Finite Group Databasebeta

← #26114024

Subgroups — #26114024

Order 12032.

554 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 93 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#20247881C211yesmaximal, characteristic
2#20245989C211yesmaximal, characteristic, complemented
3#20243466C211yesmaximal, characteristic, complemented
4#20241917C211yesmaximal, characteristic
5#20241242C211yesmaximal, characteristic, complemented
6#20241240C211yesmaximal, characteristic, complemented
7C47×#1510C211yesmaximal, characteristic
8C22×C4⋊Dic47C2211yescharacteristic
9#15689087C2211yescharacteristic
10#15689048—22no—
11#15689026C412yescomplemented
12#15689008—22no—
13#15689006—22no—
14#15689003—21no—
15#15689003—21no—
16#15688988C412yescomplemented
17#15688986C2211yescharacteristic
18#15688981—21no—
19C22⋊C4×Dic47—21no—
20C4×C22⋊Dic47C2211yescharacteristic
21#15688891—22no—
22#15688857—22no—
23C2×#121×C47C2211yescharacteristic
24C47×#407C2211yescharacteristic
25C47×#381C2211yescharacteristic
26C47×#327—22no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C2×#10625329—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47C2×C412yes—
35C2×C4⋊Dic47D412yes—
36C2×Dic47⋊C4—22no—
37C2×C22⋊Dic47—21no—
38C2×C22⋊Dic47—21no—
39C2×C22⋊Dic47—22no—
40C2×C22⋊Dic47—22no—
41C2×C4×Dic47—22no—
42C2×C4×Dic47C2×C412yes—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C47⋊#95—22no—
46C47⋊#95—22no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C47⋊#95—22no—
50C47⋊#95—22no—
51C47⋊#95D412yes—
52C22×D4×C47—21no—
53C22×D4×C47C2×C411yescharacteristic
54C23×C4×C47C2311yescharacteristic
55C2×M16×C47—21no—
56#121×C47—21no—
57#121×C47—21no—
58#121×C47—21no—
59#121×C47—21no—
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C22⋊C4×C47—21no—
62C2×C22⋊C4×C47—21no—
63C22⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65#95×C47—21no—
66#95×C47D412yes—
67C24×C47—21no—
68C24×C47C2×D411yescharacteristic
69C2×D4×C47—21no—
70C2×D4×C47—21no—
71C2×D4×C47—42no—
72C2×D4×C47—44no—
73C2×D4×C47—41no—
74C2×D4×C47—41no—
75C22×C4×C47—21no—
76C22×C4×C47—22no—
77C22×C4×C47—42no—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C22×C4×C47#4111yescharacteristic
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—22no—
89C22×Dic47—22no—
90C22×Dic47—81no—
91C22×Dic47—81no—
92C22×Dic47C22⋊C412yes—
93C22×Dic47C22⋊C412yes—
94M16×C47—42no—
95C2×C8×C47—41no—
96C4⋊C4×C47—21no—
97C4⋊C4×C47—21no—
98C22⋊C4×C47—42no—
99C22⋊C4×C47—41no—
100C22⋊C4×C47—41no—
101C22⋊Dic47—42no—
102C22⋊Dic47—42no—
103C22⋊Dic47—82no—
104C22⋊Dic47—82no—
105C4⋊Dic47—22no—
106C4⋊Dic47—22no—
107C4⋊Dic47—22no—
108C4⋊Dic47—22no—
109Dic47⋊C4—82no—
110C4×Dic47—42no—
111C4×Dic47—82no—
112#10625329—42no—
113C2×C47⋊C8—41no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—41no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47—41no—
120C23×C47—42no—
121C23×C47C2×C22⋊C411yescharacteristic
122C23×C47#12011yescharacteristic
123C23×C47#12311yescharacteristic
124D4×C47—81no—
125D4×C47—81no—
126D4×C47—82no—
127D4×C47—82no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47C4×D411yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×Dic47—22no—
144C2×Dic47—22no—
145C2×Dic47—42no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148C2×Dic47—42no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—81no—
156C2×Dic47—81no—
157C8×C47—81no—
158C47⋊C8—81no—
159#9465—471nomaximal, Sylow
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—44no—
168C22×C47—84no—
169C22×C47—81no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47#38611yescharacteristic
173C22×C47#41012yes—
174C22×C47#45412yes—
175C4×C47—21no—
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—82no—
180C4×C47—81no—
181Dic47—42no—
182Dic47—42no—
183Dic47—82no—
184Dic47—82no—
185#2504—471no—
186#2313—471no—
187#1821—471no—
188#1510—471no—
189#1510D4711yescharacteristic
190#1284—471no—
191#1284—471no—
192C2×C47—22no—
193C2×C47—22no—
194C2×C47—41no—
195C2×C47—161no—
196C2×C47#188811yescharacteristic
197C2×C47#189911yescharacteristic
198C2×C47#191111yescharacteristic
199C2×#121C2×D4711yescharacteristic
200C22×C4⋊C4—471no—
201#407—471no—
202#407C2×D4711yescharacteristic
203#397—942no—
204#390—472no—
205#387—942no—
206#387—942no—
207#386—941no—
208#386—941no—
209#382—472no—
210#381—471no—
211#381C2×D4711yescharacteristic
212#380—941no—
213C4×C22⋊C4—471no—
214C4×C22⋊C4—941no—
215#343—942no—
216#327—22no—
217#327—942no—
218C47#946511yescharacteristic, complemented, Sylow
219C22×D4—21no—
220C22×D4C4×D4711yescharacteristic
221C23×C4—471no—
222C23×C4—471no—
223C23×C4C22×D4711yescharacteristic
224C2×M16—21no—
225C2×M16—941no—
226#121—21no—
227#121—21no—
228#121—21no—
229#121—21no—
230C2×C4⋊C4—472no—
231C2×C4⋊C4—472no—
232C2×C4⋊C4—941no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—941no—
235C2×C4⋊C4—941no—
236C2×C4⋊C4—942no—
237C2×C4⋊C4C4×D4711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—942no—
241C2×C22⋊C4—942no—
242C2×C22⋊C4—941no—
243C2×C22⋊C4—941no—
244C2×C42—472no—
245C2×C42—942no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—941no—
249C22⋊C8—941no—
250#95—21no—
251#95—472no—
252#95—942no—
253#95—942no—
254#95—942no—
255#95—942no—
256#95—941no—
257#95—941no—
258#95C22⋊D4712yes—
259C24—21no—
260C24D4×D4711yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—472no—
271C22×C4—472no—
272C22×C4—941no—
273C22×C4—941no—
274C22×C4—941no—
275C22×C4—941no—
276C22×C4—942no—
277C22×C4—942no—
278C22×C4—3761no—
279C22×C4—3761no—
280C22×C4C2×C4×D4711yescharacteristic
281C22×C4#1062535011yescharacteristic
282C22×C4D4×D4711yescharacteristic
283C22×C4D4×D4711yescharacteristic
284C22×C4#1062535211yescharacteristic
285C22×C4C2×C22⋊D4711yescharacteristic
286M16—42no—
287M16—1882no—
288C2×C8—41no—
289C2×C8—1881no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—942no—
293C4⋊C4—942no—
294C4⋊C4—942no—
295C4⋊C4—942no—
296C4⋊C4—3762no—
297C22⋊C4—42no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—3762no—
301C22⋊C4—3762no—
302C22⋊C4—1882no—
303C22⋊C4—1882no—
304C42—3762no—
305C42—1882no—
306C23—22no—
307C23—22no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—42no—
313C23C22⋊C4×D4711yescharacteristic
314C23#1309399211yescharacteristic
315C23#1309400611yescharacteristic
316D4—81no—
317D4—81no—
318D4—82no—
319D4—82no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—942no—
332C2×C4—942no—
333C2×C4—3761no—
334C2×C4—3761no—
335C2×C4—1882no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4—1882no—
342C2×C4—1882no—
343C2×C4—1882no—
344C2×C4—1882no—
345C2×C4(C22⋊D47)⋊C411yescharacteristic
346C2×C4C4×C22⋊D4711yescharacteristic
347C2×C4#1309399511yescharacteristic
348C2×C4#1309399711yescharacteristic
349C8—81no—
350C8—3761no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—44no—
359C22—84no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#1568899911yescharacteristic
364C22#1568909512yes—
365C22#1568929412yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—82no—
371C4—81no—
372C4—3762no—
373C4—3762no—
374C4—1882no—
375C4—1882no—
376C2—22no—
377C2—22no—
378C2—41no—
379C2—161no—
380C2#2024378811yescharacteristic
381C2#2024384311yescharacteristic
382C2#2024391711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1510C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#1510D47
hypocenterC47#9465
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #20245989C2semidirect
3. #20243466C2semidirect
5. #20241242C2semidirect
6. #20241240C2semidirect
11. #15689026C4semidirect
16. #15688988C4semidirect
218. C47#9465semidirect

Maximal subgroups

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

subgroupindex
#202478812
#202459892
#202434662
#202419172
#202412422
#202412402
C47×#15102
#946547