Finite Group Databasebeta

← #26188772

Subgroups — #26188772

Order 12032.

930 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 183 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#20253332C212yesmaximal
2#20247675C211yesmaximal, characteristic
3#20242430C214yesmaximal, complemented
4#20242396C214yesmaximal, complemented
5#20241921C212yesmaximal
6#20241718C211yesmaximal, characteristic
7C2×C47×#544C211yesmaximal, characteristic
8C2×#13093983C2211yescharacteristic
9#15689461—24no—
10#15689456C2214yes—
11#15689451C2214yes—
12#15689163—22no—
13#15689132—22no—
14#15689087C2212yes—
15#15689062C2212yes—
16#15689053C2212yes—
17#15689004—22no—
18#15688996—21no—
19#15688986—21no—
20#15688986C2211yescharacteristic
21#15688860C2218yes—
22#15688849C2214yes—
23C47×#544—24no—
24C47×#544—22no—
25C47×#544C2214yes—
26C2×C4⋊Q8×C47—22no—
27C2×#125×C47—22no—
28C2×C22⋊Q8×C47—21no—
29C2×C22⋊Q8×C47C2212yes—
30C2×#117×C47—21no—
31C22×C4⋊C4×C47C2211yescharacteristic
32C2×#10625329—22no—
33#13093983—22no—
34#13093983—22no—
35C2×C4⋊Dic47C2311yescharacteristic
36C2×C4⋊Dic47C2311yescharacteristic
37C2×Dic47⋊C4—22no—
38C2×C22⋊Dic47—21no—
39C2×C22⋊Dic47—21no—
40C2×C4×Dic47—21no—
41#13093957—24no—
42#13093955—24no—
43#13093948—24no—
44#13093948—24no—
45C22⋊(C47⋊C8)—22no—
46C22⋊(C47⋊C8)C2314yes—
47C47⋊#95—22no—
48C47⋊#95—22no—
49C47⋊#95—21no—
50C47⋊#95—22no—
51C47⋊#95C2312yes—
52C22×Q8×C47—21no—
53C22×Q8×C47D412yes—
54C23×C4×C47D411yescharacteristic
55C23×C4×C47D411yescharacteristic
56C23×C4×C47C2311yescharacteristic
57C4⋊Q8×C47—28no—
58C4⋊Q8×C47—28no—
59#125×C47—28no—
60#125×C47—28no—
61C22⋊Q8×C47—24no—
62C22⋊Q8×C47—24no—
63C22⋊Q8×C47—22no—
64C22⋊Q8×C47—22no—
65C22⋊Q8×C47D414yes—
66C22⋊Q8×C47C2314yes—
67#117×C47—24no—
68#117×C47—24no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47—22no—
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47D412yes—
77C2×C4⋊C4×C47D412yes—
78C2×C4⋊C4×C47C2312yes—
79C2×C22⋊C4×C47—22no—
80C2×C42×C47—22no—
81C24×C47C2×D411yescharacteristic
82C2×Q8×C47—24no—
83C2×Q8×C47—22no—
84C2×Q8×C47—44no—
85C2×Q8×C47—82no—
86C2×Q8×C47C2×D414yes—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—24no—
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—42no—
95C22×C4×C47—42no—
96C22×C4×C47C2×D412yes—
97C22×C4×C47C2×D412yes—
98C22×C4×C47C2×D412yes—
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D412yes—
101C22×C4×C47C2411yescharacteristic
102C22×Dic47—22no—
103C22×Dic47—21no—
104C22×Dic47—21no—
105C4⋊C4×C47—24no—
106C4⋊C4×C47—24no—
107C4⋊C4×C47—24no—
108C4⋊C4×C47—28no—
109C4⋊C4×C47—24no—
110C4⋊C4×C47—24no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—24no—
113C4⋊C4×C47—24no—
114C4⋊C4×C47C2×D414yes—
115C22⋊C4×C47—24no—
116C22⋊C4×C47—24no—
117C42×C47—28no—
118C22⋊Dic47—42no—
119C22⋊Dic47—41no—
120C22⋊Dic47—41no—
121C4⋊Dic47—22no—
122C4⋊Dic47—22no—
123Dic47⋊C4—44no—
124C4×Dic47—42no—
125#10625329—44no—
126C2×C47⋊C8—24no—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47—41no—
130C23×C47#12011yescharacteristic
131C23×C47#12011yescharacteristic
132C23×C47C22×D411yescharacteristic
133Q8×C47—44no—
134Q8×C47—82no—
135Q8×C47—84no—
136C2×C4×C47—24no—
137C2×C4×C47—24no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—24no—
141C2×C4×C47—24no—
142C2×C4×C47—24no—
143C2×C4×C47—22no—
144C2×C4×C47—42no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—44no—
148C2×C4×C47—42no—
149C2×C4×C47—44no—
150C2×C4×C47—42no—
151C2×C4×C47—81no—
152C2×C4×C47#12012yes—
153C2×C4×C47#13714yes—
154C2×C4×C47C22×D412yes—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47—42no—
161C2×Dic47—42no—
162C47⋊C8—44no—
163#18148—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—44no—
172C22×C47#45812yes—
173C22×C47C2×#12011yescharacteristic
174C22×C47C2×#13712yes—
175C4×C47—24no—
176C4×C47—22no—
177C4×C47—22no—
178C4×C47—44no—
179C4×C47—42no—
180C4×C47—44no—
181Dic47—82no—
182Dic47—81no—
183Dic47—81no—
184C2×#544D4711yescharacteristic
185#3095—472no—
186#2484—471no—
187#1617—474no—
188#1611—474no—
189#1511—472no—
190#1463—471no—
191C2×C47—22no—
192C2×C47—22no—
193C2×C47—41no—
194C2×C47#300411yescharacteristic
195C2×C47#302011yescharacteristic
196C2×C47C2×#45811yescharacteristic
197#544—24no—
198#544—22no—
199#544C2×D4714yes—
200C2×C4⋊Q8—22no—
201C2×#125—22no—
202C2×C22⋊Q8—21no—
203C2×C22⋊Q8C2×D4712yes—
204C2×#117—21no—
205C2×#117—471no—
206C22×C4⋊C4C2×D4711yescharacteristic
207#485—944no—
208#484—474no—
209#483—474no—
210#428—942no—
211#419—942no—
212#407—472no—
213#400—472no—
214#398—472no—
215#387—942no—
216#385—941no—
217#381—471no—
218#381—941no—
219#328—478no—
220#324—474no—
221C47#1814811yescharacteristic, complemented, Sylow
222C22×Q8—21no—
223C22×Q8C22⋊D4712yes—
224C23×C4C22⋊D4711yescharacteristic
225C23×C4C22⋊D4711yescharacteristic
226C23×C4C22×D4711yescharacteristic
227C2×M16—942no—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230#125—28no—
231#125—28no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8C22⋊D4714yes—
237C22⋊Q8C22×D4714yes—
238#117—24no—
239#117—24no—
240#117—942no—
241#117—942no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—471no—
250C2×C4⋊C4—471no—
251C2×C4⋊C4—942no—
252C2×C4⋊C4C22⋊D4712yes—
253C2×C4⋊C4C22⋊D4712yes—
254C2×C4⋊C4C22×D4712yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—941no—
257C2×C22⋊C4—941no—
258C2×C42—22no—
259C2×C42—941no—
260#107—944no—
261#106—944no—
262#103—944no—
263#103—944no—
264C22⋊C8—474no—
265C22⋊C8—942no—
266#95—472no—
267#95—941no—
268#95—942no—
269#95—942no—
270#95—942no—
271C24C2×C22⋊D4711yescharacteristic
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—82no—
276C2×Q8C2×C22⋊D4714yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—942no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4C2×C22⋊D4712yes—
290C22×C4C2×C22⋊D4712yes—
291C22×C4C2×C22⋊D4712yes—
292C22×C4C2×C22⋊D4711yescharacteristic
293C22×C4C2×C22⋊D4712yes—
294C22×C4C23×D4711yescharacteristic
295M16—1884no—
296C2×C8—944no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—942no—
307C4⋊C4—942no—
308C4⋊C4—1884no—
309C4⋊C4C2×C22⋊D4714yes—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—1882no—
313C22⋊C4—1881no—
314C22⋊C4—1881no—
315C42—28no—
316C42—1882no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23#1309399311yescharacteristic
321C23#1309399311yescharacteristic
322C23C22×C22⋊D4711yescharacteristic
323Q8—44no—
324Q8—82no—
325Q8—84no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—1882no—
343C2×C4—1882no—
344C2×C4—1882no—
345C2×C4—1881no—
346C2×C4—1881no—
347C2×C4—1882no—
348C2×C4—1882no—
349C2×C4#1309399312yes—
350C2×C4#1309405514yes—
351C2×C4C22×C22⋊D4712yes—
352C8—1884no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22#1568931312yes—
362C22C2×#1309399311yescharacteristic
363C22C2×#1309405512yes—
364C4—22no—
365C4—22no—
366C4—24no—
367C4—44no—
368C4—42no—
369C4—44no—
370C4—3762no—
371C4—3761no—
372C4—3761no—
373C2—22no—
374C2—22no—
375C2—41no—
376C2#2025262211yescharacteristic
377C2#2025275211yescharacteristic
378C2C2×#1568931311yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#544C2
socleC22×C47C2×#120
prime radicalC24×C47C2×D4
hypercenterC2×#544D47
hypocenterC47#18148
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20242430C2semidirect
4. #20242396C2semidirect
221. C47#18148semidirect

Maximal subgroups

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

subgroupindex
#202533322
#202476752
#202424302
#202423962
#202419212
#202417182
C2×C47×#5442
#1814847