Finite Group Databasebeta

← #26112299

Subgroups — #26112299

Order 12032.

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

normal subgroups only

This group has 524 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
2#20249154C211yesmaximal, characteristic
3#20242059C212yesmaximal, complemented
4#20241247C212yesmaximal, complemented
5C47×#1458C211yesmaximal, characteristic
6C22×#10625329—21no—
7C2×#13094024C2211yescharacteristic
8C2×#13094003—21no—
9#15689150—24no—
10#15689147—21no—
11#15689147C2212yes—
12#15689053—21no—
13#15689053C412yes—
14#15689048—21no—
15#15689048C412yes—
16#15689029—21no—
17#15688989C2211yescharacteristic
18#15688868—24no—
19C2×#117×C47C2211yescharacteristic
20C47×#431C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C22×Dic94—21no—
24C2×#10625329—22no—
25C2×#10625329—22no—
26C2×#10625329—42no—
27C2×#10625329—42no—
28C22×C47⋊C8—22no—
29#13094024—22no—
30#13094024—22no—
31#13094003—42no—
32#13094003—42no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2×C412yes—
36C2×Dic47⋊C4—22no—
37C2×C22⋊Dic47—22no—
38C4⋊(C47⋊C8)—24no—
39C4⋊(C47⋊C8)—22no—
40C4⋊(C47⋊C8)D414yes—
41C47⋊#95—22no—
42C47⋊#95—22no—
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
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47D812yes—
66C22×C4×C47SD1612yes—
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×Dic47—22no—
71C22×Dic47—22no—
72C2×Dic94—41no—
73C2×Dic94—41no—
74C2×Dic94—41no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×C8×C47—42no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—22no—
81C22⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47C2×D412yes—
86C42×C47#4112yes—
87C22⋊Dic47—44no—
88C4⋊Dic47—24no—
89C4⋊Dic47—42no—
90C4⋊Dic47—41no—
91C4⋊Dic47—41no—
92Dic47⋊C4—44no—
93#10625329—44no—
94#10625329—42no—
95#10625329—42no—
96C2×C47⋊C8—24no—
97C2×C47⋊C8—44no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47#10212yes—
103C23×C47C2×C22⋊C411yescharacteristic
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47#10212yes—
117C2×C4×C47C4×D412yes—
118C2×C4×C47#12011yescharacteristic
119C2×C4×C47#12112yes—
120C2×C4×C47#12311yescharacteristic
121C2×C4×C47C2×D811yescharacteristic
122C2×C4×C47C2×SD1611yescharacteristic
123C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C47#13711yescharacteristic
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131Dic94—81no—
132Dic94—81no—
133Dic94—81no—
134Dic94—81no—
135C8×C47—82no—
136C47⋊C8—44no—
137#9195—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—22no—
141C22×C47—42no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—81no—
145C22×C47#38611yescharacteristic
146C22×C47C2×#10211yescharacteristic
147C22×C47#41711yescharacteristic
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47#44711yescharacteristic
154C4×C47#44811yescharacteristic
155C4×C47#48011yescharacteristic
156C4×C47#48111yescharacteristic
157Dic47—82no—
158Dic47—82no—
159#2905—471no—
160#2621—471no—
161#1546—472no—
162#1458D4711yescharacteristic
163#1286—472no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—42no—
167C2×C47#187411yescharacteristic
168C2×C47#190911yescharacteristic
169C2×C47#191511yescharacteristic
170C22×M16—941no—
171C2×C4⋊Q8—471no—
172C2×C22⋊Q8—941no—
173C2×#117C2×D4711yescharacteristic
174#431C2×D4712yes—
175#424—944no—
176#423—472no—
177#423—941no—
178#398—472no—
179#398—941no—
180#397—472no—
181#397—941no—
182#391—941no—
183#382—471no—
184#332—944no—
185#324—21no—
186#323—21no—
187C47#919511yescharacteristic, complemented, Sylow
188C22×Q8—941no—
189C23×C4D18811yescharacteristic
190C2×M16—942no—
191C2×M16—942no—
192C2×M16—1882no—
193C2×M16—1882no—
194C22×C8—942no—
195C4⋊Q8—942no—
196C4⋊Q8—942no—
197C22⋊Q8—1882no—
198C22⋊Q8—1882no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—472no—
202C2×C4⋊C4—942no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4C4×D4711yescharacteristic
206C2×C4⋊C4C22⋊D4711yescharacteristic
207C2×C22⋊C4—942no—
208C2×C22⋊C4C4×D4711yescharacteristic
209C2×C22⋊C4C22⋊D4711yescharacteristic
210C2×C42D18811yescharacteristic
211C2×C42C22×D4711yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—474no—
214C4⋊C8—944no—
215C4⋊C8—942no—
216C22⋊C8—22no—
217C4×C8—22no—
218#95—472no—
219#95—942no—
220#95—942no—
221#95—941no—
222C24(C2×D47)⋊C411yescharacteristic
223C2×Q8—1881no—
224C2×Q8—1881no—
225C2×Q8—1881no—
226C2×Q8—1881no—
227C2×Q8—1881no—
228C2×Q8—1881no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—942no—
234C22×C4—942no—
235C22×C4#1062532512yes—
236C22×C4D37612yes—
237C22×C4(C2×D47)⋊C411yescharacteristic
238C22×C4(C2×D47)⋊C411yescharacteristic
239C22×C4(C2×D47)⋊C411yescharacteristic
240C22×C4C2×C4×D4711yescharacteristic
241C22×C4C2×D18811yescharacteristic
242C22×C4C2×C22⋊D4711yescharacteristic
243M16—1884no—
244M16—1882no—
245M16—1882no—
246C2×C8—42no—
247C2×C8—944no—
248C2×C8—1884no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—944no—
252C4⋊C4—1884no—
253C4⋊C4—1882no—
254C4⋊C4—1881no—
255C4⋊C4—1881no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—1884no—
259C42—21no—
260C42—21no—
261C42D4×D4712yes—
262C42#1062535212yes—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23#1309394612yes—
268C23C2×(C2×D47)⋊C411yescharacteristic
269Q8—3761no—
270Q8—3761no—
271Q8—3761no—
272Q8—3761no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—1882no—
286C2×C4—1882no—
287C2×C4—1882no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4—1882no—
291C2×C4#1309394612yes—
292C2×C4(C22⋊D47)⋊C412yes—
293C2×C4#1309399411yescharacteristic
294C2×C4#1309399612yes—
295C2×C4#1309400811yescharacteristic
296C2×C4C2×D37611yescharacteristic
297C2×C4C2×#1062532511yescharacteristic
298C2×C4#1309405211yescharacteristic
299C2×C4#1309405711yescharacteristic
300C8—82no—
301C8—1884no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—81no—
309C22#1568900011yescharacteristic
310C22C2×#1309394611yescharacteristic
311C22#1568912511yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—3762no—
318C4—3762no—
319C4#1568925911yescharacteristic
320C4#1568926411yescharacteristic
321C4#1568943711yescharacteristic
322C4#1568944211yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2024370311yescharacteristic
327C2#2024390811yescharacteristic
328C2#2024393811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#9195
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
3. #20242059C2semidirect
4. #20241247C2semidirect
187. C47#9195semidirect

Maximal subgroups

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

subgroupindex
#202519522
#202491542
#202420592
#202412472
C47×#14582
#919547