Finite Group Databasebeta

← #26120619

Subgroups — #26120619

Order 12032.

500 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 79 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#20254139C211yesmaximal, characteristic
2#20243851C212yesmaximal, complemented
3#20241903C212yesmaximal, complemented
4#20241265C211yesmaximal, characteristic, complemented
5C47×#1851C211yesmaximal, characteristic, complemented
6C2×#13093998C2212yes—
7#15689437—22no—
8#15689359—21no—
9#15689359—21no—
10#15689259—22no—
11C2×#13093940—22no—
12#15689087C2212yes—
13#15688891—22no—
14#15688856—22no—
15#15688856—22no—
16C22×M16×C47C2211yescharacteristic
17C2×C4⋊C8×C47—21no—
18C2×C22⋊C8×C47C2211yescharacteristic
19C4×C22⋊C4×C47C2211yescharacteristic
20C47×#336—21no—
21C22×D188D412yes—
22C22×C22⋊D47—22no—
23C2×D376—21no—
24C2×D376—21no—
25C2×#10625329—22no—
26#13093998—22no—
27#13093998—22no—
28#13093998—22no—
29#13093998—22no—
30C2×C4⋊Dic47D412yes—
31C2×(C2×D47)⋊C4—22no—
32#13093958—21no—
33#13093958—21no—
34#13093946—22no—
35#13093946—22no—
36#13093940—44no—
37C22⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)—22no—
39C23×C4×C47C2311yescharacteristic
40C2×M16×C47—22no—
41C2×M16×C47—42no—
42C22×C8×C47—21no—
43C22×C8×C47D411yescharacteristic
44C22×C8×C47D411yescharacteristic
45C2×C22⋊C4×C47—21no—
46C2×C42×C47—21no—
47C4⋊C8×C47—42no—
48C22⋊C8×C47—22no—
49#95×C47D411yescharacteristic
50#95×C47D411yescharacteristic
51C24×C47#4111yescharacteristic
52C23×D47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47C2×D412yes—
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47#4112yes—
61C2×C22⋊D47—42no—
62C2×C22⋊D47—42no—
63C2×C22⋊D47—82no—
64C2×C22⋊D47—82no—
65C22×Dic47—22no—
66C2×D188—22no—
67C2×D188—22no—
68C2×D188—42no—
69C2×D188—42no—
70M16×C47—24no—
71M16×C47—42no—
72C2×C8×C47—22no—
73C2×C8×C47—21no—
74C2×C8×C47—21no—
75C2×C8×C47—41no—
76C2×C8×C47—41no—
77C2×C8×C47—41no—
78C2×C8×C47—41no—
79C22⋊C4×C47—42no—
80C42×C47—41no—
81C42×C47—41no—
82(C2×D47)⋊C4—42no—
83(C2×D47)⋊C4—42no—
84C4⋊Dic47—22no—
85C4⋊Dic47—22no—
86#10625329—42no—
87#10625329—42no—
88C2×C47⋊C8—42no—
89D376—81no—
90D376—81no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47#12111yescharacteristic
96C23×C47#12312yes—
97C22×D47—42no—
98C22×D47—44no—
99C22×D47—82no—
100C22×D47—82no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—22no—
104C2×C4×C47—21no—
105C2×C4×C47—22no—
106C2×C4×C47—41no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47#12011yescharacteristic
114C2×C4×C47#12112yes—
115C2×C4×C47#12411yescharacteristic
116C22⋊D47—84no—
117C22⋊D47—82no—
118C22⋊D47—82no—
119C2×Dic47—42no—
120C2×Dic47—82no—
121D188—42no—
122D188—42no—
123D188—82no—
124C8×C47—41no—
125C8×C47—42no—
126C8×C47—41no—
127C47⋊C8—82no—
128#10557—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—22no—
134C22×C47—81no—
135C22×C47—81no—
136C22×C47#39411yescharacteristic
137C22×C47#47211yescharacteristic
138C22×C47#47211yescharacteristic
139C2×D47—82no—
140C2×D47—82no—
141C2×D47—82no—
142C2×D47—164no—
143C4×C47—22no—
144C4×C47—21no—
145C4×C47—42no—
146C4×C47—81no—
147C4×C47B(4,𝔽2)11yescharacteristic
148C4×C47#45811yescharacteristic
149Dic47—82no—
150#3186—471no—
151#1900—472no—
152#1851D4711yescharacteristic, complemented
153#1507—472no—
154#1293—471no—
155C2×C47—21no—
156C2×C47—21no—
157C2×C47—42no—
158C2×C47#204411yescharacteristic
159C2×C47#204511yescharacteristic
160C2×C47#205011yescharacteristic
161D47—162no—
162C22×M16C2×D4711yescharacteristic
163C2×#121—472no—
164#480—942no—
165#466—941no—
166#466—941no—
167#447—942no—
168C2×C4⋊C8—21no—
169C2×#100—942no—
170#407—472no—
171C2×C22⋊C8C2×D4711yescharacteristic
172C4×C22⋊C4C2×D4711yescharacteristic
173#343—942no—
174#336—21no—
175#327—942no—
176#327—942no—
177C47#1055711yescharacteristic, complemented, Sylow
178C22×D4—472no—
179C22×D4—942no—
180C23×C4C22×D4711yescharacteristic
181C2×D8—941no—
182C2×D8—941no—
183C2×M16—22no—
184C2×M16—42no—
185C2×M16—942no—
186C22×C8—21no—
187C22×C8C22⋊D4711yescharacteristic
188C22×C8C22⋊D4711yescharacteristic
189#121—942no—
190#121—942no—
191#121—942no—
192#121—942no—
193C2×C4⋊C4—472no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—942no—
196C2×C42—21no—
197#107—941no—
198#107—941no—
199C4⋊C8—42no—
200#102—942no—
201#102—942no—
202#100—1884no—
203C22⋊C8—22no—
204C22⋊C8—942no—
205C22⋊C8—942no—
206#95D18811yescharacteristic
207#95D18811yescharacteristic
208C24—942no—
209C24#1062535011yescharacteristic
210C2×D4—942no—
211C2×D4—942no—
212C2×D4—3762no—
213C2×D4—3762no—
214C2×D4—1882no—
215C2×D4—1882no—
216C2×D4—1882no—
217C2×D4—1882no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—942no—
223C22×C4C2×D18811yescharacteristic
224C22×C4D4×D4712yes—
225C22×C4#1062535412yes—
226C22×C4C2×C22⋊D4711yescharacteristic
227D8—3761no—
228D8—3761no—
229M16—24no—
230M16—42no—
231M16—1882no—
232M16—1882no—
233C2×C8—22no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—1882no—
241C4⋊C4—942no—
242C4⋊C4—942no—
243C22⋊C4—42no—
244C22⋊C4—1882no—
245C22⋊C4—1882no—
246C42—41no—
247C42—41no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—3762no—
253C23—3762no—
254C23—1884no—
255C23—1882no—
256C23#1309399811yescharacteristic
257C23#1309400912yes—
258D4—3762no—
259D4—3764no—
260D4—3762no—
261D4—3762no—
262D4—1882no—
263D4—1882no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—3762no—
277C2×C4—1882no—
278C2×C4#1309399211yescharacteristic
279C2×C4#1309399912yes—
280C2×C4#1309401311yescharacteristic
281C8—42no—
282C8—41no—
283C8—41no—
284C8—3762no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—81no—
291C22—81no—
292C22—3762no—
293C22—3762no—
294C22—3762no—
295C22—7524no—
296C22#1568904011yescharacteristic
297C22#1568939311yescharacteristic
298C22#1568939311yescharacteristic
299C4—22no—
300C4—21no—
301C4—42no—
302C4—81no—
303C4—3762no—
304C4#1568930911yescharacteristic
305C4#1568931211yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—7522no—
310C2#2024463911yescharacteristic
311C2#2024464411yescharacteristic
312C2#2024467511yescharacteristic
subgroupG/H
centerC22#15689040
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#1851C2
socleC22×C47#394
prime radical#20254139C2
hypercenter#1851D47
hypocenterC47#10557
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. #20243851C2semidirect
3. #20241903C2semidirect
4. #20241265C2semidirect
5. C47×#1851C2semidirect
152. #1851D47semidirect
177. C47#10557semidirect

Maximal subgroups

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

subgroupindex
#202541392
#202438512
#202419032
#202412652
C47×#18512
#1055747