Finite Group Databasebeta

← #26112708

Subgroups — #26112708

Order 12032.

524 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#15689456C211yesmaximal, characteristic
2#20246267C211yesmaximal, characteristic
3#20244374C211yesmaximal, characteristic
4#20243479C211yesmaximal, characteristic
5#20241255C211yesmaximal, characteristic
6#20241234C211yesmaximal, characteristic
7C47×#1520C211yesmaximal, characteristic
8C22×C4⋊Dic47C2211yescharacteristic
9#15689456—21no—
10#15689456—21no—
11#15689456—22no—
12C2×#13093957C412yescomplemented
13C2×#13093948C412yescomplemented
14C2×C22⋊(C47⋊C8)C2211yescharacteristic
15#15689019—21no—
16#15689008—22no—
17#15688983C2211yescharacteristic
18#15688981—21no—
19C4⋊C4×Dic47—21no—
20C22⋊C4×Dic47—21no—
21C4×C22⋊Dic47C2211yescharacteristic
22#15688890—21no—
23#15688890—21no—
24#15688886—22no—
25C2×C22⋊Q8×C47C2211yescharacteristic
26C2×C22⋊C8×C47C2211yescharacteristic
27C47×#380C2211yescharacteristic
28C47×#328—22no—
29C23×Dic47—21no—
30C22×C47⋊C8C2×C412yes—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2×C412yes—
36C2×C4⋊Dic47D411yescharacteristic
37C2×C4⋊Dic47D411yescharacteristic
38C2×C4⋊Dic47D412yes—
39C2×Dic47⋊C4—22no—
40C2×C22⋊Dic47—21no—
41C2×C22⋊Dic47D411yescharacteristic
42C2×C22⋊Dic47D411yescharacteristic
43C2×C4×Dic47—21no—
44C2×C4×Dic47—22no—
45#13093957—22no—
46#13093957—22no—
47#13093948—22no—
48#13093948—22no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)—21no—
51C47⋊#95—22no—
52C47⋊#95—22no—
53C47⋊#95—21no—
54C47⋊#95—21no—
55C22×Q8×C47C2×C411yescharacteristic
56C23×C4×C47C2311yescharacteristic
57C22×C8×C47—21no—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C22⋊Q8×C47—21no—
61C22⋊Q8×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47C2×C411yescharacteristic
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67C22⋊C8×C47—21no—
68C22⋊C8×C47—21no—
69#95×C47D412yes—
70C24×C47C2×D411yescharacteristic
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—42no—
74C22×C4×C47—22no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—42no—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C22×C4×C47#4111yescharacteristic
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—22no—
88C22×Dic47—22no—
89C22×Dic47—21no—
90C22×Dic47—41no—
91C22×Dic47—82no—
92C2×C8×C47—42no—
93C2×C8×C47—41no—
94C4⋊C4×C47—21no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47—42no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C22⋊C4×C47—41no—
101C22⋊C4×C47—41no—
102C22⋊C4×C47—41no—
103C22⋊Dic47—21no—
104C22⋊Dic47—21no—
105C22⋊Dic47—21no—
106C22⋊Dic47—41no—
107C22⋊Dic47—41no—
108C22⋊Dic47Q1612yes—
109C4⋊Dic47—22no—
110C4⋊Dic47—22no—
111C4⋊Dic47—21no—
112C4⋊Dic47—21no—
113C4⋊Dic47—42no—
114C4⋊Dic47C22⋊C412yes—
115C4⋊Dic47C22⋊C412yes—
116Dic47⋊C4—82no—
117C4×Dic47—42no—
118C4×Dic47—82no—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—42no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47C4×D411yescharacteristic
124C23×C47#12011yescharacteristic
125C23×C47#12111yescharacteristic
126C23×C47C2×Q1611yescharacteristic
127C23×C47#13511yescharacteristic
128C23×C47#13711yescharacteristic
129C23×C47#13711yescharacteristic
130Q8×C47—42no—
131Q8×C47—81no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—42no—
143C2×C4×C47—81no—
144C2×C4×C47—81no—
145C2×C4×C47C2×C22⋊C411yescharacteristic
146C2×C4×C47C4×D411yescharacteristic
147C2×C4×C47#12111yescharacteristic
148C2×C4×C47#12311yescharacteristic
149C2×Dic47—22no—
150C2×Dic47—22no—
151C2×Dic47—22no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—82no—
160C2×Dic47—82no—
161C2×Dic47—81no—
162C2×Dic47—81no—
163C2×Dic47—81no—
164C8×C47—81no—
165C47⋊C8—42no—
166#9257—471nomaximal, Sylow
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—41no—
176C22×C47—41no—
177C22×C47#38611yescharacteristic
178C22×C47C4×Q1611yescharacteristic
179C22×C47#44111yescharacteristic
180C22×C47#45111yescharacteristic
181C22×C47#45211yescharacteristic
182C22×C47#46211yescharacteristic
183C22×C47#46411yescharacteristic
184C4×C47—21no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—82no—
188C4×C47—81no—
189C4×C47—81no—
190C4×C47#41012yes—
191Dic47—42no—
192Dic47—42no—
193Dic47—42no—
194Dic47—81no—
195Dic47—161no—
196C2×#484—471no—
197#2345—471no—
198#1996—471no—
199#1824—471no—
200#1520D4711yescharacteristic
201#1289—471no—
202#1282—471no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47—21no—
207C2×C47—21no—
208C2×C47—21no—
209C2×C47#187811yescharacteristic
210C2×C47#189911yescharacteristic
211C2×C47#190411yescharacteristic
212C2×C22⋊Q8C2×D4711yescharacteristic
213C22×C4⋊C4—471no—
214#484—942no—
215#484—941no—
216#484—941no—
217C2×#107—472no—
218C2×#103—472no—
219C2×C22⋊C8—471no—
220C2×C22⋊C8C2×D4711yescharacteristic
221#389—941no—
222#387—942no—
223#380—471no—
224#380—941no—
225#380C2×D4711yescharacteristic
226C4×C4⋊C4—941no—
227C4×C22⋊C4—471no—
228C4×C22⋊C4—941no—
229#342—941no—
230#342—941no—
231#340—942no—
232#328—22no—
233C47#925711yescharacteristic, complemented, Sylow
234C22×Q8C4×D4711yescharacteristic
235C23×C4—941no—
236C23×C4C22×D4711yescharacteristic
237C22×C8—21no—
238C22×C8—472no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—472no—
246C2×C4⋊C4—472no—
247C2×C4⋊C4—471no—
248C2×C4⋊C4—471no—
249C2×C4⋊C4—941no—
250C2×C4⋊C4—941no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—942no—
253C2×C4⋊C4—941no—
254C2×C4⋊C4C4×D4711yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—471no—
258C2×C22⋊C4—471no—
259C2×C22⋊C4—941no—
260C2×C42—942no—
261C2×C42—941no—
262#107—942no—
263#107—942no—
264#103—942no—
265#103—942no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—941no—
269C22⋊C8—941no—
270#95—942no—
271#95—942no—
272#95—941no—
273#95—941no—
274#95C22⋊D4712yes—
275C24D4×D4711yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—942no—
284C22×C4—942no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4—3762no—
290C22×C4—1881no—
291C22×C4C2×C4×D4711yescharacteristic
292C22×C4#1062535011yescharacteristic
293C22×C4D4×D4711yescharacteristic
294C22×C4D4×D4711yescharacteristic
295C22×C4#1062535211yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—942no—
300C2×C8—1882no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—472no—
307C4⋊C4—472no—
308C4⋊C4—942no—
309C4⋊C4—942no—
310C4⋊C4—941no—
311C4⋊C4—941no—
312C4⋊C4—3762no—
313C4⋊C4—1882no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—472no—
319C22⋊C4—941no—
320C22⋊C4—941no—
321C22⋊C4—941no—
322C22⋊C4—1881no—
323C22⋊C4—1881no—
324C42—3762no—
325C42—1882no—
326C23—41no—
327C23—41no—
328C23(C22⋊D47)⋊C411yescharacteristic
329C23#1309399211yescharacteristic
330C23#1309399711yescharacteristic
331C23Q16×D4711yescharacteristic
332C23#1309404511yescharacteristic
333C23#1309405311yescharacteristic
334C23#1309405611yescharacteristic
335Q8—42no—
336Q8—81no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—942no—
351C2×C4—942no—
352C2×C4—942no—
353C2×C4—3762no—
354C2×C4—3762no—
355C2×C4—3761no—
356C2×C4—3761no—
357C2×C4—3761no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4—1881no—
361C2×C4—1881no—
362C2×C4—1882no—
363C2×C4—1881no—
364C2×C4—1882no—
365C2×C4C22⋊C4×D4711yescharacteristic
366C2×C4C4×C22⋊D4711yescharacteristic
367C2×C4#1309399511yescharacteristic
368C2×C4#1309400611yescharacteristic
369C8—81no—
370C8—1882no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22#1568899911yescharacteristic
382C22#1568921511yescharacteristic
383C22#1568922711yescharacteristic
384C22#1568927711yescharacteristic
385C22#1568928211yescharacteristic
386C22#1568933211yescharacteristic
387C22#1568934811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—81no—
393C4—81no—
394C4—3761no—
395C4—1882no—
396C4—1882no—
397C4—1882no—
398C4—7521no—
399C4#1568909512yes—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2#2024372611yescharacteristic
407C2#2024384311yescharacteristic
408C2#2024387211yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1520C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1520D47
hypocenterC47#9257
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
12. C2×#13093957C4semidirect
13. C2×#13093948C4semidirect
233. C47#9257semidirect

Maximal subgroups

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

subgroupindex
C2×#156894562
#202462672
#202443742
#202434792
#202412552
#202412342
C47×#15202
#925747