Finite Group Databasebeta

← #26109255

Subgroups — #26109255

Order 12032.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243426C211yesmaximal, characteristic, complemented
2#20243417C211yesmaximal, characteristic, complemented
3#20241931C212yesmaximal
4#20241293C212yesmaximal, complemented
5C47×#2929C211yesmaximal, characteristic
6C2×#13093960—21no—
7C2×#13093958—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#13093958—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—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47#11711yescharacteristic
120C23×C47#12111yescharacteristic
121C23×C47C22⋊Q811yescharacteristic
122Q8×C47—41no—
123D4×C47—41no—
124D4×C47—42no—
125D4×C47—42no—
126D4×C47—41no—
127D4×C47—41no—
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—21no—
136C2×C4×C47—21no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47C2×C22⋊C411yescharacteristic
143C2×C4×C47C2×C4⋊C411yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47C22⋊Q811yescharacteristic
146C2×Dic47—82no—
147C2×Dic47—82no—
148C2×Dic47—82no—
149C8×C47—21no—
150C8×C47—21no—
151C8×C47—21no—
152C8×C47—41no—
153C8×C47—41no—
154C8×C47#9912yes—
155C47⋊C8—82no—
156#8627—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—42no—
166C22×C47—42no—
167C22×C47—41no—
168C22×C47—41no—
169C22×C47—41no—
170C22×C47#38011yescharacteristic
171C22×C47#47211yescharacteristic
172C22×C47#47311yescharacteristic
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47—41no—
177C4×C47—41no—
178C4×C47—41no—
179C4×C47C2×#9911yescharacteristic
180C4×C47#41011yescharacteristic
181Dic47—162no—
182#2929D4711yescharacteristic
183#1810—471no—
184#1808—471no—
185#1513—472no—
186#1305—472no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47—41no—
190C2×C47—41no—
191C2×C47—81no—
192C2×C47#179911yescharacteristic
193C2×C47#181411yescharacteristic
194C2×C47#181611yescharacteristic
195C2×#131Dic4711yescharacteristic, complemented
196C23×C8C2×D4711yescharacteristic
197#525C2×D4711yescharacteristic
198#435—21no—
199#435—21no—
200C8×D4—21no—
201C8×D4—21no—
202C2×#108—941no—
203C2×#107—941no—
204#424Dic4711yescharacteristic, complemented
205#408Dic4711yescharacteristic, complemented
206#407—472no—
207#407C2×D4711yescharacteristic
208#405Dic4711yescharacteristic, complemented
209#380—472no—
210#341—941no—
211#340—941no—
212#328—942no—
213#327—942no—
214C47#862711yescharacteristic, complemented, Sylow
215C2×#41C2×Dic4711yescharacteristic
216C23×C4C22×D4711yescharacteristic
217#131—42no—
218C2×M16—21no—
219C2×M16—942no—
220C2×M16C2×Dic4711yescharacteristic
221C2×M16C2×Dic4711yescharacteristic
222C22×C8—21no—
223C22×C8—42no—
224C22×C8C2×Dic4711yescharacteristic
225C22×C8C2×Dic4711yescharacteristic
226C22×C8C22⋊D4711yescharacteristic
227C22×C8C22⋊D4711yescharacteristic
228C22×C8C22⋊D4711yescharacteristic
229C22×C8C22⋊D4711yescharacteristic
230#123—21no—
231C22⋊Q8Dic9411yescharacteristic
232C22⋊Q8D18811yescharacteristic
233#121Dic9411yescharacteristic
234#121D18811yescharacteristic
235#120—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117C2×Dic4711yescharacteristic
239C2×C4⋊C4—942no—
240C2×C22⋊C4—942no—
241#108—1882no—
242#107—1882no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—942no—
250C22⋊C8—942no—
251#97—21no—
252C4×C8—21no—
253#95—942no—
254C24#1062535011yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8C4⋊Dic4711yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C4⋊Dic4711yescharacteristic
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—1882no—
266C22×C4C2×Dic9411yescharacteristic
267C22×C4C2×D18811yescharacteristic
268C22×C4#1062535011yescharacteristic
269C22×C4C22×Dic4711yescharacteristic
270C22×C4C2×C22⋊D4711yescharacteristic
271C22×C4C2×C22⋊D4711yescharacteristic
272M16—42no—
273M16—41no—
274M16—41no—
275M16—1882no—
276M16—1882no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—1882no—
287C2×C8C22⋊Dic4711yescharacteristic
288C2×C8C22⋊Dic4711yescharacteristic
289C2×C8C22⋊Dic4711yescharacteristic
290C2×C8C22⋊Dic4711yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—1882no—
294C4⋊C4—1882no—
295C4⋊C4C4⋊Dic4711yescharacteristic
296C4⋊C4C4⋊Dic4711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3762no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23#1309398311yescharacteristic
310C23#1309399811yescharacteristic
311C23#1309400311yescharacteristic
312Q8—41no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—3762no—
333C2×C4—3762no—
334C2×C4—3762no—
335C2×C4C2×C22⋊Dic4711yescharacteristic
336C2×C4C2×C4⋊Dic4711yescharacteristic
337C2×C4#1309399811yescharacteristic
338C2×C4#1309400311yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—41no—
343C8—41no—
344C8—3762no—
345C8#1309393912yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1568898311yescharacteristic
360C22#1568939311yescharacteristic
361C22#1568939811yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—7522no—
369C4C2×#1309393911yescharacteristic
370C4#1568909711yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—81no—
376C2#2024337611yescharacteristic
377C2#2024344211yescharacteristic
378C2#2024345211yescharacteristic
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#8627
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. #20243417C2semidirect
4. #20241293C2semidirect
14. C2×#131×C47C4semidirect
21. C47×#424C4semidirect
22. C47×#408C4semidirect
24. C47×#405C4semidirect
195. C2×#131Dic47semidirect
204. #424Dic47semidirect
205. #408Dic47semidirect
208. #405Dic47semidirect
214. C47#8627semidirect

Maximal subgroups

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

subgroupindex
#202434262
#202434172
#202419312
#202412932
C47×#29292
#862747