Finite Group Databasebeta

← #26114320

Subgroups — #26114320

Order 12032.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#20252436C211yesmaximal, characteristic
2#20243474C211yesmaximal, characteristic, complemented
3#20243468C211yesmaximal, characteristic, complemented
4#20241931C211yesmaximal, characteristic
5#20241295C211yesmaximal, characteristic, complemented
6#20241264C211yesmaximal, characteristic, complemented
7C47×#3092C211yesmaximal, characteristic
8#15689201—21no—
9#15689195—21no—
10#15689195—21no—
11D4×C47⋊C8—21no—
12#15689184—21no—
13#15689090—21no—
14#15689087C2211yescharacteristic
15#15689087C2211yescharacteristic
16C2×C22⋊(C47⋊C8)C2211yescharacteristic
17#15688983C2211yescharacteristic
18#15688892—21no—
19#15688892—21no—
20#15688885—21no—
21#15688885—21no—
22#15688860—21no—
23#15688857—21no—
24C47×#525C2211yescharacteristic
25C22×C4⋊C4×C47C2211yescharacteristic
26C47×#484C412yescomplemented
27C47×#483—21no—
28C47×#481C412yescomplemented
29C47×#480—21no—
30C47×#428—21no—
31C47×#417—21no—
32C47×#407C2211yescharacteristic
33C2×#10625329—21no—
34C2×#10625329—21no—
35C2×#10625329D411yescharacteristic
36C2×#10625329D411yescharacteristic
37C22×C47⋊C8—21no—
38C22×C47⋊C8D411yescharacteristic
39C22×C47⋊C8D411yescharacteristic
40C2×C4⋊Dic47—21no—
41C2×C22⋊Dic47—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52#13093935—21no—
53C4×C47⋊C8—21no—
54C47⋊#95—21no—
55C2×#41×C47D411yescharacteristic
56C23×C4×C47—21no—
57C23×C4×C47C2311yescharacteristic
58C2×M16×C47—21no—
59#123×C47—21no—
60C22⋊Q8×C47C2×C411yescharacteristic
61C22⋊Q8×C47D411yescharacteristic
62#121×C47C2×C411yescharacteristic
63#121×C47D411yescharacteristic
64#120×C47—21no—
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67#117×C47D411yescharacteristic
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47C2×C412yes—
74#107×C47—22no—
75#106×C47—22no—
76#103×C47—22no—
77#102×C47—22no—
78C22⋊C8×C47—21no—
79C22⋊C8×C47C2×C412yes—
80C24×C47#4111yescharacteristic
81#41×C47—41no—
82#41×C47—41no—
83C2×Q8×C47C22⋊C411yescharacteristic
84C2×D4×C47—21no—
85C2×D4×C47—21no—
86C2×D4×C47—21no—
87C2×D4×C47C22⋊C411yescharacteristic
88C22×C4×C47—21no—
89C22×C4×C47—22no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47—41no—
94C22×C4×C47—81no—
95C22×C4×C47—81no—
96C22×C4×C47C22×C411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47#4111yescharacteristic
102C22×Dic47—41no—
103M16×C47—41no—
104M16×C47—41no—
105C2×C8×C47—22no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47C22⋊C411yescharacteristic
113C4⋊C4×C47C22⋊C411yescharacteristic
114C22⋊C4×C47—21no—
115C22⋊C4×C47—21no—
116C22⋊C4×C47—21no—
117C22⋊C4×C47—21no—
118C22⋊C4×C47—21no—
119C42×C47—21no—
120C22⋊Dic47—81no—
121C4⋊Dic47—41no—
122C4⋊Dic47—41no—
123#10625329—41no—
124#10625329—41no—
125#10625329—41no—
126#10625329—41no—
127#10625329—81no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—21no—
131C2×C47⋊C8—21no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—81no—
136C23×C47—21no—
137C23×C47—21no—
138C23×C47—21no—
139C23×C47—41no—
140C23×C47—41no—
141C23×C47C4×D411yescharacteristic
142C23×C47#12111yescharacteristic
143C23×C47#12311yescharacteristic
144Q8×C47—41no—
145D4×C47—41no—
146D4×C47—41no—
147D4×C47—41no—
148D4×C47—81no—
149D4×C47—81no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—42no—
165C2×C4×C47—82no—
166C2×C4×C47—81no—
167C2×C4×C47—81no—
168C2×C4×C47—82no—
169C2×C4×C47C2×C22⋊C411yescharacteristic
170C2×C4×C47C4×D411yescharacteristic
171C2×C4×C47#12011yescharacteristic
172C2×C4×C47#12111yescharacteristic
173C2×Dic47—81no—
174C2×Dic47—81no—
175C2×Dic47—81no—
176C8×C47—42no—
177C47⋊C8—41no—
178C47⋊C8—41no—
179C47⋊C8—41no—
180C47⋊C8—41no—
181C47⋊C8—81no—
182#9522—471nomaximal, Sylow
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—42no—
192C22×C47—42no—
193C22×C47—41no—
194C22×C47—41no—
195C22×C47—41no—
196C22×C47#38611yescharacteristic
197C22×C47#47111yescharacteristic
198C22×C47#47211yescharacteristic
199C4×C47—21no—
200C4×C47—21no—
201C4×C47—21no—
202C4×C47—41no—
203C4×C47—41no—
204C4×C47—41no—
205C4×C47—82no—
206C4×C47#45811yescharacteristic
207C4×C47#45811yescharacteristic
208Dic47—161no—
209#3092D4711yescharacteristic
210#2976—471no—
211#1823—471no—
212#1821—471no—
213#1513—471no—
214#1305—471no—
215#1293—471no—
216C2×C47—21no—
217C2×C47—21no—
218C2×C47—41no—
219C2×C47—41no—
220C2×C47—81no—
221C2×C47#189011yescharacteristic
222C2×C47#190211yescharacteristic
223C2×C47#190711yescharacteristic
224#525C2×D4711yescharacteristic
225C22×C4⋊C4C2×D4711yescharacteristic
226#484Dic4712yescomplemented
227#483—21no—
228#481Dic4712yescomplemented
229#480—21no—
230#436—941no—
231#435—941no—
232#435—941no—
233C8×D4—941no—
234#433—941no—
235#428—21no—
236#417—21no—
237#408—941no—
238#407—471no—
239#407—471no—
240#407C2×D4711yescharacteristic
241C2×C22⋊C8—471no—
242#380—471no—
243#343—941no—
244#343—941no—
245#340—941no—
246#340—941no—
247#328—941no—
248#327—941no—
249C47#952211yescharacteristic, complemented, Sylow
250C2×#41C22⋊D4711yescharacteristic
251C23×C4—21no—
252C23×C4C22×D4711yescharacteristic
253C2×M16—21no—
254C2×M16—471no—
255C2×M16—471no—
256C2×M16—941no—
257C2×M16—941no—
258C22×C8—471no—
259C22×C8—471no—
260C22×C8—941no—
261#123—21no—
262C22⋊Q8C2×Dic4711yescharacteristic
263C22⋊Q8C22⋊D4711yescharacteristic
264#121C2×Dic4711yescharacteristic
265#121C22⋊D4711yescharacteristic
266#120—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117C22⋊D4711yescharacteristic
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—941no—
276C2×C4⋊C4C2×Dic4712yes—
277C2×C22⋊C4—941no—
278#107—22no—
279#106—22no—
280C4⋊C8—941no—
281C4⋊C8—941no—
282#103—22no—
283#102—22no—
284C22⋊C8—21no—
285C22⋊C8—941no—
286C22⋊C8—941no—
287C22⋊C8—941no—
288C22⋊C8—941no—
289C22⋊C8—941no—
290C22⋊C8—941no—
291C22⋊C8—941no—
292C22⋊C8—941no—
293C22⋊C8C2×Dic4712yes—
294#97—941no—
295C4×C8—941no—
296#95—941no—
297C24#1062535211yescharacteristic
298#41—41no—
299#41—41no—
300C2×Q8C22⋊Dic4711yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C22⋊Dic4711yescharacteristic
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—1881no—
314C22×C4D4×D4711yescharacteristic
315C22×C4D4×D4711yescharacteristic
316C22×C4#1062535211yescharacteristic
317C22×C4C22×Dic4711yescharacteristic
318C22×C4C2×C22⋊D4711yescharacteristic
319C22×C4C2×C22⋊D4711yescharacteristic
320M16—41no—
321M16—41no—
322M16—3761no—
323M16—1881no—
324M16—1881no—
325M16—1881no—
326M16—1881no—
327C2×C8—22no—
328C2×C8—941no—
329C2×C8—941no—
330C2×C8—941no—
331C2×C8—941no—
332C2×C8—3761no—
333C2×C8—1881no—
334C2×C8—1881no—
335C2×C8—1881no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—1881no—
343C4⋊C4—1881no—
344C4⋊C4C22⋊Dic4711yescharacteristic
345C4⋊C4C22⋊Dic4711yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—3761no—
352C42—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23D4×Dic4711yescharacteristic
359C23#1309399611yescharacteristic
360C23#1309400511yescharacteristic
361Q8—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—81no—
366D4—81no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—3761no—
387C2×C4—3761no—
388C2×C4—3761no—
389C2×C4C2×C22⋊Dic4711yescharacteristic
390C2×C4D4×Dic4711yescharacteristic
391C2×C4#1309399211yescharacteristic
392C2×C4#1309399611yescharacteristic
393C8—42no—
394C8—3761no—
395C8—1881no—
396C8—1881no—
397C8—1881no—
398C8—1881no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22#1568900311yescharacteristic
413C22#1568938411yescharacteristic
414C22#1568939111yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4—7521no—
423C4#1568931211yescharacteristic
424C4#1568931211yescharacteristic
425C2—21no—
426C2—21no—
427C2—41no—
428C2—41no—
429C2—81no—
430C2#2024380211yescharacteristic
431C2#2024386311yescharacteristic
432C2#2024389811yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#3092C2
socleC22×C47#386
prime radicalC47×#525C22
cosocleC23×C4×C47C23
hypercenter#3092D47
hypocenterC47#9522
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3092

Complements

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

NH ≅ G/Ntype
2. #20243474C2semidirect
3. #20243468C2semidirect
5. #20241295C2semidirect
6. #20241264C2semidirect
26. C47×#484C4semidirect
28. C47×#481C4semidirect
226. #484Dic47semidirect
228. #481Dic47semidirect
249. C47#9522semidirect

Maximal subgroups

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

subgroupindex
#202524362
#202434742
#202434682
#202419312
#202412952
#202412642
C47×#30922
#952247