Finite Group Databasebeta

← #26132298

Subgroups — #26132298

Order 12032.

420 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245029C212yesmaximal
2#20243438C211yesmaximal, characteristic, complemented
3#20243434C211yesmaximal, characteristic, complemented
4#20241517C212yesmaximal, complemented
5C47×#2930C211yesmaximal, characteristic
6C2×#13093960—21no—
7#15689161—21no—
8#15689107C2212yes—
9#15689097C2212yes—
10#15688913—22no—
11#15688910—22no—
12C47⋊#338—21no—
13#15688880—21no—
14C2×#131×C47C2211yescharacteristic
15C22×M16×C47C411yescharacteristic, complemented
16C47×#525C2211yescharacteristic
17C47×#436—21no—
18C47×#436—21no—
19C47×#435—21no—
20C47×#435—21no—
21C47×#424C411yescharacteristic, complemented
22C47×#408C2211yescharacteristic
23C47×#407C411yescharacteristic, complemented
24C47×#405C411yescharacteristic, complemented
25C2×#10625329—22no—
26#13093983—22no—
27#13093960—41no—
28#13093960—41no—
29#13093958—41no—
30#13093956—41no—
31#13093943—22no—
32#13093941—22no—
33#13093939—22no—
34#13093939—22no—
35C2×#41×C47C2311yescharacteristic
36C23×C4×C47C2×C411yescharacteristic
37#131×C47—41no—
38#131×C47—41no—
39C2×M16×C47—21no—
40C2×M16×C47—21no—
41C2×M16×C47—41no—
42C2×M16×C47—41no—
43C2×M16×C47C2×C411yescharacteristic
44C2×M16×C47C2×C411yescharacteristic
45C2×M16×C47D411yescharacteristic
46C2×M16×C47D411yescharacteristic
47C22×C8×C47C2×C411yescharacteristic
48C22×C8×C47C2×C411yescharacteristic
49C22×C8×C47D411yescharacteristic
50C22×C8×C47D411yescharacteristic
51#123×C47D411yescharacteristic
52#123×C47Q811yescharacteristic
53C22⋊Q8×C47—21no—
54#121×C47—21no—
55#120×C47D411yescharacteristic
56#120×C47Q811yescharacteristic
57C4×D4×C47—21no—
58C4×D4×C47—21no—
59#117×C47C2×C411yescharacteristic
60C4⋊C8×C47—21no—
61C4⋊C8×C47—21no—
62C22⋊C8×C47—21no—
63C22⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65C22⋊C8×C47—21no—
66#97×C47—21no—
67C4×C8×C47—21no—
68C24×C47C4⋊C411yescharacteristic
69#41×C47—41no—
70#41×C47—41no—
71C2×Q8×C47#4111yescharacteristic
72C2×D4×C47—21no—
73C2×D4×C47—21no—
74C2×D4×C47C2×D411yescharacteristic
75C2×D4×C47C2×Q811yescharacteristic
76C2×D4×C47#4111yescharacteristic
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—41no—
80C22×C4×C47—41no—
81C22×C4×C47C4⋊C411yescharacteristic
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85M16×C47—41no—
86M16×C47—41no—
87M16×C47—41no—
88M16×C47—42no—
89M16×C47—41no—
90M16×C47—41no—
91M16×C47—41no—
92C2×C8×C47—21no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47—21no—
98C2×C8×C47—21no—
99C2×C8×C47—21no—
100C2×C8×C47—41no—
101C2×C8×C47C22⋊C411yescharacteristic
102C2×C8×C47C22⋊C411yescharacteristic
103C2×C8×C47C22⋊C411yescharacteristic
104C2×C8×C47C22⋊C411yescharacteristic
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C22⋊C4×C47—21no—
109C22⋊C4×C47—21no—
110C22⋊C4×C47—21no—
111C22⋊C4×C47—21no—
112C22⋊C4×C47C4⋊C411yescharacteristic
113C22⋊C4×C47C4⋊C411yescharacteristic
114C42×C47—21no—
115C22⋊Dic47—42no—
116C4⋊Dic47—42no—
117C4×Dic47—42no—
118#10625329—42no—
119#10625329—42no—
120C2×C47⋊C8—42no—
121C23×C47—21no—
122C23×C47—41no—
123C23×C47—81no—
124C23×C47C2×C4⋊C411yescharacteristic
125C23×C47#12111yescharacteristic
126C23×C47C22⋊Q811yescharacteristic
127Q8×C47—41no—
128D4×C47—41no—
129D4×C47—41no—
130D4×C47—41no—
131D4×C47—81no—
132D4×C47—81no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—42no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47C2×C22⋊C411yescharacteristic
147C2×C4×C47#11711yescharacteristic
148C2×C4×C47#12111yescharacteristic
149C2×C4×C47C22⋊Q811yescharacteristic
150C2×Dic47—82no—
151C8×C47—21no—
152C8×C47—21no—
153C8×C47—21no—
154C8×C47—41no—
155C8×C47—41no—
156C8×C47#9911yescharacteristic
157C8×C47#9911yescharacteristic
158C47⋊C8—82no—
159#12806—471nomaximal, Sylow
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—41no—
169C22×C47—82no—
170C22×C47#38011yescharacteristic
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—41no—
175C4×C47—41no—
176C4×C47—41no—
177C4×C47C2×#9911yescharacteristic
178C4×C47#41011yescharacteristic
179Dic47—162no—
180#2930D4711yescharacteristic
181#2120—472no—
182#1813—471no—
183#1812—471no—
184#1395—472no—
185C2×C47—21no—
186C2×C47—41no—
187C2×C47—41no—
188C2×C47—41no—
189C2×C47—81no—
190C2×C47#179911yescharacteristic
191C2×#131C2×D4711yescharacteristic
192C22×M16Dic4711yescharacteristic, complemented
193#525C2×D4711yescharacteristic
194#436—21no—
195#436—21no—
196#435—21no—
197#435—21no—
198C2×#108—941no—
199#427—941no—
200#424Dic4711yescharacteristic, complemented
201#413—472no—
202#410—472no—
203#408C2×D4711yescharacteristic
204#407Dic4711yescharacteristic, complemented
205#405Dic4711yescharacteristic, complemented
206#352—942no—
207AΣL(1,𝔽16)—942no—
208#338—941no—
209H(3,ℤ/4)—941no—
210C47#1280611yescharacteristic, complemented, Sylow
211C2×#41C22×D4711yescharacteristic
212C23×C4C2×Dic4711yescharacteristic
213#131—41no—
214#131—41no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—41no—
218C2×M16—41no—
219C2×M16—942no—
220C2×M16C2×Dic4711yescharacteristic
221C2×M16C2×Dic4711yescharacteristic
222C2×M16C22⋊D4711yescharacteristic
223C2×M16C22⋊D4711yescharacteristic
224C22×C8C2×Dic4711yescharacteristic
225C22×C8C2×Dic4711yescharacteristic
226C22×C8C22⋊D4711yescharacteristic
227C22×C8C22⋊D4711yescharacteristic
228#123Dic9411yescharacteristic
229#123D18811yescharacteristic
230C22⋊Q8—21no—
231#121—21no—
232#120Dic9411yescharacteristic
233#120D18811yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236#117—942no—
237#117C2×Dic4711yescharacteristic
238#108—1881no—
239#108—1881no—
240#107—1881no—
241#106—1881no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244#101—942no—
245#100—942no—
246#99—942no—
247#99—942no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252#97—21no—
253C4×C8—21no—
254C24C4⋊Dic4711yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8#1062535011yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4C2×Dic9411yescharacteristic
261C2×D4C2×D18811yescharacteristic
262C2×D4#1062535011yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C4⋊Dic4711yescharacteristic
268C22×C4C22×Dic4711yescharacteristic
269C22×C4C2×C22⋊D4711yescharacteristic
270C22×C4C2×C22⋊D4711yescharacteristic
271M16—41no—
272M16—41no—
273M16—41no—
274M16—42no—
275M16—41no—
276M16—41no—
277M16—41no—
278M16—1882no—
279M16—1882no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—1882no—
290C2×C8C22⋊Dic4711yescharacteristic
291C2×C8C22⋊Dic4711yescharacteristic
292C2×C8C22⋊Dic4711yescharacteristic
293C2×C8C22⋊Dic4711yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—1882no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1882no—
303C22⋊C4C4⋊Dic4711yescharacteristic
304C22⋊C4C4⋊Dic4711yescharacteristic
305C42—21no—
306C42—1882no—
307C23—21no—
308C23—41no—
309C23—81no—
310C23C2×C4⋊Dic4711yescharacteristic
311C23#1309399811yescharacteristic
312C23#1309400311yescharacteristic
313Q8—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
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—21no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—3762no—
333C2×C4C2×C22⋊Dic4711yescharacteristic
334C2×C4#1309398311yescharacteristic
335C2×C4#1309399811yescharacteristic
336C2×C4#1309400311yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—41no—
341C8—41no—
342C8—3762no—
343C8#1309393911yescharacteristic
344C8#1309393911yescharacteristic
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—82no—
355C22#1568898311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—7522no—
363C4C2×#1309393911yescharacteristic
364C4#1568909711yescharacteristic
365C2—21no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2—81no—
370C2#2024337611yescharacteristic
subgroupG/H
centerC4#15689097
commutatorC22×C4×C47C22×C4
FrattiniC2×#41C22×D47
FittingC47×#2930C2
socleC2×C47#1799
prime radicalC47×#525C22
cosocleC2×#41×C47C23
hypercenter#2930D47
hypocenterC47#12806
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #2930

Complements

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

NH ≅ G/Ntype
2. #20243438C2semidirect
3. #20243434C2semidirect
4. #20241517C2semidirect
15. C22×M16×C47C4semidirect
21. C47×#424C4semidirect
23. C47×#407C4semidirect
24. C47×#405C4semidirect
192. C22×M16Dic47semidirect
200. #424Dic47semidirect
204. #407Dic47semidirect
205. #405Dic47semidirect
210. C47#12806semidirect

Maximal subgroups

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

subgroupindex
#202450292
#202434382
#202434342
#202415172
C47×#29302
#1280647