Finite Group Databasebeta

← #26107755

Subgroups — #26107755

Order 12032.

1094 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689673C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic, complemented
3C2×#15688862C214yesmaximal, complemented
4C47×#1749C211yesmaximal, characteristic, complemented
5C22×#10625350C412yes—
6#15689673—28no—
7C2×#13094024C411yescharacteristic
8C2×#13094021C411yescharacteristic
9C2×#13094012C2212yes—
10C2×#13093998—21no—
11C2×#13093998—21no—
12C2×C4×D188—21no—
13C2×C4×D188—21no—
14C2×C22⋊(C47⋊C8)—21no—
15C2×#13093935C2212yes—
16#15688878—21no—
17#15688862—28no—
18C22×C42×C47C2211yescharacteristic
19C2×C22⋊C8×C47—21no—
20C2×#97×C47C2212yes—
21C47×#336—21no—
22C2×#10625350—24no—
23C2×#10625350—24no—
24C2×#10625350—44no—
25C2×#10625350—44no—
26C22×Dic94C2×C412yes—
27C22×D188—21no—
28C22×D188—21no—
29C22×D188C2×C412yes—
30C22×C22⋊D47—22no—
31C22×C4×D47—22no—
32C22×C47⋊C8—22no—
33#13094024—24no—
34#13094021—24no—
35#13094012—24no—
36#13094012D418yes—
37#13093998—28no—
38#13093998—28no—
39C4×D188—28no—
40C4×D188—28no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×(C2×D47)⋊C4—22no—
44C2×(C2×D47)⋊C4—22no—
45C22⋊(C47⋊C8)—24no—
46#13093935—24no—
47C23×C4×C47C2×C411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47D411yescharacteristic
50C22×C8×C47—22no—
51C2×C42×C47—24no—
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47C2311yescharacteristic
56C22⋊C8×C47—24no—
57#97×C47—24no—
58C24×C47C22⋊C411yescharacteristic
59C23×D47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C2×C22⋊D47—28no—
76C2×C22⋊D47—44no—
77C2×C22⋊D47—44no—
78C22×Dic47—22no—
79#10625350—416no—
80#10625350—416no—
81C2×D188—24no—
82C2×D188—24no—
83C2×D188—22no—
84C2×D188—42no—
85C2×D188—42no—
86C2×D188—42no—
87C2×D188—42no—
88C2×D188—42no—
89C2×D188—42no—
90C2×D188C22⋊C414yes—
91C2×C4×D47—28no—
92C2×C4×D47—44no—
93C2×C4×D47—44no—
94C2×Dic94—22no—
95C2×Dic94—42no—
96C2×Dic94—42no—
97C2×Dic94C22⋊C414yes—
98C2×C8×C47—24no—
99C2×C8×C47—42no—
100C2×C8×C47—42no—
101C42×C47—22no—
102C42×C47—22no—
103C42×C47—22no—
104C42×C47C2×D414yes—
105(C2×D47)⋊C4—28no—
106(C2×D47)⋊C4—28no—
107C4⋊Dic47—24no—
108C4⋊Dic47—24no—
109C2×C47⋊C8—24no—
110C2×C47⋊C8—42no—
111C2×C47⋊C8—42no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47#10012yes—
119C23×C47C2×C22⋊C411yescharacteristic
120C22×D47—24no—
121C22×D47—42no—
122C22×D47—42no—
123C22×D47—42no—
124C22×D47—42no—
125C22×D47—42no—
126C22×D47—42no—
127C2×C4×C47—28no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—44no—
141C2×C4×C47—44no—
142C2×C4×C47#10012yes—
143C2×C4×C47#10412yes—
144C2×C4×C47#10412yes—
145C2×C4×C47C2×C22⋊C412yes—
146C2×C4×C47#12012yes—
147C2×C4×C47#12012yes—
148C22⋊D47—416no—
149C2×Dic47—24no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152D188—44no—
153D188—48no—
154D188—48no—
155D188—44no—
156C4×D47—416no—
157Dic94—44no—
158Dic94—44no—
159C8×C47—44no—
160C47⋊C8—44no—
161#8337—471nomaximal, Sylow
162C22×C47—22no—
163C22×C47—22no—
164C22×C47—22no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—41no—
169C22×C47#37911yescharacteristic
170C22×C47C2×#10011yescharacteristic
171C22×C47C2×#10011yescharacteristic
172C22×C47C2×#10411yescharacteristic
173C22×C47C2×#10411yescharacteristic
174C22×C47#42111yescharacteristic
175C22×C47#42111yescharacteristic
176C2×D47—44no—
177C2×D47—44no—
178C2×D47—44no—
179C2×D47—82no—
180C2×D47—82no—
181C2×D47—82no—
182C2×D47—82no—
183C4×C47—22no—
184C4×C47—22no—
185C4×C47—22no—
186C4×C47—22no—
187C4×C47—22no—
188C4×C47—22no—
189Dic47—44no—
190C2×#532—471no—
191#1749—471no—
192#1749D4711yescharacteristic, complemented
193C2×#329—474no—
194C2×C47—42no—
195C2×C47#152511yescharacteristic
196C2×C47#153011yescharacteristic
197C2×C47#153011yescharacteristic
198C2×C47#153611yescharacteristic
199C2×C47#178311yescharacteristic
200C2×C47#178911yescharacteristic
201C2×C47#178911yescharacteristic
202D47—84no—
203C22×#41—472no—
204#532—948no—
205C2×C4⋊Q8—471no—
206C2×#127—471no—
207C2×#124—472no—
208C2×#121—941no—
209C2×#121—941no—
210C2×C4×D4—941no—
211C2×C4×D4—941no—
212C22×C42C2×D4711yescharacteristic
213C2×C22⋊C8—21no—
214C2×C22⋊C8—941no—
215C2×#97—472no—
216C2×#97C2×D4712yes—
217#336—21no—
218#336—941no—
219#329—948no—
220C47#833711yescharacteristic, complemented, Sylow
221C2×#41—944no—
222C2×#41—944no—
223C2×#41—1884no—
224C2×#41—1884no—
225C22×Q8—472no—
226C22×D4—472no—
227C22×D4—942no—
228C22×D4—941no—
229C22×D4—941no—
230C23×C4—942no—
231C23×C4C4×D4711yescharacteristic
232C23×C4D18811yescharacteristic
233C23×C4C22⋊D4711yescharacteristic
234C22×C8—22no—
235C22×C8—942no—
236C4⋊Q8—944no—
237#127—944no—
238#124—478no—
239#124—944no—
240#121—948no—
241#121—948no—
242C4×D4—948no—
243C4×D4—948no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—941no—
246C2×C22⋊C4—942no—
247C2×C22⋊C4—942no—
248C2×C42—24no—
249C2×C42C4×D4711yescharacteristic
250C2×C42D18811yescharacteristic
251C2×C42C22⋊D4711yescharacteristic
252C2×C42C22×D4711yescharacteristic
253C22⋊C8—24no—
254C22⋊C8—944no—
255#97—24no—
256#97—944no—
257C24—942no—
258C24(C2×D47)⋊C411yescharacteristic
259#41—18816no—
260#41—18816no—
261C2×Q8—474no—
262C2×Q8—942no—
263C2×Q8—1882no—
264C2×Q8—1882no—
265C2×D4—474no—
266C2×D4—944no—
267C2×D4—944no—
268C2×D4—948no—
269C2×D4—942no—
270C2×D4—1882no—
271C2×D4—1882no—
272C2×D4—1882no—
273C2×D4—1882no—
274C2×D4—1884no—
275C2×D4—1884no—
276C2×D4—1882no—
277C2×D4—1882no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—948no—
288C22×C4—942no—
289C22×C4—1884no—
290C22×C4—1884no—
291C22×C4(C2×D47)⋊C411yescharacteristic
292C22×C4(C2×D47)⋊C411yescharacteristic
293C22×C4(C2×D47)⋊C411yescharacteristic
294C22×C4C2×C4×D4711yescharacteristic
295C22×C4C2×D18811yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297C2×C8—24no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—944no—
301C2×C8—1882no—
302C2×C8—1882no—
303C4⋊C4—944no—
304C4⋊C4—944no—
305C22⋊C4—948no—
306C22⋊C4—948no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42D4×D4714yes—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—944no—
318C23—1882no—
319C23—1882no—
320C23—1882no—
321C23—1882no—
322C23—1882no—
323C23—1882no—
324C23#1309394012yes—
325C23C2×(C2×D47)⋊C411yescharacteristic
326Q8—1884no—
327Q8—1884no—
328D4—1884no—
329D4—1888no—
330D4—1888no—
331D4—1884no—
332D4—18816no—
333C2×C4—28no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—944no—
349C2×C4—18816no—
350C2×C4—1882no—
351C2×C4—1882no—
352C2×C4#1309394012yes—
353C2×C4#1309395012yes—
354C2×C4#1309395212yes—
355C2×C4C22⋊C4×D4712yes—
356C2×C4#1309399212yes—
357C2×C4#1309399412yes—
358C8—44no—
359C8—1884no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—3762no—
368C22—3762no—
369C22—3762no—
370C22—3762no—
371C22—1884no—
372C22—1884no—
373C22—1884no—
374C22#1568897811yescharacteristic
375C22C2×#1309394011yescharacteristic
376C22C2×#1309394011yescharacteristic
377C22C2×#1309395011yescharacteristic
378C22C2×#1309395211yescharacteristic
379C22#1568913911yescharacteristic
380C22#1568914111yescharacteristic
381C4—22no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—1884no—
388C2—42no—
389C2—3764no—
390C2#2024197911yescharacteristic
391C2#2024199811yescharacteristic
392C2#2024200011yescharacteristic
393C2#2024202211yescharacteristic
394C2#2024330811yescharacteristic
395C2#2024333111yescharacteristic
396C2#2024333311yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689673C2
cosocleC2×C42×C47C23
hypercenter#1749D47
hypocenterC47#8337
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. #20243150C2semidirect
3. C2×#15688862C2semidirect
4. C47×#1749C2semidirect
192. #1749D47semidirect
220. C47#8337semidirect

Maximal subgroups

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

subgroupindex
C2×#156896732
#202431502
C2×#156888622
C47×#17492
#833747