Finite Group Databasebeta

← #26114050

Subgroups — #26114050

Order 12032.

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

normal subgroups only

This group has 598 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#20252412C211yesmaximal, characteristic
2#20247696C211yesmaximal, characteristic
3#20243472C212yesmaximal, complemented
4#20241255C212yesmaximal, complemented
5C47×#1799C211yesmaximal, characteristic
6C2×#13094005—21no—
7C22×C4⋊Dic47C2211yescharacteristic
8D4×C47⋊C8—24no—
9#15689176—21no—
10C2×C22⋊(C47⋊C8)—21no—
11C2×C22⋊(C47⋊C8)C2212yes—
12#15689053C412yescomplemented
13#15689048C412yescomplemented
14#15689021—21no—
15#15688984—21no—
16#15688983—21no—
17#15688983C2211yescharacteristic
18#15688885—24no—
19C2×C4×D4×C47C2211yescharacteristic
20C2×#99×C47—21no—
21C47×#380C2212yes—
22C47×#342—21no—
23C23×Dic47—21no—
24C22×C47⋊C8—22no—
25C22×C47⋊C8D414yes—
26#13094005—42no—
27#13094005—42no—
28C2×C4⋊Dic47—22no—
29C2×C4⋊Dic47—22no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47C2×C412yes—
33C2×Dic47⋊C4—22no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—21no—
36C4⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)—22no—
39C22⋊(C47⋊C8)—22no—
40C4×C47⋊C8—22no—
41C47⋊#95—22no—
42C47⋊#95—22no—
43C47⋊#95C2×C412yes—
44C22×D4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47C2311yescharacteristic
47C4×D4×C47—22no—
48C4×D4×C47—22no—
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C22⋊C4×C47—22no—
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55#99×C47—42no—
56#95×C47—22no—
57C24×C47C22⋊C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C2×D4×C47—22no—
60C2×D4×C47—22no—
61C2×D4×C47D812yes—
62C2×D4×C47SD1612yes—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D412yes—
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47#4112yes—
75C22×Dic47—22no—
76C22×Dic47—22no—
77C22×Dic47—42no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C22×Dic47—42no—
81C4⋊C4×C47—22no—
82C4⋊C4×C47—44no—
83C22⋊C4×C47—22no—
84C22⋊C4×C47—22no—
85C22⋊C4×C47—42no—
86C22⋊C4×C47—42no—
87C42×C47—21no—
88C42×C47—21no—
89C22⋊Dic47—44no—
90C22⋊Dic47—41no—
91C22⋊Dic47—41no—
92C4⋊Dic47—24no—
93C4⋊Dic47—42no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96Dic47⋊C4—44no—
97C2×C47⋊C8—24no—
98C2×C47⋊C8—44no—
99C2×C47⋊C8—82no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47#10212yes—
107C23×C47C2×C22⋊C411yescharacteristic
108C23×C47#12112yes—
109C23×C47C2×D811yescharacteristic
110C23×C47C2×SD1611yescharacteristic
111C23×C47#13512yes—
112D4×C47—24no—
113D4×C47—24no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—24no—
117C2×C4×C47—22no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—41no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—81no—
126C2×C4×C47—82no—
127C2×C4×C47—82no—
128C2×C4×C47—82no—
129C2×C4×C47#10212yes—
130C2×C4×C47C4×D412yes—
131C2×C4×C47#12011yescharacteristic
132C2×C4×C47#12311yescharacteristic
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—82no—
140C2×Dic47—82no—
141C47⋊C8—44no—
142#9470—471nomaximal, Sylow
143C22×C47—21no—
144C22×C47—22no—
145C22×C47—21no—
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47—81no—
155C22×C47#38611yescharacteristic
156C22×C47C2×#10211yescharacteristic
157C22×C47#41611yescharacteristic
158C22×C47#46512yes—
159C22×C47#46612yes—
160C4×C47—21no—
161C4×C47—22no—
162C4×C47—22no—
163C4×C47—41no—
164C4×C47—41no—
165C4×C47—162no—
166C4×C47#45812yes—
167Dic47—82no—
168Dic47—82no—
169#2972—471no—
170#2486—471no—
171#1823—472no—
172#1799D4711yescharacteristic
173#1289—472no—
174C2×C47—21no—
175C2×C47—22no—
176C2×C47—21no—
177C2×C47—22no—
178C2×C47—42no—
179C2×C47#189011yescharacteristic
180C2×C47#189311yescharacteristic
181C2×C47#190911yescharacteristic
182C2×#123—941no—
183C2×C4×D4C2×D4711yescharacteristic
184C22×C4⋊C4—471no—
185C8×D4—944no—
186#431—941no—
187C2×#99—21no—
188C2×C22⋊C8—472no—
189C2×C22⋊C8—941no—
190#398—472no—
191#397—472no—
192#389—941no—
193#381—941no—
194#380—471no—
195#380—941no—
196#380C2×D4712yes—
197#342—21no—
198#340—944no—
199C47#947011yescharacteristic, complemented, Sylow
200C22×D4C22⋊D4711yescharacteristic
201C23×C4—941no—
202C23×C4D18811yescharacteristic
203C23×C4C22×D4711yescharacteristic
204C22×C8—474no—
205C22×C8—942no—
206#123—1882no—
207#123—1882no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—472no—
212C2×C4⋊C4—942no—
213C2×C4⋊C4—942no—
214C2×C4⋊C4—942no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4—941no—
217C2×C4⋊C4C4×D4711yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—942no—
220C2×C22⋊C4—941no—
221C2×C22⋊C4C4×D4711yescharacteristic
222C2×C22⋊C4C22⋊D4711yescharacteristic
223C2×C42D18811yescharacteristic
224C4⋊C8—942no—
225#99—42no—
226C22⋊C8—942no—
227C22⋊C8—942no—
228C22⋊C8—942no—
229C4×C8—942no—
230#95—22no—
231#95—472no—
232#95—942no—
233#95—942no—
234C24(C2×D47)⋊C411yescharacteristic
235C24C2×C22⋊D4711yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4#1062533412yes—
239C2×D4#1062533612yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—942no—
246C22×C4—942no—
247C22×C4—1882no—
248C22×C4—1881no—
249C22×C4—1881no—
250C22×C4—1882no—
251C22×C4(C2×D47)⋊C411yescharacteristic
252C22×C4(C2×D47)⋊C411yescharacteristic
253C22×C4(C2×D47)⋊C411yescharacteristic
254C22×C4C2×C4×D4711yescharacteristic
255C22×C4C2×D18811yescharacteristic
256C22×C4D4×D4712yes—
257C22×C4#1062535212yes—
258C2×C8—944no—
259C2×C8—3762no—
260C2×C8—1884no—
261C4⋊C4—22no—
262C4⋊C4—44no—
263C4⋊C4—944no—
264C4⋊C4—1882no—
265C4⋊C4—1884no—
266C4⋊C4—1881no—
267C4⋊C4—1881no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—1884no—
273C22⋊C4—1881no—
274C22⋊C4—1881no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1309394412yes—
284C23C2×(C2×D47)⋊C411yescharacteristic
285C23#1309399612yes—
286C23C2×#1062533411yescharacteristic
287C23C2×#1062533611yescharacteristic
288C23#1309404412yes—
289D4—24no—
290D4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—3762no—
307C2×C4—3762no—
308C2×C4—1882no—
309C2×C4—1882no—
310C2×C4—1882no—
311C2×C4—1882no—
312C2×C4—1882no—
313C2×C4—1882no—
314C2×C4#1309394412yes—
315C2×C4(C22⋊D47)⋊C412yes—
316C2×C4#1309399411yescharacteristic
317C2×C4#1309400811yescharacteristic
318C8—1884no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1568900011yescharacteristic
332C22C2×#1309394411yescharacteristic
333C22#1568911711yescharacteristic
334C22#1568935412yes—
335C22#1568935712yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—162no—
342C4—3762no—
343C4—3762no—
344C4#1568931412yes—
345C2—22no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#2024379911yescharacteristic
351C2#2024381411yescharacteristic
352C2#2024390811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1799C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#1799D47
hypocenterC47#9470
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20243472C2semidirect
4. #20241255C2semidirect
12. #15689053C4semidirect
13. #15689048C4semidirect
199. C47#9470semidirect

Maximal subgroups

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

subgroupindex
#202524122
#202476962
#202434722
#202412552
C47×#17992
#947047