Finite Group Databasebeta

← #26113642

Subgroups — #26113642

Order 12032.

558 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 117 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#20252411C211yesmaximal, characteristic
2#20248383C211yesmaximal, characteristic
3#20243377C211yesmaximal, characteristic
4#20242061C211yesmaximal, characteristic, complemented
5#20242059C211yesmaximal, characteristic, complemented
6#20241291C211yesmaximal, characteristic, complemented
7C47×#1303C211yesmaximal, characteristic, complemented
8C2×#13094024C2211yescharacteristic
9C2×#13094012—21no—
10C2×C4×D188C2211yescharacteristic
11C8×D188—22no—
12#15689188—22no—
13#15689176C2211yescharacteristic
14#15689147C2211yescharacteristic
15C2×#13093940—21no—
16C22⋊C8×D47—22no—
17#15689050C414yes—
18#15689029—21no—
19#15688980—22no—
20#15688891—21no—
21#15688870—22no—
22#15688868—22no—
23C47×#431C2211yescharacteristic
24C47×#423C2211yescharacteristic
25C4×C4⋊C4×C47C2211yescharacteristic
26C22×Dic94—21no—
27C22×D188D411yescharacteristic
28C22×C4×D47D412yes—
29C2×#10625329—21no—
30C2×C8×D47—24no—
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024—21no—
35#13094012—42no—
36#13094012—42no—
37C4×D188—21no—
38C4×D188—21no—
39C4×D188—21no—
40C4×D188—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47C2×C412yes—
43C2×C4⋊Dic47D411yescharacteristic
44C2×(C2×D47)⋊C4—24no—
45C2×(C2×D47)⋊C4C2×C412yes—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)D412yes—
49#13093940—42no—
50C22⋊(C47⋊C8)—21no—
51(C2×D47)⋊C8—24no—
52C4×C47⋊C8—21no—
53C47⋊#95—24no—
54C2×M16×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C42×C47—21no—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47D412yes—
61C22⋊C8×C47—21no—
62C4×C8×C47—21no—
63#95×C47C2×C412yes—
64C23×D47C22⋊C412yes—
65C22×C4×C47—22no—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—22no—
70C22×Dic47C22⋊C412yes—
71C2×D188—21no—
72C2×D188—21no—
73C2×D188—21no—
74C2×D188—21no—
75C2×D188D812yes—
76C2×D188SD1612yes—
77C2×C4×D47—22no—
78C2×C4×D47—22no—
79C2×C4×D47—44no—
80C2×Dic94—41no—
81C2×Dic94—41no—
82C2×Dic94—41no—
83C2×Dic94—41no—
84C2×Dic94—41no—
85C2×Dic94—41no—
86M16×C47—41no—
87M16×C47—41no—
88C2×C8×C47—22no—
89C2×C8×C47—41no—
90C4⋊C4×C47—42no—
91C42×C47—41no—
92C42×C47—41no—
93C42×C47C2×D411yescharacteristic
94C42×C47C2×D411yescharacteristic
95C42×C47#4111yescharacteristic
96C42×C47#4111yescharacteristic
97(C2×D47)⋊C4—22no—
98(C2×D47)⋊C4—22no—
99(C2×D47)⋊C4—48no—
100C4⋊Dic47—21no—
101C4⋊Dic47—21no—
102C4⋊Dic47—22no—
103C4⋊Dic47—22no—
104C4⋊Dic47—41no—
105C4⋊Dic47—41no—
106#10625329—41no—
107#10625329—41no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—41no—
110C8×D47—84no—
111C23×C47C2×C22⋊C411yescharacteristic
112C22×D47—22no—
113C22×D47—22no—
114C22×D47—42no—
115C22×D47—42no—
116C22×D47#10214yes—
117C2×C4×C47—22no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47C4×D411yescharacteristic
123C2×C4×C47C4×D411yescharacteristic
124C2×C4×C47#12011yescharacteristic
125C2×C4×C47#12111yescharacteristic
126C2×C4×C47#12111yescharacteristic
127C2×C4×C47#12311yescharacteristic
128C2×C4×C47C2×D811yescharacteristic
129C2×C4×C47C2×SD1611yescharacteristic
130C2×C4×C47#13511yescharacteristic
131C2×C4×C47#13511yescharacteristic
132C2×Dic47—22no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137D188—22no—
138D188—22no—
139D188—22no—
140D188—22no—
141C4×D47—44no—
142C4×D47—82no—
143Dic94—81no—
144Dic94—81no—
145Dic94—81no—
146Dic94—81no—
147C8×C47—42no—
148C8×C47—81no—
149C47⋊C8—42no—
150C47⋊C8—81no—
151#9393—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47#38611yescharacteristic
155C22×C47C2×#10211yescharacteristic
156C22×C47#41611yescharacteristic
157C2×D47—24no—
158C2×D47—42no—
159C2×D47—42no—
160C2×D47—44no—
161C2×D47—82no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—82no—
166C4×C47#45512yes—
167C4×C47#46511yescharacteristic
168C4×C47#46511yescharacteristic
169C4×C47#46611yescharacteristic
170C4×C47#46611yescharacteristic
171Dic47—42no—
172Dic47—82no—
173#2972—471no—
174#2547—471no—
175#1800—471no—
176#1546—471no—
177#1546—471no—
178#1303—471no—
179#1303D4711yescharacteristic, complemented
180C2×C47—21no—
181C2×C47—21no—
182C2×C47#188511yescharacteristic
183C2×C47#189311yescharacteristic
184C2×C47#191511yescharacteristic
185D47—44no—
186C2×C4⋊Q8—471no—
187C2×#124—941no—
188C2×C4×D4—471no—
189C8×D4—942no—
190C8×D4—942no—
191#431—471no—
192#431C2×D4711yescharacteristic
193#423—471no—
194#423C2×D4711yescharacteristic
195C2×#100—941no—
196C2×C22⋊C8—942no—
197#397—474no—
198#391—941no—
199#380—942no—
200C4×C4⋊C4C2×D4711yescharacteristic
201#343—941no—
202#332—942no—
203#332—942no—
204C47#939311yescharacteristic, complemented, Sylow
205C22×Q8—941no—
206C22×D4—471no—
207C23×C4—472no—
208C2×M16—21no—
209C2×M16—941no—
210C22×C8—944no—
211C4⋊Q8—941no—
212C4⋊Q8—941no—
213C4⋊Q8—941no—
214C4⋊Q8—941no—
215#124—1882no—
216#124—1882no—
217C4×D4—941no—
218C4×D4—941no—
219C4×D4—941no—
220C4×D4—941no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—471no—
223C2×C4⋊C4—472no—
224C2×C4⋊C4—941no—
225C2×C22⋊C4—472no—
226C2×C22⋊C4—944no—
227C2×C42—21no—
228C2×C42C22×D4711yescharacteristic
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—472no—
232C4⋊C8—941no—
233C4⋊C8—941no—
234C4⋊C8D18812yes—
235#100—1882no—
236C22⋊C8—21no—
237C22⋊C8—944no—
238C22⋊C8—941no—
239C4×C8—21no—
240C4×C8—941no—
241#95—944no—
242#95C4×D4712yes—
243C24—472no—
244C2×Q8—1881no—
245C2×Q8—1881no—
246C2×Q8—1881no—
247C2×Q8—1881no—
248C2×Q8—1881no—
249C2×Q8—1881no—
250C2×D4—472no—
251C2×D4—472no—
252C2×D4—941no—
253C2×D4—941no—
254C2×D4—941no—
255C2×D4—941no—
256C22×C4—22no—
257C22×C4—472no—
258C22×C4—942no—
259C22×C4—942no—
260C22×C4—942no—
261C22×C4—1884no—
262C22×C4C2×C4×D4711yescharacteristic
263C22×C4D4×D4711yescharacteristic
264C22×C4D4×D4711yescharacteristic
265M16—41no—
266M16—41no—
267M16—1881no—
268M16—1881no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—942no—
272C2×C8—3764no—
273C2×C8—1881no—
274C4⋊C4—42no—
275C4⋊C4—941no—
276C4⋊C4—941no—
277C4⋊C4—942no—
278C4⋊C4—942no—
279C4⋊C4—1881no—
280C4⋊C4—1881no—
281C22⋊C4—942no—
282C22⋊C4—942no—
283C22⋊C4—1888no—
284C42—41no—
285C42—41no—
286C42C2×D18811yescharacteristic
287C42#1062535011yescharacteristic
288C42D4×D4711yescharacteristic
289C42#1062535411yescharacteristic
290C23—474no—
291C23—942no—
292C23—942no—
293C23—1882no—
294C23—1882no—
295C23C22⋊C4×D4711yescharacteristic
296Q8—3761no—
297Q8—3761no—
298Q8—3761no—
299Q8—3761no—
300D4—942no—
301D4—942no—
302D4—942no—
303D4—942no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—942no—
310C2×C4—3762no—
311C2×C4—1882no—
312C2×C4—1882no—
313C2×C4—1882no—
314C2×C4—1884no—
315C2×C4—1882no—
316C2×C4#1309398611yescharacteristic
317C2×C4C4×D18811yescharacteristic
318C2×C4#1309399411yescharacteristic
319C2×C4#1309399711yescharacteristic
320C2×C4#1309399911yescharacteristic
321C2×C4#1309400911yescharacteristic
322C2×C4D8×D4711yescharacteristic
323C2×C4SD16×D4711yescharacteristic
324C2×C4#1309404511yescharacteristic
325C2×C4#1309404711yescharacteristic
326C8—42no—
327C8—81no—
328C8—3761no—
329C8—1882no—
330C22—21no—
331C22—21no—
332C22—944no—
333C22—3762no—
334C22—1884no—
335C22—1882no—
336C22—1882no—
337C22#1568900111yescharacteristic
338C22#102×D4711yescharacteristic
339C22#1568911811yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—82no—
344C4—3762no—
345C4—1882no—
346C4#1568930212yes—
347C4#1568935311yescharacteristic
348C4#1568935511yescharacteristic
349C4#1568935811yescharacteristic
350C4#1568936011yescharacteristic
351C2—21no—
352C2—21no—
353C2—1884no—
354C2#2024377511yescharacteristic
355C2#2024381511yescharacteristic
356C2#2024393911yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1303C2
socleC22×C47#386
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1303D47
hypocenterC47#9393
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
4. #20242061C2semidirect
5. #20242059C2semidirect
6. #20241291C2semidirect
7. C47×#1303C2semidirect
179. #1303D47semidirect
204. C47#9393semidirect

Maximal subgroups

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

subgroupindex
#202524112
#202483832
#202433772
#202420612
#202420592
#202412912
C47×#13032
#939347