Finite Group Databasebeta

← #26110586

Subgroups — #26110586

Order 12032.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252056C211yesmaximal, characteristic
2#20243409C211yesmaximal, characteristic
3#20242159C211yesmaximal, characteristic, complemented
4#20242146C211yesmaximal, characteristic, complemented
5#20241768C211yesmaximal, characteristic
6#20241229C211yesmaximal, characteristic, complemented
7C47×#1303C211yesmaximal, characteristic, complemented
8C4×#10625350C2211yescharacteristic
9#15689183—22no—
10#15689179—21no—
11#15689175—21no—
12#15689147C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093952—21no—
15#15689091—21no—
16#15689081—21no—
17C2×C4×C47⋊C8C2211yescharacteristic
18#15688883—21no—
19#15688875—22no—
20#15688874—21no—
21#15688874—21no—
22#15688853C411yescharacteristic, complemented
23#15688853C411yescharacteristic, complemented
24#15688850C411yescharacteristic, complemented
25#15688850C411yescharacteristic, complemented
26C47×#431C2211yescharacteristic
27C47×#423C2211yescharacteristic
28C4×C4⋊C4×C47C2211yescharacteristic
29C2×#10625350D411yescharacteristic
30C2×#10625329—21no—
31C22×C47⋊C8—21no—
32C22×C47⋊C8—21no—
33C4×Dic94C2×C411yescharacteristic
34C4×Dic94Q811yescharacteristic
35C4×D188C2×C411yescharacteristic
36C4×D188Q811yescharacteristic
37C4×C22⋊D47—21no—
38C4×C22⋊D47—21no—
39#13093983D411yescharacteristic
40#13093981—21no—
41C42×D47—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)D411yescharacteristic
46C4⋊(C47⋊C8)D411yescharacteristic
47#13093952—42no—
48(C2×D47)⋊C8—22no—
49(C2×D47)⋊C8—22no—
50#13093934—22no—
51C4×C47⋊C8—21no—
52C4×C47⋊C8C2×C411yescharacteristic
53C4×C47⋊C8C2×C411yescharacteristic
54C8×Dic47—22no—
55C2×M16×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C42×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—21no—
60C4⋊C8×C47C2×C411yescharacteristic
61C4⋊C8×C47C2×C411yescharacteristic
62C4⋊C8×C47D412yes—
63C22⋊C8×C47—21no—
64C4×C8×C47—21no—
65#95×C47—21no—
66C22×C4×C47—41no—
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47#4111yescharacteristic
69C22×C4×C47#4111yescharacteristic
70C2×C22⋊D47—21no—
71#10625350—21no—
72#10625350—21no—
73#10625350—21no—
74#10625350—21no—
75C2×D188C4⋊C411yescharacteristic
76C2×C4×D47—21no—
77C2×C4×D47—21no—
78C2×C4×D47—21no—
79C2×Dic94C4⋊C411yescharacteristic
80M16×C47—41no—
81M16×C47—41no—
82C2×C8×C47—21no—
83C2×C8×C47—21no—
84C2×C8×C47—22no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—41no—
87C42×C47—41no—
88C42×C47—41no—
89C42×C47C22×C411yescharacteristic
90C42×C47C2×D411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47C2×Q811yescharacteristic
93C22⋊Dic47—21no—
94(C2×D47)⋊C4—21no—
95(C2×D47)⋊C4—21no—
96C4⋊Dic47C4⋊C411yescharacteristic
97C4⋊Dic47C4⋊C411yescharacteristic
98Dic47⋊C4—21no—
99Dic47⋊C4—21no—
100C4×Dic47—21no—
101C4×Dic47—21no—
102C4×Dic47—21no—
103#10625329—41no—
104#10625329—41no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—42no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—41no—
113C23×C47#12311yescharacteristic
114C22×D47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—41no—
118C2×C4×C47—42no—
119C2×C4×C47—81no—
120C2×C4×C47—81no—
121C2×C4×C47C2×C4⋊C411yescharacteristic
122C2×C4×C47C4×D411yescharacteristic
123C2×C4×C47C4×D411yescharacteristic
124C2×C4×C47#12011yescharacteristic
125C2×C4×C47C22⋊Q811yescharacteristic
126C2×C4×C47C22⋊Q811yescharacteristic
127C2×C4×C47#13511yescharacteristic
128C2×C4×C47#13511yescharacteristic
129C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C47#13711yescharacteristic
131C22⋊D47—41no—
132C22⋊D47—41no—
133C2×Dic47—21no—
134C2×Dic47—21no—
135C2×Dic47—21no—
136D188—21no—
137D188—21no—
138C4×D47—41no—
139C4×D47—41no—
140C4×D47—41no—
141C4×D47—41no—
142C4×D47—81no—
143Dic94—21no—
144Dic94—21no—
145C8×C47—41no—
146C8×C47—41no—
147C8×C47—42no—
148C47⋊C8—22no—
149C47⋊C8—41no—
150C47⋊C8—41no—
151C47⋊C8—41no—
152C47⋊C8—81no—
153#8903—471nomaximal, Sylow
154C22×C47—21no—
155C22×C47—21no—
156C22×C47#38511yescharacteristic
157C22×C47#48211yescharacteristic
158C22×C47#48511yescharacteristic
159C2×D47—41no—
160C2×D47—41no—
161C2×D47—41no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—82no—
166C4×C47#42711yescharacteristic
167C4×C47#42811yescharacteristic
168C4×C47#44311yescharacteristic
169C4×C47#44411yescharacteristic
170C4×C47#44911yescharacteristic
171C4×C47#45211yescharacteristic
172Dic47—41no—
173Dic47—41no—
174Dic47—41no—
175#2925—471no—
176#1806—471no—
177#1570—471no—
178#1567—471no—
179#1476—471no—
180#1303D4711yescharacteristic, complemented
181#1279—471no—
182C2×C47—21no—
183C2×C47—21no—
184C2×C47#185411yescharacteristic
185C2×C47#186111yescharacteristic
186C2×C47#186611yescharacteristic
187D47—81no—
188C4×#41—471no—
189#433—942no—
190#432—941no—
191#431—941no—
192#431C2×D4711yescharacteristic
193#423—471no—
194#423C2×D4711yescharacteristic
195C2×C4⋊C8—471no—
196C2×#104—941no—
197#408—941no—
198#405—941no—
199C2×C4×C8—471no—
200C4×C4⋊C4C2×D4711yescharacteristic
201#339—941no—
202#335—942no—
203#334—941no—
204#334—941no—
205Q8⋊C8—471no—
206Q8⋊C8—471no—
207D4⋊C8—471no—
208D4⋊C8—471no—
209C47#890311yescharacteristic, complemented, Sylow
210C2×#41—471no—
211C2×M16—21no—
212C2×M16—941no—
213C22×C8—941no—
214C22×C8—941no—
215C4×Q8—471no—
216C4×Q8—471no—
217C4×D4—471no—
218C4×D4—471no—
219C4×D4—941no—
220C4×D4—941no—
221#117—471no—
222#117—941no—
223C2×C4⋊C4—21no—
224C2×C42—21no—
225C2×C42—941no—
226C2×C42C22×D4711yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—471no—
229C4⋊C8—471no—
230C4⋊C8—941no—
231C4⋊C8—941no—
232C4⋊C8—941no—
233C4⋊C8C4×D4711yescharacteristic
234C4⋊C8C4×D4711yescharacteristic
235C4⋊C8D18812yes—
236#104—1882no—
237C22⋊C8—21no—
238C22⋊C8—942no—
239C22⋊C8—942no—
240#97—942no—
241C4×C8—21no—
242C4×C8—471no—
243C4×C8—471no—
244C4×C8—942no—
245C4×C8—941no—
246#95—21no—
247#41—941no—
248#41—941no—
249#41—941no—
250#41—941no—
251C2×Q8—471no—
252C2×D4—471no—
253C2×D4—941no—
254C22×C4—41no—
255C22×C4—941no—
256C22×C4—941no—
257C22×C4—941no—
258C22×C4#1062535011yescharacteristic
259C22×C4D4×D4711yescharacteristic
260C22×C4#1062535211yescharacteristic
261M16—41no—
262M16—41no—
263M16—1881no—
264M16—1881no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—941no—
269C2×C8—941no—
270C2×C8—941no—
271C2×C8—941no—
272C2×C8—1882no—
273C2×C8—1881no—
274C2×C8—1881no—
275C2×C8—1881no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—471no—
279C4⋊C4—471no—
280C4⋊C4—941no—
281C4⋊C4—941no—
282C22⋊C4—941no—
283C22⋊C4—941no—
284C22⋊C4—941no—
285C42—41no—
286C42—41no—
287C42—941no—
288C42—941no—
289C42—941no—
290C42C2×C4×D4711yescharacteristic
291C42C2×D18811yescharacteristic
292C42D4×D4711yescharacteristic
293C42Q8×D4711yescharacteristic
294C23—941no—
295C23#1309400611yescharacteristic
296Q8—941no—
297Q8—941no—
298D4—941no—
299D4—941no—
300D4—1881no—
301D4—1881no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—941no—
309C2×C4—941no—
310C2×C4—941no—
311C2×C4—3761no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4C4⋊C4×D4711yescharacteristic
317C2×C4(C22⋊D47)⋊C411yescharacteristic
318C2×C4C4×D18811yescharacteristic
319C2×C4#1309399411yescharacteristic
320C2×C4#1309400211yescharacteristic
321C2×C4#1309400411yescharacteristic
322C2×C4#1309404511yescharacteristic
323C2×C4#1309404511yescharacteristic
324C2×C4#1309404811yescharacteristic
325C2×C4#1309405611yescharacteristic
326C8—41no—
327C8—41no—
328C8—42no—
329C8—942no—
330C8—3761no—
331C8—1881no—
332C8—1881no—
333C8—1881no—
334C22—21no—
335C22—21no—
336C22—1881no—
337C22—1881no—
338C22—1881no—
339C22#1568899711yescharacteristic
340C22#1568944611yescharacteristic
341C22#1568946211yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—82no—
346C4—1881no—
347C4—1881no—
348C4—1881no—
349C4#1568915811yescharacteristic
350C4#1568916411yescharacteristic
351C4#1568923711yescharacteristic
352C4#1568924211yescharacteristic
353C4#1568926911yescharacteristic
354C4#1568928811yescharacteristic
355C2—21no—
356C2—21no—
357C2—3761no—
358C2#2024358811yescharacteristic
359C2#2024363111yescharacteristic
360C2#2024366411yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1303C2
socleC22×C47#385
prime radicalC2×#10625350D4
cosocleC2×C42×C47C23
hypercenter#1303D47
hypocenterC47#8903
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #20242159C2semidirect
4. #20242146C2semidirect
6. #20241229C2semidirect
7. C47×#1303C2semidirect
22. #15688853C4semidirect
23. #15688853C4semidirect
24. #15688850C4semidirect
25. #15688850C4semidirect
180. #1303D47semidirect
209. C47#8903semidirect

Maximal subgroups

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

subgroupindex
#202520562
#202434092
#202421592
#202421462
#202417682
#202412292
C47×#13032
#890347