Finite Group Databasebeta

← #26109653

Subgroups — #26109653

Order 12032.

940 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688983C211yesmaximal, characteristic
2#20243298C211yesmaximal, characteristic
3#20241267C214yesmaximal, complemented
4C47×C22⋊#95C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic, complemented
6C22×C4⋊Dic47C411yescharacteristic, complemented
7C22×C22⋊Dic47—21no—
8C2×C22⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)—21no—
10C2×C22⋊(C47⋊C8)C2212yes—
11#15688983—21no—
12#15688983—21no—
13#15688983C412yescomplemented
14#15688983C2212yes—
15C2×C47⋊#95—21no—
16C24×C4×C47C2211yescharacteristic
17C22×C22⋊C4×C47—21no—
18C2×#95×C47—21no—
19C2×#95×C47C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—42no—
26C2×C4⋊Dic47—42no—
27C2×C4⋊Dic47C2×C412yes—
28C2×C4⋊Dic47C2×C412yes—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—44no—
32C22⋊(C47⋊C8)—24no—
33C22⋊(C47⋊C8)—42no—
34C22⋊(C47⋊C8)—42no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C25×C47D411yescharacteristic
38C23×C4×C47—24no—
39C23×C4×C47—22no—
40C23×C4×C47—42no—
41C23×C4×C47—42no—
42C23×C4×C47C2×C411yescharacteristic
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C23×C4×C47C2311yescharacteristic
48C2×C22⋊C4×C47—24no—
49C2×C22⋊C4×C47—42no—
50C2×C22⋊C4×C47—42no—
51#95×C47—24no—
52#95×C47—42no—
53C24×C47—22no—
54C24×C47—22no—
55C24×C47—22no—
56C24×C47—22no—
57C24×C47—41no—
58C24×C47—41no—
59C24×C47C22⋊C411yescharacteristic
60C24×C47C22⋊C411yescharacteristic
61C24×C47SD1612yes—
62C24×C47Q1612yes—
63C24×C47C2×D411yescharacteristic
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—22no—
72C22×C4×C47—22no—
73C22×C4×C47—22no—
74C22×C4×C47—24no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47—41no—
79C22×C4×C47—41no—
80C22×C4×C47—41no—
81C22×C4×C47—41no—
82C22×C4×C47—416no—
83C22×C4×C47—42no—
84C22×C4×C47—42no—
85C22×C4×C47—42no—
86C22×C4×C47—42no—
87C22×C4×C47—42no—
88C22×C4×C47—42no—
89C22×C4×C47C22⋊C411yescharacteristic
90C22×C4×C47C22⋊C411yescharacteristic
91C22×C4×C47C22×C411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4114yes—
94C22×Dic47—22no—
95C22×Dic47—22no—
96C22×Dic47—42no—
97C22×Dic47—42no—
98C22×Dic47—42no—
99C22×Dic47—42no—
100C22×Dic47—42no—
101C22×Dic47—42no—
102C22⋊C4×C47—48no—
103C22⋊Dic47—44no—
104C22⋊Dic47—44no—
105C4⋊Dic47—42no—
106C4⋊Dic47—42no—
107C4⋊Dic47—42no—
108C4⋊Dic47—42no—
109C2×C47⋊C8—42no—
110C2×C47⋊C8—42no—
111C2×C47⋊C8—42no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47—24no—
121C23×C47—24no—
122C23×C47—21no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—21no—
127C23×C47—21no—
128C23×C47—24no—
129C23×C47—24no—
130C23×C47—42no—
131C23×C47—42no—
132C23×C47—44no—
133C23×C47—44no—
134C23×C47—44no—
135C23×C47—44no—
136C23×C47#9912yes—
137C23×C47#10312yes—
138C23×C47#10312yes—
139C23×C47#10412yes—
140C23×C47C2×C22⋊C411yescharacteristic
141C23×C47#12312yes—
142C23×C47C2×SD1611yescharacteristic
143C23×C47C2×Q1611yescharacteristic
144C23×C47AGL(1,ℤ/8)12yes—
145C2×C4×C47—28no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—44no—
156C2×C4×C47—44no—
157C2×C4×C47—44no—
158C2×C4×C47—44no—
159C2×C4×C47—44no—
160C2×C4×C47—44no—
161C2×C4×C47—44no—
162C2×C4×C47—44no—
163C2×C4×C47—82no—
164C2×C4×C47—82no—
165C2×C4×C47—82no—
166C2×C4×C47—82no—
167C2×C4×C47#9912yes—
168C2×C4×C47#11712yes—
169C2×C4×C47#12312yes—
170C2×Dic47—42no—
171C2×Dic47—42no—
172C2×Dic47—42no—
173C2×Dic47—42no—
174C2×Dic47—42no—
175C2×Dic47—42no—
176C2×Dic47—84no—
177C2×Dic47—84no—
178C47⋊C8—82no—
179#8713—471nomaximal, Sylow
180C22×C47—22no—
181C22×C47—22no—
182C22×C47—22no—
183C22×C47—22no—
184C22×C47—22no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—22no—
188C22×C47—22no—
189C22×C47—22no—
190C22×C47—22no—
191C22×C47—21no—
192C22×C47—22no—
193C22×C47—21no—
194C22×C47—22no—
195C22×C47—22no—
196C22×C47—22no—
197C22×C47—48no—
198C22×C47—48no—
199C22×C47—41no—
200C22×C47—41no—
201C22×C47—41no—
202C22×C47—41no—
203C22×C47#32814yes—
204C22×C47#38111yescharacteristic
205C22×C47C2×#9911yescharacteristic
206C22×C47C2×#9911yescharacteristic
207C22×C47C2×#10311yescharacteristic
208C22×C47#41811yescharacteristic
209C22×C47C2×#10411yescharacteristic
210C22×C47#42111yescharacteristic
211C22×C47#48112yes—
212C22×C47#48412yes—
213C4×C47—22no—
214C4×C47—22no—
215C4×C47—22no—
216C4×C47—41no—
217C4×C47—84no—
218Dic47—82no—
219Dic47—82no—
220C2×#380—471no—
221#1779—471no—
222C22⋊#95D4711yescharacteristic
223#1294—474no—
224C2×C47—22no—
225C2×C47—24no—
226C2×C47—22no—
227C2×C47—42no—
228C2×C47C2×#32811yescharacteristic
229C2×C47#150711yescharacteristic
230C2×C47#151011yescharacteristic
231C2×C47#152011yescharacteristic
232C2×C47#181911yescharacteristic
233C2×C47#182411yescharacteristic
234C2×C47#182611yescharacteristic
235C24×C4C2×D4711yescharacteristic
236C22×C4⋊C4—471no—
237C22×C4⋊C4—471no—
238C22×C22⋊C4—21no—
239C22×C22⋊C4—941no—
240C2×C22⋊C8—472no—
241C2×C22⋊C8—941no—
242C2×C22⋊C8—941no—
243#380—472no—
244#380—472no—
245#380—941no—
246#380—941no—
247C2×#95—21no—
248C2×#95—941no—
249C2×#95C2×D4712yes—
250C47#871311yescharacteristic, complemented, Sylow
251C25C22⋊D4711yescharacteristic
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4—941no—
257C23×C4—941no—
258C23×C4C4×D4711yescharacteristic
259C23×C4C4×D4711yescharacteristic
260C23×C4D18811yescharacteristic
261C23×C4D18811yescharacteristic
262C23×C4C22⋊D4711yescharacteristic
263C23×C4C22×D4711yescharacteristic
264C22×C8—942no—
265C2×C4⋊C4—472no—
266C2×C4⋊C4—472no—
267C2×C4⋊C4—941no—
268C2×C4⋊C4—941no—
269C2×C4⋊C4—1882no—
270C2×C4⋊C4—1882no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—942no—
275C2×C22⋊C4—942no—
276C2×C22⋊C4—1884no—
277C22⋊C8—944no—
278C22⋊C8—1882no—
279C22⋊C8—1882no—
280#95—24no—
281#95—42no—
282#95—942no—
283#95—942no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24(C2×D47)⋊C411yescharacteristic
291C24(C2×D47)⋊C411yescharacteristic
292C24#1062533512yes—
293C24#1062533712yes—
294C24C2×C22⋊D4711yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—416no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—942no—
321C22×C4—942no—
322C22×C4—1882no—
323C22×C4—1882no—
324C22×C4—1882no—
325C22×C4—1882no—
326C22×C4—1882no—
327C22×C4—1882no—
328C22×C4(C2×D47)⋊C411yescharacteristic
329C22×C4(C2×D47)⋊C411yescharacteristic
330C22×C4C2×C4×D4711yescharacteristic
331C22×C4C2×D18811yescharacteristic
332C22×C4#1062535214yes—
333C2×C8—1882no—
334C2×C8—1882no—
335C2×C8—1882no—
336C4⋊C4—1882no—
337C4⋊C4—1882no—
338C4⋊C4—1882no—
339C4⋊C4—1882no—
340C22⋊C4—48no—
341C22⋊C4—1884no—
342C22⋊C4—1884no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—24no—
355C23—24no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—24no—
360C23—24no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23#1309393812yes—
368C23#1309394712yes—
369C23#1309394712yes—
370C23#1309395212yes—
371C23C2×(C2×D47)⋊C411yescharacteristic
372C23#1309400512yes—
373C23C2×#1062533511yescharacteristic
374C23C2×#1062533711yescharacteristic
375C23#1309404912yes—
376C2×C4—28no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—3764no—
399C2×C4—3764no—
400C2×C4—1882no—
401C2×C4—1882no—
402C2×C4—1882no—
403C2×C4—1882no—
404C2×C4—1882no—
405C2×C4—1882no—
406C2×C4#1309393812yes—
407C2×C4#1309398012yes—
408C2×C4#1309400812yes—
409C8—3762no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—48no—
428C22—48no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22#1568885914yes—
434C22#1568898411yescharacteristic
435C22C2×#1309393811yescharacteristic
436C22C2×#1309393811yescharacteristic
437C22C2×#1309394711yescharacteristic
438C22#1568912711yescharacteristic
439C22C2×#1309395211yescharacteristic
440C22#1568914111yescharacteristic
441C22#1568943912yes—
442C22#1568945612yes—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—84no—
448C4—3762no—
449C4—3762no—
450C2—22no—
451C2—24no—
452C2—22no—
453C2—42no—
454C2C2×#1568885911yescharacteristic
455C2#2024190311yescharacteristic
456C2#2024191611yescharacteristic
457C2#2024195911yescharacteristic
458C2#2024346011yescharacteristic
459C2#2024347611yescharacteristic
460C2#2024348811yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×C22⋊#95C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC22⋊#95D47
hypocenterC47#8713
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. #20241267C2semidirect
5. C22×C4⋊Dic47C4semidirect
6. C22×C4⋊Dic47C4semidirect
13. #15688983C4semidirect
250. C47#8713semidirect

Maximal subgroups

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

subgroupindex
C2×#156889832
#202432982
#202412672
C47×C22⋊#952
#871347