Finite Group Databasebeta

← #26112354

Subgroups — #26112354

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#20249130C211yesmaximal, characteristic
2#20245991C211yesmaximal, characteristic, complemented
3#20241957C212yesmaximal
4#20241256C212yesmaximal, complemented
5C2×C47×#324C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8#15689053C412yescomplemented
9#15689045—22no—
10#15689040C412yescomplemented
11#15689009—24no—
12#15689002—21no—
13#15688986—21no—
14#15688983C2211yescharacteristic
15#15688983C2212yes—
16C4×C22⋊Dic47—21no—
17#15688858—24no—
18C22×C4⋊C4×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#324—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23#13094007—44no—
24#13093998—22no—
25#13093998—22no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—21no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—21no—
34C2×(C2×D47)⋊C4—22no—
35C2×C4×Dic47—22no—
36C47⋊#95—22no—
37C47⋊#95—24no—
38C47⋊#95—22no—
39C47⋊#95C2×C412yes—
40C47⋊#95D414yes—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C22×C8×C47—22no—
45C2×C4⋊C4×C47—22no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47C2×C412yes—
48C2×C4⋊C4×C47D412yes—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C24×C47C2×D411yescharacteristic
52C23×D47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47—81no—
60C22×C4×C47—81no—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22⋊C412yes—
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D412yes—
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47#4112yes—
67C2×C22⋊D47—41no—
68C2×C22⋊D47—41no—
69C2×C22⋊D47—42no—
70C2×C22⋊D47—42no—
71C22×Dic47—21no—
72C22×Dic47—22no—
73C22×Dic47—24no—
74C2×D188—22no—
75C2×D188—81no—
76C2×C8×C47—42no—
77C2×C8×C47—82no—
78C4⋊C4×C47—24no—
79C4⋊C4×C47—24no—
80C22⋊Dic47—42no—
81C22⋊Dic47—42no—
82C22⋊Dic47—42no—
83(C2×D47)⋊C4—44no—
84(C2×D47)⋊C4—42no—
85(C2×D47)⋊C4—82no—
86C4⋊Dic47—22no—
87C4⋊Dic47—82no—
88C4⋊Dic47—81no—
89Dic47⋊C4—44no—
90C4×Dic47—82no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47#12011yescharacteristic
95C23×C47#12311yescharacteristic
96C23×C47#13512yes—
97C23×C47AGL(1,ℤ/8)12yes—
98C22×D47—42no—
99C22×D47—42no—
100C22×D47—41no—
101C22×D47—41no—
102C22×D47—41no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—81no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12112yes—
118C22⋊D47—82no—
119C22⋊D47—82no—
120C2×Dic47—44no—
121C2×Dic47—42no—
122C2×Dic47—44no—
123C2×Dic47—42no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—42no—
128C2×Dic47—44no—
129D188—82no—
130C8×C47—82no—
131#9206—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47#38611yescharacteristic
141C22×C47#41611yescharacteristic
142C22×C47#41811yescharacteristic
143C22×C47#44912yes—
144C22×C47#48212yes—
145C2×D47—82no—
146C2×D47—82no—
147C2×D47—81no—
148C2×D47—81no—
149C2×D47—81no—
150C4×C47—22no—
151C4×C47—22no—
152C4×C47—42no—
153C4×C47—42no—
154Dic47—82no—
155Dic47—84no—
156Dic47—81no—
157#2618—471no—
158#2313—471no—
159#1519—472no—
160C2×#324D4711yescharacteristic, complemented
161#1289—472no—
162C2×C47—21no—
163C2×C47—22no—
164C2×C47—21no—
165C2×C47—22no—
166C2×C47#187511yescharacteristic
167C2×C47#191211yescharacteristic
168C2×C47#191311yescharacteristic
169D47—161no—
170C2×#123—941no—
171C2×#121—471no—
172C22×C4⋊C4C2×D4711yescharacteristic
173C2×C22⋊C8C2×D4712yes—
174#398—472no—
175#396—942no—
176#394—472no—
177#387—944no—
178#386—941no—
179#381—941no—
180#380—471no—
181#380—472no—
182C4×C22⋊C4—941no—
183#327—944no—
184#324—22no—
185C47#920611yescharacteristic, complemented, Sylow
186C22×D4—471no—
187C22×D4—941no—
188C23×C4D18811yescharacteristic
189C23×C4D18811yescharacteristic
190C23×C4C22×D4711yescharacteristic
191C22×C8—22no—
192#123—1884no—
193#121—942no—
194#121—942no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—471no—
198C2×C4⋊C4—942no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4—942no—
201C2×C4⋊C4C4×D4712yes—
202C2×C4⋊C4C22⋊D4712yes—
203C2×C22⋊C4—942no—
204C2×C22⋊C4—941no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4—942no—
207C2×C22⋊C4—941no—
208C2×C42—942no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—472no—
212#95—474no—
213#95—942no—
214#95—944no—
215#95—942no—
216C24—941no—
217C24C2×D18811yescharacteristic
218C2×D4—942no—
219C2×D4—3761no—
220C2×D4—1881no—
221C2×D4—1881no—
222C2×D4—1882no—
223C2×D4—1882no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—942no—
233C22×C4—941no—
234C22×C4—944no—
235C22×C4(C2×D47)⋊C412yes—
236C22×C4(C2×D47)⋊C412yes—
237C22×C4C2×C4×D4711yescharacteristic
238C22×C4D4×D4712yes—
239C22×C4#1062535212yes—
240C22×C4C2×C22⋊D4711yescharacteristic
241C2×C8—42no—
242C2×C8—82no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—942no—
246C4⋊C4—3762no—
247C4⋊C4—3761no—
248C4⋊C4—1884no—
249C22⋊C4—3762no—
250C22⋊C4—1884no—
251C22⋊C4—1882no—
252C22⋊C4—1882no—
253C22⋊C4—1882no—
254C22⋊C4—1882no—
255C42—3762no—
256C23—41no—
257C23—41no—
258C23—1882no—
259C23—1882no—
260C23—1881no—
261C23—1881no—
262C23—1881no—
263C23C2×(C2×D47)⋊C411yescharacteristic
264C23#1309399411yescharacteristic
265C23#1309400811yescharacteristic
266C23#1309404612yes—
267C23#1309405212yes—
268D4—3762no—
269D4—3762no—
270D4—3762no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—1884no—
285C2×C4—1882no—
286C2×C4—1884no—
287C2×C4—1882no—
288C2×C4—1881no—
289C2×C4—1884no—
290C2×C4—1881no—
291C2×C4—1882no—
292C2×C4—1881no—
293C2×C4(C22⋊D47)⋊C412yes—
294C2×C4#1309399612yes—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22—3762no—
305C22—3762no—
306C22—3761no—
307C22—3761no—
308C22—3761no—
309C22#1568900011yescharacteristic
310C22#1568912011yescharacteristic
311C22#1568912911yescharacteristic
312C22#1568927012yes—
313C22#1568944812yes—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—3762no—
319C4—3764no—
320C4—3761no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—7521no—
326C2#2024370811yescharacteristic
327C2#2024392311yescharacteristic
328C2#2024392811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC22×C47#386
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#9206
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20245991C2semidirect
4. #20241256C2semidirect
5. C2×C47×#324C2semidirect
8. #15689053C4semidirect
10. #15689040C4semidirect
160. C2×#324D47semidirect
185. C47#9206semidirect

Maximal subgroups

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

subgroupindex
#202491302
#202459912
#202419572
#202412562
C2×C47×#3242
#920647