Finite Group Databasebeta

← #26133153

Subgroups — #26133153

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689540C211yesmaximal, characteristic, complemented
2C2×#15689238C211yesmaximal, characteristic
3C2×#15688906C211yesmaximal, characteristic, complemented
4#20245003C211yesmaximal, characteristic, complemented
5#20241469C211yesmaximal, characteristic, complemented
6#20241465C211yesmaximal, characteristic, complemented
7C2×C47×#350C211yesmaximal, characteristic, complemented
8C2×#13094044C411yescharacteristic
9C2×#13094030C2211yescharacteristic
10#15689540—22no—
11#15689540—22no—
12C2×#13093964—21no—
13C2×#13093964C2211yescharacteristic
14C22×C47⋊C16—21no—
15C22×C47⋊C16C2211yescharacteristic
16#15689238—22no—
17#15689238—22no—
18C2×#13093959C411yescharacteristic
19C2×#13093951C411yescharacteristic
20C2×#13093950C411yescharacteristic
21C2×C8×Dic47C2211yescharacteristic
22#15688906—22no—
23(C2×D47)⋊C16—21no—
24(C2×D47)⋊C16—21no—
25C2×#131×C47C2211yescharacteristic
26C2×M32×C47C2211yescharacteristic
27C22×C16×C47C2211yescharacteristic
28C47×#350—22no—
29C2×#10625350C2×C411yescharacteristic
30#13094044—22no—
31#13094044—22no—
32C2×#10625337C811yescharacteristic
33C2×#10625335C811yescharacteristic
34C2×#10625336C811yescharacteristic
35C2×#10625334C811yescharacteristic
36#13094030—21no—
37#13094030—21no—
38#13094030—21no—
39#13094030—21no—
40C2×#10625329C2×C411yescharacteristic
41C2×#10625324—21no—
42C22×C47⋊C8C2×C411yescharacteristic
43C2×C8×D47—21no—
44C2×C4×Dic47C2×C411yescharacteristic
45#13093964—24no—
46#13093964—21no—
47#13093964D412yes—
48C2×C47⋊C16—24no—
49C2×C47⋊C16—21no—
50C2×C47⋊C16—21no—
51C2×C47⋊C16—21no—
52C2×C47⋊C16D411yescharacteristic
53C2×C47⋊C16D411yescharacteristic
54#13093959—22no—
55#13093951—22no—
56#13093950—22no—
57C8×Dic47—21no—
58C8×Dic47—21no—
59C2×#41×C47C2×C411yescharacteristic
60#131×C47—21no—
61#131×C47—21no—
62#131×C47D412yes—
63#131×C47D412yes—
64C2×M16×C47—21no—
65C2×M16×C47C2×C411yescharacteristic
66C22×C8×C47—21no—
67C22×C8×C47C2311yescharacteristic
68M32×C47—21no—
69M32×C47—21no—
70C2×C16×C47—21no—
71C2×C16×C47—21no—
72C2×C16×C47—21no—
73C2×C16×C47—21no—
74#41×C47—21no—
75#41×C47—21no—
76#41×C47C22⋊C412yes—
77#41×C47M1612yes—
78C2×Q8×C47C2×C811yescharacteristic
79C2×D4×C47—21no—
80C2×D4×C47C2×C811yescharacteristic
81C22×C4×C47—21no—
82C22×C4×C47C22×C411yescharacteristic
83C2×C22⋊D47—21no—
84C22×Dic47—21no—
85#10625350—21no—
86#10625350—21no—
87#10625350—21no—
88#10625350—21no—
89C2×D188C2×C811yescharacteristic
90C2×C4×D47—21no—
91C2×Dic94C2×C811yescharacteristic
92M16×C47—21no—
93M16×C47—22no—
94M16×C47—22no—
95M16×C47C22⋊C412yes—
96C2×C8×C47—21no—
97C2×C8×C47—22no—
98C2×C8×C47—22no—
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47C2×D411yescharacteristic
102C2×C8×C47C2×D411yescharacteristic
103C2×C8×C47#4111yescharacteristic
104C2×C8×C47#4111yescharacteristic
105#10625337—22no—
106#10625336—22no—
107#10625335—22no—
108#10625334—22no—
109C4×Dic47—21no—
110C4×Dic47—21no—
111#10625329—21no—
112#10625329—21no—
113C2×C47⋊C8—21no—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8C2×C811yescharacteristic
117C2×C47⋊C8C2×C811yescharacteristic
118#10625324—41no—
119#10625324—41no—
120C8×D47—41no—
121C8×D47—41no—
122C16×C47—41no—
123C16×C47—41no—
124C47⋊C16—21no—
125C47⋊C16—22no—
126C47⋊C16—21no—
127C23×C47—21no—
128C23×C47#13111yescharacteristic
129C22×D47—21no—
130Q8×C47—21no—
131Q8×C47C22⋊C812yes—
132D4×C47—22no—
133D4×C47—22no—
134D4×C47—21no—
135D4×C47C22⋊C812yes—
136C2×C4×C47—21no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47C2×C22⋊C411yescharacteristic
140C2×C4×C47C4×D411yescharacteristic
141C2×C4×C47C4×D411yescharacteristic
142C2×C4×C47C22×C811yescharacteristic
143C2×C4×C47C2×M1611yescharacteristic
144C2×C4×C47#13111yescharacteristic
145C22⋊D47—41no—
146C22⋊D47—41no—
147C2×Dic47—22no—
148C2×Dic47—21no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151D188—21no—
152D188—21no—
153C4×D47—41no—
154C4×D47—41no—
155Dic94—21no—
156Dic94—21no—
157C8×C47—22no—
158C8×C47#12011yescharacteristic
159C8×C47#12111yescharacteristic
160C8×C47#12111yescharacteristic
161C8×C47#12311yescharacteristic
162C47⋊C8—21no—
163C47⋊C8—21no—
164C47⋊C8—41no—
165#12980—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—22no—
169C22×C47—41no—
170C22×C47—41no—
171C22×C47#40811yescharacteristic
172C22×C47C8×D411yescharacteristic
173C22×C47#43511yescharacteristic
174C2×D47—41no—
175C2×D47—41no—
176C2×D47—41no—
177C4×C47—22no—
178C4×C47#38611yescharacteristic
179C4×C47C2×C22⋊C811yescharacteristic
180C4×C47C8×D411yescharacteristic
181C4×C47#43511yescharacteristic
182Dic47—42no—
183Dic47—41no—
184C2×#504—471no—
185C2×#443—471no—
186C2×#350—471no—
187C2×#350D4711yescharacteristic, complemented
188#2113—471no—
189#1376—471no—
190#1374—471no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—42no—
194C2×C47#186911yescharacteristic
195C2×C47#217011yescharacteristic
196C2×C47#217111yescharacteristic
197D47—81no—
198C2×#135—471no—
199C2×#131—471no—
200C2×#131C2×D4711yescharacteristic
201#504—942no—
202#504—942no—
203C2×M32—471no—
204C2×M32—941no—
205C2×M32C2×D4711yescharacteristic
206C22×C16—471no—
207C22×C16—941no—
208C22×C16C2×D4711yescharacteristic
209#443—942no—
210#443—942no—
211C2×#108—471no—
212C2×#104—471no—
213C2×#104—471no—
214C2×C4×C8—471no—
215#350—22no—
216#350—942no—
217C22⋊C16—941no—
218C22⋊C16—941no—
219C47#1298011yescharacteristic, complemented, Sylow
220C2×#41—471no—
221C2×#41C4×D4711yescharacteristic
222#135—942no—
223#135—942no—
224C2×Q16—471no—
225C2×SD16—471no—
226C2×SD16—471no—
227C2×D8—471no—
228#131—21no—
229#131—21no—
230#131—941no—
231#131—941no—
232#131—941no—
233#131—941no—
234#131D18812yes—
235#131C22⋊D4712yes—
236C2×M16—21no—
237C2×M16—471no—
238C2×M16—941no—
239C2×M16C4×D4711yescharacteristic
240C22×C8—21no—
241C22×C8—471no—
242C22×C8—941no—
243C22×C8C22×D4711yescharacteristic
244C2×C42—471no—
245M32—21no—
246M32—21no—
247M32—472no—
248M32—944no—
249M32—941no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—21no—
254C2×C16—471no—
255C2×C16—471no—
256C2×C16—944no—
257C2×C16—941no—
258C2×C16—941no—
259C2×C16—941no—
260#108—942no—
261#104—942no—
262#104—942no—
263C4×C8—941no—
264C4×C8—941no—
265#41—21no—
266#41—21no—
267#41—941no—
268#41—941no—
269#41—941no—
270#41—941no—
271#41#1062532412yes—
272#41(C2×D47)⋊C412yes—
273C2×Q8—471no—
274C2×Q8C8×D4711yescharacteristic
275C2×D4—21no—
276C2×D4—471no—
277C2×D4—941no—
278C2×D4C8×D4711yescharacteristic
279C22×C4—21no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4C2×C4×D4711yescharacteristic
283Q16—942no—
284SD16—942no—
285SD16—942no—
286D8—942no—
287M16—21no—
288M16—22no—
289M16—22no—
290M16—941no—
291M16—941no—
292M16—1881no—
293M16—1881no—
294M16(C2×D47)⋊C412yes—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—471no—
299C2×C8—471no—
300C2×C8—941no—
301C2×C8—941no—
302C2×C8—941no—
303C2×C8—1881no—
304C2×C8—1881no—
305C2×C8C2×D18811yescharacteristic
306C2×C8D4×D4711yescharacteristic
307C2×C8D4×D4711yescharacteristic
308C2×C8#1062535211yescharacteristic
309C2×C8#1062535211yescharacteristic
310C2×C8C2×C22⋊D4711yescharacteristic
311C42—941no—
312C42—941no—
313C16—41no—
314C16—41no—
315C16—941no—
316C16—941no—
317C16—942no—
318C23—21no—
319C23—941no—
320C23#1309403211yescharacteristic
321Q8—21no—
322Q8—941no—
323Q8—941no—
324Q8(C2×D47)⋊C812yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—941no—
329D4—941no—
330D4—1881no—
331D4—1881no—
332D4(C2×D47)⋊C812yes—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—942no—
337C2×C4—941no—
338C2×C4—1881no—
339C2×C4—1881no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4C2×(C2×D47)⋊C411yescharacteristic
343C2×C4(C22⋊D47)⋊C411yescharacteristic
344C2×C4(C22⋊D47)⋊C411yescharacteristic
345C2×C4C2×C8×D4711yescharacteristic
346C2×C4C2×#1062532411yescharacteristic
347C2×C4#1309403211yescharacteristic
348C8—22no—
349C8—941no—
350C8—941no—
351C8—1881no—
352C8#1309399411yescharacteristic
353C8#1309399611yescharacteristic
354C8#1309399611yescharacteristic
355C8#1309400811yescharacteristic
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—1881no—
362C22—1881no—
363C22—1881no—
364C22#1568909111yescharacteristic
365C22(C22⋊D47)⋊C811yescharacteristic
366C22#1568919111yescharacteristic
367C4—22no—
368C4—1882no—
369C4—1881no—
370C4#1568900011yescharacteristic
371C4C2×(C2×D47)⋊C811yescharacteristic
372C4(C22⋊D47)⋊C811yescharacteristic
373C4#1568919111yescharacteristic
374C2—21no—
375C2—21no—
376C2—42no—
377C2—3761no—
378C2#2024367811yescharacteristic
379C2#2024522211yescharacteristic
380C2#2024522711yescharacteristic
subgroupG/H
centerC2×C8D4×D47
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC2×C47×#350C2
socleC22×C47C8×D4
prime radicalC2×#13094044C4
cosocleC22×C8×C47C23
hypercenterC2×#350D47
hypocenterC47#12980
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
1. C2×#15689540C2semidirect
3. C2×#15688906C2semidirect
4. #20245003C2semidirect
5. #20241469C2semidirect
6. #20241465C2semidirect
7. C2×C47×#350C2semidirect
187. C2×#350D47semidirect
219. C47#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#156895402
C2×#156892382
C2×#156889062
#202450032
#202414692
#202414652
C2×C47×#3502
#1298047