×C83C2211yescharacteristic
53C22×C8×C83C2211yescharacteristic
54C22⋊Q8×C83C2211yescharacteristic
55#121×C83C2211yescharacteristic
56#106×C83C2211yescharacteristic
57#103×C83C2211yescharacteristic
58#102×C83C2211yescharacteristic
59C2×Dic166—22no—
60C2×Dic166—21no—
61C2×Dic166—21no—
62C2×Dic166—41no—
63C2×Dic166C2311yescharacteristic
64C2×Dic166C2311yescharacteristic
65Q8×D83—22no—
66Q8×D83—41no—
67C2×C22⋊D83—21no—
68C2×C22⋊D83D411yescharacteristic
69C22×Dic83—21no—
70C22×Dic83—21no—
71C22×Dic83—21no—
72C2×C4×D83—21no—
73C2×C4×D83C2311yescharacteristic
74#13050578—22no—
75#13050578—41no—
76#13050573—41no—
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C4⋊Dic83—21no—
80C4⋊Dic83—21no—
81C4⋊Dic83—21no—
82C4⋊Dic83C2311yescharacteristic
83C4⋊Dic83C2311yescharacteristic
84Dic83⋊C4—21no—
85Dic83⋊C4—21no—
86Dic83⋊C4—21no—
87Dic83⋊C4—21no—
88Dic83⋊C4D411yescharacteristic
89C22⋊Dic83—21no—
90C22⋊Dic83—21no—
91C22⋊Dic83D411yescharacteristic
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4D411yescharacteristic
95C4×Dic83—21no—
96C4×Dic83—21no—
97C4×Dic83C2311yescharacteristic
98C2×Q8×C83C2311yescharacteristic
99C2×D4×C83—21no—
100C2×D4×C83C2311yescharacteristic
101C22×C4×C83C2311yescharacteristic
102SD16×C83—41no—
103C2×C8×C83—41no—
104C2×C8×C83C2311yescharacteristic
105C2×C8×C83C2311yescharacteristic
106C4⋊C4×C83—21no—
107C4⋊C4×C83C2311yescharacteristic
108C4⋊C4×C83C2311yescharacteristic
109C22⋊C4×C83—21no—
110C22⋊C4×C83—21no—
111C23×C83—21no—
112C23×C83C2×D411yescharacteristic
113C22×D83C2×D411yescharacteristic
114Q8×C83—21no—
115Q8×C83—21no—
116D4×C83—21no—
117D4×C83—21no—
118D4×C83—42no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83C2×D411yescharacteristic
125C2×C4×C83C2411yescharacteristic
126C22⋊D83—22no—
127C22⋊D83—81no—
128C2×Dic83—21no—
129C2×Dic83—21no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—42no—
133C2×Dic83—81no—
134C2×Dic83—81no—
135C2×Dic83C2×D411yescharacteristic
136C2×Dic83C2×D411yescharacteristic
137C2×Dic83C2×D411yescharacteristic
138C4×D83—21no—
139C4×D83—21no—
140C4×D83—42no—
141Dic166—21no—
142Dic166—21no—
143Dic166—21no—
144Dic166—42no—
145Dic166—42no—
146Dic166#4112yes—
147C8×C83—21no—
148C8×C83—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83#13712yes—
157C22×C83C22×D411yescharacteristic
158C2×D83—21no—
159C2×D83—21no—
160C2×D83—21no—
161C4×C83—21no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83C2×#4111yescharacteristic
166C4×C83H(5,𝔽2)11yescharacteristic
167Dic83—22no—
168Dic83—21no—
169Dic83—21no—
170Dic83—21no—
171Dic83—41no—
172Dic83—41no—
173Dic83—41no—
174C2×C83—22no—
175C2×C83—81no—
176C2×C83#54611yescharacteristic
177C2×C83C2×#13711yescharacteristic
178C2×C83#57711yescharacteristic
179D83—41no—
180#3320—831nomaximal, Sylow
181C83#332011yescharacteristic, complemented, Sylow
182#553—831no—
183D4×Q8—831no—
184#491—831no—
185#485—831no—
186#484—831no—
187#479—831no—
188#468—831no—
189#465D8311yescharacteristic, complemented
190#461—831no—
191#448—831no—
192#448—831no—
193#440—831no—
194#435—831no—
195#419—831no—
196C2×#103—831no—
197C22×Q8—831no—
198C22×Q8—831no—
199C2×SD16—831no—
200C2×SD16—831no—
201C2×SD16C2×D8311yescharacteristic
202C2×M16—831no—
203C22×C8C2×D8311yescharacteristic
204C4⋊Q8—831no—
205C4⋊Q8—1661no—
206#126—1661no—
207#125—831no—
208#123—1661no—
209C22⋊Q8—831no—
210C22⋊Q8—1661no—
211C22⋊Q8—1661no—
212C22⋊Q8C2×D8311yescharacteristic
213#121—831no—
214#121—1661no—
215#121C2×D8311yescharacteristic
216C4×Q8—831no—
217C4×D4—831no—
218C4×D4—831no—
219C4×D4—1661no—
220C4×D4—1661no—
221#117—831no—
222C2×C4⋊C4—831no—
223#107—831no—
224#107—831no—
225#106C2×D8311yescharacteristic
226C4⋊C8—831no—
227#103—831no—
228#103—831no—
229#103—831no—
230#103—831no—
231#103—831no—
232#103—831no—
233#103—831no—
234#103—831no—
235#103C2×D8311yescharacteristic
236#102C2×D8311yescharacteristic
237C22⋊C8—831no—
238C22⋊C8—831no—
239#97—831no—
240C2×Q8—3321no—
241C2×Q8—3321no—
242C2×Q8—831no—
243C2×Q8—831no—
244C2×Q8—1662no—
245C2×Q8—1661no—
246C2×Q8—1662no—
247C2×Q8—1661no—
248C2×Q8C22×D8311yescharacteristic
249C2×D4—21no—
250C2×D4—831no—
251C2×D4—1661no—
252C2×D4C22×D8311yescharacteristic
253C22×C4—831no—
254C22×C4—1661no—
255C22×C4—1661no—
256C22×C4—1661no—
257C22×C4—1661no—
258C22×C4C22×D8311yescharacteristic
259SD16—41no—
260SD16—3321no—
261SD16—1662no—
262M16—3321no—
263C2×C8—41no—
264C2×C8—831no—
265C2×C8—831no—
266C2×C8C22×D8311yescharacteristic
267C2×C8C22×D8311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—831no—
270C4⋊C4—831no—
271C4⋊C4—831no—
272C4⋊C4—1661no—
273C4⋊C4—1661no—
274C4⋊C4—1661no—
275C4⋊C4—1661no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C4⋊C4—1661no—
279C4⋊C4C22×D8311yescharacteristic
280C4⋊C4C22×D8311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—831no—
284C22⋊C4—831no—
285C22⋊C4—1661no—
286C22⋊C4—1661no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C42—831no—
290C42—1661no—
291C42—1661no—
292C23—21no—
293C23—831no—
294C23D4×D8311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—3322no—
298Q8—3322no—
299Q8—832no—
300Q8—1661no—
301Q8—1661no—
302Q8—1661no—
303D4—21no—
304D4—21no—
305D4—42no—
306D4—6641no—
307D4—1662no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—3322no—
314C2×C4—3322no—
315C2×C4—831no—
316C2×C4—831no—
317C2×C4—831no—
318C2×C4—6641no—
319C2×C4—6641no—
320C2×C4—1661no—
321C2×C4—1661no—
322C2×C4—1661no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4—1661no—
326C2×C4D4×D8311yescharacteristic
327C2×C4C23×D8311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1661no—
331C8—1661no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—1661no—
338C22—1661no—
339C22—1661no—
340C22#1458182112yes—
341C22C2×D4×D8311yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3321no—
347C4—3321no—
348C4—3321no—
349C4—1662no—
350C4—1661no—
351C4—1661no—
352C4—1661no—
353C4#41×D8311yescharacteristic
354C4#1458183711yescharacteristic
355C2—22no—
356C2—81no—
357C2—3321no—
358C2#1912834011yescharacteristic
359C2C2×#1458182111yescharacteristic
360C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#465×C83C2
socleC22×C83C22×D4
prime radical#19128403C2
hypercenter#465D83
hypocenterC83#3320
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
2. #19128364C2semidirect
3. #19128068C2semidirect
4. #19128045C2semidirect
5. #19128034C2semidirect
6. #19128011C2semidirect
7. #19127946C2semidirect
8. #19127904C2semidirect
9. #19127839C2semidirect
10. #19127838C2semidirect
11. #19127798C2semidirect
12. #19127772C2semidirect
13. #19127708C2semidirect
14. C2×#14581717C2semidirect
15. #465×C83C2semidirect
181. C83#3320semidirect
189. #465D83semidirect

Maximal subgroups

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

subgroupindex
#191284032
#191283642
#191280682
#191280452
#191280342
#191280112
#191279462
#191279042
#191278392
#191278382
#191277982
#191277722
#191277082
C2×#145817172
#465×C832
#332083