Finite Group Databasebeta

← #26110273

Subgroups — #26110273

Order 12032.

612 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246006C211yesmaximal, characteristic
2#20243578C212yesmaximal
3#20241641C211yesmaximal, characteristic
4#20241268C212yesmaximal
5C2×C47×#324C211yesmaximal, characteristic
6C2×#13093983—21no—
7C22×C4⋊Dic47C2211yescharacteristic
8C2×#13093958—22no—
9#15689022—22no—
10#15689022—22no—
11#15688983—21no—
12#15688983C2212yes—
13C2×C47⋊#95—21no—
14C2×C47⋊#95—21no—
15C2×C47⋊#95C2211yescharacteristic
16#15688886—22no—
17C22×C4⋊C4×C47C2211yescharacteristic
18C2×C22⋊C8×C47C2212yes—
19C47×#324C414yescomplemented
20C23×Dic47—21no—
21C23×Dic47—21no—
22#13093983—42no—
23#13093983—42no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—42no—
30C2×C4⋊Dic47D412yes—
31C2×C4⋊Dic47D412yes—
32C2×C22⋊Dic47—22no—
33C2×C4×Dic47—22no—
34#13093958—44no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—42no—
39C47⋊#95—42no—
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C22×C8×C47—22no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47—22no—
46C2×C4⋊C4×C47C2×C412yes—
47C2×C4⋊C4×C47Q812yes—
48C22⋊C8×C47—22no—
49C22⋊C8×C47C2×C414yes—
50C24×C47C2×D411yescharacteristic
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47C4⋊C412yes—
62C22×C4×C47C4⋊C412yes—
63C22×C4×C47D812yes—
64C22×C4×C47Q1612yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D412yes—
67C22×C4×C47C2×Q811yescharacteristic
68C22×C4×C47#4112yes—
69C22×Dic47—22no—
70C22×Dic47—22no—
71C22×Dic47—22no—
72C22×Dic47—22no—
73C22×Dic47—41no—
74C22×Dic47—41no—
75C22×Dic47—41no—
76C22×Dic47—41no—
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C22×Dic47—41no—
81C22×Dic47—41no—
82C22×Dic47—41no—
83C22×Dic47—41no—
84C22×Dic47—41no—
85C2×C8×C47—24no—
86C2×C8×C47—42no—
87C2×C8×C47—42no—
88C4⋊C4×C47—24no—
89C4⋊C4×C47—24no—
90C22⋊Dic47—42no—
91C22⋊Dic47—42no—
92C4⋊Dic47—24no—
93C4⋊Dic47—42no—
94C4⋊Dic47—42no—
95C4⋊Dic47—42no—
96C4⋊Dic47—42no—
97C4⋊Dic47—42no—
98C4⋊Dic47—81no—
99C4⋊Dic47—81no—
100C4×Dic47—44no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47C2×C4⋊C411yescharacteristic
106C23×C47#12011yescharacteristic
107C23×C47#12311yescharacteristic
108C23×C47C2×D811yescharacteristic
109C23×C47C2×Q1611yescharacteristic
110C23×C47AGL(1,ℤ/8)11yescharacteristic
111C23×C47#13711yescharacteristic
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—81no—
128C2×C4×C47#10714yescomplemented
129C2×C4×C47C4×D412yes—
130C2×C4×C47C22⋊Q812yes—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—81no—
144C2×Dic47—81no—
145C2×Dic47—81no—
146C2×Dic47—81no—
147C2×Dic47—81no—
148C2×Dic47—81no—
149C2×Dic47—81no—
150C2×Dic47—81no—
151C8×C47—44no—
152#8856—471nomaximal, Sylow
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
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#38511yescharacteristic
166C22×C47C2×#10711yescharacteristic
167C22×C47#42811yescharacteristic
168C22×C47#44711yescharacteristic
169C22×C47#45111yescharacteristic
170C22×C47#45411yescharacteristic
171C22×C47#45611yescharacteristic
172C22×C47B(4,𝔽2)11yescharacteristic
173C22×C47#45811yescharacteristic
174C22×C47#48011yescharacteristic
175C22×C47#48411yescharacteristic
176C4×C47—24no—
177C4×C47—22no—
178C4×C47—22no—
179C4×C47—42no—
180Dic47—82no—
181Dic47—82no—
182Dic47—82no—
183Dic47—82no—
184#2316—471no—
185#1852—472no—
186C2×#324D4711yescharacteristic
187#1441—471no—
188#1294—472no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47#159811yescharacteristic
194C2×C47#160511yescharacteristic
195C2×C47#161011yescharacteristic
196C2×C47#161711yescharacteristic
197C2×C47#185211yescharacteristic
198C2×C47#185711yescharacteristic
199C2×C47#186711yescharacteristic
200C2×#117—941no—
201C22×C4⋊C4—471no—
202C22×C4⋊C4C2×D4711yescharacteristic
203C2×#107—942no—
204C2×C22⋊C8C2×D4712yes—
205#389—942no—
206#389—942no—
207#380—472no—
208#380—941no—
209C2×#95—471no—
210C2×#95—941no—
211C2×#95—941no—
212#340—942no—
213#324Dic4714yescomplemented
214C47#885611yescharacteristic, complemented, Sylow
215C23×C4—941no—
216C23×C4—941no—
217C23×C4D18811yescharacteristic
218C23×C4D18811yescharacteristic
219C23×C4C22×D4711yescharacteristic
220C22×C8—22no—
221#117—1882no—
222#117—1882no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—472no—
226C2×C4⋊C4—472no—
227C2×C4⋊C4—942no—
228C2×C4⋊C4—941no—
229C2×C4⋊C4—941no—
230C2×C4⋊C4—941no—
231C2×C4⋊C4—941no—
232C2×C4⋊C4—1882no—
233C2×C4⋊C4Dic9412yes—
234C2×C4⋊C4C2×Dic4712yes—
235C2×C22⋊C4—942no—
236C2×C42—942no—
237#107—1884no—
238C22⋊C8—22no—
239C22⋊C8C2×Dic4714yes—
240#95—942no—
241#95—942no—
242#95—942no—
243#95—1882no—
244#95—1882no—
245C24C2×D18811yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—942no—
257C22×C4—942no—
258C22×C4—942no—
259C22×C4—942no—
260C22×C4—1881no—
261C22×C4—1881no—
262C22×C4—1881no—
263C22×C4—1881no—
264C22×C4—1881no—
265C22×C4—1881no—
266C22×C4—1881no—
267C22×C4—1881no—
268C22×C4—1881no—
269C22×C4—1881no—
270C22×C4—1881no—
271C22×C4—1881no—
272C22×C4D37612yes—
273C22×C4Dic18812yes—
274C22×C4C4⋊Dic4712yes—
275C22×C4C4⋊Dic4712yes—
276C22×C4C2×Dic9411yescharacteristic
277C22×C4D4×D4712yes—
278C22×C4#1062535212yes—
279C22×C4C22×Dic4711yescharacteristic
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—944no—
286C4⋊C4—3761no—
287C4⋊C4—3761no—
288C4⋊C4—1882no—
289C4⋊C4—1882no—
290C4⋊C4—1882no—
291C4⋊C4—1882no—
292C4⋊C4—1882no—
293C22⋊C4—1882no—
294C22⋊C4—1882no—
295C42—1884no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23C2×C4⋊Dic4711yescharacteristic
301C23#1309399411yescharacteristic
302C23#1309400811yescharacteristic
303C23C2×D37611yescharacteristic
304C23C2×Dic18811yescharacteristic
305C23#1309405211yescharacteristic
306C23#1309405711yescharacteristic
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—3761no—
324C2×C4—3761no—
325C2×C4—3761no—
326C2×C4—3761no—
327C2×C4—3761no—
328C2×C4—3761no—
329C2×C4—3761no—
330C2×C4—3761no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4—1882no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4—1882no—
342C2×C4—1882no—
343C2×C4#1309395814yescomplemented
344C2×C4D4×Dic4712yes—
345C2×C4#1309400112yes—
346C8—44no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#1568899811yescharacteristic
360C22C2×#1309395811yescharacteristic
361C22#1568916511yescharacteristic
362C22#1568925911yescharacteristic
363C22#1568928111yescharacteristic
364C22#1568929811yescharacteristic
365C22#1568930811yescharacteristic
366C22#1568931111yescharacteristic
367C22#1568931411yescharacteristic
368C22#1568943711yescharacteristic
369C22#1568945911yescharacteristic
370C4—24no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—3762no—
375C4—3762no—
376C4—3762no—
377C4—3762no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#2024232511yescharacteristic
383C2#2024236411yescharacteristic
384C2#2024239311yescharacteristic
385C2#2024243211yescharacteristic
386C2#2024357811yescharacteristic
387C2#2024360511yescharacteristic
388C2#2024367111yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#8856
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
19. C47×#324C4semidirect
128. C2×C4×C47#107semidirect
213. #324Dic47semidirect
214. C47#8856semidirect
343. C2×C4#13093958semidirect

Maximal subgroups

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

subgroupindex
#202460062
#202435782
#202416412
#202412682
C2×C47×#3242
#885647