Finite Group Databasebeta

← #26127398

Subgroups — #26127398

Order 12032.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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#20243376C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241412C212yesmaximal, complemented
4#20241409C212yesmaximal, complemented
5C47×#2580C211yesmaximal, characteristic
6C2×#13093939—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688983—21no—
9#15688890C2212yes—
10#15688849—21no—
11#15688847C2212yes—
12C2×C4×Q8×C47C411yescharacteristic, complemented
13C2×C4×D4×C47C2211yescharacteristic
14C47×#399C411yescharacteristic, complemented
15C47×#399C2211yescharacteristic
16C47×#393C411yescharacteristic, complemented
17C47×#393C2211yescharacteristic
18C47×#390—21no—
19C47×#389—21no—
20C47×#385—21no—
21C47×#385—21no—
22C47×#382C411yescharacteristic, complemented
23C22×C47⋊C8—41no—
24C2×C4⋊Dic47—41no—
25C2×C22⋊Dic47—22no—
26#13093939—42no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—41no—
29C47⋊#95—41no—
30C22×Q8×C47C2×C411yescharacteristic
31C22×D4×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C4×Q8×C47—41no—
35C4×Q8×C47—41no—
36C4×D4×C47—41no—
37C4×D4×C47—41no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47C2311yescharacteristic
50C2×C42×C47—21no—
51C2×C42×C47C2×C411yescharacteristic
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C2×Q8×C47—42no—
60C2×Q8×C47—41no—
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C2×D4×C47SD1612yes—
64C2×D4×C47Q1612yes—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—81no—
74C22×C4×C47—81no—
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×Dic47—42no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—81no—
87C4⋊C4×C47—81no—
88C22⋊C4×C47—41no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—42no—
91C22⋊C4×C47—42no—
92C22⋊C4×C47—81no—
93C22⋊C4×C47—81no—
94C42×C47—21no—
95C42×C47—21no—
96C42×C47—42no—
97C22⋊Dic47—42no—
98C22⋊Dic47—42no—
99C4⋊Dic47—82no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—82no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47#10312yes—
111C23×C47C2×C22⋊C411yescharacteristic
112C23×C47C2×SD1611yescharacteristic
113C23×C47C2×Q1611yescharacteristic
114Q8×C47—42no—
115Q8×C47—42no—
116D4×C47—42no—
117D4×C47—42no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—22no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—41no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—81no—
142C2×C4×C47—81no—
143C2×C4×C47#9911yescharacteristic
144C2×C4×C47#9911yescharacteristic
145C2×C4×C47#10312yes—
146C2×Dic47—82no—
147C2×Dic47—82no—
148C2×Dic47—82no—
149C47⋊C8—82no—
150#11897—471nomaximal, Sylow
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—22no—
155C22×C47—21no—
156C22×C47—42no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47—41no—
162C22×C47—41no—
163C22×C47—81no—
164C22×C47—81no—
165C22×C47C2×#9911yescharacteristic
166C22×C47C2×#10311yescharacteristic
167C22×C47#42111yescharacteristic
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—22no—
171C4×C47—42no—
172C4×C47—41no—
173C4×C47—41no—
174C4×C47—41no—
175C4×C47—81no—
176C4×C47—81no—
177C4×C47—81no—
178Dic47—162no—
179#2580D4711yescharacteristic
180#1799—471no—
181#1459—471no—
182#1352—472no—
183#1351—472no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—41no—
187C2×C47—42no—
188C2×C47—41no—
189C2×C47#152011yescharacteristic
190C2×C47#212011yescharacteristic
191C2×C47#212711yescharacteristic
192C2×C4×Q8Dic4711yescharacteristic, complemented
193C2×C4×D4C2×D4711yescharacteristic
194C2×#99—941no—
195C2×C22⋊C8—941no—
196#399Dic4711yescharacteristic, complemented
197#399C2×D4711yescharacteristic
198#393Dic4711yescharacteristic, complemented
199#393C2×D4711yescharacteristic
200#390—21no—
201#389—21no—
202#385—21no—
203#385—21no—
204#382Dic4711yescharacteristic, complemented
205#380—941no—
206#342—472no—
207#324—941no—
208#323—472no—
209C47#1189711yescharacteristic, complemented, Sylow
210C22×Q8C2×Dic4711yescharacteristic
211C22×D4C22⋊D4711yescharacteristic
212C23×C4C22⋊D4711yescharacteristic
213C23×C4C22⋊D4711yescharacteristic
214C22×C8—1881no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×D4—41no—
218C4×D4—41no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—1881no—
224C2×C4⋊C4C2×Dic4711yescharacteristic
225C2×C4⋊C4C22⋊D4711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—942no—
231C2×C22⋊C4C2×Dic4711yescharacteristic
232C2×C22⋊C4C22×D4711yescharacteristic
233C2×C42—21no—
234C2×C42C2×Dic4711yescharacteristic
235#99—1882no—
236C22⋊C8—942no—
237C22⋊C8—1881no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—1881no—
242#95C2×Dic4711yescharacteristic
243#95C2×Dic4711yescharacteristic
244C24C22⋊Dic4711yescharacteristic
245C24C2×C22⋊D4711yescharacteristic
246C2×Q8—42no—
247C2×Q8—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4#1062533612yes—
251C2×D4#1062533712yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—1882no—
263C22×C4C22⋊Dic4711yescharacteristic
264C22×C4C22⋊Dic4711yescharacteristic
265C22×C4C22⋊Dic4711yescharacteristic
266C22×C4C22×Dic4711yescharacteristic
267C22×C4C2×C22⋊D4711yescharacteristic
268C2×C8—3762no—
269C2×C8—1882no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—3762no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—1882no—
285C22⋊C4—1882no—
286C42—21no—
287C42—21no—
288C42—42no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1309394812yes—
298C23C2×C22⋊Dic4711yescharacteristic
299C23C2×#1062533611yescharacteristic
300C23C2×#1062533711yescharacteristic
301Q8—42no—
302Q8—42no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—3762no—
331C2×C4—3762no—
332C2×C4—3762no—
333C2×C4#1309393911yescharacteristic
334C2×C4#1309393911yescharacteristic
335C2×C4#1309394812yes—
336C8—3762no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22—81no—
351C22C2×#1309393911yescharacteristic
352C22C2×#1309394811yescharacteristic
353C22#1568914011yescharacteristic
354C4—22no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—7522no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—42no—
369C2—41no—
370C2#2024196011yescharacteristic
371C2#2024502911yescharacteristic
372C2#2024505711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2580C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2580D47
hypocenterC47#11897
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #20241412C2semidirect
4. #20241409C2semidirect
12. C2×C4×Q8×C47C4semidirect
14. C47×#399C4semidirect
16. C47×#393C4semidirect
22. C47×#382C4semidirect
192. C2×C4×Q8Dic47semidirect
196. #399Dic47semidirect
198. #393Dic47semidirect
204. #382Dic47semidirect
209. C47#11897semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417062
#202414122
#202414092
C47×#25802
#1189747