Finite Group Databasebeta

← #26370998

Subgroups — #26370998

Order 12032.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253927C211yesmaximal, characteristic, complemented
2#20245552C212yesmaximal, complemented
3#20245545C212yesmaximal, complemented
4#20245535C212yesmaximal, complemented
5#20245530C212yesmaximal, complemented
6#20245174C211yesmaximal, characteristic, complemented
7C2×#15688950C211yesmaximal, characteristic, complemented
8#20245147C211yesmaximal, characteristic, complemented
9#20245100C211yesmaximal, characteristic
10#20245095C211yesmaximal, characteristic
11C47×#2951C211yesmaximal, characteristic
12#15689751—21no—
13C2×#13093964C2211yescharacteristic
14C22×C47⋊C16C2211yescharacteristic
15#15689532—22no—
16#15689528C2212yes—
17#15689525C2212yes—
18#15689165—21no—
19#15689161C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21#15688950C2212yes—
22#15688950C2212yes—
23#15688950C2212yes—
24#15688948C2212yes—
25C4⋊(C47⋊C16)C2212yes—
26C4⋊(C47⋊C16)C2212yes—
27#15688927C2211yescharacteristic
28#15688927C2212yes—
29#15688927C2211yescharacteristic
30#15688924C2211yescharacteristic
31#15688924C2212yes—
32#15688924C2211yescharacteristic
33C2×#135×C47C2211yescharacteristic
34C47×#520—21no—
35C47×#440—22no—
36C4×Q16×C47C2212yes—
37C4×D8×C47C2212yes—
38C2×#107×C47C2211yescharacteristic
39C47×#417—21no—
40C47×#405C2211yescharacteristic
41#13094024—22no—
42#13094023—41no—
43#13094016—22no—
44#13094015—41no—
45#13094001—42no—
46#13093983—21no—
47C2×C4⋊Dic47—21no—
48#13093964—22no—
49C2×C47⋊C16—22no—
50C2×C47⋊C16C2311yescharacteristic
51C2×C47⋊C16C2311yescharacteristic
52C2×C47⋊C16C2312yes—
53#13093958—22no—
54#13093958—21no—
55#13093958C2311yescharacteristic
56#13093958C2312yes—
57#13093958C2311yescharacteristic
58#13093956—22no—
59#13093956—21no—
60C2×#41×C47—21no—
61#135×C47—22no—
62#135×C47Q814yescomplemented
63C2×Q16×C47C2311yescharacteristic
64C2×SD16×C47—21no—
65C2×D8×C47C2311yescharacteristic
66C2×M16×C47D411yescharacteristic
67C22×C8×C47C2311yescharacteristic
68C4×Q8×C47—22no—
69C4×D4×C47—22no—
70C4×D4×C47—42no—
71#117×C47—41no—
72#117×C47D411yescharacteristic
73C2×C4⋊C4×C47—21no—
74C2×C4⋊C4×C47—41no—
75#107×C47—21no—
76#107×C47C2312yes—
77#103×C47—22no—
78#102×C47—22no—
79#97×C47D412yes—
80C4×C8×C47C2312yes—
81#41×C47—24no—
82#41×C47—42no—
83C2×Q8×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47—42no—
89C22×C4×C47C2×D411yescharacteristic
90C22×Dic47—41no—
91C2×Dic94—42no—
92Q16×C47—21no—
93Q16×C47C2×Q812yes—
94SD16×C47—24no—
95D8×C47—21no—
96D8×C47C2×Q812yes—
97M16×C47—22no—
98C2×C8×C47C2×D412yes—
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×Q812yes—
101C2×C8×C47#4112yes—
102C2×C8×C47C2411yescharacteristic
103C4⋊C4×C47—22no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—42no—
106C4⋊C4×C47D811yescharacteristic
107C4⋊C4×C47D811yescharacteristic
108C22⋊C4×C47—42no—
109C22⋊C4×C47D812yes—
110C42×C47—42no—
111C42×C47C2×D412yes—
112C22⋊Dic47—41no—
113C4⋊Dic47—21no—
114C4⋊Dic47—22no—
115C4⋊Dic47—21no—
116C4⋊Dic47—41no—
117C4⋊Dic47—42no—
118Dic47⋊C4—42no—
119Dic47⋊C4—42no—
120C4×Dic47—41no—
121C47⋊C16—21no—
122C47⋊C16—21no—
123C47⋊C16—22no—
124C23×C47—41no—
125C23×C47C2×D811yescharacteristic
126Q8×C47—22no—
127Q8×C47—41no—
128D4×C47—22no—
129D4×C47—44no—
130D4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—44no—
133C2×C4×C47—42no—
134C2×C4×C47—162no—
135C2×C4×C47—161no—
136C2×C4×C47C22⋊Q812yes—
137C2×C4×C47C2×D812yes—
138C2×C4×C47C2×D812yes—
139C2×C4×C47C2×D811yescharacteristic
140C2×C4×C47#13712yes—
141C2×C4×C47C22×D411yescharacteristic
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—42no—
145C2×Dic47—161no—
146Dic94—162no—
147C8×C47—22no—
148C8×C47C22⋊Q812yes—
149C8×C47C22×Q811yescharacteristic
150C8×C47C2×#4111yescharacteristic
151#33634—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—42no—
155C22×C47—81no—
156C22×C47—81no—
157C22×C47#47412yes—
158C22×C47C22×D811yescharacteristic
159C4×C47—22no—
160C4×C47—22no—
161C4×C47—42no—
162C4×C47—82no—
163C4×C47#47412yes—
164C4×C47C2×C22⋊Q811yescharacteristic
165C4×C47C2×#13711yescharacteristic
166Dic47—81no—
167Dic47—82no—
168Dic47—81no—
169#3164—471no—
170#2951D4711yescharacteristic
171#2229—472no—
172#2228—472no—
173#2226—472no—
174#2225—472no—
175#2159—471no—
176C2×#367—471no—
177#2151—471no—
178#2139—471no—
179#2138—471no—
180C2×C47—22no—
181C2×C47—82no—
182C2×C47C2×#47411yescharacteristic
183C2×C47#341611yescharacteristic
184C2×C47#341611yescharacteristic
185C2×#135C2×D4711yescharacteristic
186#544—941no—
187#520—21no—
188C2×M32—471no—
189C22×C16—471no—
190#501—942no—
191#500—472no—
192#499—472no—
193#440—22no—
194C4×Q16C2×D4712yes—
195C4×D8C2×D4712yes—
196#428—941no—
197#427—471no—
198C2×#107—471no—
199C2×#107C2×D4711yescharacteristic
200#417—21no—
201#405C2×D4711yescharacteristic
202#367—472no—
203#367—472no—
204#367—472no—
205#366—472no—
206C4⋊C16—472no—
207C4⋊C16—472no—
208#358—471no—
209#358—471no—
210#358—472no—
211#357—471no—
212#357—471no—
213#357—472no—
214C47#3363411yescharacteristic, complemented, Sylow
215C2×#41—21no—
216#135—22no—
217#135Dic9414yescomplemented
218C2×Q16C22×D4711yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D4711yescharacteristic
221C2×M16C22⋊D4711yescharacteristic
222C22×C8C22×D4711yescharacteristic
223C4⋊Q8—942no—
224C4⋊Q8—1881no—
225#125—942no—
226#125—1881no—
227C22⋊Q8—1882no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—42no—
231#117—41no—
232#117—941no—
233#117C22⋊D4711yescharacteristic
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—941no—
237M32—942no—
238C2×C16—472no—
239C2×C16—471no—
240C2×C16—471no—
241C2×C16—942no—
242#107—21no—
243#107—471no—
244#107—472no—
245#107—471no—
246#107—941no—
247#107—942no—
248#107C22×D4712yes—
249#106—941no—
250#106—942no—
251#103—22no—
252#102—22no—
253#97C22⋊D4712yes—
254C4×C8C22×D4712yes—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—1882no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—1881no—
265C22×C4C2×C22⋊D4711yescharacteristic
266Q16—21no—
267Q16C2×Dic9412yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic9412yes—
271M16—22no—
272C2×C8C2×Dic9412yes—
273C2×C8#1062535012yes—
274C2×C8C2×C22⋊D4712yes—
275C2×C8C2×C22⋊D4711yescharacteristic
276C2×C8C23×D4711yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—941no—
281C4⋊C4—942no—
282C4⋊C4—941no—
283C4⋊C4—1882no—
284C4⋊C4—1882no—
285C4⋊C4—1881no—
286C4⋊C4—1882no—
287C4⋊C4#1062533411yescharacteristic
288C4⋊C4#1062533411yescharacteristic
289C22⋊C4—42no—
290C22⋊C4—1881no—
291C22⋊C4#1062533412yes—
292C42—42no—
293C42—1881no—
294C42C2×C22⋊D4712yes—
295C16—941no—
296C16—941no—
297C16—942no—
298C23—41no—
299C23C2×#1062533411yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—7522no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—162no—
310C2×C4—161no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1882no—
314C2×C4—7521no—
315C2×C4#1309400312yes—
316C2×C4C2×#1062533412yes—
317C2×C4C2×#1062533412yes—
318C2×C4C2×#1062533411yescharacteristic
319C2×C4#1309405412yes—
320C2×C4C22×C22⋊D4711yescharacteristic
321C8—22no—
322C8#1309400312yes—
323C8C22×Dic9411yescharacteristic
324C8C2×#1062535011yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1568940312yes—
331C22C22×#1062533411yescharacteristic
332C4—22no—
333C4—22no—
334C4—42no—
335C4—82no—
336C4—3762no—
337C4—3761no—
338C4—3761no—
339C4#1568940312yes—
340C4C2×#1309400311yescharacteristic
341C4C2×#1309405411yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1568940311yescharacteristic
345C2#2025653211yescharacteristic
346C2#2025653511yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2951C2
socleC22×C47C22×D8
prime radicalC2×#135×C47C22
hypercenter#2951D47
hypocenterC47#33634
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #20253927C2semidirect
2. #20245552C2semidirect
3. #20245545C2semidirect
4. #20245535C2semidirect
5. #20245530C2semidirect
6. #20245174C2semidirect
7. C2×#15688950C2semidirect
8. #20245147C2semidirect
62. #135×C47Q8semidirect
214. C47#33634semidirect
217. #135Dic94semidirect

Maximal subgroups

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

subgroupindex
#202539272
#202455522
#202455452
#202455352
#202455302
#202451742
C2×#156889502
#202451472
#202451002
#202450952
C47×#29512
#3363447