Finite Group Databasebeta

← #26113998

Subgroups — #26113998

Order 12032.

904 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245966C211yesmaximal, characteristic
2#20243309C211yesmaximal, characteristic
3#20241921C212yesmaximal, complemented
4#20241241C212yesmaximal, complemented
5C47×#2435C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7C2×#13093941—22no—
8#15689087C2212yes—
9#15689003—24no—
10#15688986C2212yes—
11#15688979—21no—
12#15688860—24no—
13C23×D4×C47C2211yescharacteristic
14C2×#124×C47C412yescomplemented
15C2×C22⋊Q8×C47C2211yescharacteristic
16C47×#394—22no—
17C47×#390C412yescomplemented
18C47×#379—21no—
19C47×#379—21no—
20C4×C22⋊C4×C47C2211yescharacteristic
21C23×Dic47—22no—
22C2×#10625329—22no—
23C2×C22⋊Dic47—22no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—22no—
27C2×C22⋊Dic47—44no—
28#13093941—44no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C47⋊#95—22no—
32C47⋊#95D414yes—
33C25×C47D411yescharacteristic
34C25×C47D411yescharacteristic
35C22×Q8×C47C2×C411yescharacteristic, complemented
36C22×D4×C47—22no—
37C22×D4×C47—22no—
38C22×D4×C47—44no—
39C22×D4×C47C2×C412yes—
40C22×D4×C47D412yes—
41C23×C4×C47C2311yescharacteristic
42#124×C47—44no—
43C22⋊Q8×C47—22no—
44C22⋊Q8×C47—22no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47C2×C411yescharacteristic, complemented
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—22no—
52C2×C22⋊C4×C47—22no—
53C2×C42×C47C2×C412yescomplemented
54#95×C47—21no—
55C24×C47—21no—
56C24×C47—24no—
57C24×C47—21no—
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—42no—
61C24×C47—42no—
62C24×C47—42no—
63C24×C47—81no—
64C24×C47—81no—
65C24×C47C22⋊C412yes—
66C24×C47C22⋊C412yes—
67C24×C47C2×D411yescharacteristic
68C2×Q8×C47—22no—
69C2×Q8×C47—42no—
70C2×D4×C47—24no—
71C2×D4×C47—24no—
72C2×D4×C47—24no—
73C2×D4×C47—24no—
74C2×D4×C47—42no—
75C2×D4×C47—42no—
76C2×D4×C47—44no—
77C2×D4×C47—44no—
78C2×D4×C47—48no—
79C22×C4×C47—22no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—42no—
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D412yes—
86C22×C4×C47#4112yes—
87C22×Dic47—24no—
88C22×Dic47—44no—
89C22×Dic47—82no—
90C22×Dic47—82no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—42no—
93C22⋊C4×C47—42no—
94C22⋊C4×C47—44no—
95C22⋊C4×C47—44no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—82no—
99C22⋊C4×C47—82no—
100C42×C47—42no—
101C22⋊Dic47—44no—
102C22⋊Dic47—44no—
103C22⋊Dic47—82no—
104C22⋊Dic47—82no—
105#10625329—44no—
106C2×C47⋊C8—42no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—24no—
110C23×C47—22no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47—41no—
118C23×C47—42no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—42no—
122C23×C47—42no—
123C23×C47—42no—
124C23×C47—42no—
125C23×C47—42no—
126C23×C47—42no—
127C23×C47—42no—
128C23×C47—81no—
129C23×C47—81no—
130C23×C47—81no—
131C23×C47—81no—
132C23×C47—88no—
133C23×C47—88no—
134C23×C47#9914yes—
135C23×C47C2×C22⋊C411yescharacteristic
136C23×C47#12011yescharacteristic
137C23×C47#12311yescharacteristic
138Q8×C47—82no—
139D4×C47—48no—
140D4×C47—48no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47C4×D412yes—
153C2×C4×C47#12112yes—
154C2×Dic47—44no—
155C2×Dic47—44no—
156C2×Dic47—44no—
157C2×Dic47—88no—
158C47⋊C8—82no—
159#9459—471nomaximal, Sylow
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—24no—
167C22×C47—21no—
168C22×C47—44no—
169C22×C47—44no—
170C22×C47—44no—
171C22×C47—44no—
172C22×C47—44no—
173C22×C47—42no—
174C22×C47—42no—
175C22×C47—42no—
176C22×C47—42no—
177C22×C47—42no—
178C22×C47—82no—
179C22×C47—82no—
180C22×C47—84no—
181C22×C47—84no—
182C22×C47—84no—
183C22×C47#38611yescharacteristic
184C22×C47C2×#9912yes—
185C22×C47#45512yes—
186C4×C47—22no—
187C4×C47—22no—
188C4×C47—82no—
189C4×C47—81no—
190C4×C47—81no—
191Dic47—84no—
192#2435D4711yescharacteristic
193#2308—471no—
194#1783—471no—
195#1511—472no—
196#1284—472no—
197C2×C47—22no—
198C2×C47—22no—
199C2×C47—41no—
200C2×C47—44no—
201C2×C47—82no—
202C2×C47#188511yescharacteristic
203C2×C47#189811yescharacteristic
204C2×C47#191111yescharacteristic
205C23×D4C2×D4711yescharacteristic
206C2×#124Dic4712yescomplemented
207C2×C22⋊Q8C2×D4711yescharacteristic
208C22×C22⋊C4—941no—
209C2×#100—942no—
210#407—472no—
211#394—22no—
212#390Dic4712yescomplemented
213#386—944no—
214#381—472no—
215#379—21no—
216#379—21no—
217#379—941no—
218C4×C22⋊C4C2×D4711yescharacteristic
219#328—944no—
220C47#945911yescharacteristic, complemented, Sylow
221C25C22⋊D4711yescharacteristic
222C25C22⋊D4711yescharacteristic
223C22×Q8C2×Dic4711yescharacteristic, complemented
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4C2×Dic4712yes—
228C22×D4C22⋊D4712yes—
229C23×C4—942no—
230C23×C4C22×D4711yescharacteristic
231C2×M16—942no—
232#124—44no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×Dic4711yescharacteristic, complemented
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—942no—
244C2×C22⋊C4—942no—
245C2×C22⋊C4—942no—
246C2×C22⋊C4—942no—
247C2×C22⋊C4—1884no—
248C2×C42C2×Dic4712yescomplemented
249#100—1884no—
250C22⋊C8—942no—
251C22⋊C8—942no—
252#95—21no—
253#95—474no—
254#95—942no—
255C24—21no—
256C24—24no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—81no—
264C24—81no—
265C24C22⋊Dic4712yes—
266C24C22⋊Dic4712yes—
267C24C2×C22⋊D4711yescharacteristic
268C2×Q8—22no—
269C2×Q8—42no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—48no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—944no—
284C22×C4—3762no—
285C22×C4—3762no—
286C22×C4—1884no—
287C22×C4D4×D4712yes—
288C22×C4#1062535212yes—
289C22×C4C22×Dic4711yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291M16—1884no—
292C2×C8—1882no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—3762no—
303C22⋊C4—3762no—
304C22⋊C4—1884no—
305C22⋊C4—1884no—
306C42—42no—
307C23—22no—
308C23—22no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—88no—
333C23—88no—
334C23#1309393914yes—
335C23C2×C22⋊Dic4711yescharacteristic
336C23#1309399211yescharacteristic
337C23#1309400511yescharacteristic
338Q8—82no—
339D4—48no—
340D4—48no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3768no—
353C2×C4—1884no—
354C2×C4—1884no—
355C2×C4—1884no—
356C2×C4D4×Dic4712yes—
357C2×C4#1309399612yes—
358C8—3762no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—24no—
366C22—21no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22#1568900311yescharacteristic
383C22C2×#1309393912yes—
384C22#1568930012yes—
385C4—22no—
386C4—22no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—3764no—
391C2—22no—
392C2—22no—
393C2—41no—
394C2—44no—
395C2—82no—
396C2#2024377711yescharacteristic
397C2#2024384211yescharacteristic
398C2#2024392111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2435C2
socleC22×C47#386
prime radicalC23×D4×C47C22
cosocleC23×C4×C47C23
hypercenter#2435D47
hypocenterC47#9459
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
3. #20241921C2semidirect
4. #20241241C2semidirect
14. C2×#124×C47C4semidirect
17. C47×#390C4semidirect
35. C22×Q8×C47C2×C4semidirect
46. C2×C4⋊C4×C47C2×C4semidirect
53. C2×C42×C47C2×C4semidirect
206. C2×#124Dic47semidirect
212. #390Dic47semidirect
220. C47#9459semidirect
223. C22×Q8C2×Dic47semidirect
236. C2×C4⋊C4C2×Dic47semidirect
248. C2×C42C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202459662
#202433092
#202419212
#202412412
C47×#24352
#945947