Finite Group Databasebeta

← #26111522

Subgroups — #26111522

Order 12032.

432 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#20252337C211yesmaximal, characteristic
2#20244333C211yesmaximal, characteristic
3#20241887C211yesmaximal, characteristic
4#20241791C211yesmaximal, characteristic
5#20241228C211yesmaximal, characteristic
6#20241225C211yesmaximal, characteristic
7C47×#3170C211yesmaximal, characteristic
8#15689254—21no—
9#15689254—22no—
10Q8×C47⋊C8—21no—
11#15689176C2211yescharacteristic
12#15689150—21no—
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14#15689079—21no—
15C4×#10625329C2211yescharacteristic
16C4×C4⋊Dic47C2211yescharacteristic
17#15688882—21no—
18#15688882—21no—
19#15688860—22no—
20Q8⋊(C47⋊C8)—21no—
21Q8⋊(C47⋊C8)—21no—
22C47×#533C2211yescharacteristic
23C47×#533C2211yescharacteristic
24C47×#500C411yescharacteristic, complemented
25C47×#494C411yescharacteristic, complemented
26C47×#487C411yescharacteristic, complemented
27C47×#487C411yescharacteristic, complemented
28C47×#467—22no—
29C4×Q16×C47—22no—
30C2×C4×C8×C47C2211yescharacteristic
31C2×#10625329D411yescharacteristic
32C2×#10625329D411yescharacteristic
33C22×C47⋊C8—21no—
34C2×C4⋊Dic47—21no—
35C2×C4×Dic47—21no—
36C4⋊(C47⋊C8)—22no—
37C4⋊(C47⋊C8)—22no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)D411yescharacteristic
41C4⋊(C47⋊C8)D411yescharacteristic
42C22⋊(C47⋊C8)—21no—
43#13093935—22no—
44#13093935—21no—
45C4×C47⋊C8—22no—
46C4×C47⋊C8—21no—
47C4×C47⋊C8—21no—
48C47⋊#95—21no—
49C2×Q16×C47—22no—
50C22×C8×C47—21no—
51C4⋊Q8×C47C2×C411yescharacteristic
52C4⋊Q8×C47C2×C411yescharacteristic
53C4⋊Q8×C47C2×C411yescharacteristic
54C4⋊Q8×C47C2×C411yescharacteristic
55#125×C47—21no—
56#125×C47—21no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C22⋊Q8×C47D412yes—
61C4×Q8×C47—21no—
62C4×Q8×C47—21no—
63C4×Q8×C47—21no—
64C4×Q8×C47D412yes—
65#117×C47—21no—
66#117×C47—21no—
67C2×C42×C47C2311yescharacteristic
68#107×C47—22no—
69#103×C47—22no—
70#103×C47—22no—
71C4×C8×C47—21no—
72C4×C8×C47C2×C411yescharacteristic
73C4×C8×C47C2×C411yescharacteristic
74C2×Q8×C47—21no—
75C2×Q8×C47—21no—
76C2×Q8×C47—21no—
77C2×Q8×C47C22⋊C412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—41no—
82Q16×C47—44no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—42no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—22no—
90C4⋊C4×C47—22no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—21no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47C22⋊C412yes—
98C22⋊C4×C47—22no—
99C22⋊C4×C47—41no—
100C42×C47—22no—
101C42×C47—41no—
102C42×C47C22×C411yescharacteristic
103C42×C47C2×D411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106C4⋊Dic47—22no—
107C4⋊Dic47—41no—
108C4×Dic47—41no—
109C4×Dic47—41no—
110#10625329—21no—
111#10625329—21no—
112#10625329—21no—
113#10625329Q1612yescomplemented
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8—21no—
117C2×C47⋊C8—21no—
118C2×C47⋊C8—41no—
119C2×C47⋊C8—41no—
120C2×C47⋊C8—41no—
121C23×C47#12111yescharacteristic
122Q8×C47—42no—
123Q8×C47—42no—
124Q8×C47—41no—
125Q8×C47—41no—
126Q8×C47—41no—
127Q8×C47—41no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47C2×C22⋊C411yescharacteristic
136C2×C4×C47C4×D411yescharacteristic
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47#12011yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47#12311yescharacteristic
141C2×C4×C47C2×Q1611yescharacteristic
142C2×C4×C47#13511yescharacteristic
143C2×C4×C47#13711yescharacteristic
144C2×C4×C47#13711yescharacteristic
145C2×Dic47—42no—
146C2×Dic47—81no—
147C2×Dic47—81no—
148C8×C47—22no—
149C8×C47—41no—
150C47⋊C8—22no—
151C47⋊C8—41no—
152C47⋊C8—41no—
153C47⋊C8—41no—
154C47⋊C8—81no—
155#9056—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47#38611yescharacteristic
159C22×C47#46211yescharacteristic
160C22×C47#46411yescharacteristic
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—42no—
165C4×C47—42no—
166C4×C47—81no—
167C4×C47—81no—
168C4×C47C4×Q1611yescharacteristic
169C4×C47#44111yescharacteristic
170C4×C47#44311yescharacteristic
171C4×C47#44411yescharacteristic
172C4×C47#45111yescharacteristic
173C4×C47#45211yescharacteristic
174Dic47—82no—
175#3170D4711yescharacteristic
176#2962—471no—
177#1986—471no—
178#1503—471no—
179#1481—471no—
180#1279—471no—
181#1278—471no—
182C2×C47—21no—
183C2×C47—21no—
184C2×C47#187611yescharacteristic
185C2×C47#187811yescharacteristic
186C2×C47#188411yescharacteristic
187#533C2×D4711yescharacteristic
188#533C2×D4711yescharacteristic
189#500Dic4711yescharacteristic, complemented
190#494Dic4711yescharacteristic, complemented
191#487Dic4711yescharacteristic, complemented
192#487Dic4711yescharacteristic, complemented
193#467—22no—
194#446—941no—
195#446—942no—
196C8×Q8—941no—
197C4×Q16—22no—
198#431—471no—
199#424—941no—
200C2×C4⋊C8—471no—
201#405—941no—
202C4×M16—471no—
203C2×C4×C8C2×D4711yescharacteristic
204C4×C4⋊C4—471no—
205#339—941no—
206#339—941no—
207#328—942no—
208Q8⋊C8—941no—
209Q8⋊C8—941no—
210C47#905611yescharacteristic, complemented, Sylow
211C2×Q16—22no—
212C2×M16—471no—
213C2×M16—471no—
214C22×C8—21no—
215C22×C8—941no—
216C4⋊Q8C2×Dic4711yescharacteristic
217C4⋊Q8C2×Dic4711yescharacteristic
218C4⋊Q8C2×Dic4711yescharacteristic
219C4⋊Q8C2×Dic4711yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8C22⋊D4712yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C22⋊D4712yes—
230#117—21no—
231#117—21no—
232C2×C4⋊C4—941no—
233C2×C42—941no—
234C2×C42C22×D4711yescharacteristic
235#107—22no—
236C4⋊C8—471no—
237C4⋊C8—471no—
238C4⋊C8—942no—
239C4⋊C8—942no—
240C4⋊C8—941no—
241C4⋊C8—941no—
242#103—22no—
243#103—22no—
244C22⋊C8—941no—
245#97—942no—
246#97—941no—
247C4×C8—21no—
248C4×C8—942no—
249C4×C8—941no—
250C4×C8—941no—
251C4×C8C2×Dic4711yescharacteristic
252C4×C8C2×Dic4711yescharacteristic
253#95—941no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8C22⋊Dic4712yes—
258C22×C4—1881no—
259C22×C4D4×D4711yescharacteristic
260C22×C4#1062535211yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262Q16—44no—
263M16—472no—
264M16—941no—
265M16—941no—
266M16—941no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—941no—
271C2×C8—941no—
272C2×C8—941no—
273C2×C8—941no—
274C2×C8—1881no—
275C2×C8—1881no—
276C2×C8—1881no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—942no—
289C4⋊C4—1881no—
290C4⋊C4C22⋊Dic4712yes—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—22no—
294C42—41no—
295C42—1881no—
296C42—1881no—
297C42D4×D4711yescharacteristic
298C42#1062535211yescharacteristic
299C42C22×Dic4711yescharacteristic
300C42C2×C22⋊D4711yescharacteristic
301C23#1309399611yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—3761no—
316C2×C4—3761no—
317C2×C4—1882no—
318C2×C4C2×C22⋊Dic4711yescharacteristic
319C2×C4D4×Dic4711yescharacteristic
320C2×C4D4×Dic4711yescharacteristic
321C2×C4#1309399211yescharacteristic
322C2×C4#1309399611yescharacteristic
323C2×C4#1309400511yescharacteristic
324C2×C4Q16×D4711yescharacteristic
325C2×C4#1309404711yescharacteristic
326C2×C4#1309405611yescharacteristic
327C2×C4#1309405611yescharacteristic
328C8—22no—
329C8—41no—
330C8—942no—
331C8—3761no—
332C8—1881no—
333C8—1881no—
334C8—1881no—
335C22—21no—
336C22—21no—
337C22#1568900311yescharacteristic
338C22#1568933111yescharacteristic
339C22#1568934711yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4—3762no—
348C4Q16×Dic4711yescharacteristic
349C4#1568923011yescharacteristic
350C4#1568924011yescharacteristic
351C4#1568924511yescharacteristic
352C4#1568927711yescharacteristic
353C4#1568928411yescharacteristic
354C2—21no—
355C2—21no—
356C2#2024371711yescharacteristic
357C2#2024373111yescharacteristic
358C2#2024377211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3170C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#3170D47
hypocenterC47#9056
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
24. C47×#500C4semidirect
25. C47×#494C4semidirect
26. C47×#487C4semidirect
27. C47×#487C4semidirect
113. #10625329Q16semidirect
189. #500Dic47semidirect
190. #494Dic47semidirect
191. #487Dic47semidirect
192. #487Dic47semidirect
210. C47#9056semidirect

Maximal subgroups

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

subgroupindex
#202523372
#202443332
#202418872
#202417912
#202412282
#202412252
C47×#31702
#905647