Finite Group Databasebeta

← #26112314

Subgroups — #26112314

Order 12032.

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

normal subgroups only

This group has 578 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#20252026C211yesmaximal, characteristic, complemented
2#20249138C211yesmaximal, characteristic
3#20242055C212yesmaximal, complemented
4#20241247C212yesmaximal, complemented
5C47×#1458C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7C2×#13093998—21no—
8#15689184—24no—
9#15689147C2212yes—
10#15689087—21no—
11C4×#10625329—21no—
12#15689040—21no—
13#15689040C412yes—
14#15689032—21no—
15#15689032C412yes—
16#15689025—21no—
17#15688989C2211yescharacteristic
18#15688866—24no—
19C2×#117×C47C2211yescharacteristic
20C47×#431C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C22×D188—21no—
24C22×D188—21no—
25C22×D188C2×C412yes—
26C22×C22⋊D47—22no—
27C2×#10625329—22no—
28#13094021—22no—
29#13094021—22no—
30#13093998—44no—
31C2×C4⋊Dic47—21no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—22no—
35C4⋊(C47⋊C8)—22no—
36C4⋊(C47⋊C8)D414yescomplemented
37C22⋊(C47⋊C8)—24no—
38#13093935—22no—
39C4×C47⋊C8—22no—
40C47⋊#95—21no—
41C47⋊#95C2×C412yes—
42C23×C4×C47D411yescharacteristic
43#117×C47—22no—
44#117×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47C2311yescharacteristic
51C4⋊C8×C47—22no—
52C22⋊C8×C47—22no—
53C4×C8×C47—22no—
54C24×C47C22⋊C411yescharacteristic
55C23×D47—22no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C2×C22⋊D47—44no—
67C2×C22⋊D47—82no—
68C2×C22⋊D47—82no—
69C22×Dic47—22no—
70C2×D188—24no—
71C2×D188—41no—
72C2×D188—42no—
73C2×D188—41no—
74C2×D188—42no—
75C2×D188—42no—
76C2×D188—82no—
77C2×D188—82no—
78C2×C8×C47—42no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—22no—
81C22⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47C2×D412yes—
86C42×C47#4112yes—
87(C2×D47)⋊C4—44no—
88(C2×D47)⋊C4—82no—
89(C2×D47)⋊C4—82no—
90C4⋊Dic47—42no—
91#10625329—42no—
92#10625329—42no—
93C2×C47⋊C8—24no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47C2×C22⋊C411yescharacteristic
100C22×D47—42no—
101C22×D47—42no—
102C22×D47—42no—
103C22×D47—42no—
104C22×D47—42no—
105C22×D47—42no—
106C22×D47—42no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47C4×D412yes—
119C2×C4×C47#12011yescharacteristic
120C2×C4×C47#12112yes—
121C2×C4×C47#12311yescharacteristic
122C2×C4×C47#13512yes—
123C2×C4×C47AGL(1,ℤ/8)12yes—
124C22⋊D47—88no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128D188—84no—
129D188—84no—
130D188—82no—
131D188—82no—
132C8×C47—82no—
133C47⋊C8—44no—
134#9198—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47#38611yescharacteristic
143C22×C47#41611yescharacteristic
144C22×C47#41811yescharacteristic
145C2×D47—82no—
146C2×D47—82no—
147C2×D47—82no—
148C2×D47—82no—
149C2×D47—82no—
150C2×D47—82no—
151C2×D47—82no—
152C4×C47—22no—
153C4×C47—22no—
154C4×C47—41no—
155C4×C47—41no—
156C4×C47—41no—
157C4×C47#44912yes—
158C4×C47#48212yes—
159Dic47—82no—
160#2921—471no—
161#2619—471no—
162#1545—472no—
163#1458D4711yescharacteristic, complemented
164#1286—472no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47—81no—
168C2×C47#187511yescharacteristic
169C2×C47#191011yescharacteristic
170C2×C47#191411yescharacteristic
171D47—162no—
172C2×#127—471no—
173C2×#121—941no—
174C2×#117C2×D4711yescharacteristic
175#433—944no—
176#431C2×D4712yes—
177#423—472no—
178#407—941no—
179C4×M16—941no—
180#394—472no—
181#394—941no—
182#392—472no—
183#392—941no—
184#390—941no—
185#382—471no—
186#331—944no—
187#324—21no—
188#323—21no—
189C47#919811yescharacteristic, complemented, Sylow
190C22×D4—472no—
191C22×D4—942no—
192C22×D4—941no—
193C22×D4—941no—
194C23×C4D18811yescharacteristic
195C2×M16—942no—
196#127—942no—
197#127—942no—
198#121—1884no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—941no—
202C2×C4⋊C4C4×D4711yescharacteristic
203C2×C4⋊C4C22⋊D4711yescharacteristic
204C2×C22⋊C4—942no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4—942no—
207C2×C22⋊C4C4×D4711yescharacteristic
208C2×C22⋊C4C22⋊D4711yescharacteristic
209C2×C42D18811yescharacteristic
210C2×C42C22×D4711yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—474no—
213C4⋊C8—942no—
214C22⋊C8—22no—
215C22⋊C8—944no—
216#97—942no—
217C4×C8—22no—
218C4×C8—942no—
219#95—472no—
220#95—941no—
221C24—942no—
222C24(C2×D47)⋊C411yescharacteristic
223C2×D4—944no—
224C2×D4—3762no—
225C2×D4—3762no—
226C2×D4—3762no—
227C2×D4—3762no—
228C2×D4—1884no—
229C2×D4—1881no—
230C2×D4—1882no—
231C2×D4—1881no—
232C2×D4—1882no—
233C2×D4—1882no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—942no—
239C22×C4(C2×D47)⋊C411yescharacteristic
240C22×C4(C2×D47)⋊C411yescharacteristic
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4C2×D18811yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245M16—1882no—
246M16—1882no—
247C2×C8—42no—
248C2×C8—944no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—1882no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—3762no—
255C22⋊C4—3762no—
256C22⋊C4—1884no—
257C42—21no—
258C42—21no—
259C42D4×D4712yes—
260C42#1062535212yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—1882no—
267C23—1882no—
268C23—1882no—
269C23—1882no—
270C23—1882no—
271C23—1882no—
272C23—1882no—
273C23C2×(C2×D47)⋊C411yescharacteristic
274D4—3764no—
275D4—3764no—
276D4—3768no—
277D4—3762no—
278D4—3762no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—1882no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4(C22⋊D47)⋊C412yes—
294C2×C4#1309399411yescharacteristic
295C2×C4#1309399612yes—
296C2×C4#1309400811yescharacteristic
297C2×C4#1309404612yes—
298C2×C4#1309405212yes—
299C8—82no—
300C8—1884no—
301C22—21no—
302C22—21no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—3762no—
309C22—3762no—
310C22—3762no—
311C22—3762no—
312C22—3762no—
313C22—3762no—
314C22—3762no—
315C22#1568900011yescharacteristic
316C22#1568912011yescharacteristic
317C22#1568912911yescharacteristic
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—3762no—
324C4#1568927012yes—
325C4#1568944812yes—
326C2—21no—
327C2—21no—
328C2—81no—
329C2—7522no—
330C2#2024370811yescharacteristic
331C2#2024391311yescharacteristic
332C2#2024393311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#386
prime radical#20249138C2
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#9198
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
1. #20252026C2semidirect
3. #20242055C2semidirect
4. #20241247C2semidirect
5. C47×#1458C2semidirect
36. C4⋊(C47⋊C8)D4semidirect
163. #1458D47semidirect
189. C47#9198semidirect

Maximal subgroups

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

subgroupindex
#202520262
#202491382
#202420552
#202412472
C47×#14582
#919847