Finite Group Databasebeta

← #26112970

Subgroups — #26112970

Order 12032.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254326C211yesmaximal, characteristic
2#20244212C211yesmaximal, characteristic
3#20242057C211yesmaximal, characteristic
4#20241952C211yesmaximal, characteristic, complemented
5#20241248C211yesmaximal, characteristic, complemented
6#20241219C211yesmaximal, characteristic, complemented
7C47×#2961C211yesmaximal, characteristic, complemented
8C2×#13094021C2211yescharacteristic
9#15689497C412yescomplemented
10#15689471C412yescomplemented
11#15689323—22no—
12#15689273—22no—
13#15689176C2211yescharacteristic
14#15689176C2211yescharacteristic
15C2×#13093946—21no—
16C2×#13093946—21no—
17#15688989C2211yescharacteristic
18#15688866—22no—
19#15688860—22no—
20C47×#533C2211yescharacteristic
21C47×#446—22no—
22C47×#446—22no—
23C47×#424—21no—
24C47×#423C2211yescharacteristic
25C47×#405—21no—
26C2×#97×C47C2211yescharacteristic
27C22×D188—21no—
28C22×D188—21no—
29C22×D188—21no—
30C2×#10625336—24no—
31#13094021—21no—
32#13094021—21no—
33#13094021—21no—
34#13094021C2×C412yes—
35C2×C4⋊Dic47—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38#13093946—42no—
39#13093946—42no—
40#13093944—24no—
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43C4×C47⋊C8—21no—
44C4×C47⋊C8C2×C412yes—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C2×M16×C47—21no—
48C22×C8×C47D411yescharacteristic
49C22×C8×C47D411yescharacteristic
50C4⋊Q8×C47C2×C411yescharacteristic
51C4⋊Q8×C47C2×C411yescharacteristic
52#125×C47—21no—
53C22⋊Q8×C47—21no—
54C22⋊Q8×C47D412yes—
55C4×Q8×C47—21no—
56C4×Q8×C47D412yes—
57#117×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—24no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47D412yes—
62#97×C47—22no—
63#97×C47—21no—
64#97×C47—21no—
65C4×C8×C47—22no—
66C23×D47—41no—
67C2×Q8×C47—21no—
68C2×Q8×C47C22⋊C412yes—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—41no—
73C2×D188—22no—
74C2×D188—22no—
75C2×D188—22no—
76C2×D188—41no—
77C2×D188—41no—
78C2×D188—41no—
79C2×D188—41no—
80C2×D188—41no—
81C2×D188—41no—
82C2×D188—41no—
83C2×D188—82no—
84C2×D188—82no—
85M16×C47—41no—
86M16×C47—41no—
87C2×C8×C47—21no—
88C2×C8×C47—21no—
89C2×C8×C47—21no—
90C2×C8×C47—22no—
91C2×C8×C47—21no—
92C2×C8×C47—21no—
93C2×C8×C47SD1612yescomplemented
94C4⋊C4×C47—21no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—21no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47C22⋊C412yes—
100C22⋊C4×C47—22no—
101C42×C47—22no—
102C42×C47C22×C411yescharacteristic
103C42×C47C2×D411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106#10625336—84no—
107C4⋊Dic47—41no—
108C4⋊Dic47—41no—
109C2×C47⋊C8—22no—
110C2×C47⋊C8—41no—
111C23×C47#12111yescharacteristic
112C22×D47—42no—
113C22×D47—82no—
114C22×D47—82no—
115C22×D47—81no—
116C22×D47—81no—
117Q8×C47—42no—
118Q8×C47—81no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47C2×C22⋊C411yescharacteristic
124C2×C4×C47C4×D411yescharacteristic
125C2×C4×C47C4×D411yescharacteristic
126C2×C4×C47#12011yescharacteristic
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47#12311yescharacteristic
129C2×C4×C47C2×SD1611yescharacteristic
130C2×C4×C47#13511yescharacteristic
131C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C47#13711yescharacteristic
133C2×Dic47—81no—
134C2×Dic47—81no—
135C2×Dic47—81no—
136D188—44no—
137D188—42no—
138D188—42no—
139D188—82no—
140D188—82no—
141D188—82no—
142D188—81no—
143D188—81no—
144C8×C47—22no—
145C8×C47—41no—
146C8×C47—42no—
147C47⋊C8—42no—
148C47⋊C8—81no—
149#9294—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47#38611yescharacteristic
153C22×C47#46011yescharacteristic
154C22×C47#46411yescharacteristic
155C2×D47—82no—
156C2×D47—82no—
157C2×D47—82no—
158C2×D47—162no—
159C2×D47—162no—
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—42no—
165C4×C47—42no—
166C4×C47C4×SD1611yescharacteristic
167C4×C47#44011yescharacteristic
168C4×C47#44811yescharacteristic
169C4×C47#44911yescharacteristic
170Dic47—161no—
171#3208—471no—
172#2961D4711yescharacteristic, complemented
173#1961—471no—
174#1545—471no—
175#1518—471no—
176#1286—471no—
177#1275—471no—
178C2×C47—21no—
179C2×C47—21no—
180C2×C47#188011yescharacteristic
181C2×C47#190311yescharacteristic
182C2×C47#191011yescharacteristic
183D47—162no—
184#533C2×D4711yescharacteristic
185C2×#127—471no—
186#492—472no—
187#486—472no—
188#460—942no—
189#450—942no—
190#446—22no—
191#446—22no—
192#431—471no—
193#431—471no—
194#424—21no—
195#423C2×D4711yescharacteristic
196C2×#102—941no—
197C2×#102—941no—
198#405—21no—
199C2×#97C2×D4711yescharacteristic
200#382—471no—
201#331—942no—
202#328—942no—
203C47#929411yescharacteristic, complemented, Sylow
204C22×D4—941no—
205C22×D4—941no—
206C22×D4—941no—
207C2×SD16—944no—
208C2×M16—21no—
209C22×C8C22⋊D4711yescharacteristic
210C22×C8C22⋊D4711yescharacteristic
211C4⋊Q8C4×D4711yescharacteristic
212C4⋊Q8C4×D4711yescharacteristic
213#127—472no—
214#127—941no—
215#127—941no—
216#127—941no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D18812yes—
220C4×Q8—21no—
221C4×Q8C22⋊D4712yes—
222#117—21no—
223C2×C4⋊C4—941no—
224C2×C42C22×D4711yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—941no—
228C4⋊C8—941no—
229C4⋊C8C22⋊D4712yes—
230#102—944no—
231#102—1882no—
232#102—1882no—
233C22⋊C8—941no—
234C22⋊C8—941no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—472no—
240C4×C8—941no—
241#95—941no—
242#95—941no—
243C24—1881no—
244C2×Q8—21no—
245C2×Q8(C2×D47)⋊C412yes—
246C2×D4—942no—
247C2×D4—942no—
248C2×D4—942no—
249C2×D4—3762no—
250C2×D4—3762no—
251C2×D4—1881no—
252C2×D4—1881no—
253C2×D4—1881no—
254C2×D4—1881no—
255C2×D4—1881no—
256C2×D4—1881no—
257C2×D4—1881no—
258C22×C4—1881no—
259C22×C4C2×D18811yescharacteristic
260C22×C4#1062535011yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262SD16—3764no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—942no—
272C2×C8—1881no—
273C2×C8#1062533612yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—1881no—
280C4⋊C4—1881no—
281C4⋊C4(C2×D47)⋊C412yes—
282C22⋊C4—22no—
283C42—22no—
284C42C2×C4×D4711yescharacteristic
285C42#1062535011yescharacteristic
286C42C2×C22⋊D4711yescharacteristic
287C42C2×C22⋊D4711yescharacteristic
288C23—3762no—
289C23—3762no—
290C23—3761no—
291C23—3761no—
292C23—1882no—
293C23#1309399811yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—3762no—
297D4—3762no—
298D4—3762no—
299D4—3761no—
300D4—3761no—
301D4—1884no—
302D4—1882no—
303D4—1882no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—3761no—
309C2×C4—3761no—
310C2×C4—3761no—
311C2×C4C2×(C2×D47)⋊C411yescharacteristic
312C2×C4C4×C22⋊D4711yescharacteristic
313C2×C4C4×C22⋊D4711yescharacteristic
314C2×C4#1309399311yescharacteristic
315C2×C4#1309399811yescharacteristic
316C2×C4#1309400711yescharacteristic
317C2×C4C2×#1062533611yescharacteristic
318C2×C4#1309404411yescharacteristic
319C2×C4#1309404911yescharacteristic
320C2×C4#1309405511yescharacteristic
321C8—22no—
322C8—41no—
323C8—42no—
324C8—3761no—
325C8—1882no—
326C22—21no—
327C22—21no—
328C22—3762no—
329C22—3762no—
330C22—3762no—
331C22—7522no—
332C22—7522no—
333C22#1568900211yescharacteristic
334C22#1568932311yescharacteristic
335C22#1568934511yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—7521no—
343C4C4×#1062533611yescharacteristic
344C4#1568922111yescharacteristic
345C4#1568926211yescharacteristic
346C4#1568926811yescharacteristic
347C2—21no—
348C2—21no—
349C2—7522no—
350C2#2024374311yescharacteristic
351C2#2024386811yescharacteristic
352C2#2024391511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2961C2
socleC22×C47#386
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#2961D47
hypocenterC47#9294
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
4. #20241952C2semidirect
5. #20241248C2semidirect
6. #20241219C2semidirect
7. C47×#2961C2semidirect
9. #15689497C4semidirect
10. #15689471C4semidirect
93. C2×C8×C47SD16semidirect
172. #2961D47semidirect
203. C47#9294semidirect
273. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
#202543262
#202442122
#202420572
#202419522
#202412482
#202412192
C47×#29612
#929447