Finite Group Databasebeta

← #26109250

Subgroups — #26109250

Order 12032.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243426C211yesmaximal, characteristic, complemented
2#20243413C211yesmaximal, characteristic, complemented
3#20241931C212yesmaximal
4#20241293C212yesmaximal, complemented
5C47×#2929C211yesmaximal, characteristic
6C2×#13093960—21no—
7C2×#13093956—21no—
8#15689087C2212yes—
9#15688983C2212yes—
10#15688888—21no—
11#15688886—21no—
12#15688860—22no—
13#15688857—22no—
14C2×#131×C47C411yescharacteristic, complemented
15C23×C8×C47C2211yescharacteristic
16C47×#525C2211yescharacteristic
17C47×#435—21no—
18C47×#435—21no—
19C8×D4×C47—21no—
20C8×D4×C47—21no—
21C47×#424C411yescharacteristic, complemented
22C47×#408C411yescharacteristic, complemented
23C47×#407C2211yescharacteristic
24C47×#405C411yescharacteristic, complemented
25C2×#10625329—22no—
26C2×C4⋊Dic47—22no—
27C2×C22⋊Dic47—22no—
28#13093960—42no—
29#13093956—42no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32C47⋊#95—22no—
33C2×#41×C47C2×C411yescharacteristic
34C23×C4×C47C2311yescharacteristic
35#131×C47—42no—
36C2×M16×C47—21no—
37C2×M16×C47C2×C411yescharacteristic
38C2×M16×C47C2×C411yescharacteristic
39C22×C8×C47—21no—
40C22×C8×C47—42no—
41C22×C8×C47C2×C411yescharacteristic
42C22×C8×C47C2×C411yescharacteristic
43C22×C8×C47D411yescharacteristic
44C22×C8×C47D411yescharacteristic
45C22×C8×C47D411yescharacteristic
46C22×C8×C47D411yescharacteristic
47#123×C47—21no—
48C22⋊Q8×C47D411yescharacteristic
49C22⋊Q8×C47Q811yescharacteristic
50#121×C47D411yescharacteristic
51#121×C47Q811yescharacteristic
52#120×C47—21no—
53C4×D4×C47—21no—
54C4×D4×C47—21no—
55#117×C47C2×C411yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C22⋊C8×C47—21no—
59C22⋊C8×C47—21no—
60C22⋊C8×C47—21no—
61C22⋊C8×C47—21no—
62#97×C47—21no—
63C4×C8×C47—21no—
64C24×C47#4111yescharacteristic
65#41×C47—41no—
66#41×C47—41no—
67C2×Q8×C47C4⋊C411yescharacteristic
68C2×D4×C47—21no—
69C2×D4×C47—21no—
70C2×D4×C47—21no—
71C2×D4×C47C4⋊C411yescharacteristic
72C22×C4×C47—21no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×Q811yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—42no—
82M16×C47—42no—
83M16×C47—41no—
84M16×C47—41no—
85C2×C8×C47—22no—
86C2×C8×C47—22no—
87C2×C8×C47—22no—
88C2×C8×C47—21no—
89C2×C8×C47—21no—
90C2×C8×C47—41no—
91C2×C8×C47—41no—
92C2×C8×C47—41no—
93C2×C8×C47—42no—
94C2×C8×C47C22⋊C411yescharacteristic
95C2×C8×C47C22⋊C411yescharacteristic
96C2×C8×C47C22⋊C411yescharacteristic
97C2×C8×C47C22⋊C411yescharacteristic
98C4⋊C4×C47—21no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47C4⋊C411yescharacteristic
101C4⋊C4×C47C4⋊C411yescharacteristic
102C22⋊C4×C47—21no—
103C22⋊C4×C47—21no—
104C22⋊C4×C47—21no—
105C22⋊C4×C47—21no—
106C22⋊C4×C47—21no—
107C42×C47—21no—
108C22⋊Dic47—82no—
109C4⋊Dic47—42no—
110C4⋊Dic47—42no—
111#10625329—42no—
112#10625329—42no—
113C2×C47⋊C8—42no—
114C23×C47—22no—
115C23×C47—21no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47#11711yescharacteristic
119C23×C47#12111yescharacteristic
120C23×C47C22⋊Q811yescharacteristic
121Q8×C47—41no—
122D4×C47—41no—
123D4×C47—42no—
124D4×C47—42no—
125D4×C47—42no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47C2×C4⋊C411yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47C22⋊Q811yescharacteristic
144C2×Dic47—82no—
145C2×Dic47—82no—
146C2×Dic47—82no—
147C8×C47—21no—
148C8×C47—21no—
149C8×C47—21no—
150C8×C47—41no—
151C8×C47—41no—
152C8×C47#9912yes—
153C47⋊C8—82no—
154#8626—471nomaximal, Sylow
155C22×C47—21no—
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—41no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47#38011yescharacteristic
168C22×C47#47112yes—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—41no—
173C4×C47—41no—
174C4×C47—41no—
175C4×C47C2×#9911yescharacteristic
176C4×C47#41011yescharacteristic
177Dic47—162no—
178#2929D4711yescharacteristic
179#1810—471no—
180#1807—471no—
181#1513—472no—
182#1305—472no—
183C2×C47—22no—
184C2×C47—41no—
185C2×C47—41no—
186C2×C47—81no—
187C2×C47#179911yescharacteristic
188C2×C47#181411yescharacteristic
189C2×C47#181511yescharacteristic
190C2×#131Dic4711yescharacteristic, complemented
191C23×C8C2×D4711yescharacteristic
192#525C2×D4711yescharacteristic
193#435—21no—
194#435—21no—
195C8×D4—21no—
196C8×D4—21no—
197C2×#108—941no—
198C2×#106—941no—
199#424Dic4711yescharacteristic, complemented
200#408Dic4711yescharacteristic, complemented
201#407—472no—
202#407C2×D4711yescharacteristic
203#405Dic4711yescharacteristic, complemented
204#380—472no—
205#341—941no—
206#340—941no—
207#328—942no—
208#327—942no—
209C47#862611yescharacteristic, complemented, Sylow
210C2×#41C2×Dic4711yescharacteristic
211C23×C4C22×D4711yescharacteristic
212#131—42no—
213C2×M16—21no—
214C2×M16—942no—
215C2×M16C2×Dic4711yescharacteristic
216C2×M16C2×Dic4711yescharacteristic
217C22×C8—21no—
218C22×C8—42no—
219C22×C8C2×Dic4711yescharacteristic
220C22×C8C2×Dic4711yescharacteristic
221C22×C8C22⋊D4711yescharacteristic
222C22×C8C22⋊D4711yescharacteristic
223C22×C8C22⋊D4711yescharacteristic
224C22×C8C22⋊D4711yescharacteristic
225#123—21no—
226C22⋊Q8Dic9411yescharacteristic
227C22⋊Q8D18811yescharacteristic
228#121Dic9411yescharacteristic
229#121D18811yescharacteristic
230#120—21no—
231C4×D4—21no—
232C4×D4—21no—
233#117C2×Dic4711yescharacteristic
234C2×C4⋊C4—942no—
235C2×C22⋊C4—942no—
236#108—1882no—
237#106—1882no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—942no—
245C22⋊C8—942no—
246#97—21no—
247C4×C8—21no—
248#95—942no—
249C24#1062535011yescharacteristic
250#41—41no—
251#41—41no—
252C2×Q8C4⋊Dic4711yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C4⋊Dic4711yescharacteristic
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—1882no—
261C22×C4C2×Dic9411yescharacteristic
262C22×C4C2×D18811yescharacteristic
263C22×C4#1062535011yescharacteristic
264C22×C4C22×Dic4711yescharacteristic
265C22×C4C2×C22⋊D4711yescharacteristic
266C22×C4C2×C22⋊D4711yescharacteristic
267M16—42no—
268M16—41no—
269M16—41no—
270M16—1882no—
271M16—1882no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—1882no—
282C2×C8C22⋊Dic4711yescharacteristic
283C2×C8C22⋊Dic4711yescharacteristic
284C2×C8C22⋊Dic4711yescharacteristic
285C2×C8C22⋊Dic4711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—1882no—
289C4⋊C4—1882no—
290C4⋊C4C4⋊Dic4711yescharacteristic
291C4⋊C4C4⋊Dic4711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—3762no—
298C42—21no—
299C23—22no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23#1309398311yescharacteristic
304C23#1309399811yescharacteristic
305C23#1309400311yescharacteristic
306Q8—41no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—3762no—
326C2×C4—3762no—
327C2×C4—3762no—
328C2×C4C2×C22⋊Dic4711yescharacteristic
329C2×C4C2×C4⋊Dic4711yescharacteristic
330C2×C4#1309399811yescharacteristic
331C2×C4#1309400311yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—41no—
336C8—41no—
337C8—3762no—
338C8#1309393912yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#1568898311yescharacteristic
352C22#1568938612yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—7522no—
360C4C2×#1309393911yescharacteristic
361C4#1568909711yescharacteristic
362C2—22no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2#2024337611yescharacteristic
367C2#2024344211yescharacteristic
368C2#2024344711yescharacteristic
subgroupG/H
centerC22#15688983
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2929C2
socleC22×C47#380
prime radicalC47×#525C22
cosocleC23×C4×C47C23
hypercenter#2929D47
hypocenterC47#8626
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #20243426C2semidirect
2. #20243413C2semidirect
4. #20241293C2semidirect
14. C2×#131×C47C4semidirect
21. C47×#424C4semidirect
22. C47×#408C4semidirect
24. C47×#405C4semidirect
190. C2×#131Dic47semidirect
199. #424Dic47semidirect
200. #408Dic47semidirect
203. #405Dic47semidirect
209. C47#8626semidirect

Maximal subgroups

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

subgroupindex
#202434262
#202434132
#202419312
#202412932
C47×#29292
#862647