Finite Group Databasebeta

← #26114098

Subgroups — #26114098

Order 12032.

838 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 107 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#20243332C212yesmaximal, complemented
3#20241715C211yesmaximal, characteristic, complemented
4#20241240C212yesmaximal, complemented
5C47×#2404C211yesmaximal, characteristic
6C2×#13093951—24no—
7#15689087C2212yes—
8#15689016—24no—
9#15688983—21no—
10#15688979—21no—
11C4×C22⋊Dic47C2212yes—
12#15688847—22no—
13C22×#41×C47C2211yescharacteristic
14C2×#120×C47—21no—
15C22×C22⋊C4×C47C2211yescharacteristic
16C47×#393C412yescomplemented
17C47×#392C412yescomplemented
18C47×#386—21no—
19C47×#386—22no—
20C47×#381C2211yescharacteristic
21C2×#10625329—22no—
22C2×C4⋊Dic47—22no—
23C2×C22⋊Dic47—22no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C4×Dic47D414yescomplemented
27#13093951—48no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C47⋊#95—22no—
31C47⋊#95—22no—
32C25×C47D411yescharacteristic
33C2×#41×C47—22no—
34C2×#41×C47—22no—
35C2×#41×C47—44no—
36C22×Q8×C47C2×C411yescharacteristic
37C22×D4×C47—21no—
38C22×D4×C47—22no—
39C22×D4×C47C2×C411yescharacteristic
40C23×C4×C47—21no—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43#120×C47—44no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47C2×C412yes—
50C2×C22⋊C4×C47D412yes—
51#95×C47—22no—
52#95×C47—21no—
53#95×C47C2×C412yes—
54C24×C47—21no—
55C24×C47—21no—
56C24×C47—21no—
57C24×C47—42no—
58C24×C47—42no—
59C24×C47—81no—
60C24×C47C22⋊C412yes—
61C24×C47C2×D411yescharacteristic
62#41×C47—28no—
63#41×C47—44no—
64#41×C47—44no—
65#41×C47—44no—
66C2×Q8×C47—22no—
67C2×Q8×C47—42no—
68C2×D4×C47—24no—
69C2×D4×C47—22no—
70C2×D4×C47—42no—
71C2×D4×C47—42no—
72C2×D4×C47—41no—
73C2×D4×C47—41no—
74C2×D4×C47—44no—
75C2×D4×C47—44no—
76C2×D4×C47—82no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—22no—
81C22×C4×C47—42no—
82C22×C4×C47—42no—
83C22×C4×C47—42no—
84C22×C4×C47—42no—
85C22×C4×C47—81no—
86C22×C4×C47C22⋊C412yes—
87C22×C4×C47C22×C411yescharacteristic
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47#4112yes—
91C22×Dic47—24no—
92C22⋊C4×C47—24no—
93C22⋊C4×C47—24no—
94C22⋊C4×C47—44no—
95C22⋊C4×C47—42no—
96C22⋊C4×C47—82no—
97C22⋊Dic47—44no—
98C22⋊Dic47—82no—
99C22⋊Dic47—82no—
100C4⋊Dic47—82no—
101C4×Dic47—44no—
102#10625329—44no—
103C2×C47⋊C8—42no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—22no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47—41no—
111C23×C47—42no—
112C23×C47—42no—
113C23×C47—42no—
114C23×C47—42no—
115C23×C47—41no—
116C23×C47—42no—
117C23×C47—42no—
118C23×C47—41no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—81no—
122C23×C47—81no—
123C23×C47—88no—
124C23×C47C2×C22⋊C411yescharacteristic
125C23×C47#12011yescharacteristic
126C23×C47#12311yescharacteristic
127Q8×C47—24no—
128Q8×C47—42no—
129D4×C47—24no—
130D4×C47—42no—
131D4×C47—44no—
132D4×C47—44no—
133D4×C47—88no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—42no—
140C2×C4×C47—48no—
141C2×C4×C47—42no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—81no—
149C2×C4×C47C4×D412yes—
150C2×C4×C47#12112yes—
151C2×Dic47—44no—
152C2×Dic47—44no—
153C2×Dic47—44no—
154C47⋊C8—82no—
155#9479—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—44no—
164C22×C47—44no—
165C22×C47—44no—
166C22×C47—42no—
167C22×C47—42no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—82no—
171C22×C47—82no—
172C22×C47—84no—
173C22×C47—84no—
174C22×C47—81no—
175C22×C47—81no—
176C22×C47#38611yescharacteristic
177C22×C47#42112yes—
178C22×C47B(4,𝔽2)12yes—
179C4×C47—22no—
180C4×C47—22no—
181C4×C47—42no—
182C4×C47—42no—
183C4×C47—82no—
184Dic47—84no—
185#2404D4711yescharacteristic
186#2312—471no—
187#1789—472no—
188#1462—471no—
189#1284—472no—
190C2×C47—22no—
191C2×C47—22no—
192C2×C47—41no—
193C2×C47—82no—
194C2×C47—82no—
195C2×C47#188911yescharacteristic
196C2×C47#189711yescharacteristic
197C2×C47#191111yescharacteristic
198C22×#41C2×D4711yescharacteristic
199C2×#120—21no—
200C22×C22⋊C4C2×D4711yescharacteristic
201C2×#104—944no—
202#407—472no—
203#393Dic4712yescomplemented
204#392Dic4712yescomplemented
205#388—944no—
206#386—21no—
207#386—22no—
208#381C2×D4711yescharacteristic
209#380—941no—
210#379—941no—
211C4×C22⋊C4—472no—
212#323—942no—
213C47#947911yescharacteristic, complemented, Sylow
214C25C22⋊D4711yescharacteristic
215C2×#41—22no—
216C2×#41—22no—
217C2×#41—44no—
218C22×Q8C2×Dic4711yescharacteristic
219C22×D4—21no—
220C22×D4—22no—
221C22×D4C2×Dic4711yescharacteristic
222C23×C4—21no—
223C23×C4C22⋊D4711yescharacteristic
224C23×C4C22×D4711yescharacteristic
225C2×M16—942no—
226#120—44no—
227C2×C4⋊C4—942no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—942no—
234C2×C22⋊C4—942no—
235C2×C22⋊C4—942no—
236C2×C22⋊C4C2×Dic4712yes—
237C2×C22⋊C4C22⋊D4712yes—
238C2×C42—474no—
239#104—1888no—
240C22⋊C8—942no—
241C22⋊C8—942no—
242#95—22no—
243#95—21no—
244#95—942no—
245#95—942no—
246#95C2×Dic4712yes—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—42no—
251C24—42no—
252C24—81no—
253C24C22⋊Dic4712yes—
254C24C2×C22⋊D4711yescharacteristic
255#41—28no—
256#41—44no—
257#41—44no—
258#41—44no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—944no—
280C22×C4C22⋊Dic4712yes—
281C22×C4D4×D4712yes—
282C22×C4#1062535212yes—
283C22×C4C22×Dic4711yescharacteristic
284C22×C4C2×C22⋊D4711yescharacteristic
285M16—1884no—
286C2×C8—1882no—
287C4⋊C4—3762no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—82no—
293C22⋊C4—3762no—
294C22⋊C4—3762no—
295C22⋊C4—1884no—
296C42—1884no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—41no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—88no—
317C23C2×C22⋊Dic4711yescharacteristic
318C23#1309399211yescharacteristic
319C23#1309400511yescharacteristic
320Q8—24no—
321Q8—42no—
322D4—24no—
323D4—42no—
324D4—44no—
325D4—44no—
326D4—88no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—48no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—1884no—
343C2×C4—1884no—
344C2×C4—1884no—
345C2×C4D4×Dic4712yes—
346C2×C4#1309399612yes—
347C8—3762no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—84no—
364C22—82no—
365C22—84no—
366C22—81no—
367C22—81no—
368C22#1568900311yescharacteristic
369C22#1568914012yes—
370C22#1568930912yes—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—3764no—
377C2—22no—
378C2—22no—
379C2—41no—
380C2—82no—
381C2—82no—
382C2#2024379711yescharacteristic
383C2#2024383711yescharacteristic
384C2#2024392111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC47×#2404C2
socleC22×C47#386
cosocleC23×C4×C47C23
hypercenter#2404D47
hypocenterC47#9479
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. #20245986C2semidirect
2. #20243332C2semidirect
3. #20241715C2semidirect
4. #20241240C2semidirect
16. C47×#393C4semidirect
17. C47×#392C4semidirect
26. C2×C4×Dic47D4semidirect
203. #393Dic47semidirect
204. #392Dic47semidirect
213. C47#9479semidirect

Maximal subgroups

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

subgroupindex
#202459862
#202433322
#202417152
#202412402
C47×#24042
#947947