Finite Group Databasebeta

← #26112309

Subgroups — #26112309

Order 12032.

624 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251952C211yesmaximal, characteristic, complemented
2#20249138C211yesmaximal, characteristic
3#20242055C212yesmaximal, complemented
4#20241247C212yesmaximal, complemented
5C47×#1458C211yesmaximal, characteristic, complemented
6C22×#10625329—21no—
7C2×#13094021C2211yescharacteristic
8C2×#13093998—21no—
9#15689150—24no—
10#15689147—21no—
11#15689147C2212yes—
12#15689040—21no—
13#15689040C412yes—
14#15689032—21no—
15#15689032C412yes—
16#15689025—21no—
17#15688989C2211yescharacteristic
18#15688866—24no—
19C2×#117×C47C2211yescharacteristic
20C47×#431C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C22×D188—21no—
24C22×D188—21no—
25C22×D188C2×C412yes—
26C22×C22⋊D47—22no—
27C2×#10625329—22no—
28C2×#10625329—22no—
29C2×#10625329—42no—
30C2×#10625329—42no—
31C22×C47⋊C8—22no—
32#13094021—22no—
33#13094021—22no—
34#13093998—42no—
35#13093998—42no—
36C2×C4⋊Dic47—21no—
37C2×(C2×D47)⋊C4—22no—
38C2×(C2×D47)⋊C4—22no—
39C2×(C2×D47)⋊C4—22no—
40C4⋊(C47⋊C8)—24no—
41C4⋊(C47⋊C8)—22no—
42C4⋊(C47⋊C8)D414yescomplemented
43C47⋊#95—21no—
44C47⋊#95C2×C412yes—
45C23×C4×C47D411yescharacteristic
46#117×C47—22no—
47#117×C47—22no—
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47C2×C411yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47C2311yescharacteristic
54C4⋊C8×C47—22no—
55C22⋊C8×C47—22no—
56C4×C8×C47—22no—
57C24×C47C22⋊C411yescharacteristic
58C23×D47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47D812yes—
67C22×C4×C47SD1612yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C2×C22⋊D47—44no—
72C2×C22⋊D47—82no—
73C2×C22⋊D47—82no—
74C22×Dic47—22no—
75C2×D188—24no—
76C2×D188—41no—
77C2×D188—42no—
78C2×D188—41no—
79C2×D188—41no—
80C2×D188—41no—
81C2×D188—41no—
82C2×D188—41no—
83C2×D188—82no—
84C2×D188—82no—
85C2×C8×C47—42no—
86C4⋊C4×C47—22no—
87C4⋊C4×C47—22no—
88C22⋊C4×C47—22no—
89C22⋊C4×C47—22no—
90C42×C47—21no—
91C42×C47—21no—
92C42×C47C2×D412yes—
93C42×C47#4112yes—
94(C2×D47)⋊C4—44no—
95(C2×D47)⋊C4—82no—
96(C2×D47)⋊C4—82no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99#10625329—44no—
100#10625329—42no—
101#10625329—42no—
102C2×C47⋊C8—24no—
103C2×C47⋊C8—44no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47#10212yes—
109C23×C47C2×C22⋊C411yescharacteristic
110C22×D47—42no—
111C22×D47—42no—
112C22×D47—42no—
113C22×D47—42no—
114C22×D47—42no—
115C22×D47—42no—
116C22×D47—42no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47#10212yes—
130C2×C4×C47C4×D412yes—
131C2×C4×C47#12011yescharacteristic
132C2×C4×C47#12112yes—
133C2×C4×C47#12311yescharacteristic
134C2×C4×C47C2×D811yescharacteristic
135C2×C4×C47C2×SD1611yescharacteristic
136C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C47#13711yescharacteristic
138C22⋊D47—88no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—42no—
142D188—84no—
143D188—84no—
144D188—81no—
145D188—81no—
146D188—81no—
147D188—81no—
148C8×C47—82no—
149C47⋊C8—44no—
150#9197—471nomaximal, Sylow
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—22no—
154C22×C47—42no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—81no—
158C22×C47#38611yescharacteristic
159C22×C47C2×#10211yescharacteristic
160C22×C47#41711yescharacteristic
161C2×D47—82no—
162C2×D47—82no—
163C2×D47—82no—
164C2×D47—82no—
165C2×D47—82no—
166C2×D47—82no—
167C2×D47—82no—
168C4×C47—22no—
169C4×C47—22no—
170C4×C47—22no—
171C4×C47—41no—
172C4×C47—41no—
173C4×C47#44711yescharacteristic
174C4×C47#44811yescharacteristic
175C4×C47#48011yescharacteristic
176C4×C47#48111yescharacteristic
177Dic47—82no—
178#2905—471no—
179#2619—471no—
180#1545—472no—
181#1458D4711yescharacteristic, complemented
182#1286—472no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—42no—
186C2×C47#187411yescharacteristic
187C2×C47#191011yescharacteristic
188C2×C47#191411yescharacteristic
189D47—162no—
190C22×M16—941no—
191C2×#127—471no—
192C2×#121—941no—
193C2×#117C2×D4711yescharacteristic
194#431C2×D4712yes—
195#424—944no—
196#423—472no—
197#423—941no—
198#394—472no—
199#394—941no—
200#392—472no—
201#392—941no—
202#390—941no—
203#382—471no—
204#331—944no—
205#324—21no—
206#323—21no—
207C47#919711yescharacteristic, complemented, Sylow
208C22×D4—472no—
209C22×D4—942no—
210C22×D4—941no—
211C22×D4—941no—
212C23×C4D18811yescharacteristic
213C2×M16—942no—
214C2×M16—942no—
215C2×M16—1882no—
216C2×M16—1882no—
217C22×C8—942no—
218#127—942no—
219#127—942no—
220#121—1882no—
221#121—1882no—
222#117—22no—
223#117—22no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4C4×D4711yescharacteristic
226C2×C4⋊C4C22⋊D4711yescharacteristic
227C2×C22⋊C4—942no—
228C2×C22⋊C4—942no—
229C2×C22⋊C4—942no—
230C2×C22⋊C4C4×D4711yescharacteristic
231C2×C22⋊C4C22⋊D4711yescharacteristic
232C2×C42D18811yescharacteristic
233C2×C42C22×D4711yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—474no—
236C4⋊C8—944no—
237C4⋊C8—942no—
238C22⋊C8—22no—
239C4×C8—22no—
240#95—472no—
241#95—941no—
242C24—942no—
243C24(C2×D47)⋊C411yescharacteristic
244C2×D4—944no—
245C2×D4—3762no—
246C2×D4—3762no—
247C2×D4—3762no—
248C2×D4—3762no—
249C2×D4—1884no—
250C2×D4—1881no—
251C2×D4—1882no—
252C2×D4—1881no—
253C2×D4—1881no—
254C2×D4—1881no—
255C2×D4—1881no—
256C2×D4—1881no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—942no—
262C22×C4#1062532512yes—
263C22×C4D37612yes—
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4(C2×D47)⋊C411yescharacteristic
266C22×C4(C2×D47)⋊C411yescharacteristic
267C22×C4C2×C4×D4711yescharacteristic
268C22×C4C2×D18811yescharacteristic
269C22×C4C2×C22⋊D4711yescharacteristic
270M16—1884no—
271M16—1882no—
272M16—1882no—
273C2×C8—42no—
274C2×C8—944no—
275C2×C8—1884no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—1881no—
279C4⋊C4—1881no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—3762no—
283C22⋊C4—3762no—
284C22⋊C4—1884no—
285C42—21no—
286C42—21no—
287C42D4×D4712yes—
288C42#1062535212yes—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—1882no—
294C23—1882no—
295C23—1882no—
296C23—1882no—
297C23—1882no—
298C23—1882no—
299C23—1882no—
300C23#1309394612yes—
301C23C2×(C2×D47)⋊C411yescharacteristic
302D4—3764no—
303D4—3764no—
304D4—3768no—
305D4—3761no—
306D4—3761no—
307D4—3761no—
308D4—3761no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—1882no—
322C2×C4—1882no—
323C2×C4—1882no—
324C2×C4#1309394612yes—
325C2×C4(C22⋊D47)⋊C412yes—
326C2×C4#1309399411yescharacteristic
327C2×C4#1309399612yes—
328C2×C4#1309400811yescharacteristic
329C2×C4C2×D37611yescharacteristic
330C2×C4C2×#1062532511yescharacteristic
331C2×C4#1309405211yescharacteristic
332C2×C4#1309405711yescharacteristic
333C8—82no—
334C8—1884no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—3762no—
343C22—3762no—
344C22—3762no—
345C22—3762no—
346C22—3762no—
347C22—3762no—
348C22—3762no—
349C22#1568900011yescharacteristic
350C22C2×#1309394611yescharacteristic
351C22#1568912511yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—3762no—
358C4#1568925911yescharacteristic
359C4#1568926411yescharacteristic
360C4#1568943711yescharacteristic
361C4#1568944211yescharacteristic
362C2—21no—
363C2—21no—
364C2—42no—
365C2—7522no—
366C2#2024370311yescharacteristic
367C2#2024391311yescharacteristic
368C2#2024393311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#386
prime radical#20249138C2
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#9197
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
1. #20251952C2semidirect
3. #20242055C2semidirect
4. #20241247C2semidirect
5. C47×#1458C2semidirect
42. C4⋊(C47⋊C8)D4semidirect
181. #1458D47semidirect
207. C47#9197semidirect

Maximal subgroups

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

subgroupindex
#202519522
#202491382
#202420552
#202412472
C47×#14582
#919747