Finite Group Databasebeta

← #26112343

Subgroups — #26112343

Order 12032.

622 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 113 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#20249132C211yesmaximal, characteristic
2#20246005C211yesmaximal, characteristic, complemented
3#20241956C211yesmaximal, characteristic
4C2×#15688848C211yesmaximal, characteristic, complemented
5#20241256C211yesmaximal, characteristic, complemented
6#20241255C211yesmaximal, characteristic, complemented
7C47×#1519C211yesmaximal, characteristic
8C2×#13094005—21no—
9C2×#13093980—21no—
10C22×Dic47⋊C4C2211yescharacteristic
11C2×C22⋊(C47⋊C8)C2211yescharacteristic
12#15689054C412yescomplemented
13#15689046—22no—
14#15689037C412yescomplemented
15#15689020—22no—
16#15689018—22no—
17#15689003—21no—
18#15688983C2211yescharacteristic
19#15688983C2211yescharacteristic
20#15688981—21no—
21#15688857—22no—
22#15688848—22no—
23C2×#121×C47C2211yescharacteristic
24C2×C22⋊C8×C47C2211yescharacteristic
25C47×#380C2211yescharacteristic
26C47×#327—22no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C22×C47⋊C8—21no—
30#13094005—42no—
31#13094005—42no—
32#13093980—42no—
33#13093980—42no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47—21no—
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—21no—
38C2×Dic47⋊C4—21no—
39C2×Dic47⋊C4—21no—
40C2×Dic47⋊C4—22no—
41C2×Dic47⋊C4—22no—
42C2×Dic47⋊C4—22no—
43C2×Dic47⋊C4C2×C412yes—
44C2×Dic47⋊C4D412yes—
45C2×C22⋊Dic47—21no—
46C2×C22⋊Dic47—21no—
47C2×C22⋊Dic47—22no—
48C2×C22⋊Dic47—22no—
49C2×C4×Dic47—22no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52C47⋊#95—22no—
53C47⋊#95—22no—
54C47⋊#95C2×C412yes—
55C47⋊#95D412yes—
56C22×D4×C47—21no—
57C22×D4×C47C2×C411yescharacteristic
58C23×C4×C47C2311yescharacteristic
59C22×C8×C47—21no—
60#121×C47—21no—
61#121×C47—21no—
62#121×C47—21no—
63#121×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47C2×C411yescharacteristic
66C2×C22⋊C4×C47—21no—
67C2×C22⋊C4×C47—21no—
68C22⋊C8×C47—21no—
69C22⋊C8×C47—21no—
70#95×C47D412yes—
71C24×C47—21no—
72C24×C47C2×D411yescharacteristic
73C2×D4×C47—21no—
74C2×D4×C47—21no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47—41no—
80C2×D4×C47—41no—
81C2×D4×C47—81no—
82C22×C4×C47—21no—
83C22×C4×C47—22no—
84C22×C4×C47—81no—
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47#4111yescharacteristic
90C22×C4×C47#4111yescharacteristic
91C22×Dic47—21no—
92C22×Dic47—21no—
93C22×Dic47—21no—
94C22×Dic47—21no—
95C22×Dic47—21no—
96C22×Dic47—21no—
97C22×Dic47—22no—
98C22×Dic47—22no—
99C22×Dic47—81no—
100C22×Dic47C22⋊C412yes—
101C22×Dic47C22⋊C412yes—
102C22×Dic47D812yes—
103C22×Dic47SD1612yes—
104C2×C8×C47—41no—
105C2×C8×C47—81no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—81no—
109C22⋊C4×C47—41no—
110C22⋊C4×C47—41no—
111C22⋊C4×C47—41no—
112C22⋊C4×C47—41no—
113C22⋊Dic47—41no—
114C22⋊Dic47—41no—
115C22⋊Dic47—41no—
116C22⋊Dic47—41no—
117C22⋊Dic47—44no—
118C22⋊Dic47—82no—
119C4⋊Dic47—81no—
120C4⋊Dic47—81no—
121Dic47⋊C4—22no—
122Dic47⋊C4—22no—
123Dic47⋊C4—22no—
124Dic47⋊C4—22no—
125Dic47⋊C4—44no—
126Dic47⋊C4—44no—
127Dic47⋊C4—82no—
128C4×Dic47—44no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—81no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—42no—
134C23×C47—42no—
135C23×C47—41no—
136C23×C47—41no—
137C23×C47—41no—
138C23×C47C2×C22⋊C411yescharacteristic
139C23×C47#12011yescharacteristic
140C23×C47#12311yescharacteristic
141C23×C47C2×D811yescharacteristic
142C23×C47C2×SD1611yescharacteristic
143C23×C47AGL(1,ℤ/8)11yescharacteristic
144C23×C47#13711yescharacteristic
145D4×C47—81no—
146D4×C47—81no—
147D4×C47—81no—
148D4×C47—81no—
149D4×C47—81no—
150D4×C47—81no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—42no—
158C2×C4×C47—42no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—42no—
163C2×C4×C47C4×D411yescharacteristic
164C2×C4×C47C4×D411yescharacteristic
165C2×C4×C47#12111yescharacteristic
166C2×C4×C47#12111yescharacteristic
167C2×Dic47—22no—
168C2×Dic47—22no—
169C2×Dic47—22no—
170C2×Dic47—22no—
171C2×Dic47—22no—
172C2×Dic47—22no—
173C2×Dic47—22no—
174C2×Dic47—42no—
175C2×Dic47—42no—
176C2×Dic47—42no—
177C2×Dic47—42no—
178C2×Dic47—42no—
179C2×Dic47—42no—
180C2×Dic47—42no—
181C2×Dic47—42no—
182C2×Dic47—81no—
183C2×Dic47#10214yes—
184C8×C47—81no—
185C47⋊C8—81no—
186#9204—471nomaximal, Sylow
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—41no—
194C22×C47—41no—
195C22×C47—41no—
196C22×C47—41no—
197C22×C47—82no—
198C22×C47—82no—
199C22×C47—81no—
200C22×C47—81no—
201C22×C47—81no—
202C22×C47#38611yescharacteristic
203C22×C47C2×#10211yescharacteristic
204C22×C47#41711yescharacteristic
205C22×C47#44711yescharacteristic
206C22×C47#44811yescharacteristic
207C22×C47#48011yescharacteristic
208C22×C47#48111yescharacteristic
209C4×C47—21no—
210C4×C47—21no—
211C4×C47—21no—
212C4×C47—21no—
213C4×C47—82no—
214C4×C47—81no—
215Dic47—24no—
216Dic47—42no—
217Dic47—82no—
218Dic47—82no—
219#2618—471no—
220#2316—471no—
221#1519—471no—
222#1519D4711yescharacteristic
223C2×#324—471no—
224#1289—471no—
225#1289—471no—
226C2×C47—21no—
227C2×C47—21no—
228C2×C47—21no—
229C2×C47—21no—
230C2×C47—21no—
231C2×C47—21no—
232C2×C47—161no—
233C2×C47#187411yescharacteristic
234C2×C47#191211yescharacteristic
235C2×C47#191311yescharacteristic
236C2×#123—941no—
237C2×#121C2×D4711yescharacteristic
238C2×#117—941no—
239C22×C4⋊C4—471no—
240C2×C22⋊C8—471no—
241C2×C22⋊C8C2×D4711yescharacteristic
242#398—472no—
243#396—942no—
244#394—472no—
245#389—942no—
246#389—942no—
247#386—941no—
248#380—471no—
249#380—471no—
250#380—941no—
251#380C2×D4711yescharacteristic
252#327—22no—
253#327—942no—
254#324—942no—
255C47#920411yescharacteristic, complemented, Sylow
256C22×D4—21no—
257C22×D4C4×D4711yescharacteristic
258C23×C4—471no—
259C23×C4—471no—
260C23×C4C22×D4711yescharacteristic
261C22×C8—21no—
262C22×C8—941no—
263#123—1882no—
264#123—1882no—
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#117—1882no—
270#117—1882no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—472no—
273C2×C4⋊C4—472no—
274C2×C4⋊C4—941no—
275C2×C4⋊C4—941no—
276C2×C4⋊C4—941no—
277C2×C4⋊C4—941no—
278C2×C4⋊C4—942no—
279C2×C4⋊C4—941no—
280C2×C4⋊C4—941no—
281C2×C4⋊C4—942no—
282C2×C4⋊C4—942no—
283C2×C4⋊C4C4×D4711yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—942no—
287C2×C22⋊C4—942no—
288C2×C22⋊C4—941no—
289C2×C22⋊C4—941no—
290C2×C42—942no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—941no—
294C22⋊C8—941no—
295#95—472no—
296#95—472no—
297#95—942no—
298#95—942no—
299#95C22⋊D4712yes—
300C24—21no—
301C24D4×D4711yescharacteristic
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—81no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—81no—
314C22×C4—472no—
315C22×C4—472no—
316C22×C4—472no—
317C22×C4—472no—
318C22×C4—941no—
319C22×C4—941no—
320C22×C4—941no—
321C22×C4—941no—
322C22×C4—941no—
323C22×C4—941no—
324C22×C4—942no—
325C22×C4—942no—
326C22×C4—3761no—
327C22×C4C2×C4×D4711yescharacteristic
328C22×C4#1062535011yescharacteristic
329C22×C4D4×D4711yescharacteristic
330C22×C4D4×D4711yescharacteristic
331C22×C4#1062535211yescharacteristic
332C22×C4C2×C22⋊D4711yescharacteristic
333C2×C8—41no—
334C2×C8—81no—
335C2×C8—3761no—
336C2×C8—1881no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—81no—
340C4⋊C4—942no—
341C4⋊C4—942no—
342C4⋊C4—942no—
343C4⋊C4—942no—
344C4⋊C4—3762no—
345C4⋊C4—3761no—
346C4⋊C4—3761no—
347C4⋊C4—1884no—
348C4⋊C4—1884no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—3762no—
354C22⋊C4—1884no—
355C22⋊C4—1881no—
356C22⋊C4—1881no—
357C22⋊C4—1881no—
358C22⋊C4—1881no—
359C42—1884no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23C22⋊C4×D4711yescharacteristic
368C23#1309399211yescharacteristic
369C23#1309400611yescharacteristic
370C23D8×D4711yescharacteristic
371C23SD16×D4711yescharacteristic
372C23#1309404811yescharacteristic
373C23#1309405311yescharacteristic
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—81no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—474no—
393C2×C4—942no—
394C2×C4—942no—
395C2×C4—942no—
396C2×C4—942no—
397C2×C4—942no—
398C2×C4—942no—
399C2×C4—942no—
400C2×C4—3761no—
401C2×C4—1882no—
402C2×C4—1882no—
403C2×C4—1882no—
404C2×C4—1882no—
405C2×C4—1882no—
406C2×C4—1882no—
407C2×C4—1882no—
408C2×C4—1882no—
409C2×C4(C22⋊D47)⋊C411yescharacteristic
410C2×C4C4×C22⋊D4711yescharacteristic
411C2×C4#1309399511yescharacteristic
412C2×C4#1309399711yescharacteristic
413C8—81no—
414C8—3761no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—82no—
426C22—82no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22#1568899911yescharacteristic
431C22#102×D4711yescharacteristic
432C22#1568912111yescharacteristic
433C22#1568925511yescharacteristic
434C22#1568926011yescharacteristic
435C22#1568943511yescharacteristic
436C22#1568944011yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—81no—
443C4—944no—
444C4—3762no—
445C4—3762no—
446C4—1882no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—161no—
454C2#2024370211yescharacteristic
455C2#2024392211yescharacteristic
456C2#2024392711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1519C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#1519D47
hypocenterC47#9204
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #20246005C2semidirect
4. C2×#15688848C2semidirect
5. #20241256C2semidirect
6. #20241255C2semidirect
12. #15689054C4semidirect
14. #15689037C4semidirect
255. C47#9204semidirect

Maximal subgroups

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

subgroupindex
#202491322
#202460052
#202419562
C2×#156888482
#202412562
#202412552
C47×#15192
#920447