Finite Group Databasebeta

← #26115413

Subgroups — #26115413

Order 12032.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251740C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic
3C2×#15688859C214yesmaximal
4C47×#1434C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13094003C2212yes—
7C2×C4⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689029C412yescomplemented
10#15689021—21no—
11#15689021—21no—
12#15688993C411yescharacteristic, complemented
13#15688983—21no—
14#15688983—21no—
15#15688878—21no—
16#15688859—28no—
17C22×C42×C47C2211yescharacteristic
18C2×#95×C47—21no—
19C2×#95×C47—21no—
20C2×#95×C47C2212yes—
21C22×Dic94C2×C412yes—
22C22×C47⋊C8—22no—
23#13094024—42no—
24#13094003—24no—
25#13094003—24no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47C2×C412yes—
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—22no—
33C4⋊(C47⋊C8)—42no—
34C22⋊(C47⋊C8)—24no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C23×C4×C47—22no—
38C23×C4×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C2×C42×C47—42no—
42C2×C42×C47C2×C411yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46#95×C47—22no—
47#95×C47—42no—
48#95×C47—42no—
49#95×C47D414yes—
50C24×C47C2×D411yescharacteristic
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—24no—
54C22×C4×C47—41no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47—42no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47SD1612yes—
70C22×C4×C47Q1612yes—
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D412yes—
74C22×C4×C47#4112yes—
75C22×Dic47—22no—
76C22×Dic47—22no—
77C2×Dic94—24no—
78C2×Dic94—42no—
79C2×Dic94—42no—
80C42×C47—22no—
81C42×C47—22no—
82C42×C47—41no—
83C42×C47—41no—
84C22⋊Dic47—44no—
85C4⋊Dic47—24no—
86C4⋊Dic47—42no—
87C4⋊Dic47—42no—
88C4⋊Dic47—42no—
89C4⋊Dic47—42no—
90Dic47⋊C4—42no—
91Dic47⋊C4—42no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—42no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47C2×C22⋊C411yescharacteristic
100C23×C47#12112yes—
101C23×C47C2×SD1611yescharacteristic
102C23×C47C2×Q1611yescharacteristic
103C23×C47#13712yes—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—44no—
122C2×C4×C47—44no—
123C2×C4×C47—44no—
124C2×C4×C47—44no—
125C2×C4×C47—82no—
126C2×C4×C47—82no—
127C2×C4×C47—82no—
128C2×C4×C47—82no—
129C2×C4×C47#9912yes—
130C2×C4×C47#10312yes—
131C2×C4×C47#10312yes—
132C2×C4×C47C4×D412yes—
133C2×C4×C47#12411yescharacteristic
134C2×C4×C47#12711yescharacteristic
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141Dic94—84no—
142C47⋊C8—82no—
143#9714—471nomaximal, Sylow
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47#39011yescharacteristic
153C22×C47C2×#9911yescharacteristic
154C22×C47#41011yescharacteristic
155C22×C47C2×#10311yescharacteristic
156C22×C47#41911yescharacteristic
157C22×C47#42111yescharacteristic
158C22×C47#42111yescharacteristic
159C22×C47#46112yes—
160C22×C47#46212yes—
161C4×C47—21no—
162C4×C47—22no—
163C4×C47—21no—
164C4×C47—22no—
165C4×C47—22no—
166C4×C47—22no—
167C4×C47—41no—
168C4×C47—84no—
169Dic47—82no—
170Dic47—82no—
171#2857—471no—
172#1797—471no—
173C2×#328—474no—
174#1434D4711yescharacteristic
175C2×C47—22no—
176C2×C47—22no—
177C2×C47#151111yescharacteristic
178C2×C47#151511yescharacteristic
179C2×C47#151811yescharacteristic
180C2×C47#152011yescharacteristic
181C2×C47#196311yescharacteristic
182C2×C47#196411yescharacteristic
183C2×C47#198111yescharacteristic
184C2×C4⋊Q8—471no—
185C2×C22⋊Q8—472no—
186C22×C42C2×D4711yescharacteristic
187C2×C4⋊C8—941no—
188C2×C22⋊C8—472no—
189#391—472no—
190#389—941no—
191#389—941no—
192#384—471no—
193#380—941no—
194#380—941no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95C2×D4712yes—
198#336—941no—
199#328—948no—
200C47#971411yescharacteristic, complemented, Sylow
201C22×Q8—472no—
202C23×C4—22no—
203C23×C4C22⋊D4711yescharacteristic
204C23×C4C22⋊D4711yescharacteristic
205C23×C4C22×D4711yescharacteristic
206C22×C8—942no—
207C4⋊Q8—1882no—
208C22⋊Q8—944no—
209C22⋊Q8—944no—
210C2×C4⋊C4—472no—
211C2×C4⋊C4—941no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4—942no—
216C2×C22⋊C4—942no—
217C2×C42—42no—
218C2×C42C4×D4711yescharacteristic
219C2×C42C4×D4711yescharacteristic
220C2×C42D18811yescharacteristic
221C2×C42D18811yescharacteristic
222C4⋊C8—1882no—
223C22⋊C8—944no—
224#95—22no—
225#95—42no—
226#95—42no—
227#95—942no—
228#95—942no—
229#95D18814yes—
230C24C2×C22⋊D4711yescharacteristic
231C2×Q8—944no—
232C2×Q8—1882no—
233C2×Q8—1882no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—942no—
249C22×C4—942no—
250C22×C4(C2×D47)⋊C411yescharacteristic
251C22×C4(C2×D47)⋊C411yescharacteristic
252C22×C4(C2×D47)⋊C411yescharacteristic
253C22×C4(C2×D47)⋊C411yescharacteristic
254C22×C4#1062533512yes—
255C22×C4#1062533712yes—
256C22×C4C2×C4×D4711yescharacteristic
257C22×C4C2×D18811yescharacteristic
258C22×C4C2×D18812yes—
259C22×C4#1062535012yes—
260C2×C8—1882no—
261C2×C8—1882no—
262C2×C8—1882no—
263C4⋊C4—944no—
264C4⋊C4—1882no—
265C4⋊C4—1882no—
266C4⋊C4—1882no—
267C4⋊C4—1882no—
268C4⋊C4—1882no—
269C4⋊C4—1882no—
270C22⋊C4—1884no—
271C42—22no—
272C42—22no—
273C42—41no—
274C42—41no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×(C2×D47)⋊C411yescharacteristic
280C23#1309399812yes—
281C23C2×#1062533511yescharacteristic
282C23C2×#1062533711yescharacteristic
283C23#1309405412yes—
284Q8—3764no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—1882no—
311C2×C4—1882no—
312C2×C4—1882no—
313C2×C4—1882no—
314C2×C4—1882no—
315C2×C4—1882no—
316C2×C4#1309393812yes—
317C2×C4#1309394712yes—
318C2×C4#1309394712yes—
319C2×C4C4×D18812yes—
320C2×C4#1309401211yescharacteristic
321C2×C4#1309402111yescharacteristic
322C8—3762no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22#1568902511yescharacteristic
332C22C2×#1309393811yescharacteristic
333C22#1568909611yescharacteristic
334C22C2×#1309394711yescharacteristic
335C22#1568913111yescharacteristic
336C22#1568914111yescharacteristic
337C22#1568914111yescharacteristic
338C22#1568932812yes—
339C22#1568933312yes—
340C4—22no—
341C4—21no—
342C4—22no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—84no—
348C4—3762no—
349C4—3762no—
350C2—22no—
351C2—22no—
352C2#2024192011yescharacteristic
353C2#2024193811yescharacteristic
354C2#2024195111yescharacteristic
355C2#2024195911yescharacteristic
356C2#2024422011yescharacteristic
357C2#2024422411yescharacteristic
358C2#2024431111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1434C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1434D47
hypocenterC47#9714
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
5. C2×#13094024C4semidirect
9. #15689029C4semidirect
12. #15688993C4semidirect
200. C47#9714semidirect

Maximal subgroups

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

subgroupindex
#202517402
#202433682
C2×#156888592
C47×#14342
#971447