Finite Group Databasebeta

← #26108802

Subgroups — #26108802

Order 12032.

720 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 131 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#20242105C212yesmaximal, complemented
2#20242012C212yesmaximal, complemented
3#20241722C211yesmaximal, characteristic, complemented
4#20241722C211yesmaximal, characteristic, complemented
5C2×C47×#542C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689180C2212yes—
8#15688869—24no—
9#15688863—24no—
10#15688849—21no—
11#15688849—21no—
12#15688847—21no—
13#15688847—21no—
14C47×#542—24no—
15C47×#542—24no—
16C2×C4⋊Q8×C47C2211yescharacteristic
17C2×#126×C47—21no—
18C2×#124×C47—21no—
19C2×#124×C47C2211yescharacteristic
20C2×#123×C47—21no—
21C2×C22⋊Q8×C47—21no—
22C2×C22⋊Q8×C47C411yescharacteristic
23C2×#121×C47C411yescharacteristic
24C2×C4×Q8×C47C411yescharacteristic
25C2×C4×D4×C47C411yescharacteristic
26C2×#117×C47C2211yescharacteristic
27C4⋊(C47⋊C8)—22no—
28C4⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32#13093935—22no—
33C22×Q8×C47—21no—
34C22×Q8×C47C2×C411yescharacteristic
35C22×D4×C47—21no—
36C22×D4×C47C2×C411yescharacteristic
37C23×C4×C47—21no—
38C23×C4×C47C2×C411yescharacteristic
39C4⋊Q8×C47—22no—
40C4⋊Q8×C47D414yes—
41#126×C47—28no—
42#124×C47—28no—
43#124×C47—22no—
44#124×C47D414yes—
45#123×C47—28no—
46C22⋊Q8×C47—28no—
47C22⋊Q8×C47—22no—
48C22⋊Q8×C47—22no—
49#121×C47—22no—
50#121×C47—22no—
51C4×Q8×C47—22no—
52C4×Q8×C47—22no—
53C4×D4×C47—22no—
54C4×D4×C47—22no—
55#117×C47—22no—
56#117×C47—22no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47C2×C411yescharacteristic
61C2×C4⋊C4×C47C2×C411yescharacteristic
62C2×C4⋊C4×C47D411yescharacteristic
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47—21no—
69C2×C22⋊C4×C47D411yescharacteristic
70C2×C22⋊C4×C47D411yescharacteristic
71C2×C42×C47—21no—
72C2×C42×C47C2×C411yescharacteristic
73C2×C42×C47C2311yescharacteristic
74C24×C47—21no—
75C24×C47C22⋊C411yescharacteristic
76C2×Q8×C47—21no—
77C2×Q8×C47—24no—
78C2×Q8×C47—42no—
79C2×Q8×C47—82no—
80C2×Q8×C47C22⋊C412yes—
81C2×D4×C47—24no—
82C2×D4×C47—21no—
83C2×D4×C47—42no—
84C2×D4×C47—82no—
85C2×D4×C47C22⋊C412yes—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—24no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—42no—
93C22×C4×C47—82no—
94C22×C4×C47C22⋊C411yescharacteristic
95C22×C4×C47C22⋊C411yescharacteristic
96C22×C4×C47C22⋊C411yescharacteristic
97C22×C4×C47C22×C411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C4⋊C4×C47—24no—
101C4⋊C4×C47—24no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—24no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47—22no—
107C4⋊C4×C47C22⋊C412yes—
108C4⋊C4×C47C22⋊C412yes—
109C22⋊C4×C47—24no—
110C22⋊C4×C47—24no—
111C22⋊C4×C47—24no—
112C22⋊C4×C47—24no—
113C22⋊C4×C47—24no—
114C22⋊C4×C47—22no—
115C22⋊C4×C47—22no—
116C42×C47—24no—
117C42×C47—21no—
118C42×C47—21no—
119C42×C47C2×D412yes—
120C42×C47C2×D412yes—
121C2×C47⋊C8—42no—
122C2×C47⋊C8—42no—
123C23×C47—21no—
124C23×C47—22no—
125C23×C47—22no—
126C23×C47—42no—
127C23×C47—42no—
128C23×C47—41no—
129C23×C47—41no—
130C23×C47—42no—
131C23×C47C2×C22⋊C411yescharacteristic
132Q8×C47—42no—
133Q8×C47—42no—
134Q8×C47—84no—
135D4×C47—42no—
136D4×C47—42no—
137D4×C47—84no—
138C2×C4×C47—22no—
139C2×C4×C47—21no—
140C2×C4×C47—22no—
141C2×C4×C47—21no—
142C2×C4×C47—22no—
143C2×C4×C47—21no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—42no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—84no—
156C2×C4×C47C2×C22⋊C411yescharacteristic
157C2×C4×C47C2×C22⋊C411yescharacteristic
158C2×C4×C47#12012yes—
159C2×C4×C47#12012yes—
160C47⋊C8—82no—
161C47⋊C8—82no—
162#8523—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—42no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—42no—
169C22×C47—41no—
170C22×C47—42no—
171C22×C47—82no—
172C22×C47—84no—
173C22×C47#37911yescharacteristic
174C22×C47#41312yes—
175C4×C47—22no—
176C4×C47—22no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—42no—
180C4×C47—42no—
181C4×C47—41no—
182C4×C47—41no—
183C4×C47—41no—
184C4×C47—42no—
185C2×#542D4711yescharacteristic
186#1556—472no—
187#1533—472no—
188#1464—471no—
189#1464—471no—
190C2×C47—22no—
191C2×C47—81no—
192C2×C47—82no—
193C2×C47#178511yescharacteristic
194C2×C47#179511yescharacteristic
195C2×C47#179611yescharacteristic
196#542—24no—
197#542—24no—
198C2×C4⋊Q8C2×D4711yescharacteristic
199C2×#126—21no—
200C2×#124—21no—
201C2×#124C2×D4711yescharacteristic
202C2×#123—21no—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8Dic4711yescharacteristic
205C2×#121Dic4711yescharacteristic
206C2×C4×Q8Dic4711yescharacteristic
207C2×C4×D4Dic4711yescharacteristic
208C2×#117C2×D4711yescharacteristic
209#432—472no—
210#432—472no—
211#332—944no—
212#329—944no—
213#324—941no—
214#324—941no—
215#323—941no—
216#323—941no—
217C47#852311yescharacteristic, complemented, Sylow
218C22×Q8—21no—
219C22×Q8C2×Dic4711yescharacteristic
220C22×D4—21no—
221C22×D4C2×Dic4711yescharacteristic
222C23×C4—21no—
223C23×C4C2×Dic4711yescharacteristic
224C4⋊Q8—22no—
225C4⋊Q8C22⋊D4714yes—
226#126—28no—
227#124—28no—
228#124—22no—
229#124C22⋊D4714yes—
230#123—28no—
231C22⋊Q8—28no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234#121—22no—
235#121—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×D4—22no—
239C4×D4—22no—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×Dic4711yescharacteristic
246C2×C4⋊C4C2×Dic4711yescharacteristic
247C2×C4⋊C4C22⋊D4711yescharacteristic
248C2×C4⋊C4C22⋊D4711yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4C22⋊D4711yescharacteristic
255C2×C22⋊C4C22⋊D4711yescharacteristic
256C2×C42—21no—
257C2×C42C2×Dic4711yescharacteristic
258C2×C42C22×D4711yescharacteristic
259C4⋊C8—942no—
260C4⋊C8—942no—
261C22⋊C8—942no—
262C22⋊C8—942no—
263#97—942no—
264#97—942no—
265C24—21no—
266C24C22⋊Dic4711yescharacteristic
267C2×Q8—21no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—82no—
271C2×Q8C22⋊Dic4712yes—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—42no—
275C2×D4—82no—
276C2×D4C22⋊Dic4712yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—82no—
285C22×C4C22⋊Dic4711yescharacteristic
286C22×C4C22⋊Dic4711yescharacteristic
287C22×C4C22⋊Dic4711yescharacteristic
288C22×C4C22×Dic4711yescharacteristic
289C22×C4C2×C22⋊D4711yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291C2×C8—1882no—
292C2×C8—1882no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4C22⋊Dic4712yes—
301C4⋊C4C22⋊Dic4712yes—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C42—24no—
310C42—21no—
311C42—21no—
312C42C2×C22⋊D4712yes—
313C42C2×C22⋊D4712yes—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23C2×C22⋊Dic4711yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—84no—
326D4—42no—
327D4—42no—
328D4—84no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—84no—
347C2×C4C2×C22⋊Dic4711yescharacteristic
348C2×C4C2×C22⋊Dic4711yescharacteristic
349C2×C4#1309399312yes—
350C2×C4#1309399312yes—
351C8—3762no—
352C8—3762no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—82no—
362C22—84no—
363C22#1568897911yescharacteristic
364C22#1568910712yes—
365C4—21no—
366C4—21no—
367C4—22no—
368C4—22no—
369C4—41no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—41no—
374C4—41no—
375C2—22no—
376C2—82no—
377C2—81no—
378C2#2024331611yescharacteristic
379C2#2024335811yescharacteristic
380C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#542C2
socleC22×C47#379
prime radicalC2×#121×C47C4
cosocleC2×C42×C47C23
hypercenterC2×#542D47
hypocenterC47#8523
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. #20242105C2semidirect
2. #20242012C2semidirect
3. #20241722C2semidirect
4. #20241722C2semidirect
217. C47#8523semidirect

Maximal subgroups

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

subgroupindex
#202421052
#202420122
#202417222
C2×C47×#5422
#852347