Finite Group Databasebeta

← #26113446

Subgroups — #26113446

Order 12032.

500 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 103 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#20253336C211yesmaximal, characteristic
2#20244372C211yesmaximal, characteristic
3#20243487C211yesmaximal, characteristic
4#20241907C211yesmaximal, characteristic
5#20241294C211yesmaximal, characteristic
6#20241274C211yesmaximal, characteristic
7C47×#2347C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689464—22no—
10#15689458—21no—
11#15689453—21no—
12#15689166C412yescomplemented
13#15689131C412yescomplemented
14#15689087C2211yescharacteristic
15#15689087C2211yescharacteristic
16C4×C22⋊Dic47C2211yescharacteristic
17#15688889—22no—
18#15688882—22no—
19#15688859—21no—
20#15688859—21no—
21C2×#117×C47C2211yescharacteristic
22C47×#389—21no—
23C47×#388—22no—
24C47×#383—21no—
25C47×#380C2211yescharacteristic
26C4×C4⋊C4×C47—21no—
27C4×C4⋊C4×C47—21no—
28C4×C22⋊C4×C47C2211yescharacteristic
29C22×Dic94C2×C411yescharacteristic
30C2×#10625329—21no—
31C2×#10625329C2×C412yes—
32#13094003—21no—
33#13094003—21no—
34#13094003—21no—
35#13094003—21no—
36C2×C4⋊Dic47C2×C411yescharacteristic
37C2×Dic47⋊C4—21no—
38C2×C22⋊Dic47—21no—
39C2×C22⋊Dic47—21no—
40C2×C4×Dic47—21no—
41#13093957—22no—
42#13093955—22no—
43#13093947—22no—
44#13093947—22no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)—21no—
49C47⋊#95—21no—
50C23×C4×C47C2311yescharacteristic
51#117×C47—21no—
52#117×C47—21no—
53#117×C47C2×C412yes—
54#117×C47D412yes—
55C2×C4⋊C4×C47—22no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47D411yescharacteristic
58C2×C4⋊C4×C47D411yescharacteristic
59C2×C22⋊C4×C47—21no—
60C2×C22⋊C4×C47—21no—
61C2×C22⋊C4×C47D412yes—
62C2×C42×C47—21no—
63C2×C42×C47—21no—
64C2×C42×C47—22no—
65#95×C47—22no—
66#95×C47—22no—
67#95×C47—21no—
68#95×C47D411yescharacteristic
69#95×C47D411yescharacteristic
70C24×C47C2×D411yescharacteristic
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—22no—
76C22×C4×C47—42no—
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47#4111yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—41no—
86C2×Dic94—21no—
87C2×Dic94—21no—
88C2×Dic94—42no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—42no—
92C4⋊C4×C47—44no—
93C4⋊C4×C47C22⋊C412yes—
94C4⋊C4×C47C22⋊C412yes—
95C22⋊C4×C47—22no—
96C22⋊C4×C47—22no—
97C22⋊C4×C47—42no—
98C22⋊C4×C47—42no—
99C42×C47—22no—
100C42×C47—22no—
101C42×C47—42no—
102C42×C47—82no—
103C22⋊Dic47—22no—
104C22⋊Dic47—41no—
105C22⋊Dic47—41no—
106C22⋊Dic47—41no—
107C4⋊Dic47—21no—
108C4⋊Dic47—21no—
109Dic47⋊C4—41no—
110Dic47⋊C4—41no—
111C4×Dic47—42no—
112#10625329—42no—
113#10625329—42no—
114C2×C47⋊C8—22no—
115C2×C47⋊C8—41no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47C4×D411yescharacteristic
121C23×C47#12011yescharacteristic
122C23×C47#12111yescharacteristic
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—41no—
143C2×C4×C47C2×C22⋊C411yescharacteristic
144C2×C4×C47C4×D411yescharacteristic
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47#12311yescharacteristic
147C2×Dic47—42no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—81no—
151C2×Dic47—81no—
152C2×Dic47—81no—
153C2×Dic47—81no—
154Dic94—42no—
155Dic94—81no—
156C47⋊C8—42no—
157C47⋊C8—81no—
158#9362—471nomaximal, Sylow
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—81no—
168C22×C47—81no—
169C22×C47#38611yescharacteristic
170C22×C47#45511yescharacteristic
171C22×C47#45611yescharacteristic
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—42no—
176C4×C47—42no—
177C4×C47—42no—
178C4×C47—82no—
179C4×C47—81no—
180C4×C47#41012yes—
181Dic47—82no—
182Dic47—81no—
183Dic47—81no—
184#3095—471no—
185#2347D4711yescharacteristic
186#1996—471no—
187#1826—471no—
188#1508—471no—
189#1305—471no—
190#1297—471no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—41no—
194C2×C47—41no—
195C2×C47#188211yescharacteristic
196C2×C47#189911yescharacteristic
197C2×C47#190611yescharacteristic
198C2×C22⋊Q8—471no—
199C2×#117C2×D4711yescharacteristic
200#485—942no—
201#484—941no—
202#483—941no—
203#428—472no—
204#419—472no—
205#407—471no—
206#407—471no—
207#389—21no—
208#388—22no—
209#383—21no—
210#380C2×D4711yescharacteristic
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C22⋊C4—471no—
214C4×C22⋊C4C2×D4711yescharacteristic
215#342—942no—
216#339—942no—
217#328—941no—
218#328—941no—
219C47#936211yescharacteristic, complemented, Sylow
220C22×Q8—471no—
221C23×C4C22×D4711yescharacteristic
222C2×M16—472no—
223C2×M16—941no—
224C22⋊Q8—941no—
225C22⋊Q8—941no—
226C22⋊Q8—941no—
227C22⋊Q8—941no—
228#117—21no—
229#117—21no—
230#117C4×D4712yes—
231#117D18812yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—471no—
235C2×C4⋊C4—941no—
236C2×C4⋊C4C22⋊D4711yescharacteristic
237C2×C4⋊C4C22⋊D4711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—941no—
241C2×C22⋊C4—941no—
242C2×C22⋊C4C22⋊D4712yes—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—941no—
247#107—942no—
248#106—942no—
249#103—942no—
250#103—942no—
251C22⋊C8—941no—
252C22⋊C8—941no—
253C22⋊C8—941no—
254C22⋊C8—941no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—941no—
259#95C22⋊D4711yescharacteristic
260#95C22⋊D4711yescharacteristic
261C24C2×C22⋊D4711yescharacteristic
262C2×Q8—941no—
263C2×Q8—941no—
264C2×Q8—1882no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4—1881no—
274C22×C4C2×C4×D4711yescharacteristic
275C22×C4C2×D18811yescharacteristic
276C22×C4#1062535011yescharacteristic
277C22×C4#1062535011yescharacteristic
278C22×C4C2×C22⋊D4711yescharacteristic
279C22×C4C2×C22⋊D4711yescharacteristic
280M16—1882no—
281M16—1882no—
282C2×C8—942no—
283C2×C8—1881no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—941no—
289C4⋊C4—941no—
290C4⋊C4—1881no—
291C4⋊C4—1881no—
292C4⋊C4(C2×D47)⋊C412yes—
293C4⋊C4(C2×D47)⋊C412yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—942no—
299C22⋊C4—1881no—
300C22⋊C4—1881no—
301C22⋊C4—1881no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—82no—
306C42—1882no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23C4×C22⋊D4711yescharacteristic
312C23#1309399311yescharacteristic
313C23#1309399811yescharacteristic
314Q8—3761no—
315Q8—1882no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—3761no—
337C2×C4—3761no—
338C2×C4—3761no—
339C2×C4—3761no—
340C2×C4—1882no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4C2×(C2×D47)⋊C411yescharacteristic
344C2×C4C4×C22⋊D4711yescharacteristic
345C2×C4#1309399811yescharacteristic
346C2×C4#1309400711yescharacteristic
347C8—3761no—
348C8—1882no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1568900211yescharacteristic
360C22#1568930111yescharacteristic
361C22#1568930611yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—81no—
370C4—3762no—
371C4—3761no—
372C4—3761no—
373C4#1568909612yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2024375711yescharacteristic
379C2#2024384611yescharacteristic
380C2#2024388911yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2347C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2347D47
hypocenterC47#9362
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
12. #15689166C4semidirect
13. #15689131C4semidirect
219. C47#9362semidirect

Maximal subgroups

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

subgroupindex
#202533362
#202443722
#202434872
#202419072
#202412942
#202412742
C47×#23472
#936247