Finite Group Databasebeta

← #26111783

Subgroups — #26111783

Order 12032.

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

normal subgroups only

This group has 630 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
1#20246110C211yesmaximal, characteristic
2#20241780C212yesmaximal, complemented
3#20241709C211yesmaximal, characteristic, complemented
4#20241216C212yesmaximal, complemented
5C47×#2972C211yesmaximal, characteristic, complemented
6C2×#13094020C411yescharacteristic
7C2×C4×D188C2211yescharacteristic
8#15689176C2212yes—
9#15689026C411yescharacteristic
10#15689024C411yescharacteristic
11#15689002—22no—
12#15688999—22no—
13#15688988C411yescharacteristic
14#15688969C2211yescharacteristic
15#15688852—24no—
16#15688847—22no—
17C2×C4×D4×C47C2211yescharacteristic
18C8×D4×C47—24no—
19C47×#431C2212yes—
20C2×C22⋊C8×C47—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—22no—
23C22×C4×D47—21no—
24#13094020—22no—
25#13094020C814yescomplemented
26C4×D188—22no—
27C4×D188—22no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C22⋊Dic47—22no—
30C2×(C2×D47)⋊C4—21no—
31C2×(C2×D47)⋊C4—22no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47C2×C411yescharacteristic
34C4⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)—22no—
36C4×C47⋊C8—22no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C22×D4×C47C2×C411yescharacteristic
41C23×C4×C47D412yes—
42C22×C8×C47—24no—
43C4×D4×C47—22no—
44C4×D4×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D412yes—
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47—22no—
49C22⋊C8×C47—24no—
50C22⋊C8×C47—22no—
51C4×C8×C47D414yescomplemented
52C24×C47C22⋊C412yes—
53C23×D47—21no—
54C2×D4×C47—22no—
55C2×D4×C47—22no—
56C2×D4×C47C2×C812yes—
57C2×D4×C47M1612yes—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—82no—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C2×C22⋊D47—28no—
66C2×C22⋊D47—82no—
67C2×C22⋊D47—82no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C2×D188—22no—
72C2×D188—22no—
73C2×D188—21no—
74C2×D188C2×C812yescomplemented
75C2×C4×D47—42no—
76C2×C4×D47—82no—
77C2×C8×C47—24no—
78C2×C8×C47—84no—
79C4⋊C4×C47—22no—
80C22⋊C4×C47—24no—
81C42×C47C2×D412yes—
82C42×C47#4112yes—
83C22⋊Dic47—82no—
84(C2×D47)⋊C4—41no—
85(C2×D47)⋊C4—41no—
86(C2×D47)⋊C4—82no—
87C4⋊Dic47—22no—
88C4×Dic47—21no—
89C4×Dic47—22no—
90C4×Dic47C2×C812yescomplemented
91C2×C47⋊C8—42no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—42no—
95C23×C47—42no—
96C23×C47C22⋊C814yes—
97C23×C47C2×C22⋊C411yescharacteristic
98C22×D47—22no—
99C22×D47—42no—
100C22×D47—42no—
101C22×D47—41no—
102C22×D47—41no—
103D4×C47—24no—
104D4×C47—24no—
105C2×C4×C47—22no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—44no—
109C2×C4×C47—42no—
110C2×C4×C47—82no—
111C2×C4×C47C4×D412yes—
112C2×C4×C47#12011yescharacteristic
113C2×C4×C47#12112yes—
114C2×C4×C47#12311yescharacteristic
115C2×C4×C47C22×C811yescharacteristic
116C2×C4×C47C2×M1611yescharacteristic
117C2×C4×C47#13112yes—
118C22⋊D47—88no—
119C2×Dic47—22no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—42no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127D188—24no—
128D188—42no—
129C4×D47—84no—
130C8×C47—44no—
131C47⋊C8—82no—
132#9101—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—24no—
136C22×C47—44no—
137C22×C47—42no—
138C22×C47—42no—
139C22×C47—82no—
140C22×C47#38611yescharacteristic
141C22×C47C2×C22⋊C811yescharacteristic
142C22×C47#40811yescharacteristic
143C2×D47—44no—
144C2×D47—42no—
145C2×D47—82no—
146C2×D47—81no—
147C2×D47—81no—
148C4×C47—22no—
149C4×C47—21no—
150C4×C47—21no—
151C4×C47—42no—
152C4×C47C8×D412yes—
153C4×C47#43512yes—
154Dic47—42no—
155Dic47—81no—
156Dic47—81no—
157#2972D4711yescharacteristic, complemented
158#2328—471no—
159#1478—472no—
160#1460—471no—
161#1274—472no—
162C2×C47—21no—
163C2×C47—21no—
164C2×C47—44no—
165C2×C47#186911yescharacteristic
166C2×C47#191011yescharacteristic
167C2×C47#191111yescharacteristic
168D47—82no—
169C2×#127—471no—
170C2×C4×D4—471no—
171C2×C4×D4C2×D4711yescharacteristic
172C8×D4—24no—
173#431—472no—
174#431C2×D4712yes—
175C2×C22⋊C8—22no—
176#390—471no—
177#390—471no—
178#386—942no—
179#386—942no—
180#382—471no—
181#376—471no—
182D4⋊C8—944no—
183#323—942no—
184C47#910111yescharacteristic, complemented, Sylow
185C22×D4—471no—
186C22×D4—942no—
187C22×D4C4×D4711yescharacteristic
188C23×C4—941no—
189C23×C4C22⋊D4712yes—
190C22×C8—24no—
191#127—474no—
192#127—942no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—942no—
196C4×D4—942no—
197C2×C4⋊C4—471no—
198C2×C4⋊C4C4×D4711yescharacteristic
199C2×C22⋊C4—942no—
200C2×C22⋊C4—942no—
201C2×C22⋊C4—941no—
202C2×C22⋊C4D18812yes—
203C2×C42—471no—
204C2×C42—471no—
205C2×C42C22×D4711yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—942no—
208C22⋊C8—24no—
209C22⋊C8—22no—
210C22⋊C8—942no—
211C4×C8—942no—
212C4×C8C22⋊D4714yescomplemented
213#95—942no—
214#95—941no—
215#95—941no—
216C24—941no—
217C24(C2×D47)⋊C412yes—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—472no—
221C2×D4—948no—
222C2×D4—942no—
223C2×D4—941no—
224C2×D4—942no—
225C2×D4—3762no—
226C2×D4—3762no—
227C2×D4C8×D4712yes—
228C2×D4#1062532412yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—941no—
233C22×C4—941no—
234C22×C4—941no—
235C22×C4—3762no—
236C22×C4—1882no—
237C22×C4(C2×D47)⋊C412yes—
238C22×C4C2×C4×D4711yescharacteristic
239C22×C4C2×D18811yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241C2×C8—24no—
242C2×C8—84no—
243C2×C8—1882no—
244C4⋊C4—22no—
245C4⋊C4—942no—
246C22⋊C4—24no—
247C22⋊C4—3762no—
248C22⋊C4—3762no—
249C22⋊C4—1881no—
250C22⋊C4—1881no—
251C42—472no—
252C42—941no—
253C42—942no—
254C42#1062535012yes—
255C42C2×C22⋊D4712yes—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—942no—
261C23—1882no—
262C23—1882no—
263C23—1881no—
264C23—1881no—
265C23(C2×D47)⋊C814yes—
266C23C2×(C2×D47)⋊C411yescharacteristic
267D4—24no—
268D4—24no—
269D4—944no—
270D4—3768no—
271D4—1882no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—942no—
279C2×C4—3764no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1882no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4C4×C22⋊D4712yes—
288C2×C4#1309399311yescharacteristic
289C2×C4#1309399812yes—
290C2×C4#1309400711yescharacteristic
291C2×C4C2×C8×D4711yescharacteristic
292C2×C4C2×#1062532411yescharacteristic
293C2×C4#1309403012yes—
294C8—44no—
295C8—3762no—
296C22—21no—
297C22—21no—
298C22—24no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—3762no—
304C22—3761no—
305C22—3761no—
306C22—1884no—
307C22—1882no—
308C22#1568900211yescharacteristic
309C22C2×(C2×D47)⋊C811yescharacteristic
310C22#1568908911yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—3761no—
316C4—3761no—
317C4—1882no—
318C4C8×C22⋊D4712yes—
319C4#1568919412yes—
320C2—21no—
321C2—21no—
322C2—44no—
323C2—3762no—
324C2#2024368011yescharacteristic
325C2#2024391511yescharacteristic
326C2#2024392011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#2972C2
socleC22×C47#386
prime radicalC2×#13094020C4
cosocleC2×C42×C47C23
hypercenter#2972D47
hypocenterC47#9101
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20241780C2semidirect
3. #20241709C2semidirect
4. #20241216C2semidirect
5. C47×#2972C2semidirect
25. #13094020C8semidirect
51. C4×C8×C47D4semidirect
74. C2×D188C2×C8semidirect
90. C4×Dic47C2×C8semidirect
157. #2972D47semidirect
184. C47#9101semidirect
212. C4×C8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202461102
#202417802
#202417092
#202412162
C47×#29722
#910147