Finite Group Databasebeta

← #26119887

Subgroups — #26119887

Order 12032.

408 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 99 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×#15689333C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic
3#20242163C211yesmaximal, characteristic
4#20242059C211yesmaximal, characteristic
5#20242059C211yesmaximal, characteristic
6#20241284C211yesmaximal, characteristic
7C47×#1479C211yesmaximal, characteristic
8C22×#10625337—21no—
9C2×#13094024C2211yescharacteristic
10#15689333—22no—
11#15689333—22no—
12#15689147C2211yescharacteristic
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093947—21no—
16C2×#13093943—21no—
17#15688892—21no—
18#15688876—22no—
19#15688868—22no—
20#15688868—22no—
21C2×C4×Q8×C47C2211yescharacteristic
22C47×#431C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C47×Q8⋊C8—22no—
25C22×Dic94—21no—
26C2×#10625337—24no—
27C2×#10625337—42no—
28C2×#10625337—42no—
29C2×#10625329—21no—
30C2×#10625329—21no—
31C22×C47⋊C8—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024—21no—
35#13094024D412yes—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)D412yes—
44#13093947—24no—
45#13093943—42no—
46C22×Q8×C47D411yescharacteristic
47C4×Q8×C47—21no—
48C4×Q8×C47—21no—
49C4×Q8×C47—21no—
50C4×Q8×C47—21no—
51C2×C4⋊C4×C47—21no—
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C42×C47—21no—
54C2×C42×C47C2311yescharacteristic
55C4⋊C8×C47—21no—
56C4⋊C8×C47D412yes—
57C22⋊C8×C47—21no—
58C22⋊C8×C47—21no—
59C4×C8×C47—21no—
60C4×C8×C47—21no—
61C2×Q8×C47—21no—
62C2×Q8×C47—21no—
63C2×Q8×C47—21no—
64C2×Q8×C47—21no—
65C2×Q8×C47—21no—
66C2×Q8×C47D812yes—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4111yescharacteristic
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—41no—
73C2×Dic94—22no—
74C2×Dic94—41no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×C8×C47—22no—
80C2×C8×C47—41no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C42×C47—41no—
86C42×C47—41no—
87C42×C47C2×D411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47C2×D411yescharacteristic
90C42×C47#4111yescharacteristic
91#10625337—48no—
92C4⋊Dic47—22no—
93C4⋊Dic47—22no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96#10625329—41no—
97#10625329—41no—
98#10625329—41no—
99#10625329—41no—
100C2×C47⋊C8—22no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C23×C47#12311yescharacteristic
106Q8×C47—22no—
107Q8×C47—22no—
108Q8×C47—41no—
109Q8×C47—41no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—81no—
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12111yescharacteristic
119C2×C4×C47#12111yescharacteristic
120C2×C4×C47#12111yescharacteristic
121C2×C4×C47#12311yescharacteristic
122C2×C4×C47#12411yescharacteristic
123C2×C4×C47C2×D811yescharacteristic
124C2×C4×C47#13511yescharacteristic
125C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C47#13711yescharacteristic
127C2×Dic47—42no—
128C2×Dic47—81no—
129C2×Dic47—81no—
130Dic94—42no—
131Dic94—42no—
132Dic94—81no—
133Dic94—81no—
134C8×C47—42no—
135C8×C47—81no—
136C47⋊C8—42no—
137C47⋊C8—81no—
138C47⋊C8—81no—
139#10446—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47#39411yescharacteristic
143C22×C47#48011yescharacteristic
144C22×C47#48211yescharacteristic
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—42no—
149C4×C47—81no—
150C4×C47#44711yescharacteristic
151C4×C47#45211yescharacteristic
152C4×C47#46611yescharacteristic
153C4×C47#46811yescharacteristic
154C4×C47#47312yes—
155Dic47—82no—
156C2×#462—471no—
157#1801—471no—
158#1571—471no—
159#1546—471no—
160#1546—471no—
161#1479D4711yescharacteristic
162#1301—471no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47#203311yescharacteristic
166C2×C47#204711yescharacteristic
167C2×C47#204911yescharacteristic
168C22×Q16—941no—
169C2×C4⋊Q8—471no—
170C2×C4×Q8C2×D4711yescharacteristic
171#462—942no—
172#462—942no—
173#431C2×D4711yescharacteristic
174#431C2×D4711yescharacteristic
175#423—471no—
176#423—471no—
177C2×C4⋊C8—471no—
178C2×#103—941no—
179C2×#101—941no—
180#343—941no—
181#335—942no—
182#332—942no—
183#332—942no—
184Q8⋊C8—22no—
185C47#1044611yescharacteristic, complemented, Sylow
186C22×Q8—941no—
187C22×Q8D18811yescharacteristic
188C2×Q16—944no—
189C2×Q16—1882no—
190C2×Q16—1882no—
191C2×M16—941no—
192C2×M16—941no—
193C22×C8—941no—
194C4⋊Q8—472no—
195C4⋊Q8—941no—
196C4⋊Q8—941no—
197C4⋊Q8—941no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4D18811yescharacteristic
206C2×C42—21no—
207C2×C42C22×D4711yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—472no—
210C4⋊C8—941no—
211C4⋊C8—941no—
212C4⋊C8—941no—
213C4⋊C8—941no—
214C4⋊C8—941no—
215C4⋊C8C22⋊D4712yes—
216#103—944no—
217#101—1882no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—942no—
228C2×Q8—1881no—
229C2×Q8—1881no—
230C2×Q8—1881no—
231C2×Q8—1881no—
232C2×Q8—1881no—
233C2×Q8D37612yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1881no—
237C22×C4C2×D18811yescharacteristic
238C22×C4#1062535211yescharacteristic
239C22×C4#1062535211yescharacteristic
240Q16—1888no—
241M16—1881no—
242M16—1881no—
243M16—1881no—
244M16—1881no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—942no—
248C2×C8—1881no—
249C2×C8—1881no—
250C2×C8—1881no—
251C2×C8—1881no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—942no—
257C4⋊C4—942no—
258C4⋊C4—1881no—
259C4⋊C4—1881no—
260C42—41no—
261C42—41no—
262C42#1062535011yescharacteristic
263C42D4×D4711yescharacteristic
264C42D4×D4711yescharacteristic
265C42C2×C22⋊D4711yescharacteristic
266C23#1309400811yescharacteristic
267Q8—22no—
268Q8—22no—
269Q8—41no—
270Q8—41no—
271Q8—3761no—
272Q8—3761no—
273Q8—1882no—
274Q8—1882no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—3761no—
283C2×C4—3761no—
284C2×C4—1882no—
285C2×C4#1309399411yescharacteristic
286C2×C4#1309399511yescharacteristic
287C2×C4#1309399611yescharacteristic
288C2×C4#1309399811yescharacteristic
289C2×C4#1309400611yescharacteristic
290C2×C4#1309401011yescharacteristic
291C2×C4C2×D37611yescharacteristic
292C2×C4#1309404611yescharacteristic
293C2×C4#1309405211yescharacteristic
294C2×C4#1309405711yescharacteristic
295C8—42no—
296C8—81no—
297C8—3761no—
298C8—3761no—
299C8—1882no—
300C22—21no—
301C22—21no—
302C22#1568903911yescharacteristic
303C22#1568943711yescharacteristic
304C22#1568944811yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—81no—
310C4—3762no—
311C4#1568925911yescharacteristic
312C4#1568928711yescharacteristic
313C4#1568935911yescharacteristic
314C4#1568936911yescharacteristic
315C4#1568939612yes—
316C2—21no—
317C2—21no—
318C2#2024457111yescharacteristic
319C2#2024465511yescharacteristic
320C2#2024466911yescharacteristic
subgroupG/H
centerC22#15689039
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1479C2
socleC22×C47#394
prime radicalC23×C47#123
hypercenter#1479D47
hypocenterC47#10446
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
185. C47#10446semidirect

Maximal subgroups

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

subgroupindex
C2×#156893332
#202433842
#202421632
#202420592
#202412842
C47×#14792
#1044647