Finite Group Databasebeta

← #26110978

Subgroups — #26110978

Order 12032.

600 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C4×#13094005C211yesmaximal, characteristic
2#20243454C212yesmaximal, complemented
3#20241706C211yesmaximal, characteristic
4#20241235C212yesmaximal, complemented
5C47×#2973C211yesmaximal, characteristic
6C2×#13094005C411yescharacteristic
7C22×C4×Dic47C2211yescharacteristic
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689011—21no—
10#15689004—21no—
11#15689003C411yescharacteristic
12#15688995C411yescharacteristic
13#15688984C411yescharacteristic
14C4×Dic47⋊C4—21no—
15C4×C22⋊Dic47—21no—
16C4×C22⋊Dic47C2211yescharacteristic
17#15688877—24no—
18#15688849—21no—
19#15688847—21no—
20C2×C4×D4×C47C2211yescharacteristic
21C47×#435—24no—
22C47×#432—21no—
23C47×#407—21no—
24C2×C22⋊C8×C47C2212yes—
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C22×C47⋊C8—22no—
28#13094005—22no—
29#13094005—22no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47C2×C411yescharacteristic
35C2×C22⋊Dic47C2×C411yescharacteristic
36C2×C4×Dic47—21no—
37C2×C4×Dic47—21no—
38C2×C4×Dic47—21no—
39C2×C4×Dic47—22no—
40C2×C4×Dic47—22no—
41C22⋊(C47⋊C8)—22no—
42C22⋊(C47⋊C8)—22no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C22×D4×C47C2×C411yescharacteristic
49C23×C4×C47D411yescharacteristic
50C23×C4×C47C2311yescharacteristic
51C2×M16×C47—22no—
52C22×C8×C47D414yes—
53C4×D4×C47—22no—
54C4×D4×C47—22no—
55C2×C4⋊C4×C47D411yescharacteristic
56C2×C22⋊C4×C47C2×C411yescharacteristic
57C2×C22⋊C4×C47D411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C4⋊C8×C47—22no—
60C22⋊C8×C47—22no—
61C22⋊C8×C47—22no—
62C22⋊C8×C47—22no—
63#97×C47—22no—
64C24×C47C22⋊C411yescharacteristic
65C24×C47C22×C411yescharacteristic
66C2×D4×C47—22no—
67C2×D4×C47—21no—
68C2×D4×C47—21no—
69C2×D4×C47—22no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4112yes—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—22no—
85C22×Dic47—22no—
86C22×Dic47—21no—
87C22×Dic47—22no—
88C22×Dic47—21no—
89C22×Dic47—42no—
90C22×Dic47M1612yescomplemented
91M16×C47—82no—
92C2×C8×C47—24no—
93C2×C8×C47—44no—
94C4⋊C4×C47—22no—
95C22⋊C4×C47—22no—
96C22⋊C4×C47—22no—
97C42×C47—21no—
98C42×C47—21no—
99C22⋊Dic47—24no—
100C22⋊Dic47—21no—
101C22⋊Dic47—22no—
102C22⋊Dic47—42no—
103C22⋊Dic47M1612yescomplemented
104Dic47⋊C4—24no—
105Dic47⋊C4—42no—
106C4×Dic47—42no—
107C4×Dic47—41no—
108C4×Dic47—42no—
109C4×Dic47—41no—
110C2×C47⋊C8—44no—
111C2×C47⋊C8—42no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47C2×C22⋊C411yescharacteristic
120C23×C47C4×D412yes—
121C23×C47C2×M1611yescharacteristic
122C23×C47C2×M1611yescharacteristic
123C23×C47#13112yes—
124D4×C47—42no—
125D4×C47—42no—
126C2×C4×C47—24no—
127C2×C4×C47—22no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—41no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—81no—
139C2×C4×C47#12011yescharacteristic
140C2×C4×C47#12112yes—
141C2×C4×C47#12311yescharacteristic
142C2×Dic47—22no—
143C2×Dic47—24no—
144C2×Dic47—22no—
145C2×Dic47—22no—
146C2×Dic47—24no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—42no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—42no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C8×C47—44no—
159C47⋊C8—82no—
160#8968—471nomaximal, Sylow
161C22×C47—21no—
162C22×C47—22no—
163C22×C47—21no—
164C22×C47—22no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—42no—
168C22×C47—42no—
169C22×C47—41no—
170C22×C47—41no—
171C22×C47—41no—
172C22×C47—81no—
173C22×C47#38611yescharacteristic
174C22×C47#40711yescharacteristic
175C22×C47#40811yescharacteristic
176C22×C47#43512yes—
177C22×C47#43612yes—
178C4×C47—22no—
179C4×C47—21no—
180C4×C47—41no—
181C4×C47—41no—
182C4×C47—41no—
183C4×C47#45812yes—
184Dic47—42no—
185Dic47—42no—
186Dic47—42no—
187Dic47—81no—
188#2973D4711yescharacteristic
189C4×#123—471no—
190#1817—472no—
191#1459—471no—
192#1282—472no—
193C2×C47—21no—
194C2×C47—21no—
195C2×C47—22no—
196C2×C47—22no—
197C2×C47—81no—
198C2×C47#187011yescharacteristic
199C2×C47#189011yescharacteristic
200C2×C47#189111yescharacteristic
201C2×#123—471no—
202C2×C4×D4C2×D4711yescharacteristic
203C22×C42—471no—
204#435—24no—
205#432—21no—
206#407—21no—
207C2×C22⋊C8—472no—
208C2×C22⋊C8C2×D4712yes—
209#388—941no—
210#387—941no—
211#386—471no—
212#385—471no—
213#381—471no—
214C4×C4⋊C4—941no—
215C4×C22⋊C4—471no—
216C4×C22⋊C4—941no—
217#336—944no—
218#324—941no—
219#323—941no—
220C47#896811yescharacteristic, complemented, Sylow
221C22×D4C4×D4711yescharacteristic
222C23×C4—471no—
223C23×C4—471no—
224C23×C4C22⋊D4711yescharacteristic
225C23×C4C22×D4711yescharacteristic
226C2×M16—22no—
227C22×C8—942no—
228C22×C8C22⋊D4714yes—
229#123—942no—
230#123—942no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—941no—
235C2×C4⋊C4D18811yescharacteristic
236C2×C22⋊C4—471no—
237C2×C22⋊C4—471no—
238C2×C22⋊C4—941no—
239C2×C22⋊C4—941no—
240C2×C22⋊C4C4×D4711yescharacteristic
241C2×C22⋊C4D18811yescharacteristic
242C2×C42—941no—
243C2×C42—942no—
244C2×C42—942no—
245C2×C42—941no—
246C2×C42—941no—
247C2×C42C22⋊D4711yescharacteristic
248C4⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—942no—
253C22⋊C8—942no—
254#97—22no—
255#95—941no—
256#95—941no—
257#95—941no—
258#95—941no—
259#95—941no—
260C24(C2×D47)⋊C411yescharacteristic
261C24C2×C4×D4711yescharacteristic
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—472no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4—941no—
274C22×C4—942no—
275C22×C4—942no—
276C22×C4—941no—
277C22×C4—942no—
278C22×C4—941no—
279C22×C4—1882no—
280C22×C4(C2×D47)⋊C411yescharacteristic
281C22×C4(C2×D47)⋊C411yescharacteristic
282C22×C4(C2×D47)⋊C411yescharacteristic
283C22×C4C2×D18811yescharacteristic
284C22×C4#1062535012yes—
285C22×C4C2×C22⋊D4712yes—
286C22×C4C2×C22⋊D4711yescharacteristic
287M16—82no—
288C2×C8—24no—
289C2×C8—44no—
290C2×C8—1884no—
291C2×C8—1882no—
292C4⋊C4—22no—
293C4⋊C4—944no—
294C4⋊C4—1882no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—472no—
298C22⋊C4—944no—
299C22⋊C4—941no—
300C22⋊C4—942no—
301C22⋊C4—1882no—
302C42—21no—
303C42—21no—
304C42—1882no—
305C42—1881no—
306C42—1882no—
307C42—1881no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23C2×(C2×D47)⋊C411yescharacteristic
316C23C4×C22⋊D4712yes—
317C23M16×D4711yescharacteristic
318C23M16×D4711yescharacteristic
319C23#1309403212yes—
320D4—42no—
321D4—42no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—942no—
336C2×C4—944no—
337C2×C4—942no—
338C2×C4—942no—
339C2×C4—944no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4—1881no—
345C2×C4—1881no—
346C2×C4—1882no—
347C2×C4—1881no—
348C2×C4—1881no—
349C2×C4—1881no—
350C2×C4—1882no—
351C2×C4#1309399311yescharacteristic
352C2×C4#1309399812yes—
353C2×C4#1309400711yescharacteristic
354C8—44no—
355C8—3762no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22#1568900211yescharacteristic
369C22#1568908611yescharacteristic
370C22#1568909111yescharacteristic
371C22#1568919212yes—
372C22#1568919912yes—
373C4—22no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—3761no—
379C4—1882no—
380C4—1882no—
381C4—1882no—
382C4#1568931312yes—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—22no—
387C2—81no—
388C2#2024368511yescharacteristic
389C2#2024380111yescharacteristic
390C2#2024380611yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2973C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#2973D47
hypocenterC47#8968
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20243454C2semidirect
4. #20241235C2semidirect
90. C22×Dic47M16semidirect
103. C22⋊Dic47M16semidirect
220. C47#8968semidirect

Maximal subgroups

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

subgroupindex
C4×#130940052
#202434542
#202417062
#202412352
C47×#29732
#896847