Finite Group Databasebeta

← #26112332

Subgroups — #26112332

Order 12032.

612 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 119 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#20245986C211yesmaximal, characteristic
2#20241960C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic
4#20241255C212yesmaximal, complemented
5C47×#2626C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)C2212yes—
7#15689016—24no—
8#15688983C2212yes—
9#15688979—21no—
10C4×C22⋊Dic47—21no—
11#15688860—24no—
12#15688847—22no—
13C2×C22⋊Q8×C47C2211yescharacteristic
14C22×C22⋊C4×C47C2211yescharacteristic
15C47×#399C412yescomplemented
16C47×#396—22no—
17C47×#393C412yescomplemented
18C47×#385—21no—
19C47×#380C2211yescharacteristic
20C47×#379—21no—
21C22×C47⋊C8—22no—
22C2×C4⋊Dic47—22no—
23C2×C22⋊Dic47—22no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C4×Dic47—22no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C47⋊#95—22no—
30C47⋊#95D414yes—
31C25×C47D411yescharacteristic
32C22×Q8×C47C2×C411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47C2311yescharacteristic
35C22⋊Q8×C47—22no—
36C22⋊Q8×C47—22no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47C2×C412yes—
48C2×C22⋊C4×C47D412yes—
49#95×C47—22no—
50#95×C47C2×C412yes—
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—22no—
54C24×C47—22no—
55C24×C47—81no—
56C24×C47—81no—
57C24×C47C22⋊C412yes—
58C24×C47C2×D411yescharacteristic
59C2×Q8×C47—22no—
60C2×Q8×C47—81no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—81no—
67C22×C4×C47—81no—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D412yes—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47#4112yes—
73C22×Dic47—24no—
74C4⋊C4×C47—22no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—81no—
77C4⋊C4×C47—82no—
78C22⋊C4×C47—24no—
79C22⋊C4×C47—24no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—42no—
83C22⋊C4×C47—82no—
84C22⋊C4×C47—82no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C22⋊Dic47—82no—
88C22⋊Dic47—82no—
89C4⋊Dic47—82no—
90C4×Dic47—82no—
91C2×C47⋊C8—42no—
92C2×C47⋊C8—82no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47—22no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47—42no—
104C23×C47—81no—
105C23×C47—81no—
106C23×C47—84no—
107C23×C47—84no—
108C23×C47C2×C22⋊C411yescharacteristic
109C23×C47#12011yescharacteristic
110C23×C47#12311yescharacteristic
111C23×C47#13512yes—
112C23×C47#13712yes—
113Q8×C47—82no—
114C2×C4×C47—22no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47—81no—
129C2×C4×C47C4×D412yes—
130C2×C4×C47#12112yes—
131C2×Dic47—44no—
132C2×Dic47—44no—
133C2×Dic47—44no—
134C47⋊C8—82no—
135#9201—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—42no—
143C22×C47—42no—
144C22×C47—44no—
145C22×C47—42no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—82no—
149C22×C47—84no—
150C22×C47—84no—
151C22×C47#38611yescharacteristic
152C22×C47#41611yescharacteristic
153C22×C47#41911yescharacteristic
154C22×C47#45212yes—
155C22×C47#48512yes—
156C4×C47—22no—
157C4×C47—22no—
158C4×C47—42no—
159C4×C47—82no—
160C4×C47—81no—
161C4×C47—81no—
162Dic47—84no—
163#2626D4711yescharacteristic
164#2312—471no—
165#1520—472no—
166C2×#323—471no—
167#1289—472no—
168C2×C47—22no—
169C2×C47—21no—
170C2×C47—21no—
171C2×C47—22no—
172C2×C47—82no—
173C2×C47#187311yescharacteristic
174C2×C47#191111yescharacteristic
175C2×C47#191311yescharacteristic
176C2×C22⋊Q8C2×D4711yescharacteristic
177C22×C22⋊C4C2×D4711yescharacteristic
178C2×C22⋊C8—472no—
179#399Dic4712yescomplemented
180#396—22no—
181#393Dic4712yescomplemented
182#388—944no—
183#385—21no—
184#380—472no—
185#380C2×D4711yescharacteristic
186#379—21no—
187#379—941no—
188C4×C22⋊C4—941no—
189#328—944no—
190#323—942no—
191C47#920111yescharacteristic, complemented, Sylow
192C25C22⋊D4711yescharacteristic
193C22×Q8C2×Dic4711yescharacteristic
194C23×C4C22⋊D4711yescharacteristic
195C23×C4C22×D4711yescharacteristic
196C22×C8—942no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—942no—
203C2×C4⋊C4C2×Dic4711yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—942no—
211C2×C22⋊C4—942no—
212C2×C22⋊C4—942no—
213C2×C22⋊C4C2×Dic4712yes—
214C2×C22⋊C4C22⋊D4712yes—
215C2×C42—942no—
216C22⋊C8—942no—
217C22⋊C8—942no—
218#95—22no—
219#95—474no—
220#95—942no—
221#95C2×Dic4712yes—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24C22⋊Dic4712yes—
229C24C2×C22⋊D4711yescharacteristic
230C2×Q8—22no—
231C2×Q8—81no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—944no—
240C22×C4C22⋊Dic4712yes—
241C22×C4D4×D4712yes—
242C22×C4#1062535212yes—
243C22×C4C22×Dic4711yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245C2×C8—3762no—
246C2×C8—1882no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C4⋊C4—3762no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—3762no—
260C22⋊C4—3762no—
261C22⋊C4—1882no—
262C22⋊C4—1882no—
263C42—3762no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×C22⋊Dic4711yescharacteristic
280C23#1309399211yescharacteristic
281C23#1309400511yescharacteristic
282C23#1309404412yes—
283C23#1309405512yes—
284Q8—82no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4—1884no—
301C2×C4—1884no—
302C2×C4—1884no—
303C2×C4D4×Dic4712yes—
304C2×C4#1309399612yes—
305C8—3762no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1568900311yescharacteristic
322C22#1568911911yescharacteristic
323C22#1568913211yescharacteristic
324C22#1568928312yes—
325C22#1568946112yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—3764no—
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#2024370111yescharacteristic
339C2#2024392111yescharacteristic
340C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2626C2
socleC22×C47#386
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2626D47
hypocenterC47#9201
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #20241960C2semidirect
4. #20241255C2semidirect
15. C47×#399C4semidirect
17. C47×#393C4semidirect
179. #399Dic47semidirect
181. #393Dic47semidirect
191. C47#9201semidirect

Maximal subgroups

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

subgroupindex
#202459862
#202419602
C2×#156888472
#202412552
C47×#26262
#920147