Finite Group Databasebeta

← #26114045

Subgroups — #26114045

Order 12032.

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

normal subgroups only

This group has 578 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#20252412C211yesmaximal, characteristic
2#20247705C211yesmaximal, characteristic
3#20243476C212yesmaximal, complemented
4#20241255C212yesmaximal, complemented
5C47×#1799C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7D4×C47⋊C8—24no—
8#15689176—21no—
9C2×C22⋊(C47⋊C8)—21no—
10C2×C22⋊(C47⋊C8)C2212yes—
11#15689062C412yescomplemented
12#15689059C412yescomplemented
13#15689021—21no—
14#15689003—21no—
15#15688995—21no—
16#15688983—21no—
17#15688983C2211yescharacteristic
18#15688885—24no—
19C2×C4×D4×C47C2211yescharacteristic
20C2×#99×C47—21no—
21C47×#380C2212yes—
22C47×#342—21no—
23C23×Dic47—21no—
24C22×C47⋊C8—22no—
25C22×C47⋊C8D414yes—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—22no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47C2×C412yes—
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—22no—
34C4⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38C4×C47⋊C8—22no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C47⋊#95C2×C412yes—
42C22×D4×C47D411yescharacteristic
43C23×C4×C47D411yescharacteristic
44C23×C4×C47C2311yescharacteristic
45C4×D4×C47—22no—
46C4×D4×C47—22no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47—22no—
50C2×C22⋊C4×C47C2×C411yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C42×C47D411yescharacteristic
53#99×C47—42no—
54#95×C47—22no—
55C24×C47C22⋊C411yescharacteristic
56C24×C47C2×D411yescharacteristic
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47SD1612yes—
60C2×D4×C47Q1612yes—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D412yes—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47#4112yes—
73C22×Dic47—22no—
74C22×Dic47—22no—
75C22×Dic47—42no—
76C22×Dic47—82no—
77C4⋊C4×C47—22no—
78C4⋊C4×C47—44no—
79C22⋊C4×C47—22no—
80C22⋊C4×C47—22no—
81C22⋊C4×C47—42no—
82C22⋊C4×C47—42no—
83C42×C47—21no—
84C42×C47—21no—
85C22⋊Dic47—42no—
86C22⋊Dic47—82no—
87C4⋊Dic47—24no—
88C4⋊Dic47—42no—
89C4⋊Dic47—41no—
90C4⋊Dic47—41no—
91Dic47⋊C4—82no—
92C2×C47⋊C8—24no—
93C2×C47⋊C8—44no—
94C2×C47⋊C8—82no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47#10312yes—
102C23×C47C2×C22⋊C411yescharacteristic
103C23×C47#12112yes—
104C23×C47C2×SD1611yescharacteristic
105C23×C47C2×Q1611yescharacteristic
106C23×C47#13512yes—
107D4×C47—24no—
108D4×C47—24no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—24no—
112C2×C4×C47—22no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—41no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—81no—
121C2×C4×C47—82no—
122C2×C4×C47—82no—
123C2×C4×C47—82no—
124C2×C4×C47#10312yes—
125C2×C4×C47C4×D412yes—
126C2×C4×C47#12011yescharacteristic
127C2×C4×C47#12311yescharacteristic
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—84no—
135C47⋊C8—44no—
136#9469—471nomaximal, Sylow
137C22×C47—21no—
138C22×C47—22no—
139C22×C47—21no—
140C22×C47—22no—
141C22×C47—22no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—42no—
145C22×C47—42no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—81no—
149C22×C47#38611yescharacteristic
150C22×C47C2×#10311yescharacteristic
151C22×C47#41611yescharacteristic
152C22×C47#46512yes—
153C22×C47#46712yes—
154C4×C47—21no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—41no—
158C4×C47—41no—
159C4×C47—162no—
160C4×C47#45812yes—
161Dic47—82no—
162Dic47—82no—
163#2972—471no—
164#2487—471no—
165#1824—472no—
166#1799D4711yescharacteristic
167#1289—472no—
168C2×C47—21no—
169C2×C47—22no—
170C2×C47—21no—
171C2×C47—22no—
172C2×C47—42no—
173C2×C47#189011yescharacteristic
174C2×C47#189211yescharacteristic
175C2×C47#191011yescharacteristic
176C2×C4×D4C2×D4711yescharacteristic
177C22×C4⋊C4—471no—
178C8×D4—944no—
179#431—941no—
180C2×#99—21no—
181C2×C22⋊C8—472no—
182C2×C22⋊C8—941no—
183#400—472no—
184#399—472no—
185#389—941no—
186#386—941no—
187#385—941no—
188#380—471no—
189#380—941no—
190#380C2×D4712yes—
191#342—21no—
192#340—944no—
193C47#946911yescharacteristic, complemented, Sylow
194C22×D4C22⋊D4711yescharacteristic
195C23×C4—941no—
196C23×C4D18811yescharacteristic
197C23×C4C22×D4711yescharacteristic
198C22×C8—474no—
199C22×C8—942no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—472no—
204C2×C4⋊C4—942no—
205C2×C4⋊C4—942no—
206C2×C4⋊C4—942no—
207C2×C4⋊C4—941no—
208C2×C4⋊C4—941no—
209C2×C4⋊C4C4×D4711yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—942no—
212C2×C22⋊C4—941no—
213C2×C22⋊C4C4×D4711yescharacteristic
214C2×C22⋊C4C22⋊D4711yescharacteristic
215C2×C42D18811yescharacteristic
216C4⋊C8—942no—
217#99—42no—
218C22⋊C8—942no—
219C22⋊C8—942no—
220C22⋊C8—942no—
221C4×C8—942no—
222#95—22no—
223#95—472no—
224#95—942no—
225#95—942no—
226C24(C2×D47)⋊C411yescharacteristic
227C24C2×C22⋊D4711yescharacteristic
228C2×D4—22no—
229C2×D4—22no—
230C2×D4#1062533512yes—
231C2×D4#1062533712yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—942no—
238C22×C4—942no—
239C22×C4—3762no—
240C22×C4—1882no—
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4(C2×D47)⋊C411yescharacteristic
243C22×C4(C2×D47)⋊C411yescharacteristic
244C22×C4C2×C4×D4711yescharacteristic
245C22×C4C2×D18811yescharacteristic
246C22×C4D4×D4712yes—
247C22×C4#1062535212yes—
248C2×C8—944no—
249C2×C8—3762no—
250C2×C8—1884no—
251C4⋊C4—22no—
252C4⋊C4—44no—
253C4⋊C4—944no—
254C4⋊C4—3762no—
255C4⋊C4—1882no—
256C4⋊C4—1881no—
257C4⋊C4—1881no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—3762no—
263C22⋊C4—1882no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#1309394712yes—
273C23C2×(C2×D47)⋊C411yescharacteristic
274C23#1309399612yes—
275C23C2×#1062533511yescharacteristic
276C23C2×#1062533711yescharacteristic
277C23#1309404412yes—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—3764no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4—1882no—
302C2×C4#1309394712yes—
303C2×C4(C22⋊D47)⋊C412yes—
304C2×C4#1309399411yescharacteristic
305C2×C4#1309400811yescharacteristic
306C8—1884no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22#1568900011yescharacteristic
320C22C2×#1309394711yescharacteristic
321C22#1568911711yescharacteristic
322C22#1568935012yes—
323C22#1568936212yes—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—162no—
330C4—3762no—
331C4—3762no—
332C4#1568931412yes—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#2024379911yescharacteristic
339C2#2024380911yescharacteristic
340C2#2024391311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1799C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#1799D47
hypocenterC47#9469
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20243476C2semidirect
4. #20241255C2semidirect
11. #15689062C4semidirect
12. #15689059C4semidirect
193. C47#9469semidirect

Maximal subgroups

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

subgroupindex
#202524122
#202477052
#202434762
#202412552
C47×#17992
#946947