Finite Group Databasebeta

← #26112344

Subgroups — #26112344

Order 12032.

622 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 127 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#20249130C211yesmaximal, characteristic
2#20246006C211yesmaximal, characteristic, complemented
3#20241957C212yesmaximal
4#20241256C212yesmaximal, complemented
5C2×C47×#324C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8C2×#13093983—21no—
9#15689053C412yescomplemented
10#15689045—22no—
11#15689040C412yescomplemented
12#15689022—24no—
13#15689002—21no—
14#15688983—21no—
15#15688983C2211yescharacteristic
16#15688983C2212yes—
17#15688858—24no—
18C22×C4⋊C4×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#324—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23#13094007—42no—
24#13094007—42no—
25#13093998—22no—
26#13093998—22no—
27#13093983—42no—
28#13093983—42no—
29C2×C4⋊Dic47—22no—
30C2×C4⋊Dic47—22no—
31C2×C4⋊Dic47—22no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47C2×C411yescharacteristic
34C2×Dic47⋊C4—22no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47—21no—
37C2×(C2×D47)⋊C4—22no—
38C2×(C2×D47)⋊C4—21no—
39C2×(C2×D47)⋊C4—22no—
40C2×C4×Dic47—22no—
41C47⋊#95—22no—
42C47⋊#95—22no—
43C47⋊#95C2×C412yes—
44C47⋊#95D414yes—
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C23×C4×C47C2311yescharacteristic
48C22×C8×C47—22no—
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47—22no—
51C2×C4⋊C4×C47C2×C412yes—
52C2×C4⋊C4×C47D412yes—
53C22⋊C8×C47—22no—
54C22⋊C8×C47—22no—
55C24×C47C2×D411yescharacteristic
56C23×D47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—81no—
64C22×C4×C47—81no—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47D812yes—
68C22×C4×C47SD1612yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D412yes—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47#4112yes—
73C2×C22⋊D47—41no—
74C2×C22⋊D47—41no—
75C2×C22⋊D47—41no—
76C2×C22⋊D47—41no—
77C2×C22⋊D47—41no—
78C2×C22⋊D47—41no—
79C22×Dic47—21no—
80C22×Dic47—22no—
81C22×Dic47—24no—
82C2×D188—22no—
83C2×D188—81no—
84C2×C8×C47—42no—
85C2×C8×C47—82no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47—24no—
88C22⋊Dic47—42no—
89C22⋊Dic47—41no—
90C22⋊Dic47—41no—
91C22⋊Dic47—42no—
92(C2×D47)⋊C4—44no—
93(C2×D47)⋊C4—42no—
94(C2×D47)⋊C4—82no—
95C4⋊Dic47—22no—
96C4⋊Dic47—44no—
97C4⋊Dic47—82no—
98C4⋊Dic47—82no—
99C4⋊Dic47—81no—
100Dic47⋊C4—44no—
101C4×Dic47—44no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47#12011yescharacteristic
106C23×C47#12311yescharacteristic
107C23×C47C2×D811yescharacteristic
108C23×C47C2×SD1611yescharacteristic
109C23×C47AGL(1,ℤ/8)11yescharacteristic
110C23×C47#13711yescharacteristic
111C22×D47—42no—
112C22×D47—42no—
113C22×D47—41no—
114C22×D47—41no—
115C22×D47—41no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—42no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—81no—
130C2×C4×C47#10214yes—
131C2×C4×C47C4×D412yes—
132C2×C4×C47#12112yes—
133C22⋊D47—81no—
134C22⋊D47—81no—
135C22⋊D47—81no—
136C22⋊D47—81no—
137C2×Dic47—44no—
138C2×Dic47—42no—
139C2×Dic47—44no—
140C2×Dic47—42no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—42no—
144C2×Dic47—41no—
145C2×Dic47—44no—
146D188—82no—
147C8×C47—82no—
148#9204—471nomaximal, Sylow
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47#38611yescharacteristic
160C22×C47C2×#10211yescharacteristic
161C22×C47#41711yescharacteristic
162C22×C47#44711yescharacteristic
163C22×C47#44811yescharacteristic
164C22×C47#48011yescharacteristic
165C22×C47#48111yescharacteristic
166C2×D47—82no—
167C2×D47—82no—
168C2×D47—81no—
169C2×D47—81no—
170C2×D47—81no—
171C4×C47—24no—
172C4×C47—22no—
173C4×C47—22no—
174C4×C47—42no—
175Dic47—84no—
176Dic47—82no—
177Dic47—81no—
178#2618—471no—
179#2316—471no—
180#1519—472no—
181C2×#324D4711yescharacteristic, complemented
182#1289—472no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47#187411yescharacteristic
190C2×C47#191211yescharacteristic
191C2×C47#191311yescharacteristic
192D47—161no—
193C2×#123—941no—
194C2×#121—471no—
195C2×#117—941no—
196C22×C4⋊C4C2×D4711yescharacteristic
197C2×C22⋊C8C2×D4712yes—
198#398—472no—
199#396—942no—
200#394—472no—
201#389—944no—
202#386—941no—
203#380—471no—
204#380—472no—
205#380—941no—
206#327—944no—
207#324—22no—
208C47#920411yescharacteristic, complemented, Sylow
209C22×D4—471no—
210C22×D4—941no—
211C23×C4D18811yescharacteristic
212C23×C4D18811yescharacteristic
213C23×C4C22×D4711yescharacteristic
214C22×C8—22no—
215#123—1882no—
216#123—1882no—
217#121—942no—
218#121—942no—
219#117—1882no—
220#117—1882no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—471no—
224C2×C4⋊C4—942no—
225C2×C4⋊C4—942no—
226C2×C4⋊C4—941no—
227C2×C4⋊C4—942no—
228C2×C4⋊C4—942no—
229C2×C4⋊C4C4×D4712yes—
230C2×C4⋊C4C22⋊D4712yes—
231C2×C22⋊C4—942no—
232C2×C22⋊C4—942no—
233C2×C22⋊C4—941no—
234C2×C22⋊C4—942no—
235C2×C22⋊C4—941no—
236C2×C42—942no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239#95—472no—
240#95—474no—
241#95—942no—
242#95—942no—
243C24—941no—
244C24C2×D18811yescharacteristic
245C2×D4—942no—
246C2×D4—3761no—
247C2×D4—1881no—
248C2×D4—1881no—
249C2×D4—1881no—
250C2×D4—1881no—
251C2×D4—1881no—
252C2×D4—1881no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—942no—
262C22×C4—941no—
263C22×C4—944no—
264C22×C4#1062532512yes—
265C22×C4D37612yes—
266C22×C4(C2×D47)⋊C412yes—
267C22×C4(C2×D47)⋊C412yes—
268C22×C4C2×C4×D4711yescharacteristic
269C22×C4D4×D4712yes—
270C22×C4#1062535212yes—
271C22×C4C2×C22⋊D4711yescharacteristic
272C2×C8—42no—
273C2×C8—82no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—942no—
277C4⋊C4—3762no—
278C4⋊C4—3761no—
279C4⋊C4—3762no—
280C4⋊C4—1884no—
281C4⋊C4—1884no—
282C22⋊C4—3762no—
283C22⋊C4—1882no—
284C22⋊C4—1881no—
285C22⋊C4—1881no—
286C22⋊C4—1882no—
287C22⋊C4—1884no—
288C22⋊C4—1882no—
289C42—1884no—
290C23—41no—
291C23—41no—
292C23—1882no—
293C23—1882no—
294C23—1881no—
295C23—1881no—
296C23—1881no—
297C23C2×(C2×D47)⋊C411yescharacteristic
298C23#1309399411yescharacteristic
299C23#1309400811yescharacteristic
300C23C2×D37611yescharacteristic
301C23C2×#1062532511yescharacteristic
302C23#1309405211yescharacteristic
303C23#1309405711yescharacteristic
304D4—3762no—
305D4—3761no—
306D4—3761no—
307D4—3761no—
308D4—3761no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—1882no—
324C2×C4—1884no—
325C2×C4—1884no—
326C2×C4—1882no—
327C2×C4—1881no—
328C2×C4—1884no—
329C2×C4—1882no—
330C2×C4—1881no—
331C2×C4—1881no—
332C2×C4#1309394614yes—
333C2×C4(C22⋊D47)⋊C412yes—
334C2×C4#1309399612yes—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3762no—
347C22—3762no—
348C22—3761no—
349C22—3761no—
350C22—3761no—
351C22#1568900011yescharacteristic
352C22C2×#1309394611yescharacteristic
353C22#1568912511yescharacteristic
354C22#1568925911yescharacteristic
355C22#1568926411yescharacteristic
356C22#1568943711yescharacteristic
357C22#1568944211yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—3764no—
363C4—3762no—
364C4—3761no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—7521no—
372C2#2024370311yescharacteristic
373C2#2024392311yescharacteristic
374C2#2024392811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC22×C47#386
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#9204
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20246006C2semidirect
4. #20241256C2semidirect
5. C2×C47×#324C2semidirect
9. #15689053C4semidirect
11. #15689040C4semidirect
181. C2×#324D47semidirect
208. C47#9204semidirect

Maximal subgroups

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

subgroupindex
#202491302
#202460062
#202419572
#202412562
C2×C47×#3242
#920447