Finite Group Databasebeta

← #26112701

Subgroups — #26112701

Order 12032.

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

normal subgroups only

This group has 552 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×#15689451C211yesmaximal, characteristic
2#20246259C211yesmaximal, characteristic
3#20244374C211yesmaximal, characteristic
4#20243475C211yesmaximal, characteristic
5#20241255C211yesmaximal, characteristic
6#20241234C211yesmaximal, characteristic
7C47×#1520C211yesmaximal, characteristic
8C2×#13093980—21no—
9C22×C4⋊Dic47C2211yescharacteristic
10#15689451—21no—
11#15689451—21no—
12#15689451—22no—
13C2×#13093955C412yescomplemented
14C2×#13093948C412yescomplemented
15C2×C22⋊(C47⋊C8)C2211yescharacteristic
16#15689019—21no—
17#15689006—22no—
18#15688984—21no—
19#15688983C2211yescharacteristic
20C4⋊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—
31#13093980—42no—
32#13093980—42no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47C2×C412yes—
38C2×C4⋊Dic47D411yescharacteristic
39C2×C4⋊Dic47D411yescharacteristic
40C2×C4⋊Dic47D412yes—
41C2×Dic47⋊C4—22no—
42C2×C22⋊Dic47—21no—
43C2×C22⋊Dic47D411yescharacteristic
44C2×C22⋊Dic47D411yescharacteristic
45C2×C4×Dic47—22no—
46C2×C4×Dic47—21no—
47#13093955—22no—
48#13093955—22no—
49#13093948—22no—
50#13093948—22no—
51C22⋊(C47⋊C8)—21no—
52C22⋊(C47⋊C8)—21no—
53C47⋊#95—22no—
54C47⋊#95—22no—
55C47⋊#95—21no—
56C47⋊#95—21no—
57C22×Q8×C47C2×C411yescharacteristic
58C23×C4×C47C2311yescharacteristic
59C22×C8×C47—21no—
60C22⋊Q8×C47—21no—
61C22⋊Q8×C47—21no—
62C22⋊Q8×C47—21no—
63C22⋊Q8×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47C2×C411yescharacteristic
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47—21no—
69C22⋊C8×C47—21no—
70C22⋊C8×C47—21no—
71#95×C47D412yes—
72C24×C47C2×D411yescharacteristic
73C2×Q8×C47—21no—
74C2×Q8×C47—21no—
75C2×Q8×C47—42no—
76C22×C4×C47—22no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—42no—
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47#4111yescharacteristic
85C22×C4×C47#4111yescharacteristic
86C22×Dic47—22no—
87C22×Dic47—21no—
88C22×Dic47—21no—
89C22×Dic47—21no—
90C22×Dic47—22no—
91C22×Dic47—21no—
92C22×Dic47—41no—
93C22×Dic47—41no—
94C22×Dic47—41no—
95C22×Dic47—41no—
96C22×Dic47—41no—
97C2×C8×C47—42no—
98C2×C8×C47—41no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—42no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104C22⋊C4×C47—41no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊Dic47—21no—
109C22⋊Dic47—21no—
110C22⋊Dic47—21no—
111C22⋊Dic47—41no—
112C22⋊Dic47—41no—
113C22⋊Dic47SD1612yes—
114C4⋊Dic47—22no—
115C4⋊Dic47—22no—
116C4⋊Dic47—21no—
117C4⋊Dic47—21no—
118C4⋊Dic47—42no—
119C4⋊Dic47C22⋊C412yes—
120C4⋊Dic47C22⋊C412yes—
121Dic47⋊C4—44no—
122C4×Dic47—44no—
123C4×Dic47—42no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—42no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47C4×D411yescharacteristic
129C23×C47#12011yescharacteristic
130C23×C47#12111yescharacteristic
131C23×C47C2×SD1611yescharacteristic
132C23×C47#13511yescharacteristic
133C23×C47AGL(1,ℤ/8)11yescharacteristic
134C23×C47#13711yescharacteristic
135Q8×C47—42no—
136Q8×C47—81no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—81no—
149C2×C4×C47—81no—
150C2×C4×C47C2×C22⋊C411yescharacteristic
151C2×C4×C47C4×D411yescharacteristic
152C2×C4×C47#12111yescharacteristic
153C2×C4×C47#12311yescharacteristic
154C2×Dic47—22no—
155C2×Dic47—22no—
156C2×Dic47—22no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—42no—
160C2×Dic47—42no—
161C2×Dic47—41no—
162C2×Dic47—41no—
163C2×Dic47—41no—
164C2×Dic47—41no—
165C2×Dic47—42no—
166C2×Dic47—42no—
167C2×Dic47—41no—
168C2×Dic47—41no—
169C2×Dic47—41no—
170C2×Dic47—81no—
171C2×Dic47—81no—
172C2×Dic47—81no—
173C8×C47—81no—
174C47⋊C8—42no—
175#9256—471nomaximal, Sylow
176C22×C47—21no—
177C22×C47—21no—
178C22×C47—21no—
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—41no—
183C22×C47—41no—
184C22×C47—41no—
185C22×C47—41no—
186C22×C47#38611yescharacteristic
187C22×C47C4×SD1611yescharacteristic
188C22×C47#44011yescharacteristic
189C22×C47#45011yescharacteristic
190C22×C47#45211yescharacteristic
191C22×C47#46111yescharacteristic
192C22×C47#46311yescharacteristic
193C4×C47—21no—
194C4×C47—21no—
195C4×C47—21no—
196C4×C47—82no—
197C4×C47—81no—
198C4×C47—81no—
199C4×C47#41012yes—
200Dic47—42no—
201Dic47—42no—
202Dic47—42no—
203Dic47—81no—
204Dic47—161no—
205C2×#483—471no—
206#2344—471no—
207#1996—471no—
208#1823—471no—
209#1520D4711yescharacteristic
210#1289—471no—
211#1282—471no—
212C2×C47—21no—
213C2×C47—21no—
214C2×C47—21no—
215C2×C47—21no—
216C2×C47—21no—
217C2×C47—21no—
218C2×C47#187711yescharacteristic
219C2×C47#189911yescharacteristic
220C2×C47#190511yescharacteristic
221C2×C22⋊Q8C2×D4711yescharacteristic
222C2×#117—941no—
223C22×C4⋊C4—471no—
224#483—942no—
225#483—941no—
226#483—941no—
227C2×#106—472no—
228C2×#103—472no—
229C2×C22⋊C8—471no—
230C2×C22⋊C8C2×D4711yescharacteristic
231#389—941no—
232#387—942no—
233#381—941no—
234#380—471no—
235#380C2×D4711yescharacteristic
236C4×C4⋊C4—941no—
237C4×C22⋊C4—471no—
238#342—941no—
239#342—941no—
240#340—942no—
241#328—22no—
242C47#925611yescharacteristic, complemented, Sylow
243C22×Q8C4×D4711yescharacteristic
244C23×C4—941no—
245C23×C4C22×D4711yescharacteristic
246C22×C8—21no—
247C22×C8—472no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252#117—1882no—
253#117—1882no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—472no—
257C2×C4⋊C4—472no—
258C2×C4⋊C4—471no—
259C2×C4⋊C4—471no—
260C2×C4⋊C4—941no—
261C2×C4⋊C4—941no—
262C2×C4⋊C4—941no—
263C2×C4⋊C4—942no—
264C2×C4⋊C4—941no—
265C2×C4⋊C4C4×D4711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—471no—
269C2×C22⋊C4—471no—
270C2×C22⋊C4—941no—
271C2×C42—942no—
272C2×C42—941no—
273#106—942no—
274#106—942no—
275#103—942no—
276#103—942no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—941no—
280C22⋊C8—941no—
281#95—942no—
282#95—942no—
283#95—941no—
284#95—941no—
285#95C22⋊D4712yes—
286C24D4×D4711yescharacteristic
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—42no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—942no—
295C22×C4—941no—
296C22×C4—941no—
297C22×C4—941no—
298C22×C4—942no—
299C22×C4—941no—
300C22×C4—1881no—
301C22×C4—1881no—
302C22×C4—1881no—
303C22×C4—1881no—
304C22×C4—1881no—
305C22×C4C2×C4×D4711yescharacteristic
306C22×C4#1062535011yescharacteristic
307C22×C4D4×D4711yescharacteristic
308C22×C4D4×D4711yescharacteristic
309C22×C4#1062535211yescharacteristic
310C22×C4C2×C22⋊D4711yescharacteristic
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—942no—
314C2×C8—1882no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—472no—
321C4⋊C4—472no—
322C4⋊C4—942no—
323C4⋊C4—942no—
324C4⋊C4—941no—
325C4⋊C4—941no—
326C4⋊C4—1882no—
327C4⋊C4—1884no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—472no—
333C22⋊C4—941no—
334C22⋊C4—941no—
335C22⋊C4—941no—
336C22⋊C4—1881no—
337C22⋊C4—1881no—
338C42—1882no—
339C42—1884no—
340C23—41no—
341C23—41no—
342C23(C22⋊D47)⋊C411yescharacteristic
343C23#1309399211yescharacteristic
344C23#1309399711yescharacteristic
345C23SD16×D4711yescharacteristic
346C23#1309404711yescharacteristic
347C23#1309405111yescharacteristic
348C23#1309405611yescharacteristic
349Q8—42no—
350Q8—81no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—942no—
365C2×C4—942no—
366C2×C4—942no—
367C2×C4—3761no—
368C2×C4—3761no—
369C2×C4—3761no—
370C2×C4—1881no—
371C2×C4—1881no—
372C2×C4—1881no—
373C2×C4—1881no—
374C2×C4—1881no—
375C2×C4—1882no—
376C2×C4—1881no—
377C2×C4—1881no—
378C2×C4—1881no—
379C2×C4—1881no—
380C2×C4—1882no—
381C2×C4—1882no—
382C2×C4—1882no—
383C2×C4C22⋊C4×D4711yescharacteristic
384C2×C4C4×C22⋊D4711yescharacteristic
385C2×C4#1309399511yescharacteristic
386C2×C4#1309400611yescharacteristic
387C8—81no—
388C8—1882no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22#1568899911yescharacteristic
400C22#1568920811yescharacteristic
401C22#1568922611yescharacteristic
402C22#1568927211yescharacteristic
403C22#1568928411yescharacteristic
404C22#1568932711yescharacteristic
405C22#1568934111yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4—3761no—
413C4—1882no—
414C4—1882no—
415C4—1882no—
416C4—7521no—
417C4#1568909512yes—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2#2024371911yescharacteristic
425C2#2024384311yescharacteristic
426C2#2024387911yescharacteristic
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#9256
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
13. C2×#13093955C4semidirect
14. C2×#13093948C4semidirect
242. C47#9256semidirect

Maximal subgroups

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

subgroupindex
C2×#156894512
#202462592
#202443742
#202434752
#202412552
#202412342
C47×#15202
#925647