Finite Group Databasebeta

← #26114022

Subgroups — #26114022

Order 12032.

610 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 103 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#20247855C211yesmaximal, characteristic
2#20243468C211yesmaximal, characteristic, complemented
3#20241916C212yesmaximal
4#20241240C212yesmaximal, complemented
5C47×#2313C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13094007—21no—
8C2×#13093998C2211yescharacteristic
9#15689087C2212yes—
10#15689058—22no—
11#15689024C412yescomplemented
12#15688992C412yescomplemented
13C4×C22⋊Dic47C2211yescharacteristic
14#15688892—22no—
15#15688856—24no—
16C22×C4⋊C4×C47C2211yescharacteristic
17C47×#387—24no—
18C47×#381C2212yes—
19C47×#380—21no—
20C4×C22⋊C4×C47—21no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23C2×#10625329—22no—
24#13094007—44no—
25#13094007—44no—
26#13093998—22no—
27#13093998—22no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—22no—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—21no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—21no—
35C2×C4×Dic47C2×C412yes—
36C22⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38C47⋊#95—21no—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C23×C4×C47C2311yescharacteristic
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47C2×C412yes—
46C2×C4⋊C4×C47D412yes—
47C2×C22⋊C4×C47—22no—
48C2×C42×C47—22no—
49#95×C47—22no—
50#95×C47—22no—
51#95×C47—22no—
52#95×C47D414yes—
53C24×C47C2×D411yescharacteristic
54C23×D47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—24no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47C22⋊C412yes—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47C2×D412yes—
68C22×C4×C47#4112yes—
69C2×C22⋊D47—42no—
70C2×C22⋊D47—42no—
71C2×C22⋊D47—41no—
72C2×C22⋊D47—41no—
73C22×Dic47—22no—
74C22×Dic47—21no—
75C2×D188—22no—
76C2×D188—42no—
77C4⋊C4×C47—24no—
78C4⋊C4×C47—24no—
79C4⋊C4×C47—82no—
80C22⋊C4×C47—44no—
81C42×C47—82no—
82C22⋊Dic47—24no—
83(C2×D47)⋊C4—44no—
84(C2×D47)⋊C4—44no—
85(C2×D47)⋊C4—42no—
86C4⋊Dic47—22no—
87Dic47⋊C4—44no—
88Dic47⋊C4—44no—
89C4×Dic47—42no—
90#10625329—44no—
91C2×C47⋊C8—42no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—41no—
95C23×C47C2×C22⋊C411yescharacteristic
96C23×C47#12011yescharacteristic
97C23×C47#12311yescharacteristic
98C22×D47—41no—
99C22×D47—41no—
100C22×D47—42no—
101C22×D47—41no—
102C22×D47—42no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—44no—
114C2×C4×C47—44no—
115C2×C4×C47—44no—
116C2×C4×C47—81no—
117C2×C4×C47—81no—
118C2×C4×C47C4×D412yes—
119C2×C4×C47#12112yes—
120C22⋊D47—82no—
121C22⋊D47—82no—
122C2×Dic47—24no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128D188—82no—
129C47⋊C8—82no—
130#9464—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—44no—
139C22×C47#38611yescharacteristic
140C22×C47#41012yes—
141C22×C47#45312yes—
142C2×D47—82no—
143C2×D47—82no—
144C2×D47—81no—
145C2×D47—81no—
146C2×D47—81no—
147C4×C47—22no—
148C4×C47—22no—
149C4×C47—42no—
150C4×C47—42no—
151C4×C47—84no—
152Dic47—44no—
153Dic47—81no—
154#2501—471no—
155#2313D4711yescharacteristic, complemented
156#1821—471no—
157#1510—472no—
158#1284—472no—
159C2×C47—22no—
160C2×C47—22no—
161C2×C47—41no—
162C2×C47#188711yescharacteristic
163C2×C47#189911yescharacteristic
164C2×C47#191211yescharacteristic
165D47—161no—
166C2×#123—941no—
167C2×#123—941no—
168C2×#121—471no—
169C22×C4⋊C4C2×D4711yescharacteristic
170#407—472no—
171#399—942no—
172#390—472no—
173#387—24no—
174#384—472no—
175#381C2×D4712yes—
176#380—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—471no—
179#343—942no—
180#327—944no—
181C47#946411yescharacteristic, complemented, Sylow
182C22×D4—471no—
183C22×D4—941no—
184C23×C4C22⋊D4711yescharacteristic
185C23×C4C22⋊D4711yescharacteristic
186C23×C4C22×D4711yescharacteristic
187C2×M16—942no—
188#123—1884no—
189#123—1884no—
190#121—942no—
191#121—942no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—471no—
196C2×C4⋊C4—942no—
197C2×C4⋊C4—942no—
198C2×C4⋊C4C4×D4712yes—
199C2×C4⋊C4D18812yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—941no—
202C2×C22⋊C4—942no—
203C2×C22⋊C4—942no—
204C2×C22⋊C4—941no—
205C2×C42—22no—
206C2×C42—472no—
207C22⋊C8—942no—
208C22⋊C8—942no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—941no—
213#95C22⋊D4714yes—
214C24—941no—
215C24C2×C22⋊D4711yescharacteristic
216C2×D4—942no—
217C2×D4—1882no—
218C2×D4—1882no—
219C2×D4—1882no—
220C2×D4—1881no—
221C2×D4—1881no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—942no—
231C22×C4—941no—
232C22×C4(C2×D47)⋊C412yes—
233C22×C4(C2×D47)⋊C412yes—
234C22×C4C2×C4×D4711yescharacteristic
235C22×C4C2×D18811yescharacteristic
236C22×C4#1062535012yes—
237C22×C4C2×C22⋊D4712yes—
238M16—1884no—
239C2×C8—1882no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—82no—
243C4⋊C4—942no—
244C4⋊C4—1884no—
245C4⋊C4—1884no—
246C22⋊C4—44no—
247C22⋊C4—944no—
248C22⋊C4—1884no—
249C22⋊C4—1884no—
250C22⋊C4—1882no—
251C42—82no—
252C42—1882no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—1881no—
257C23—1881no—
258C23—1882no—
259C23—1881no—
260C23—1882no—
261C23C2×(C2×D47)⋊C411yescharacteristic
262C23#1309399311yescharacteristic
263C23#1309400711yescharacteristic
264D4—3762no—
265D4—3762no—
266D4—3762no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—944no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1882no—
287C2×C4—1882no—
288C2×C4C4×C22⋊D4712yes—
289C2×C4#1309399812yes—
290C8—3762no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—3762no—
300C22—3762no—
301C22—3761no—
302C22—3761no—
303C22—3761no—
304C22#1568900211yescharacteristic
305C22#1568909612yes—
306C22#1568929112yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—84no—
312C4—3761no—
313C4—1884no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—7521no—
318C2#2024378611yescharacteristic
319C2#2024384611yescharacteristic
320C2#2024392511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2313C2
socleC22×C47#386
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#2313D47
hypocenterC47#9464
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
2. #20243468C2semidirect
4. #20241240C2semidirect
5. C47×#2313C2semidirect
11. #15689024C4semidirect
12. #15688992C4semidirect
155. #2313D47semidirect
181. C47#9464semidirect

Maximal subgroups

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

subgroupindex
#202478552
#202434682
#202419162
#202412402
C47×#23132
#946447