Finite Group Databasebeta

← #26112352

Subgroups — #26112352

Order 12032.

690 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 119 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#20245986C211yesmaximal, characteristic, complemented
2#20241956C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic, complemented
4#20241255C212yesmaximal, complemented
5C47×#2617C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)C2212yes—
7#15689016—24no—
8#15688983C2212yes—
9#15688979—21no—
10C4×C22⋊Dic47—21no—
11#15688857—24no—
12#15688847—22no—
13C2×#121×C47C2211yescharacteristic
14C2×#120×C47—21no—
15C22×C22⋊C4×C47C2211yescharacteristic
16C47×#397C412yescomplemented
17C47×#394—22no—
18C47×#392C412yescomplemented
19C47×#386—21no—
20C47×#380C2211yescharacteristic
21C22×C47⋊C8—22no—
22C2×C4⋊Dic47—22no—
23C2×C22⋊Dic47—22no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C4×Dic47—22no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C47⋊#95—22no—
30C47⋊#95D414yescomplemented
31C25×C47D411yescharacteristic
32C22×D4×C47—21no—
33C22×D4×C47—22no—
34C22×D4×C47C2×C411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37#121×C47—22no—
38#121×C47—22no—
39#120×C47—44no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47C2×C412yes—
49C2×C22⋊C4×C47D412yes—
50#95×C47—22no—
51#95×C47C2×C412yes—
52C24×C47—21no—
53C24×C47—21no—
54C24×C47—22no—
55C24×C47—22no—
56C24×C47—21no—
57C24×C47—81no—
58C24×C47—81no—
59C24×C47C22⋊C412yes—
60C24×C47C2×D411yescharacteristic
61C2×D4×C47—22no—
62C2×D4×C47—44no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
65C2×D4×C47—42no—
66C2×D4×C47—42no—
67C2×D4×C47—82no—
68C2×D4×C47—82no—
69C2×D4×C47—81no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—22no—
73C22×C4×C47—21no—
74C22×C4×C47—81no—
75C22×C4×C47—81no—
76C22×C4×C47C22⋊C412yes—
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D412yes—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4112yes—
81C22×Dic47—24no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—81no—
84C22⋊C4×C47—24no—
85C22⋊C4×C47—24no—
86C22⋊C4×C47—44no—
87C22⋊C4×C47—42no—
88C22⋊C4×C47—42no—
89C22⋊C4×C47—82no—
90C22⋊Dic47—42no—
91C22⋊Dic47—42no—
92C22⋊Dic47—82no—
93C22⋊Dic47—82no—
94C4⋊Dic47—82no—
95C4×Dic47—82no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—82no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—42no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47—42no—
109C23×C47—42no—
110C23×C47—42no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—81no—
115C23×C47—81no—
116C23×C47—84no—
117C23×C47—84no—
118C23×C47C2×C22⋊C411yescharacteristic
119C23×C47#12011yescharacteristic
120C23×C47#12311yescharacteristic
121C23×C47#13512yes—
122C23×C47AGL(1,ℤ/8)12yes—
123D4×C47—82no—
124D4×C47—88no—
125D4×C47—82no—
126D4×C47—82no—
127C2×C4×C47—22no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47—41no—
140C2×C4×C47—81no—
141C2×C4×C47C4×D412yes—
142C2×C4×C47#12112yes—
143C2×Dic47—44no—
144C2×Dic47—44no—
145C2×Dic47—44no—
146C47⋊C8—82no—
147#9205—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—42no—
155C22×C47—42no—
156C22×C47—44no—
157C22×C47—42no—
158C22×C47—42no—
159C22×C47—42no—
160C22×C47—82no—
161C22×C47—84no—
162C22×C47—84no—
163C22×C47—82no—
164C22×C47—82no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47#38611yescharacteristic
169C22×C47#41611yescharacteristic
170C22×C47#41811yescharacteristic
171C22×C47#44912yes—
172C22×C47#48212yes—
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—42no—
176C4×C47—82no—
177C4×C47—81no—
178Dic47—84no—
179#2617D4711yescharacteristic
180#2312—471no—
181#1519—472no—
182C2×#323—471no—
183#1289—472no—
184C2×C47—22no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47—22no—
188C2×C47—82no—
189C2×C47—161no—
190C2×C47#187511yescharacteristic
191C2×C47#191111yescharacteristic
192C2×C47#191311yescharacteristic
193C2×#121C2×D4711yescharacteristic
194C2×#120—21no—
195C22×C22⋊C4C2×D4711yescharacteristic
196C2×C22⋊C8—472no—
197#397Dic4712yescomplemented
198#394—22no—
199#392Dic4712yescomplemented
200#388—944no—
201#386—21no—
202#380—472no—
203#380C2×D4711yescharacteristic
204#379—941no—
205C4×C22⋊C4—941no—
206#327—944no—
207#323—942no—
208C47#920511yescharacteristic, complemented, Sylow
209C25C22⋊D4711yescharacteristic
210C22×D4—21no—
211C22×D4—22no—
212C22×D4C2×Dic4711yescharacteristic
213C23×C4C22⋊D4711yescharacteristic
214C23×C4C22×D4711yescharacteristic
215C22×C8—942no—
216#121—22no—
217#121—22no—
218#120—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—942no—
221C2×C4⋊C4C2×Dic4711yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—942no—
229C2×C22⋊C4—942no—
230C2×C22⋊C4—942no—
231C2×C22⋊C4C2×Dic4712yes—
232C2×C22⋊C4C22⋊D4712yes—
233C2×C42—942no—
234C22⋊C8—942no—
235C22⋊C8—942no—
236#95—22no—
237#95—474no—
238#95—942no—
239#95C2×Dic4712yes—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—21no—
245C24—81no—
246C24—81no—
247C24C22⋊Dic4712yes—
248C24C2×C22⋊D4711yescharacteristic
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—944no—
265C22×C4C22⋊Dic4712yes—
266C22×C4D4×D4712yes—
267C22×C4#1062535212yes—
268C22×C4C22×Dic4711yescharacteristic
269C22×C4C2×C22⋊D4711yescharacteristic
270C2×C8—3762no—
271C2×C8—1882no—
272C4⋊C4—22no—
273C4⋊C4—81no—
274C4⋊C4—3762no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—82no—
281C22⋊C4—3762no—
282C22⋊C4—3762no—
283C22⋊C4—1882no—
284C22⋊C4—1882no—
285C42—3762no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—81no—
303C23—81no—
304C23—84no—
305C23—84no—
306C23C2×C22⋊Dic4711yescharacteristic
307C23#1309399211yescharacteristic
308C23#1309400511yescharacteristic
309C23#1309404412yes—
310C23#1309405012yes—
311D4—82no—
312D4—88no—
313D4—82no—
314D4—82no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—1884no—
330C2×C4—1884no—
331C2×C4—1884no—
332C2×C4D4×Dic4712yes—
333C2×C4#1309399612yes—
334C8—3762no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—84no—
349C22—84no—
350C22—82no—
351C22—82no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#1568900311yescharacteristic
356C22#1568911911yescharacteristic
357C22#1568912811yescharacteristic
358C22#1568926612yes—
359C22#1568944412yes—
360C4—22no—
361C4—22no—
362C4—42no—
363C4—82no—
364C4—81no—
365C4—3764no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2—22no—
370C2—82no—
371C2—161no—
372C2#2024371111yescharacteristic
373C2#2024392111yescharacteristic
374C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC47×#2617C2
socleC22×C47#386
cosocleC23×C4×C47C23
hypercenter#2617D47
hypocenterC47#9205
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. #20245986C2semidirect
2. #20241956C2semidirect
3. C2×#15688847C2semidirect
4. #20241255C2semidirect
16. C47×#397C4semidirect
18. C47×#392C4semidirect
30. C47⋊#95D4semidirect
197. #397Dic47semidirect
199. #392Dic47semidirect
208. C47#9205semidirect

Maximal subgroups

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

subgroupindex
#202459862
#202419562
C2×#156888472
#202412552
C47×#26172
#920547