Finite Group Databasebeta

← #26111325

Subgroups — #26111325

Order 12032.

768 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#15689048C211yesmaximal, characteristic
2#20246002C211yesmaximal, characteristic
3#20243475C212yesmaximal, complemented
4#20241268C212yesmaximal, complemented
5C2×C47×#323C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7C22×C22⋊Dic47—21no—
8C22×C22⋊Dic47—21no—
9#15689048—22no—
10#15689048C414yescomplemented
11#15688998—24no—
12#15688983—21no—
13#15688983C2212yes—
14C2×C47⋊#95—21no—
15C2×C47⋊#95C2211yescharacteristic
16#15688886—24no—
17C22×C22⋊C4×C47C2211yescharacteristic
18C2×C22⋊C8×C47C2212yes—
19C47×#323—22no—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C23×Dic47—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47C2×C412yescomplemented
28C2×C4⋊Dic47D414yes—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—24no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—42no—
33C2×C22⋊Dic47—42no—
34C2×C22⋊Dic47—42no—
35C2×C22⋊Dic47—42no—
36C47⋊#95—22no—
37C47⋊#95—24no—
38C47⋊#95—21no—
39C47⋊#95—22no—
40C47⋊#95C2×C412yescomplemented
41C25×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C22×C8×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47C2×C412yes—
48C2×C22⋊C4×C47D412yes—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—21no—
54C24×C47—21no—
55C24×C47—42no—
56C24×C47—42no—
57C24×C47C22⋊C412yes—
58C24×C47D812yes—
59C24×C47SD1612yes—
60C24×C47C2×D411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D412yes—
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4112yes—
70C22×Dic47—22no—
71C22×Dic47—22no—
72C22×Dic47—24no—
73C22×Dic47—44no—
74C22×Dic47—42no—
75C22×Dic47—42no—
76C22×Dic47—41no—
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C22×Dic47—41no—
81C22×Dic47—41no—
82C22×Dic47—41no—
83C22×Dic47—41no—
84C22×Dic47—82no—
85C22×Dic47—82no—
86C2×C8×C47—44no—
87C2×C8×C47—42no—
88C22⋊C4×C47—24no—
89C22⋊C4×C47—24no—
90C22⋊Dic47—44no—
91C22⋊Dic47—48no—
92C22⋊Dic47—42no—
93C22⋊Dic47—42no—
94C4⋊Dic47—24no—
95C4⋊Dic47—42no—
96C4⋊Dic47—44no—
97C4⋊Dic47—82no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47—44no—
109C23×C47—44no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47—82no—
113C23×C47—82no—
114C23×C47—81no—
115C23×C47—81no—
116C23×C47#10214yes—
117C23×C47C2×C22⋊C411yescharacteristic
118C23×C47#12011yescharacteristic
119C23×C47#12311yescharacteristic
120C23×C47C2×D811yescharacteristic
121C23×C47C2×SD1611yescharacteristic
122C23×C47AGL(1,ℤ/8)11yescharacteristic
123C23×C47#13711yescharacteristic
124C2×C4×C47—24no—
125C2×C4×C47—22no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—81no—
132C2×C4×C47C4×D412yes—
133C2×C4×C47#12112yes—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—44no—
141C2×Dic47—44no—
142C2×Dic47—44no—
143C2×Dic47—82no—
144C2×Dic47—82no—
145C2×Dic47—88no—
146C2×Dic47—81no—
147C2×Dic47—81no—
148C2×Dic47—81no—
149C2×Dic47—81no—
150C8×C47—82no—
151#9025—471nomaximal, Sylow
152C22×C47—22no—
153C22×C47—22no—
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—24no—
161C22×C47—44no—
162C22×C47—44no—
163C22×C47—44no—
164C22×C47—42no—
165C22×C47—42no—
166C22×C47—82no—
167C22×C47—82no—
168C22×C47—82no—
169C22×C47#38611yescharacteristic
170C22×C47C2×#10211yescharacteristic
171C22×C47#41711yescharacteristic
172C22×C47#44711yescharacteristic
173C22×C47#44811yescharacteristic
174C22×C47#45512yes—
175C22×C47#45812yes—
176C22×C47#48011yescharacteristic
177C22×C47#48111yescharacteristic
178C4×C47—22no—
179C4×C47—22no—
180C4×C47—42no—
181Dic47—82no—
182Dic47—84no—
183Dic47—82no—
184C2×#397—471no—
185#2315—471no—
186#1823—472no—
187C2×#323D4711yescharacteristic
188#1294—472no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—44no—
194C2×C47#160712yes—
195C2×C47#160812yes—
196C2×C47#187411yescharacteristic
197C2×C47#188511yescharacteristic
198C2×C47#189011yescharacteristic
199C22×C4⋊C4—471no—
200C22×C22⋊C4—941no—
201C22×C22⋊C4—941no—
202C22×C22⋊C4C2×D4711yescharacteristic
203C2×C22⋊C8C2×D4712yes—
204#397—474no—
205#397—942no—
206#385—944no—
207#380—472no—
208#380—941no—
209C2×#95—471no—
210C2×#95—941no—
211#340—944no—
212#323—22no—
213C47#902511yescharacteristic, complemented, Sylow
214C25D18811yescharacteristic
215C23×C4—942no—
216C23×C4—941no—
217C23×C4—941no—
218C23×C4D18811yescharacteristic
219C23×C4C22×D4711yescharacteristic
220C22×C8—22no—
221C2×C4⋊C4—472no—
222C2×C4⋊C4—474no—
223C2×C4⋊C4—942no—
224C2×C4⋊C4—942no—
225C2×C4⋊C4—942no—
226C2×C4⋊C4—941no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—942no—
230C2×C22⋊C4—944no—
231C2×C22⋊C4—942no—
232C2×C22⋊C4—1882no—
233C2×C22⋊C4—1882no—
234C2×C22⋊C4—1882no—
235C2×C22⋊C4—1882no—
236C2×C22⋊C4C4×D4712yes—
237C2×C22⋊C4C22⋊D4712yes—
238C22⋊C8—22no—
239C22⋊C8—22no—
240#95—472no—
241#95—942no—
242#95—944no—
243#95—942no—
244#95—941no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24#1062532512yes—
252C24D37612yes—
253C24(C2×D47)⋊C412yes—
254C24C2×D18811yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—942no—
260C22×C4—942no—
261C22×C4—944no—
262C22×C4—3762no—
263C22×C4—3762no—
264C22×C4—1884no—
265C22×C4—1882no—
266C22×C4—1882no—
267C22×C4—1881no—
268C22×C4—1881no—
269C22×C4—1881no—
270C22×C4—1881no—
271C22×C4—1881no—
272C22×C4—1881no—
273C22×C4—1881no—
274C22×C4—1881no—
275C22×C4(C2×D47)⋊C412yes—
276C22×C4C2×C4×D4711yescharacteristic
277C22×C4D4×D4712yes—
278C22×C4#1062535212yes—
279C22×C4C2×C22⋊D4711yescharacteristic
280C2×C8—44no—
281C2×C8—42no—
282C4⋊C4—944no—
283C4⋊C4—3762no—
284C4⋊C4—1884no—
285C4⋊C4—1882no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—1884no—
289C22⋊C4—1888no—
290C22⋊C4—1882no—
291C22⋊C4—1882no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—44no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—82no—
309C23—82no—
310C23#1309394614yes—
311C23C2×(C2×D47)⋊C411yescharacteristic
312C23#1309399411yescharacteristic
313C23#1309400811yescharacteristic
314C23C2×D37611yescharacteristic
315C23C2×#1062532511yescharacteristic
316C23#1309405211yescharacteristic
317C23#1309405711yescharacteristic
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—3762no—
327C2×C4—3762no—
328C2×C4—3768no—
329C2×C4—3761no—
330C2×C4—3761no—
331C2×C4—3761no—
332C2×C4—3761no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4—1882no—
336C2×C4—1884no—
337C2×C4—1884no—
338C2×C4—1884no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4—1882no—
342C2×C4(C22⋊D47)⋊C412yes—
343C2×C4#1309399612yes—
344C8—82no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—44no—
358C22—44no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22#1568900011yescharacteristic
363C22C2×#1309394611yescharacteristic
364C22#1568912511yescharacteristic
365C22#1568925911yescharacteristic
366C22#1568926411yescharacteristic
367C22#1568930312yes—
368C22#1568931412yes—
369C22#1568943711yescharacteristic
370C22#1568944211yescharacteristic
371C4—22no—
372C4—22no—
373C4—42no—
374C4—3762no—
375C4—3764no—
376C4—3762no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2024237612yes—
383C2#2024238112yes—
384C2#2024370311yescharacteristic
385C2#2024377411yescharacteristic
386C2#2024379911yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#323C2
socleC23×C47#120
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC2×#323D47
hypocenterC47#9025
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20243475C2semidirect
4. #20241268C2semidirect
10. #15689048C4semidirect
27. C2×C4⋊Dic47C2×C4semidirect
40. C47⋊#95C2×C4semidirect
213. C47#9025semidirect

Maximal subgroups

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

subgroupindex
C2×#156890482
#202460022
#202434752
#202412682
C2×C47×#3232
#902547