Finite Group Databasebeta

← #26114127

Subgroups — #26114127

Order 12032.

626 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 131 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#20248285C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic, complemented
3#20241814C212yesmaximal, complemented
4#20241247C212yesmaximal, complemented
5C47×#2972C211yesmaximal, characteristic, complemented
6C2×#13093995—21no—
7C2×C4×D188C2211yescharacteristic
8#15689147C2212yes—
9C2×#13093941—21no—
10#15689048C412yes—
11#15689032C412yes—
12#15689002—22no—
13#15688989C2211yescharacteristic
14#15688980—21no—
15#15688892—21no—
16#15688852—24no—
17C2×C4×D4×C47C2211yescharacteristic
18C8×D4×C47—24no—
19C47×#431C2212yes—
20C2×C22⋊C8×C47—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—22no—
23C22×C4×D47—21no—
24C2×#10625329—22no—
25#13093995—42no—
26#13093995—42no—
27C4×D188—22no—
28C4×D188—22no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C22⋊Dic47—22no—
32C2×(C2×D47)⋊C4—21no—
33C2×(C2×D47)⋊C4—22no—
34C4⋊(C47⋊C8)—22no—
35C4⋊(C47⋊C8)—22no—
36#13093941—42no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95C2×C412yes—
40C22×D4×C47D411yescharacteristic
41C23×C4×C47D412yes—
42C22×C8×C47—24no—
43C4×D4×C47—22no—
44C4×D4×C47—22no—
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47C2×C412yes—
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47—22no—
49C22⋊C8×C47—24no—
50C22⋊C8×C47—22no—
51C4×C8×C47D414yescomplemented
52C24×C47C22⋊C412yes—
53C23×D47—21no—
54C2×D4×C47—22no—
55C2×D4×C47—22no—
56C2×D4×C47D812yes—
57C2×D4×C47SD1612yes—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—82no—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C2×C22⋊D47—44no—
66C2×C22⋊D47—44no—
67C2×C22⋊D47—82no—
68C22×Dic47—22no—
69C22×Dic47—21no—
70C2×D188—22no—
71C2×D188—42no—
72C2×C4×D47—42no—
73C2×C4×D47—41no—
74C2×C4×D47—41no—
75C2×C4×D47—42no—
76C2×C8×C47—24no—
77C2×C8×C47—84no—
78C4⋊C4×C47—22no—
79C22⋊C4×C47—24no—
80C42×C47C2×D412yes—
81C42×C47#4112yes—
82C22⋊Dic47—44no—
83(C2×D47)⋊C4—41no—
84(C2×D47)⋊C4—41no—
85(C2×D47)⋊C4—82no—
86C4⋊Dic47—22no—
87C4⋊Dic47—41no—
88C4⋊Dic47—41no—
89#10625329—42no—
90#10625329—42no—
91C2×C47⋊C8—42no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—42no—
95C23×C47—42no—
96C23×C47#10214yes—
97C23×C47C2×C22⋊C411yescharacteristic
98C22×D47—22no—
99C22×D47—41no—
100C22×D47—41no—
101C22×D47—41no—
102C22×D47—41no—
103C22×D47—41no—
104C22×D47—41no—
105D4×C47—24no—
106D4×C47—24no—
107C2×C4×C47—22no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—44no—
111C2×C4×C47—42no—
112C2×C4×C47—82no—
113C2×C4×C47C4×D412yes—
114C2×C4×C47#12011yescharacteristic
115C2×C4×C47#12112yes—
116C2×C4×C47#12311yescharacteristic
117C2×C4×C47C2×D811yescharacteristic
118C2×C4×C47C2×SD1611yescharacteristic
119C2×C4×C47#13512yes—
120C22⋊D47—88no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—41no—
124C2×Dic47—42no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127D188—24no—
128D188—42no—
129C4×D47—82no—
130C4×D47—82no—
131C8×C47—44no—
132C47⋊C8—82no—
133#9485—471nomaximal, Sylow
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—24no—
137C22×C47—44no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47—82no—
141C22×C47#38611yescharacteristic
142C22×C47C2×#10211yescharacteristic
143C22×C47#41611yescharacteristic
144C2×D47—42no—
145C2×D47—42no—
146C2×D47—42no—
147C2×D47—81no—
148C2×D47—81no—
149C2×D47—81no—
150C2×D47—81no—
151C4×C47—22no—
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—42no—
155C4×C47#46512yes—
156C4×C47#46612yes—
157Dic47—82no—
158Dic47—81no—
159#2972D4711yescharacteristic, complemented
160#2535—471no—
161#1800—471no—
162#1486—472no—
163#1286—472no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—44no—
167C2×C47#189111yescharacteristic
168C2×C47#189311yescharacteristic
169C2×C47#191111yescharacteristic
170D47—82no—
171C2×#121—941no—
172C2×C4×D4—471no—
173C2×C4×D4C2×D4711yescharacteristic
174C8×D4—24no—
175#431C2×D4712yes—
176#423—472no—
177C2×#100—941no—
178C2×C22⋊C8—22no—
179#397—472no—
180#392—472no—
181#386—942no—
182#382—471no—
183#380—941no—
184#343—941no—
185D4⋊C8—944no—
186C47#948511yescharacteristic, complemented, Sylow
187C22×D4—471no—
188C22×D4—942no—
189C22×D4D18811yescharacteristic
190C23×C4—941no—
191C23×C4C22⋊D4712yes—
192C2×M16—942no—
193C22×C8—24no—
194#121—1882no—
195#121—1882no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—942no—
199C4×D4—942no—
200C2×C4⋊C4—471no—
201C2×C4⋊C4—941no—
202C2×C4⋊C4D18811yescharacteristic
203C2×C22⋊C4—942no—
204C2×C22⋊C4—942no—
205C2×C22⋊C4—941no—
206C2×C22⋊C4C4×D4712yes—
207C2×C42C22×D4711yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—942no—
210C4⋊C8—942no—
211#100—1882no—
212C22⋊C8—24no—
213C22⋊C8—22no—
214C4×C8C22⋊D4714yescomplemented
215#95—472no—
216#95—942no—
217#95—941no—
218C24—941no—
219C24(C2×D47)⋊C412yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—942no—
223C2×D4—3762no—
224C2×D4—1882no—
225C2×D4—1884no—
226C2×D4—1884no—
227C2×D4#1062532512yes—
228C2×D4D37612yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—942no—
233C22×C4—941no—
234C22×C4—1882no—
235C22×C4—1881no—
236C22×C4—1881no—
237C22×C4—1882no—
238C22×C4(C2×D47)⋊C412yes—
239C22×C4C2×C4×D4711yescharacteristic
240C22×C4C2×D18811yescharacteristic
241C22×C4C2×C22⋊D4711yescharacteristic
242M16—1882no—
243M16—1882no—
244C2×C8—24no—
245C2×C8—84no—
246C2×C8—1882no—
247C4⋊C4—22no—
248C4⋊C4—942no—
249C4⋊C4—1881no—
250C4⋊C4—1881no—
251C22⋊C4—24no—
252C22⋊C4—3762no—
253C22⋊C4—1884no—
254C22⋊C4—1881no—
255C22⋊C4—1881no—
256C42#1062535012yes—
257C42C2×C22⋊D4712yes—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—942no—
263C23—1881no—
264C23—1881no—
265C23—1881no—
266C23—1881no—
267C23—1881no—
268C23—1881no—
269C23#1309394614yes—
270C23C2×(C2×D47)⋊C411yescharacteristic
271D4—24no—
272D4—24no—
273D4—944no—
274D4—3768no—
275D4—1882no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—3762no—
283C2×C4—3762no—
284C2×C4—1882no—
285C2×C4—1882no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1882no—
289C2×C4—1881no—
290C2×C4C4×C22⋊D4712yes—
291C2×C4#1309399311yescharacteristic
292C2×C4#1309399812yes—
293C2×C4#1309400711yescharacteristic
294C2×C4C2×D37611yescharacteristic
295C2×C4C2×#1062532511yescharacteristic
296C2×C4#1309404612yes—
297C8—44no—
298C8—3762no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—3761no—
307C22—3761no—
308C22—3761no—
309C22—3761no—
310C22—1882no—
311C22—1882no—
312C22—1882no—
313C22#1568900211yescharacteristic
314C22C2×#1309394611yescharacteristic
315C22#1568912011yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—3762no—
321C4—3761no—
322C4#1568935212yes—
323C4#1568935912yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—3762no—
328C2#2024380611yescharacteristic
329C2#2024381611yescharacteristic
330C2#2024392011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2972C2
socleC22×C47#386
prime radical#20248285C2
cosocleC2×C42×C47C23
hypercenter#2972D47
hypocenterC47#9485
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20243380C2semidirect
3. #20241814C2semidirect
4. #20241247C2semidirect
5. C47×#2972C2semidirect
51. C4×C8×C47D4semidirect
159. #2972D47semidirect
186. C47#9485semidirect
214. C4×C8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202482852
#202433802
#202418142
#202412472
C47×#29722
#948547