Finite Group Databasebeta

← #26112049

Subgroups — #26112049

Order 12032.

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

normal subgroups only

This group has 902 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×#15688986C211yesmaximal, characteristic
2#20243298C211yesmaximal, characteristic
3C2×#15688860C212yesmaximal, complemented
4#20241266C212yesmaximal, complemented
5C47×#2848C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7C2×C22⋊(C47⋊C8)—22no—
8C2×C22⋊(C47⋊C8)C2212yes—
9#15688986—24no—
10C2×C47⋊#95—21no—
11C2×C47⋊#95C2212yes—
12#15688860—24no—
13C24×C4×C47C2211yescharacteristic
14C2×C22⋊Q8×C47C412yescomplemented
15C2×C22⋊Q8×C47C2211yescharacteristic
16C47×#385—22no—
17C47×#385—21no—
18C47×#380C412yescomplemented
19C47×#380C2211yescharacteristic
20C47×#379—21no—
21C23×Dic47—22no—
22C22×C47⋊C8—22no—
23C2×C22⋊Dic47—24no—
24C2×C22⋊Dic47—44no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—44no—
28C47⋊#95—24no—
29C47⋊#95—22no—
30C47⋊#95D414yes—
31C25×C47D411yescharacteristic
32C22×Q8×C47C2×C411yescharacteristic
33C23×C4×C47—22no—
34C23×C4×C47—22no—
35C23×C4×C47—44no—
36C23×C4×C47C2×C412yes—
37C23×C4×C47D412yes—
38C23×C4×C47D411yescharacteristic
39C23×C4×C47C2311yescharacteristic
40C22⋊Q8×C47—22no—
41C22⋊Q8×C47—22no—
42C22⋊Q8×C47—44no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47C2×C412yes—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51#95×C47—22no—
52#95×C47—21no—
53C24×C47—24no—
54C24×C47—21no—
55C24×C47—21no—
56C24×C47—22no—
57C24×C47—22no—
58C24×C47—42no—
59C24×C47C22⋊C412yes—
60C24×C47C2×D411yescharacteristic
61C2×Q8×C47—22no—
62C2×Q8×C47—42no—
63C22×C4×C47—24no—
64C22×C4×C47—24no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—24no—
68C22×C4×C47—24no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—22no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47—42no—
75C22×C4×C47—42no—
76C22×C4×C47—48no—
77C22×C4×C47—48no—
78C22×C4×C47C22⋊C412yes—
79C22×C4×C47C22×C411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D412yes—
82C22×C4×C47#4112yes—
83C22×Dic47—24no—
84C22×Dic47—44no—
85C22×Dic47—44no—
86C22×Dic47—42no—
87C22×Dic47—42no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—44no—
90C4⋊C4×C47—42no—
91C4⋊C4×C47—42no—
92C22⋊C4×C47—44no—
93C22⋊C4×C47—44no—
94C22⋊C4×C47—41no—
95C22⋊C4×C47—41no—
96C22⋊C4×C47—42no—
97C22⋊Dic47—48no—
98C2×C47⋊C8—44no—
99C2×C47⋊C8—42no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—24no—
103C23×C47—24no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—22no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47—48no—
111C23×C47—48no—
112C23×C47—42no—
113C23×C47—42no—
114C23×C47—42no—
115C23×C47—42no—
116C23×C47#9914yes—
117C23×C47C2×C22⋊C411yescharacteristic
118C23×C47#12011yescharacteristic
119C23×C47#12311yescharacteristic
120C23×C47#13512yes—
121C23×C47#13712yes—
122Q8×C47—82no—
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—48no—
131C2×C4×C47—48no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47C4×D412yes—
142C2×C4×C47#12112yes—
143C2×Dic47—44no—
144C2×Dic47—44no—
145C2×Dic47—44no—
146C2×Dic47—84no—
147C2×Dic47—84no—
148C47⋊C8—82no—
149#9154—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—24no—
155C22×C47—24no—
156C22×C47—24no—
157C22×C47—24no—
158C22×C47—24no—
159C22×C47—48no—
160C22×C47—48no—
161C22×C47—42no—
162C22×C47—42no—
163C22×C47—42no—
164C22×C47—42no—
165C22×C47#38611yescharacteristic
166C22×C47C2×#9912yes—
167C22×C47#41611yescharacteristic
168C22×C47#41911yescharacteristic
169C22×C47#42112yes—
170C22×C47#45212yes—
171C22×C47#48512yes—
172C4×C47—22no—
173C4×C47—22no—
174C4×C47—42no—
175C4×C47—81no—
176C4×C47—81no—
177C4×C47—82no—
178Dic47—84no—
179#2848D4711yescharacteristic
180C2×#381—471no—
181#1779—471no—
182C2×#328—472no—
183#1294—472no—
184C2×C47—22no—
185C2×C47—22no—
186C2×C47—44no—
187C2×C47#151112yes—
188C2×C47#151612yes—
189C2×C47#187311yescharacteristic
190C2×C47#189711yescharacteristic
191C2×C47#189811yescharacteristic
192C24×C4C2×D4711yescharacteristic
193C2×C22⋊Q8Dic4712yescomplemented
194C2×C22⋊Q8C2×D4711yescharacteristic
195C22×C22⋊C4—941no—
196C2×C22⋊C8—472no—
197C2×C22⋊C8—942no—
198#385—21no—
199#385—22no—
200#381—944no—
201#380Dic4712yescomplemented
202#380C2×D4711yescharacteristic
203#379—21no—
204C2×#95—472no—
205C2×#95—941no—
206#328—944no—
207C47#915411yescharacteristic, complemented, Sylow
208C25C22⋊D4711yescharacteristic
209C22×Q8C2×Dic4711yescharacteristic
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4—942no—
214C23×C4C2×Dic4712yes—
215C23×C4C22⋊D4712yes—
216C23×C4C22⋊D4711yescharacteristic
217C23×C4C22×D4711yescharacteristic
218C22×C8—942no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C2×Dic4711yescharacteristic
225C2×C4⋊C4C2×Dic4712yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—944no—
231C2×C22⋊C4—1884no—
232C22⋊C8—942no—
233C22⋊C8—942no—
234C22⋊C8—1884no—
235#95—22no—
236#95—21no—
237#95—474no—
238#95—944no—
239#95—942no—
240C24—24no—
241C24—21no—
242C24—21no—
243C24—22no—
244C24—22no—
245C24—42no—
246C24C22⋊Dic4712yes—
247C24C2×C22⋊D4711yescharacteristic
248C2×Q8—22no—
249C2×Q8—42no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—48no—
264C22×C4—48no—
265C22×C4—944no—
266C22×C4—1884no—
267C22×C4—1884no—
268C22×C4—1882no—
269C22×C4—1882no—
270C22×C4C22⋊Dic4712yes—
271C22×C4D4×D4712yes—
272C22×C4#1062535212yes—
273C22×C4C22×Dic4711yescharacteristic
274C22×C4C2×C22⋊D4711yescharacteristic
275C2×C8—1884no—
276C2×C8—1882no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—1888no—
287C23—22no—
288C23—22no—
289C23—24no—
290C23—24no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—48no—
298C23—48no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23#1309393914yes—
304C23C2×C22⋊Dic4711yescharacteristic
305C23#1309399211yescharacteristic
306C23#1309400511yescharacteristic
307C23#1309404412yes—
308C23#1309405512yes—
309Q8—82no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—3764no—
329C2×C4—3764no—
330C2×C4—1884no—
331C2×C4—1884no—
332C2×C4—1884no—
333C2×C4D4×Dic4712yes—
334C2×C4#1309399612yes—
335C8—3762no—
336C22—24no—
337C22—24no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—48no—
346C22—48no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22#1568900311yescharacteristic
352C22C2×#1309393912yes—
353C22#1568911911yescharacteristic
354C22#1568913211yescharacteristic
355C22#1568914012yes—
356C22#1568928312yes—
357C22#1568946112yes—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—3764no—
365C2—22no—
366C2—22no—
367C2—44no—
368C2#2024192112yes—
369C2#2024194412yes—
370C2#2024370111yescharacteristic
371C2#2024383711yescharacteristic
372C2#2024384211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2848C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2848D47
hypocenterC47#9154
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
3. C2×#15688860C2semidirect
4. #20241266C2semidirect
14. C2×C22⋊Q8×C47C4semidirect
18. C47×#380C4semidirect
193. C2×C22⋊Q8Dic47semidirect
201. #380Dic47semidirect
207. C47#9154semidirect

Maximal subgroups

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

subgroupindex
C2×#156889862
#202432982
C2×#156888602
#202412662
C47×#28482
#915447