Finite Group Databasebeta

← #26104864

Subgroups — #26104864

Order 12032.

570 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#20251689C211yesmaximal, characteristic
2#20243082C211yesmaximal, characteristic, complemented
3#20241213C212yesmaximal, complemented
4#20241208C212yesmaximal, complemented
5C47×#1797C211yesmaximal, characteristic, complemented
6C22×#10625329—21no—
7C2×#13094012C411yescharacteristic
8C2×#13094003C411yescharacteristic
9C2×#13093998C411yescharacteristic
10#15689029C2211yescharacteristic
11#15689025C2211yescharacteristic
12#15689014—21no—
13#15689014—21no—
14#15689002—21no—
15#15689002—21no—
16#15688989C411yescharacteristic
17#15688878—21no—
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C4⋊C8×C47—21no—
21C2×C22⋊C8×C47—21no—
22C47×#336C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C22×D188C2×C411yescharacteristic
25C22×C22⋊D47—21no—
26C2×#10625329—24no—
27C2×#10625329—42no—
28C2×#10625329—42no—
29C22×C47⋊C8—22no—
30#13094012—42no—
31#13094003—42no—
32#13093998—42no—
33C2×C4⋊Dic47C2×C411yescharacteristic
34C2×C4⋊Dic47C2×C411yescharacteristic
35C2×Dic47⋊C4—21no—
36C2×C22⋊Dic47—21no—
37C2×(C2×D47)⋊C4—21no—
38C2×(C2×D47)⋊C4—21no—
39C2×(C2×D47)⋊C4—21no—
40C2×(C2×D47)⋊C4—21no—
41C2×(C2×D47)⋊C4—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C22×C8×C47—22no—
50C2×C42×C47—42no—
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C2×C42×C47C2311yescharacteristic
55C4⋊C8×C47—42no—
56C22⋊C8×C47—42no—
57C24×C47C22⋊C411yescharacteristic
58C23×D47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C2×C22⋊D47—42no—
77C2×C22⋊D47—42no—
78C2×C22⋊D47—42no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C2×D188—22no—
83C2×D188—41no—
84C2×D188—41no—
85C2×Dic94—22no—
86C2×Dic94—41no—
87C2×Dic94—41no—
88C2×C8×C47—42no—
89C2×C8×C47—42no—
90C2×C8×C47—42no—
91C42×C47—22no—
92C42×C47—22no—
93C42×C47—41no—
94C42×C47—41no—
95C22⋊Dic47—42no—
96(C2×D47)⋊C4—42no—
97(C2×D47)⋊C4—42no—
98(C2×D47)⋊C4—42no—
99(C2×D47)⋊C4—41no—
100(C2×D47)⋊C4—41no—
101(C2×D47)⋊C4—41no—
102(C2×D47)⋊C4—41no—
103C4⋊Dic47—41no—
104C4⋊Dic47—41no—
105Dic47⋊C4—42no—
106#10625329—48no—
107C2×C47⋊C8—24no—
108C2×C47⋊C8—42no—
109C2×C47⋊C8—42no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47#10412yes—
117C23×C47C2×C22⋊C411yescharacteristic
118C22×D47—41no—
119C22×D47—41no—
120C22×D47—41no—
121C22×D47—41no—
122C22×D47—41no—
123C22×D47—41no—
124C22×D47—41no—
125C2×C4×C47—24no—
126C2×C4×C47—24no—
127C2×C4×C47—21no—
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—42no—
142C2×C4×C47—42no—
143C2×C4×C47#9912yes—
144C2×C4×C47#10012yes—
145C2×C4×C47#13112yes—
146C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C47#13711yescharacteristic
149C2×C4×C47#13711yescharacteristic
150C22⋊D47—84no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160D188—82no—
161Dic94—82no—
162C8×C47—82no—
163C47⋊C8—44no—
164#7710—471nomaximal, Sylow
165C22×C47—22no—
166C22×C47—22no—
167C22×C47—22no—
168C22×C47—41no—
169C22×C47—41no—
170C22×C47—41no—
171C22×C47—41no—
172C22×C47#40811yescharacteristic
173C22×C47C2×#9911yescharacteristic
174C22×C47C2×#10011yescharacteristic
175C22×C47#41711yescharacteristic
176C22×C47#41711yescharacteristic
177C22×C47C2×#10411yescharacteristic
178C22×C47#42111yescharacteristic
179C2×D47—81no—
180C2×D47—81no—
181C2×D47—81no—
182C2×D47—81no—
183C2×D47—81no—
184C2×D47—81no—
185C2×D47—81no—
186C4×C47—22no—
187C4×C47—22no—
188C4×C47—21no—
189C4×C47—22no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—21no—
193C4×C47—41no—
194Dic47—81no—
195Dic47—81no—
196Dic47—81no—
197#2850—471no—
198#1797D4711yescharacteristic, complemented
199#1728—471no—
200#1272—472no—
201#1270—472no—
202C2×C47—42no—
203C2×C47#147111yescharacteristic
204C2×C47#148311yescharacteristic
205C2×C47#149511yescharacteristic
206C2×C47#150411yescharacteristic
207C2×C47#151311yescharacteristic
208C2×C47#152711yescharacteristic
209C2×C47#156011yescharacteristic
210D47—161no—
211C22×M16—941no—
212C2×#124—471no—
213C2×C22⋊Q8—471no—
214C2×#121—471no—
215C22×C42C2×D4711yescharacteristic
216C2×C4⋊C8—21no—
217C2×C22⋊C8—21no—
218#391—471no—
219#390—471no—
220#388—941no—
221#388—941no—
222#386—941no—
223#386—941no—
224#382—471no—
225#336—472no—
226#336—941no—
227#336C2×D4712yes—
228C47#771011yescharacteristic, complemented, Sylow
229C22×Q8—471no—
230C22×D4—471no—
231C22×D4—941no—
232C23×C4C4×D4711yescharacteristic
233C23×C4D18811yescharacteristic
234C23×C4C22⋊D4711yescharacteristic
235C2×M16—944no—
236C2×M16—1882no—
237C2×M16—1882no—
238C22×C8—22no—
239C22×C8—942no—
240#124—1882no—
241C22⋊Q8—1882no—
242#121—1882no—
243C2×C4⋊C4—471no—
244C2×C4⋊C4—471no—
245C2×C4⋊C4—941no—
246C2×C22⋊C4—941no—
247C2×C22⋊C4—941no—
248C2×C22⋊C4—941no—
249C2×C22⋊C4—941no—
250C2×C22⋊C4—941no—
251C2×C22⋊C4—941no—
252C2×C42—42no—
253C2×C42C4×D4711yescharacteristic
254C2×C42D18811yescharacteristic
255C2×C42C22⋊D4711yescharacteristic
256C2×C42C22×D4711yescharacteristic
257C4⋊C8—42no—
258C22⋊C8—42no—
259#95—941no—
260#95—941no—
261#95—941no—
262#95—941no—
263C24—941no—
264C24(C2×D47)⋊C411yescharacteristic
265C2×Q8—942no—
266C2×Q8—1881no—
267C2×Q8—1881no—
268C2×D4—942no—
269C2×D4—1881no—
270C2×D4—1881no—
271C2×D4—1882no—
272C2×D4—1882no—
273C2×D4—1882no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4(C2×D47)⋊C411yescharacteristic
289C22×C4(C2×D47)⋊C411yescharacteristic
290C22×C4(C2×D47)⋊C411yescharacteristic
291C22×C4C2×C4×D4711yescharacteristic
292C22×C4C2×D18811yescharacteristic
293C22×C4C2×C22⋊D4711yescharacteristic
294M16—1888no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—944no—
299C2×C8—1882no—
300C2×C8—1882no—
301C4⋊C4—1881no—
302C4⋊C4—1881no—
303C4⋊C4—1882no—
304C22⋊C4—1882no—
305C22⋊C4—1882no—
306C22⋊C4—1882no—
307C22⋊C4—1882no—
308C22⋊C4—1881no—
309C22⋊C4—1881no—
310C22⋊C4—1881no—
311C22⋊C4—1881no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—1881no—
323C23—1881no—
324C23—1881no—
325C23—1881no—
326C23—1881no—
327C23—1881no—
328C23—1881no—
329C23#1309395012yes—
330C23C2×(C2×D47)⋊C411yescharacteristic
331Q8—3762no—
332D4—3762no—
333D4—3764no—
334C2×C4—24no—
335C2×C4—24no—
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—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—1881no—
353C2×C4—1881no—
354C2×C4—1881no—
355C2×C4—1881no—
356C2×C4—1881no—
357C2×C4—1881no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4—1881no—
361C2×C4#1309393812yes—
362C2×C4#1309394012yes—
363C2×C4#1309403012yes—
364C2×C4#1309405011yescharacteristic
365C2×C4#1309405211yescharacteristic
366C2×C4#1309405511yescharacteristic
367C2×C4#1309405711yescharacteristic
368C8—82no—
369C8—1884no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3761no—
378C22—3761no—
379C22—3761no—
380C22—3761no—
381C22—3761no—
382C22—3761no—
383C22—3761no—
384C22#1568908911yescharacteristic
385C22C2×#1309393811yescharacteristic
386C22C2×#1309394011yescharacteristic
387C22#1568912211yescharacteristic
388C22#1568912511yescharacteristic
389C22C2×#1309395011yescharacteristic
390C22#1568914111yescharacteristic
391C4—21no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—21no—
396C4—22no—
397C4—21no—
398C4—41no—
399C4—3761no—
400C4—3761no—
401C4—3761no—
402C2—42no—
403C2—7521no—
404C2#2024174711yescharacteristic
405C2#2024179911yescharacteristic
406C2#2024185411yescharacteristic
407C2#2024189311yescharacteristic
408C2#2024192911yescharacteristic
409C2#2024198711yescharacteristic
410C2#2024212111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C4
cosocleC2×C42×C47C23
hypercenter#1797D47
hypocenterC47#7710
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #20243082C2semidirect
3. #20241213C2semidirect
4. #20241208C2semidirect
5. C47×#1797C2semidirect
198. #1797D47semidirect
228. C47#7710semidirect

Maximal subgroups

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

subgroupindex
#202516892
#202430822
#202412132
#202412082
C47×#17972
#771047