Finite Group Databasebeta

← #26111473

Subgroups — #26111473

Order 12032.

544 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252106C211yesmaximal, characteristic, complemented
2#20244187C211yesmaximal, characteristic, complemented
3#20242037C211yesmaximal, characteristic, complemented
4#20241813C211yesmaximal, characteristic, complemented
5#20241284C211yesmaximal, characteristic, complemented
6#20241232C211yesmaximal, characteristic, complemented
7C47×#3100C211yesmaximal, characteristic
8C2×#13094031—21no—
9#15689195—22no—
10#15689195—22no—
11#15689176C2211yescharacteristic
12#15689147C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14#15689090—21no—
15C2×C22⋊(C47⋊C8)—21no—
16C2×C22⋊(C47⋊C8)—21no—
17C2×#13093935C2211yescharacteristic
18#15688869—21no—
19#15688867—21no—
20D4⋊(C47⋊C8)—22no—
21C47×#532C2211yescharacteristic
22C2×C4×D4×C47C2211yescharacteristic
23C47×#480C412yes—
24C47×#474—21no—
25C47×#459—21no—
26C47×#447C412yes—
27C4×D8×C47—22no—
28C47×#431C2211yescharacteristic
29#13094031—42no—
30#13094031—42no—
31C2×#10625329—22no—
32C2×#10625329—21no—
33C22×C47⋊C8—22no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8D411yescharacteristic
36C22×C47⋊C8D411yescharacteristic
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)D412yes—
42C22⋊(C47⋊C8)—24no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—42no—
45C22⋊(C47⋊C8)—42no—
46#13093935—21no—
47#13093935—21no—
48C4×C47⋊C8—21no—
49C2×#41×C47—21no—
50C22×D4×C47C2×C411yescharacteristic
51C23×C4×C47—21no—
52C2×D8×C47—22no—
53C4⋊Q8×C47D411yescharacteristic
54#127×C47D411yescharacteristic
55#124×C47—21no—
56#121×C47—21no—
57#121×C47C2×C412yes—
58C4×D4×C47—21no—
59C4×D4×C47—21no—
60C4×D4×C47—21no—
61C4×D4×C47—21no—
62C4×D4×C47—21no—
63C4×D4×C47D412yes—
64C2×C4⋊C4×C47C2×C411yescharacteristic
65C2×C22⋊C4×C47—21no—
66C2×C42×C47C2311yescharacteristic
67#107×C47—22no—
68C4⋊C8×C47—21no—
69#102×C47—22no—
70#102×C47—22no—
71C22⋊C8×C47C2×C412yes—
72C4×C8×C47—21no—
73C24×C47—21no—
74#41×C47—41no—
75#41×C47—41no—
76#41×C47—41no—
77#41×C47—41no—
78C2×Q8×C47—21no—
79C2×D4×C47—22no—
80C2×D4×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47—21no—
83C2×D4×C47—21no—
84C2×D4×C47—42no—
85C2×D4×C47C22⋊C412yes—
86C22×C4×C47—22no—
87C22×C4×C47—21no—
88C22×C4×C47—41no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47—41no—
94C22×C4×C47C22×C411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47#4111yescharacteristic
97D8×C47—44no—
98C2×C8×C47—22no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47C22⋊C412yes—
103C22⋊C4×C47—22no—
104C22⋊C4×C47—22no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C42×C47C2×D411yescharacteristic
108C42×C47C2×D411yescharacteristic
109C42×C47C2×D411yescharacteristic
110C42×C47#4111yescharacteristic
111#10625329—44no—
112#10625329—41no—
113#10625329—41no—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8—21no—
117C2×C47⋊C8—22no—
118C2×C47⋊C8—21no—
119C2×C47⋊C8—21no—
120C2×C47⋊C8—44no—
121C2×C47⋊C8—41no—
122C2×C47⋊C8—41no—
123C2×C47⋊C8—41no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8D812yescomplemented
126C23×C47—22no—
127C23×C47—22no—
128C23×C47—41no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—41no—
134C23×C47C4×D411yescharacteristic
135Q8×C47—41no—
136Q8×C47—41no—
137D4×C47—22no—
138D4×C47—22no—
139D4×C47—42no—
140D4×C47—41no—
141D4×C47—41no—
142D4×C47—42no—
143D4×C47—42no—
144D4×C47—41no—
145D4×C47—41no—
146D4×C47—81no—
147C2×C4×C47—22no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—81no—
156C2×C4×C47—81no—
157C2×C4×C47—81no—
158C2×C4×C47—81no—
159C2×C4×C47C2×C22⋊C411yescharacteristic
160C2×C4×C47C4×D411yescharacteristic
161C2×C4×C47#12011yescharacteristic
162C2×C4×C47#12111yescharacteristic
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47#12311yescharacteristic
165C2×C4×C47C2×D811yescharacteristic
166C2×C4×C47#13511yescharacteristic
167C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
169C8×C47—42no—
170C47⋊C8—22no—
171C47⋊C8—42no—
172C47⋊C8—41no—
173C47⋊C8—81no—
174C47⋊C8—81no—
175#9049—471nomaximal, Sylow
176C22×C47—21no—
177C22×C47—21no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—42no—
181C22×C47—42no—
182C22×C47—42no—
183C22×C47—42no—
184C22×C47—81no—
185C22×C47—81no—
186C22×C47—81no—
187C22×C47—81no—
188C22×C47#38611yescharacteristic
189C22×C47C4×D811yescharacteristic
190C22×C47#44211yescharacteristic
191C4×C47—21no—
192C4×C47—21no—
193C4×C47—21no—
194C4×C47—21no—
195C4×C47—42no—
196C4×C47—81no—
197C4×C47#44711yescharacteristic
198C4×C47#44911yescharacteristic
199C4×C47#45911yescharacteristic
200C4×C47#46311yescharacteristic
201#3100D4711yescharacteristic
202#2932—471no—
203#1955—471no—
204#1540—471no—
205#1486—471no—
206#1301—471no—
207#1281—471no—
208C2×C47—21no—
209C2×C47—21no—
210C2×C47—82no—
211C2×C47—82no—
212C2×C47#187911yescharacteristic
213C2×C47#188111yescharacteristic
214C2×C47#189111yescharacteristic
215C2×#131—941no—
216#532C2×D4711yescharacteristic
217C2×C4×D4C2×D4711yescharacteristic
218#480Dic4712yes—
219#474—21no—
220#459—21no—
221#447Dic4712yes—
222C4×D8—22no—
223#435—942no—
224#435—942no—
225#431—471no—
226#431C2×D4711yescharacteristic
227#423—471no—
228C2×C4⋊C8—471no—
229#408—941no—
230C2×C22⋊C8—941no—
231C2×C22⋊C8—941no—
232C2×#97—471no—
233#332—941no—
234#331—941no—
235D4⋊C8—942no—
236C47#904911yescharacteristic, complemented, Sylow
237C2×#41—21no—
238C22×D4C2×Dic4711yescharacteristic
239C23×C4—21no—
240C2×D8—22no—
241#131—1882no—
242#131—1882no—
243C2×M16—942no—
244C2×M16—941no—
245C22×C8—471no—
246C22×C8—471no—
247C22×C8—942no—
248C22×C8—941no—
249C4⋊Q8C22⋊D4711yescharacteristic
250#127C22⋊D4711yescharacteristic
251#124—21no—
252#121—21no—
253#121C2×Dic4712yes—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4C22⋊D4712yes—
260C2×C4⋊C4C2×Dic4711yescharacteristic
261C2×C22⋊C4—21no—
262C2×C42C22×D4711yescharacteristic
263#107—22no—
264C4⋊C8—21no—
265C4⋊C8—472no—
266C4⋊C8—941no—
267C4⋊C8—941no—
268C4⋊C8—941no—
269C4⋊C8—941no—
270#102—22no—
271#102—22no—
272C22⋊C8—944no—
273C22⋊C8—941no—
274C22⋊C8—1882no—
275C22⋊C8—1882no—
276C22⋊C8C2×Dic4712yes—
277#97—941no—
278#97—941no—
279C4×C8—21no—
280C4×C8—941no—
281C24—21no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4C22⋊Dic4712yes—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4D4×D4711yescharacteristic
303C22×C4#1062535211yescharacteristic
304C22×C4C22×Dic4711yescharacteristic
305D8—44no—
306M16—1884no—
307M16—1881no—
308M16—1881no—
309C2×C8—22no—
310C2×C8—472no—
311C2×C8—941no—
312C2×C8—941no—
313C2×C8—941no—
314C2×C8—942no—
315C2×C8—941no—
316C2×C8—941no—
317C2×C8—1884no—
318C2×C8—1881no—
319C2×C8—1881no—
320C2×C8—1881no—
321C2×C8—1881no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4C22⋊Dic4712yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42D4×D4711yescharacteristic
331C42#1062535211yescharacteristic
332C42C2×C22⋊D4711yescharacteristic
333C42C2×C22⋊D4711yescharacteristic
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23D4×Dic4711yescharacteristic
343Q8—41no—
344Q8—41no—
345D4—22no—
346D4—22no—
347D4—42no—
348D4—41no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—41no—
353D4—41no—
354D4—81no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4C2×C22⋊Dic4711yescharacteristic
368C2×C4D4×Dic4711yescharacteristic
369C2×C4#1309399211yescharacteristic
370C2×C4#1309399611yescharacteristic
371C2×C4#1309399611yescharacteristic
372C2×C4#1309400511yescharacteristic
373C2×C4D8×D4711yescharacteristic
374C2×C4#1309404311yescharacteristic
375C2×C4#1309404811yescharacteristic
376C2×C4#1309404811yescharacteristic
377C8—42no—
378C8—942no—
379C8—3761no—
380C8—3761no—
381C8—1882no—
382C8—1881no—
383C22—21no—
384C22—21no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22#1568900311yescharacteristic
396C22D8×Dic4711yescharacteristic
397C22#1568923511yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4#1568925511yescharacteristic
405C4#1568926511yescharacteristic
406C4#1568931611yescharacteristic
407C4#1568933611yescharacteristic
408C2—21no—
409C2—21no—
410C2—82no—
411C2—82no—
412C2#2024373811yescharacteristic
413C2#2024375211yescharacteristic
414C2#2024380711yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3100C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3100D47
hypocenterC47#9049
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #20252106C2semidirect
2. #20244187C2semidirect
3. #20242037C2semidirect
4. #20241813C2semidirect
5. #20241284C2semidirect
6. #20241232C2semidirect
125. C2×C47⋊C8D8semidirect
236. C47#9049semidirect

Maximal subgroups

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

subgroupindex
#202521062
#202441872
#202420372
#202418132
#202412842
#202412322
C47×#31002
#904947