Finite Group Databasebeta

← #26316835

Subgroups — #26316835

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246113C211yesmaximal, characteristic
2#20244773C212yesmaximal, complemented
3#20244759C212yesmaximal, complemented
4#20244211C211yesmaximal, characteristic
5#20244193C211yesmaximal, characteristic
6#20243648C212yesmaximal, complemented
7#20243634C212yesmaximal, complemented
8#20243554C211yesmaximal, characteristic, complemented
9#20243518C211yesmaximal, characteristic, complemented
10#20243245C211yesmaximal, characteristic, complemented
11C47×#2647C211yesmaximal, characteristic, complemented
12C2×#13094021C2211yescharacteristic
13C2×C4×D188C2211yescharacteristic
14C2×C4×D188C2211yescharacteristic
15C2×#13093957C2212yes—
16C2×#13093955C2212yes—
17C2×C4⋊(C47⋊C8)C2212yes—
18C2×#13093944C2212yes—
19C2×#13093944C2212yes—
20C2×#13093944C2212yes—
21C2×#13093944C2212yes—
22C2×#13093935C2211yescharacteristic
23C2×C4×C47⋊C8C2211yescharacteristic
24#15689025C2211yescharacteristic
25#15689025C2211yescharacteristic
26#15689001—22no—
27#15689001—22no—
28#15688989C2211yescharacteristic
29#15688885C2212yes—
30#15688885C2212yes—
31#15688878C2212yes—
32C47×#400—22no—
33C47×#398—22no—
34C47×#389C2212yes—
35C47×#389C2212yes—
36C47×#384C2211yescharacteristic
37C47×#382C2211yescharacteristic
38C47×#376C2211yescharacteristic
39C22×D188—22no—
40C22×D188C2311yescharacteristic
41C22×D188C2311yescharacteristic
42C22×C4×D47—21no—
43C22×C4×D47—21no—
44C22×C47⋊C8C2312yes—
45C22×C47⋊C8C2312yes—
46#13094021—22no—
47#13094021Q814yes—
48C4×D188—42no—
49C4×D188—42no—
50C2×C4⋊Dic47C2311yescharacteristic
51C2×C4⋊Dic47C2311yescharacteristic
52C2×(C2×D47)⋊C4—21no—
53C2×(C2×D47)⋊C4—21no—
54C2×(C2×D47)⋊C4—22no—
55C2×(C2×D47)⋊C4—22no—
56#13093957—42no—
57#13093955—42no—
58C4⋊(C47⋊C8)—24no—
59#13093944—24no—
60#13093944—24no—
61#13093944—42no—
62#13093944—42no—
63#13093935—22no—
64C4×C47⋊C8—22no—
65C47⋊#95—22no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47C2312yes—
70C2×C4⋊C4×C47C2312yes—
71C2×C42×C47C2311yescharacteristic
72C2×C42×C47C2311yescharacteristic
73C2×C42×C47C2311yescharacteristic
74#95×C47—22no—
75#95×C47—22no—
76#95×C47—22no—
77#95×C47D412yes—
78#95×C47D412yes—
79C23×D47—21no—
80C23×D47—21no—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2411yescharacteristic
88C22×Dic47—21no—
89C22×Dic47—21no—
90C2×D188—22no—
91C2×D188—22no—
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—21no—
95C2×D188—21no—
96C2×D188—28no—
97C2×D188—82no—
98C2×D188—82no—
99C2×D188C2×Q812yes—
100C2×D188C2×Q812yes—
101C2×C4×D47—42no—
102C2×C4×D47—42no—
103C2×C4×D47—82no—
104C2×C4×D47—82no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47—22no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—82no—
110C4⋊C4×C47—82no—
111C4⋊C4×C47—82no—
112C42×C47—22no—
113C42×C47—22no—
114C42×C47C2×Q812yes—
115C42×C47#4112yes—
116(C2×D47)⋊C4—42no—
117(C2×D47)⋊C4—42no—
118(C2×D47)⋊C4—82no—
119(C2×D47)⋊C4—82no—
120C4⋊Dic47—22no—
121C4⋊Dic47—22no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8#4114yes—
128C23×C47C22×D411yescharacteristic
129C22×D47—22no—
130C22×D47—22no—
131C22×D47—41no—
132C22×D47—41no—
133C22×D47—41no—
134C22×D47—41no—
135C22×D47—42no—
136C22×D47—42no—
137C22×D47—42no—
138C22×D47—42no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—22no—
147C2×C4×C47—22no—
148C2×C4×C47—42no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47C22⋊Q814yes—
155C2×C4×C47C22×Q811yescharacteristic
156C2×C4×C47C2×#4112yes—
157C2×C4×C47C2×#4111yescharacteristic
158C2×C4×C47H(5,𝔽2)12yes—
159C2×Dic47—41no—
160C2×Dic47—41no—
161C2×Dic47—42no—
162C2×Dic47—42no—
163D188—24no—
164D188—24no—
165D188—42no—
166D188—42no—
167D188—88no—
168C4×D47—84no—
169C4×D47—84no—
170C47⋊C8—24no—
171C47⋊C8—42no—
172#29204—471nomaximal, Sylow
173C22×C47C2×C22⋊Q811yescharacteristic
174C22×C47#53211yescharacteristic
175C22×C47#53511yescharacteristic
176C22×C47#57612yes—
177C22×C47#57712yes—
178C2×D47—42no—
179C2×D47—42no—
180C2×D47—44no—
181C2×D47—44no—
182C2×D47—82no—
183C2×D47—82no—
184C2×D47—81no—
185C2×D47—81no—
186C2×D47—81no—
187C2×D47—81no—
188C4×C47—24no—
189C4×C47—42no—
190C4×C47—41no—
191C4×C47—41no—
192C4×C47—82no—
193C4×C47—82no—
194C4×C47#55312yes—
195C4×C47#55412yes—
196Dic47—81no—
197Dic47—81no—
198#2647D4711yescharacteristic, complemented
199#2328—471no—
200#2067—472no—
201#2065—472no—
202#1961—471no—
203#1957—471no—
204#1864—472no—
205#1862—472no—
206#1847—471no—
207#1835—471no—
208#1768—471no—
209C2×C47#256611yescharacteristic
210C2×C47#308112yes—
211C2×C47#317912yes—
212C2×C47#322011yescharacteristic
213C2×C47#322211yescharacteristic
214D47—82no—
215D47—82no—
216C2×#127—471no—
217C2×C4×D4—471no—
218C2×C4×D4—471no—
219C2×#107—472no—
220C2×#106—472no—
221C2×C4⋊C8—472no—
222C2×#102—472no—
223C2×#102—472no—
224C2×#102—472no—
225C2×#102—472no—
226C2×#97—471no—
227C2×C4×C8—471no—
228#400—22no—
229#398—22no—
230#390—471no—
231#390—471no—
232#389C2×D4712yes—
233#389C2×D4712yes—
234#386—942no—
235#386—942no—
236#384C2×D4711yescharacteristic
237#382—471no—
238#382C2×D4711yescharacteristic
239#376C2×D4711yescharacteristic
240#340—472no—
241#340—472no—
242#336—472no—
243C47#2920411yescharacteristic, complemented, Sylow
244C22×D4—471no—
245C22×D4—471no—
246C22×D4—942no—
247C23×C4—941no—
248C23×C4—941no—
249C22×C8—472no—
250C22×C8—472no—
251#127—474no—
252#127—942no—
253C4×D4—1882no—
254C4×D4—1882no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—471no—
259C2×C4⋊C4—471no—
260C2×C4⋊C4C22×D4712yes—
261C2×C4⋊C4C22×D4712yes—
262C2×C22⋊C4—942no—
263C2×C22⋊C4—942no—
264C2×C22⋊C4—941no—
265C2×C22⋊C4—941no—
266C2×C42C22×D4711yescharacteristic
267C2×C42C22×D4711yescharacteristic
268C2×C42C22×D4711yescharacteristic
269#107—1882no—
270#106—1882no—
271C4⋊C8—944no—
272#102—944no—
273#102—944no—
274#102—1882no—
275#102—1882no—
276#97—942no—
277C4×C8—942no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—942no—
282#95C22⋊D4712yes—
283#95C22⋊D4712yes—
284C24—941no—
285C24—941no—
286C2×D4—472no—
287C2×D4—472no—
288C2×D4—948no—
289C2×D4—942no—
290C2×D4—942no—
291C2×D4—941no—
292C2×D4—941no—
293C2×D4—942no—
294C2×D4—942no—
295C2×D4—3762no—
296C2×D4—3762no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—941no—
300C22×C4—941no—
301C22×C4—3762no—
302C22×C4—3762no—
303C22×C4—1882no—
304C22×C4—1882no—
305C22×C4C2×C22⋊D4711yescharacteristic
306C22×C4C2×C22⋊D4712yes—
307C22×C4C2×C22⋊D4711yescharacteristic
308C22×C4C2×C22⋊D4712yes—
309C22×C4C23×D4711yescharacteristic
310C2×C8—474no—
311C2×C8—942no—
312C2×C8—942no—
313C2×C8—942no—
314C2×C8—942no—
315C2×C8—942no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—942no—
324C4⋊C4—942no—
325C22⋊C4—3762no—
326C22⋊C4—3762no—
327C22⋊C4—1882no—
328C22⋊C4—1882no—
329C42—22no—
330C42—22no—
331C42Q8×D4712yes—
332C42#1062535412yes—
333C23—942no—
334C23—942no—
335C23—1881no—
336C23—1881no—
337C23—1882no—
338C23—1882no—
339C23—1882no—
340C23—1882no—
341C23—1881no—
342C23—1881no—
343C23C22×C22⋊D4711yescharacteristic
344D4—944no—
345D4—944no—
346D4—3768no—
347D4—1882no—
348D4—1882no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—3764no—
365C2×C4—3764no—
366C2×C4—1882no—
367C2×C4—1882no—
368C2×C4—1881no—
369C2×C4—1881no—
370C2×C4#1309400014yes—
371C2×C4C2×Q8×D4711yescharacteristic
372C2×C4#41×D4712yes—
373C2×C4C2×#1062535411yescharacteristic
374C2×C4#1309407012yes—
375C8—944no—
376C8—1882no—
377C22—3762no—
378C22—3762no—
379C22—3761no—
380C22—3761no—
381C22—3761no—
382C22—3761no—
383C22—1882no—
384C22—1882no—
385C22—1884no—
386C22—1884no—
387C22C2×#1309400011yescharacteristic
388C22#1568966911yescharacteristic
389C22#1568968711yescharacteristic
390C22#1568992612yes—
391C22#1568993112yes—
392C4—24no—
393C4—42no—
394C4—41no—
395C4—41no—
396C4—82no—
397C4—82no—
398C4—3761no—
399C4—3761no—
400C4#1568982612yes—
401C4#1568982912yes—
402C2—3762no—
403C2—3762no—
404C2#2024858911yescharacteristic
405C2#2025321312yes—
406C2#2025406512yes—
407C2#2025443211yescharacteristic
408C2#2025445411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2647C2
socleC23×C47C22×D4
prime radicalC2×#13094021C22
hypercenter#2647D47
hypocenterC47#29204
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2647

Complements

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

NH ≅ G/Ntype
2. #20244773C2semidirect
3. #20244759C2semidirect
6. #20243648C2semidirect
7. #20243634C2semidirect
8. #20243554C2semidirect
9. #20243518C2semidirect
10. #20243245C2semidirect
11. C47×#2647C2semidirect
198. #2647D47semidirect
243. C47#29204semidirect

Maximal subgroups

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

subgroupindex
#202461132
#202447732
#202447592
#202442112
#202441932
#202436482
#202436342
#202435542
#202435182
#202432452
C47×#26472
#2920447