Finite Group Databasebeta

← #26112963

Subgroups — #26112963

Order 12032.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254303C211yesmaximal, characteristic
2#20244212C211yesmaximal, characteristic, complemented
3#20242057C211yesmaximal, characteristic, complemented
4#20241948C211yesmaximal, characteristic, complemented
5#20241248C211yesmaximal, characteristic, complemented
6#20241219C211yesmaximal, characteristic, complemented
7C47×#2932C211yesmaximal, characteristic, complemented
8C2×#13094021C2211yescharacteristic
9#15689499C412yescomplemented
10#15689471C412yescomplemented
11#15689318—22no—
12#15689256—22no—
13#15689176C2211yescharacteristic
14#15689176C2211yescharacteristic
15C2×#13093946—21no—
16C2×#13093946—21no—
17#15688989C2211yescharacteristic
18#15688866—22no—
19#15688857—22no—
20C2×#131×C47—21no—
21C47×#532C2211yescharacteristic
22C47×#435—22no—
23C47×#435—22no—
24C47×#423C2211yescharacteristic
25C47×#408—21no—
26C2×#97×C47C2211yescharacteristic
27C22×D188—21no—
28C22×D188—21no—
29C22×D188—21no—
30C2×#10625334—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×#41×C47—21no—
48#131×C47—42no—
49#131×C47—42no—
50C2×M16×C47—22no—
51C2×M16×C47—21no—
52C22×C8×C47—22no—
53C22×C8×C47D411yescharacteristic
54C22×C8×C47D411yescharacteristic
55C4⋊Q8×C47C2×C411yescharacteristic
56#127×C47C2×C411yescharacteristic
57#124×C47—21no—
58#121×C47—21no—
59#121×C47D412yes—
60C4×D4×C47—21no—
61C4×D4×C47D412yes—
62C2×C42×C47C2311yescharacteristic
63C4⋊C8×C47—21no—
64C4⋊C8×C47D412yes—
65C22⋊C8×C47—24no—
66#97×C47—21no—
67#97×C47—21no—
68C23×D47—41no—
69#41×C47—41no—
70#41×C47—41no—
71#41×C47—41no—
72#41×C47—41no—
73C2×Q8×C47—21no—
74C2×D4×C47—22no—
75C2×D4×C47—21no—
76C2×D4×C47—21no—
77C2×D4×C47C22⋊C412yes—
78C22×C4×C47—22no—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47#4111yescharacteristic
82C22×Dic47—41no—
83C2×D188—22no—
84C2×D188—22no—
85C2×D188—22no—
86C2×D188—41no—
87C2×D188—41no—
88C2×D188—41no—
89C2×D188—41no—
90C2×D188—41no—
91C2×D188—41no—
92C2×D188—41no—
93C2×D188—82no—
94C2×D188—82no—
95M16×C47—44no—
96M16×C47—41no—
97M16×C47—41no—
98C2×C8×C47—21no—
99C2×C8×C47—21no—
100C2×C8×C47—22no—
101C2×C8×C47—21no—
102C2×C8×C47—21no—
103C2×C8×C47—21no—
104C2×C8×C47—44no—
105C2×C8×C47D812yescomplemented
106C4⋊C4×C47—21no—
107C4⋊C4×C47C22⋊C412yes—
108C22⋊C4×C47—22no—
109C22⋊C4×C47—22no—
110C42×C47C22×C411yescharacteristic
111C42×C47C2×D411yescharacteristic
112C42×C47C2×D411yescharacteristic
113C42×C47#4111yescharacteristic
114#10625334—84no—
115C4⋊Dic47—41no—
116C4⋊Dic47—41no—
117C2×C47⋊C8—22no—
118C2×C47⋊C8—41no—
119C23×C47—22no—
120C23×C47#12111yescharacteristic
121C22×D47—42no—
122C22×D47—82no—
123C22×D47—82no—
124C22×D47—81no—
125C22×D47—81no—
126Q8×C47—41no—
127Q8×C47—41no—
128D4×C47—42no—
129D4×C47—42no—
130D4×C47—42no—
131D4×C47—41no—
132D4×C47—41no—
133D4×C47—81no—
134C2×C4×C47—22no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47C2×C22⋊C411yescharacteristic
140C2×C4×C47C4×D411yescharacteristic
141C2×C4×C47C4×D411yescharacteristic
142C2×C4×C47#12011yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12311yescharacteristic
145C2×C4×C47C2×D811yescharacteristic
146C2×C4×C47#13511yescharacteristic
147C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
149C2×Dic47—81no—
150C2×Dic47—81no—
151C2×Dic47—81no—
152D188—44no—
153D188—42no—
154D188—42no—
155D188—82no—
156D188—82no—
157D188—82no—
158D188—81no—
159D188—81no—
160C8×C47—22no—
161C8×C47—42no—
162C8×C47—41no—
163C47⋊C8—42no—
164C47⋊C8—81no—
165#9293—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47#38611yescharacteristic
172C22×C47#45911yescharacteristic
173C22×C47#46311yescharacteristic
174C2×D47—82no—
175C2×D47—82no—
176C2×D47—82no—
177C2×D47—162no—
178C2×D47—162no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—42no—
184C4×C47C4×D811yescharacteristic
185C4×C47#44211yescharacteristic
186C4×C47#44711yescharacteristic
187C4×C47#44911yescharacteristic
188Dic47—161no—
189#3205—471no—
190#2932D4711yescharacteristic, complemented
191#1961—471no—
192#1545—471no—
193#1517—471no—
194#1286—471no—
195#1275—471no—
196C2×C47—21no—
197C2×C47—21no—
198C2×C47—82no—
199C2×C47#187911yescharacteristic
200C2×C47#190311yescharacteristic
201C2×C47#191011yescharacteristic
202D47—162no—
203C2×#131—21no—
204#532C2×D4711yescharacteristic
205C2×#127—471no—
206#493—472no—
207#486—472no—
208#459—942no—
209#447—942no—
210#435—22no—
211#435—22no—
212#431—471no—
213#431—471no—
214#423C2×D4711yescharacteristic
215C2×#102—941no—
216C2×#102—941no—
217#408—21no—
218C2×#97C2×D4711yescharacteristic
219#382—471no—
220#331—942no—
221#327—942no—
222C47#929311yescharacteristic, complemented, Sylow
223C2×#41—21no—
224C22×D4—941no—
225C22×D4—941no—
226C22×D4—941no—
227C2×D8—944no—
228#131—42no—
229#131—42no—
230C2×M16—22no—
231C2×M16—21no—
232C22×C8—22no—
233C22×C8C22⋊D4711yescharacteristic
234C22×C8C22⋊D4711yescharacteristic
235C4⋊Q8C4×D4711yescharacteristic
236#127—472no—
237#127—941no—
238#127—941no—
239#127—941no—
240#127C4×D4711yescharacteristic
241#124—21no—
242#121—21no—
243#121D18812yes—
244C4×D4—21no—
245C4×D4C22⋊D4712yes—
246C2×C4⋊C4—941no—
247C2×C42C22×D4711yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—941no—
250C4⋊C8—941no—
251C4⋊C8C22⋊D4712yes—
252#102—944no—
253#102—1882no—
254#102—1882no—
255C22⋊C8—24no—
256C22⋊C8—941no—
257C22⋊C8—941no—
258#97—21no—
259#97—21no—
260C4×C8—472no—
261C4×C8—941no—
262#95—941no—
263#95—941no—
264C24—1881no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—942no—
274C2×D4—942no—
275C2×D4—942no—
276C2×D4—3762no—
277C2×D4—3762no—
278C2×D4—1881no—
279C2×D4—1881no—
280C2×D4—1881no—
281C2×D4—1881no—
282C2×D4—1881no—
283C2×D4—1881no—
284C2×D4—1881no—
285C2×D4(C2×D47)⋊C412yes—
286C22×C4—22no—
287C22×C4—1881no—
288C22×C4C2×D18811yescharacteristic
289C22×C4#1062535011yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291D8—3764no—
292M16—44no—
293M16—41no—
294M16—41no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—44no—
302C2×C8—942no—
303C2×C8—1881no—
304C2×C8#1062533412yescomplemented
305C4⋊C4—21no—
306C4⋊C4—1881no—
307C4⋊C4—1881no—
308C4⋊C4(C2×D47)⋊C412yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42C2×C4×D4711yescharacteristic
312C42#1062535011yescharacteristic
313C42C2×C22⋊D4711yescharacteristic
314C42C2×C22⋊D4711yescharacteristic
315C23—22no—
316C23—3762no—
317C23—3762no—
318C23—3761no—
319C23—3761no—
320C23—1882no—
321C23#1309399811yescharacteristic
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—3762no—
331D4—3762no—
332D4—3762no—
333D4—3761no—
334D4—3761no—
335D4—1884no—
336D4—1882no—
337D4—1882no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—3761no—
344C2×C4—3761no—
345C2×C4—3761no—
346C2×C4C2×(C2×D47)⋊C411yescharacteristic
347C2×C4C4×C22⋊D4711yescharacteristic
348C2×C4C4×C22⋊D4711yescharacteristic
349C2×C4#1309399311yescharacteristic
350C2×C4#1309399811yescharacteristic
351C2×C4#1309400711yescharacteristic
352C2×C4C2×#1062533411yescharacteristic
353C2×C4#1309404411yescharacteristic
354C2×C4#1309404911yescharacteristic
355C2×C4#1309405011yescharacteristic
356C8—22no—
357C8—42no—
358C8—41no—
359C8—3761no—
360C8—1882no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—3762no—
367C22—3762no—
368C22—3762no—
369C22—7522no—
370C22—7522no—
371C22#1568900211yescharacteristic
372C22#1568931811yescharacteristic
373C22#1568933811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—7521no—
380C4C4×#1062533411yescharacteristic
381C4#1568923311yescharacteristic
382C4#1568925711yescharacteristic
383C4#1568926811yescharacteristic
384C2—21no—
385C2—21no—
386C2—82no—
387C2—7522no—
388C2#2024373611yescharacteristic
389C2#2024386811yescharacteristic
390C2#2024391511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2932C2
socleC22×C47#386
prime radical#20254303C2
cosocleC2×C42×C47C23
hypercenter#2932D47
hypocenterC47#9293
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #20244212C2semidirect
3. #20242057C2semidirect
4. #20241948C2semidirect
5. #20241248C2semidirect
6. #20241219C2semidirect
7. C47×#2932C2semidirect
9. #15689499C4semidirect
10. #15689471C4semidirect
105. C2×C8×C47D8semidirect
190. #2932D47semidirect
222. C47#9293semidirect
304. C2×C8#10625334semidirect

Maximal subgroups

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

subgroupindex
#202543032
#202442122
#202420572
#202419482
#202412482
#202412192
C47×#29322
#929347