Finite Group Databasebeta

← #26119899

Subgroups — #26119899

Order 12032.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#15689400C211yesmaximal, characteristic
2#20243377C211yesmaximal, characteristic
3#20242159C211yesmaximal, characteristic, complemented
4#20242069C211yesmaximal, characteristic, complemented
5#20241778C211yesmaximal, characteristic
6#20241283C211yesmaximal, characteristic, complemented
7C47×#1549C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9#15689400—22no—
10#15689400—22no—
11#15689176C2211yescharacteristic
12C2×#13093957—21no—
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093944—21no—
16C2×#13093940—21no—
17#15688891—21no—
18#15688874—22no—
19#15688872—22no—
20#15688850—22no—
21C2×C4⋊Q8×C47C2211yescharacteristic
22C47×#431C2211yescharacteristic
23C47×#423C2211yescharacteristic
24C47×#332—22no—
25C22×D188D411yescharacteristic
26C22×C4×D47—21no—
27C2×#10625329—21no—
28C22×C47⋊C8—21no—
29C4×D188—21no—
30C4×D188—21no—
31C4×D188—21no—
32C4×D188—21no—
33C2×C4⋊Dic47D411yescharacteristic
34C2×(C2×D47)⋊C4—21no—
35#13093957—24no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)D412yes—
41#13093944—24no—
42#13093940—42no—
43C22⋊(C47⋊C8)—21no—
44C4×C47⋊C8—21no—
45C22×Q8×C47—21no—
46C2×M16×C47—21no—
47C4⋊Q8×C47—21no—
48C4⋊Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47D412yes—
51C2×C4⋊C4×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C42×C47C2311yescharacteristic
54C4⋊C8×C47—21no—
55C4⋊C8×C47—21no—
56C4⋊C8×C47D412yes—
57C22⋊C8×C47—21no—
58C4×C8×C47—21no—
59C23×D47—21no—
60C2×Q8×C47—22no—
61C2×Q8×C47—41no—
62C2×Q8×C47—82no—
63C22×C4×C47—41no—
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47#4111yescharacteristic
66C22×C4×C47#4111yescharacteristic
67C22×Dic47—21no—
68C2×D188—21no—
69C2×D188—21no—
70C2×D188—21no—
71C2×D188—21no—
72C2×D188—21no—
73C2×D188SD1612yes—
74C2×C4×D47—41no—
75C2×C4×D47—41no—
76C2×C4×D47—82no—
77M16×C47—41no—
78M16×C47—41no—
79C2×C8×C47—22no—
80C2×C8×C47—41no—
81C4⋊C4×C47—22no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C42×C47C2×D411yescharacteristic
86C42×C47C2×D411yescharacteristic
87C42×C47C2×D411yescharacteristic
88C42×C47#4111yescharacteristic
89(C2×D47)⋊C4—41no—
90(C2×D47)⋊C4—41no—
91C4⋊Dic47—21no—
92C4⋊Dic47—21no—
93#10625329—41no—
94#10625329—41no—
95C2×C47⋊C8—22no—
96C2×C47⋊C8—41no—
97C2×C47⋊C8—41no—
98C2×C47⋊C8—41no—
99C2×C47⋊C8—41no—
100C23×C47#12311yescharacteristic
101C22×D47—22no—
102C22×D47—41no—
103C22×D47—41no—
104C22×D47—81no—
105C22×D47—81no—
106Q8×C47—82no—
107Q8×C47—82no—
108C2×C4×C47—22no—
109C2×C4×C47—41no—
110C2×C4×C47—42no—
111C2×C4×C47—82no—
112C2×C4×C47#12011yescharacteristic
113C2×C4×C47#12111yescharacteristic
114C2×C4×C47#12111yescharacteristic
115C2×C4×C47#12111yescharacteristic
116C2×C4×C47#12311yescharacteristic
117C2×C4×C47#12411yescharacteristic
118C2×C4×C47C2×SD1611yescharacteristic
119C2×C4×C47#13511yescharacteristic
120C2×C4×C47#13711yescharacteristic
121C2×C4×C47#13711yescharacteristic
122C2×Dic47—41no—
123C2×Dic47—81no—
124D188—22no—
125D188—22no—
126D188—41no—
127D188—41no—
128C4×D47—82no—
129C4×D47—82no—
130C8×C47—42no—
131C8×C47—81no—
132C47⋊C8—42no—
133C47⋊C8—81no—
134C47⋊C8—81no—
135#10448—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47#39411yescharacteristic
139C22×C47#48311yescharacteristic
140C22×C47#48511yescharacteristic
141C2×D47—42no—
142C2×D47—82no—
143C2×D47—81no—
144C2×D47—81no—
145C2×D47—82no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—82no—
150C4×C47#44811yescharacteristic
151C4×C47#45211yescharacteristic
152C4×C47#46511yescharacteristic
153C4×C47#47011yescharacteristic
154C4×C47#47212yes—
155Dic47—81no—
156C2×#474—471no—
157#1800—471no—
158#1570—471no—
159#1549D4711yescharacteristic, complemented
160#1548—471no—
161#1478—471no—
162#1301—471no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47#203511yescharacteristic
166C2×C47#204511yescharacteristic
167C2×C47#204911yescharacteristic
168D47—82no—
169C2×C4⋊Q8C2×D4711yescharacteristic
170C2×C4×D4—471no—
171#474—942no—
172#474—942no—
173#431—471no—
174#431C2×D4711yescharacteristic
175C2×#107—941no—
176#423—471no—
177#423C2×D4711yescharacteristic
178C2×C4⋊C8—471no—
179C2×#102—941no—
180C2×#100—941no—
181#343—941no—
182#334—942no—
183#333—942no—
184#332—22no—
185D4⋊C8—942no—
186C47#1044811yescharacteristic, complemented, Sylow
187C22×Q8—21no—
188C22×D4—471no—
189C23×C4—941no—
190C2×M16—21no—
191C2×M16—941no—
192C22×C8—941no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8D18812yes—
197C4×D4—941no—
198C4×D4—941no—
199C4×D4—941no—
200C4×D4—941no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—471no—
204C2×C22⋊C4—941no—
205C2×C42C22×D4711yescharacteristic
206#107—944no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—472no—
210C4⋊C8—941no—
211C4⋊C8—941no—
212C4⋊C8—941no—
213C4⋊C8—941no—
214C4⋊C8C22⋊D4712yes—
215#102—944no—
216#100—1882no—
217C22⋊C8—21no—
218C22⋊C8—941no—
219C4×C8—21no—
220C4×C8—941no—
221C24—941no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×D4—472no—
226C2×D4—941no—
227C2×D4—941no—
228C2×D4—941no—
229C2×D4—941no—
230C2×D4—941no—
231C22×C4—41no—
232C22×C4—941no—
233C22×C4—3762no—
234C22×C4—1881no—
235C22×C4—1881no—
236C22×C4#1062535011yescharacteristic
237C22×C4D4×D4711yescharacteristic
238C22×C4#1062535211yescharacteristic
239M16—41no—
240M16—41no—
241M16—1881no—
242M16—1881no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—942no—
246C2×C8—1881no—
247C2×C8—1881no—
248C2×C8—1881no—
249C2×C8—1881no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—941no—
255C4⋊C4—941no—
256C22⋊C4—1881no—
257C22⋊C4—1881no—
258C42C2×D18811yescharacteristic
259C42D4×D4711yescharacteristic
260C42#1062535211yescharacteristic
261C42C2×C22⋊D4711yescharacteristic
262C23—942no—
263C23—3761no—
264C23—3761no—
265C23—1881no—
266C23—1881no—
267C23#1309400611yescharacteristic
268Q8—82no—
269Q8—82no—
270D4—942no—
271D4—942no—
272D4—1881no—
273D4—1881no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—3762no—
279C2×C4—3762no—
280C2×C4—3761no—
281C2×C4—1881no—
282C2×C4#1309399411yescharacteristic
283C2×C4#1309399511yescharacteristic
284C2×C4#1309399611yescharacteristic
285C2×C4#1309399811yescharacteristic
286C2×C4#1309400811yescharacteristic
287C2×C4#1309401011yescharacteristic
288C2×C4SD16×D4711yescharacteristic
289C2×C4#1309404511yescharacteristic
290C2×C4#1309405611yescharacteristic
291C2×C4#1309405611yescharacteristic
292C8—42no—
293C8—81no—
294C8—3761no—
295C8—3761no—
296C8—1882no—
297C22—21no—
298C22—21no—
299C22—3762no—
300C22—3762no—
301C22—3761no—
302C22—3761no—
303C22—1882no—
304C22#1568903911yescharacteristic
305C22#1568945211yescharacteristic
306C22#1568946211yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—82no—
311C4—3761no—
312C4#1568926311yescharacteristic
313C4#1568928811yescharacteristic
314C4#1568934911yescharacteristic
315C4#1568937811yescharacteristic
316C4#1568939112yes—
317C2—21no—
318C2—21no—
319C2—3762no—
320C2#2024458311yescharacteristic
321C2#2024464311yescharacteristic
322C2#2024466711yescharacteristic
subgroupG/H
centerC22#15689039
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1549C2
socleC22×C47#394
prime radicalC22×D188D4
hypercenter#1549D47
hypocenterC47#10448
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. #20242159C2semidirect
4. #20242069C2semidirect
6. #20241283C2semidirect
7. C47×#1549C2semidirect
159. #1549D47semidirect
186. C47#10448semidirect

Maximal subgroups

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

subgroupindex
C2×#156894002
#202433772
#202421592
#202420692
#202417782
#202412832
C47×#15492
#1044847