Finite Group Databasebeta

← #26114052

Subgroups — #26114052

Order 12032.

598 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 135 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#20247696C211yesmaximal, characteristic
2#20243475C212yesmaximal, complemented
3#20243376C211yesmaximal, characteristic
4#20241256C212yesmaximal, complemented
5C47×#2972C211yesmaximal, characteristic
6C2×#13094005—21no—
7C22×C4⋊Dic47C2211yescharacteristic
8C2×#13093939—21no—
9#15689053C412yescomplemented
10#15689048C412yescomplemented
11#15689021—21no—
12#15688984—21no—
13#15688983—21no—
14#15688983C2212yes—
15#15688983C2211yescharacteristic
16#15688890—21no—
17#15688886—24no—
18C2×C4×D4×C47C2211yescharacteristic
19C8×D4×C47—24no—
20C47×#431—21no—
21C2×C22⋊C8×C47—21no—
22C2×C22⋊C8×C47C2212yes—
23C23×Dic47—21no—
24#13094005—42no—
25#13094005—42no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—22no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47—22no—
31C2×C4⋊Dic47C2×C412yes—
32C2×Dic47⋊C4—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—22no—
36#13093939—42no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C47⋊#95C2×C412yes—
41C22×D4×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C22×C8×C47—22no—
45C22×C8×C47D414yes—
46C4×D4×C47—22no—
47C4×D4×C47—22no—
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47C2×C411yescharacteristic
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C4⋊C8×C47—22no—
53C22⋊C8×C47—22no—
54C22⋊C8×C47—22no—
55C22⋊C8×C47—22no—
56C4×C8×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×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D412yes—
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4112yes—
74C22×Dic47—22no—
75C22×Dic47—22no—
76C22×Dic47—42no—
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47—42no—
80C22×Dic47—42no—
81C2×C8×C47—24no—
82C2×C8×C47—44no—
83C2×C8×C47—82no—
84C4⋊C4×C47—22no—
85C22⋊C4×C47—22no—
86C22⋊C4×C47—22no—
87C42×C47—21no—
88C42×C47—21no—
89C22⋊Dic47—44no—
90C22⋊Dic47—41no—
91C22⋊Dic47—41no—
92C22⋊Dic47—42no—
93C22⋊Dic47—42no—
94C4⋊Dic47—24no—
95C4⋊Dic47—42no—
96C4⋊Dic47—41no—
97C4⋊Dic47—41no—
98C4⋊Dic47—44no—
99Dic47⋊C4—44no—
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#10212yes—
127C2×C4×C47C4×D412yes—
128C2×C4×C47#12011yescharacteristic
129C2×C4×C47#12311yescharacteristic
130C2×Dic47—42no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—82no—
137C2×Dic47—82no—
138C2×Dic47—82no—
139C2×Dic47—82no—
140C2×Dic47—82no—
141C8×C47—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#45812yes—
166Dic47—82no—
167Dic47—82no—
168Dic47—162no—
169#2972D4711yescharacteristic
170#2486—471no—
171#1823—472no—
172#1799—471no—
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—24no—
186#431—21no—
187C2×#99—941no—
188C2×C22⋊C8—21no—
189C2×C22⋊C8C2×D4712yes—
190#398—472no—
191#397—472no—
192#389—941no—
193#381—941no—
194#380—471no—
195#380—472no—
196#380—941no—
197#342—941no—
198#340—944no—
199C47#947011yescharacteristic, complemented, Sylow
200C22×D4D18811yescharacteristic
201C23×C4—941no—
202C23×C4C22⋊D4711yescharacteristic
203C23×C4C22×D4711yescharacteristic
204C22×C8—22no—
205C22×C8C22⋊D4714yes—
206#123—1882no—
207#123—1882no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—472no—
211C2×C4⋊C4—942no—
212C2×C4⋊C4—942no—
213C2×C4⋊C4—942no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4—942no—
217C2×C4⋊C4C4×D4711yescharacteristic
218C2×C22⋊C4—942no—
219C2×C22⋊C4—941no—
220C2×C22⋊C4—942no—
221C2×C22⋊C4C4×D4711yescharacteristic
222C2×C22⋊C4D18811yescharacteristic
223C2×C42C22⋊D4711yescharacteristic
224C4⋊C8—22no—
225#99—1882no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C4×C8—22no—
230#95—472no—
231#95—942no—
232#95—942no—
233#95—942no—
234C24(C2×D47)⋊C411yescharacteristic
235C24C2×D18811yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4#1062532512yes—
239C2×D4D37612yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—942no—
245C22×C4—942no—
246C22×C4—1882no—
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×C4#1062535012yes—
256C22×C4C2×C22⋊D4712yes—
257C22×C4C2×C22⋊D4711yescharacteristic
258C2×C8—24no—
259C2×C8—44no—
260C2×C8—82no—
261C4⋊C4—22no—
262C4⋊C4—944no—
263C4⋊C4—1884no—
264C4⋊C4—1882no—
265C4⋊C4—1884no—
266C4⋊C4—1881no—
267C4⋊C4—1881no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—1882no—
271C22⋊C4—1882no—
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#1309394612yes—
284C23C2×(C2×D47)⋊C411yescharacteristic
285C23#1309399812yes—
286C23C2×D37611yescharacteristic
287C23C2×#1062532511yescharacteristic
288C23#1309404612yes—
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—3762no—
304C2×C4—3762no—
305C2×C4—3762no—
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#1309394612yes—
315C2×C4C4×C22⋊D4712yes—
316C2×C4#1309399311yescharacteristic
317C2×C4#1309400711yescharacteristic
318C8—44no—
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#1568900211yescharacteristic
332C22C2×#1309394611yescharacteristic
333C22#1568912011yescharacteristic
334C22#1568935212yes—
335C22#1568935912yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—3762no—
342C4—3762no—
343C4—7522no—
344C4#1568931312yes—
345C2—21no—
346C2—22no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#2024380111yescharacteristic
351C2#2024381611yescharacteristic
352C2#2024391011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2972C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#2972D47
hypocenterC47#9470
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20243475C2semidirect
4. #20241256C2semidirect
9. #15689053C4semidirect
10. #15689048C4semidirect
199. C47#9470semidirect

Maximal subgroups

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

subgroupindex
#202476962
#202434752
#202433762
#202412562
C47×#29722
#947047