Finite Group Databasebeta

← #26126937

Subgroups — #26126937

Order 12032.

572 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15689048C211yesmaximal, characteristic
2#20241715C211yesmaximal, characteristic
3#20241387C214yesmaximal, complemented
4C47×#1819C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic, complemented
6C22×C22⋊Dic47—21no—
7#15689087—21no—
8#15689048—21no—
9#15689048—21no—
10#15689048C412yescomplemented
11#15689048C2212yes—
12C2×C47⋊#95—21no—
13C2×C47⋊#95C411yescharacteristic, complemented
14#15688847—21no—
15#15688847C2212yes—
16C22×C22⋊C4×C47C2211yescharacteristic
17C47×#379—21no—
18C47×#342—21no—
19C47×#342C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C2×#10625329—41no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yes—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—44no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—41no—
32C47⋊#95—22no—
33C47⋊#95—21no—
34C47⋊#95—22no—
35C47⋊#95C2×C412yes—
36C25×C47D411yescharacteristic
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—41no—
41C2×C22⋊C4×C47—41no—
42C2×C22⋊C4×C47—41no—
43C2×C22⋊C4×C47—41no—
44C2×C22⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47C2311yescharacteristic
48C24×C47—22no—
49C24×C47—21no—
50C24×C47—21no—
51C24×C47—41no—
52C24×C47—42no—
53C24×C47—42no—
54C24×C47C22⋊C411yescharacteristic
55C24×C47C22⋊C411yescharacteristic
56C24×C47C2×D411yescharacteristic
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—22no—
69C22×Dic47—22no—
70C22×Dic47—42no—
71C22×Dic47—42no—
72C22×Dic47—42no—
73C22×Dic47—42no—
74C22×Dic47—42no—
75C22×Dic47—41no—
76C22×Dic47—41no—
77C22⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—42no—
82C22⋊C4×C47—42no—
83C22⋊C4×C47—82no—
84C22⋊C4×C47—82no—
85C22⋊Dic47—44no—
86C22⋊Dic47—44no—
87C4⋊Dic47—42no—
88C4⋊Dic47—42no—
89#10625329—82no—
90C2×C47⋊C8—42no—
91C23×C47—22no—
92C23×C47—22no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—48no—
107C23×C47—42no—
108C23×C47—42no—
109C23×C47—81no—
110C23×C47—81no—
111C23×C47—84no—
112C23×C47#9911yescharacteristic
113C23×C47#9911yescharacteristic
114C23×C47C2×C22⋊C411yescharacteristic
115C23×C47AGL(1,ℤ/8)12yes—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—81no—
128C2×C4×C47—82no—
129C2×C4×C47—82no—
130C2×C4×C47—82no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—84no—
138C2×Dic47—82no—
139C2×Dic47—82no—
140C47⋊C8—82no—
141#11797—471nomaximal, Sylow
142C22×C47—24no—
143C22×C47—24no—
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—22no—
151C22×C47—42no—
152C22×C47—42no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—81no—
159C22×C47—84no—
160C22×C47—84no—
161C22×C47—81no—
162C22×C47#35212yes—
163C22×C47#35412yes—
164C22×C47C2×#9911yescharacteristic
165C22×C47#41811yescharacteristic
166C22×C47#42111yescharacteristic
167C4×C47—41no—
168C4×C47—41no—
169C4×C47—41no—
170C4×C47—41no—
171C4×C47—162no—
172Dic47—82no—
173Dic47—82no—
174C2×#397—471no—
175#1819D4711yescharacteristic
176#1462—471no—
177#1344—474no—
178C2×C47—21no—
179C2×C47—22no—
180C2×C47—22no—
181C2×C47—21no—
182C2×C47—42no—
183C2×C47—81no—
184C2×C47#151011yescharacteristic
185C2×C47C2×#35211yescharacteristic
186C2×C47C2×#35411yescharacteristic
187C22×C4⋊C4—471no—
188C22×C22⋊C4—941no—
189C22×C22⋊C4C2×D4711yescharacteristic
190#407—941no—
191#397—472no—
192#397—472no—
193#397—941no—
194#397—941no—
195#379—21no—
196C2×#95—471no—
197C2×#95—941no—
198#342—21no—
199#342C2×D4712yes—
200#323—472no—
201#323—941no—
202C47#1179711yescharacteristic, complemented, Sylow
203C25C22⋊D4711yescharacteristic
204C23×C4—941no—
205C23×C4—941no—
206C23×C4C4×D4711yescharacteristic
207C23×C4D18811yescharacteristic
208C2×M16—1881no—
209C2×C4⋊C4—472no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4—1881no—
212C2×C4⋊C4—1881no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—942no—
219C2×C22⋊C4—942no—
220C2×C22⋊C4—1884no—
221C2×C22⋊C4C4×D4711yescharacteristic
222C2×C22⋊C4D18811yescharacteristic
223C2×C22⋊C4C22⋊D4711yescharacteristic
224C2×C22⋊C4C22×D4711yescharacteristic
225C22⋊C8—942no—
226C22⋊C8—1881no—
227#95—472no—
228#95—941no—
229#95—942no—
230#95—942no—
231C24—22no—
232C24—21no—
233C24—21no—
234C24—41no—
235C24—42no—
236C24—42no—
237C24(C2×D47)⋊C411yescharacteristic
238C24(C2×D47)⋊C411yescharacteristic
239C24C2×C22⋊D4711yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—942no—
248C22×C4—942no—
249C22×C4—1882no—
250C22×C4—1882no—
251C22×C4—1882no—
252C22×C4—1882no—
253C22×C4—1882no—
254C22×C4—1881no—
255C22×C4—1881no—
256C22×C4(C2×D47)⋊C411yescharacteristic
257C22×C4(C2×D47)⋊C411yescharacteristic
258C22×C4C2×C4×D4711yescharacteristic
259C22×C4C2×D18811yescharacteristic
260M16—3762no—
261C2×C8—1882no—
262C4⋊C4—1882no—
263C4⋊C4—1882no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—1884no—
273C22⋊C4—1884no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—42no—
284C23—42no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—48no—
290C23—42no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23—84no—
295C23#1309393811yescharacteristic
296C23#1309393811yescharacteristic
297C23C2×(C2×D47)⋊C411yescharacteristic
298C23#1309404912yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—3764no—
315C2×C4—3762no—
316C2×C4—3762no—
317C2×C4—1882no—
318C2×C4—1882no—
319C2×C4—1882no—
320C2×C4—1882no—
321C2×C4—1882no—
322C2×C4—1882no—
323C8—3762no—
324C22—24no—
325C22—24no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—84no—
342C22—84no—
343C22—81no—
344C22#1568891212yes—
345C22#1568891712yes—
346C22C2×#1309393811yescharacteristic
347C22#1568912711yescharacteristic
348C22#1568914111yescharacteristic
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—162no—
354C4—3762no—
355C4—3762no—
356C2—21no—
357C2—21no—
358C2—22no—
359C2—22no—
360C2—42no—
361C2—81no—
362C2#2024191611yescharacteristic
363C2C2×#1568891211yescharacteristic
364C2C2×#1568891711yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1819C2
socleC22×C47C2×#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1819D47
hypocenterC47#11797
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #20241387C2semidirect
5. C22×C4⋊Dic47C4semidirect
10. #15689048C4semidirect
13. C2×C47⋊#95C4semidirect
202. C47#11797semidirect

Maximal subgroups

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

subgroupindex
C2×#156890482
#202417152
#202413872
C47×#18192
#1179747