Finite Group Databasebeta

← #26120059

Subgroups — #26120059

Order 12032.

366 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 91 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#20253189C211yesmaximal, characteristic
2#20243404C211yesmaximal, characteristic
3#20242162C211yesmaximal, characteristic
4#20241952C211yesmaximal, characteristic
5#20241711C211yesmaximal, characteristic
6#20241247C211yesmaximal, characteristic
7C47×#1542C211yesmaximal, characteristic
8C2×C4×Dic94C2211yescharacteristic
9#15689427—22no—
10#15689415—21no—
11#15689405—21no—
12#15689176C2211yescharacteristic
13#15689166—21no—
14#15689147C2211yescharacteristic
15#15689131—21no—
16C2×#13093949—21no—
17#15688989C2211yescharacteristic
18#15688886—21no—
19#15688875—22no—
20#15688860—22no—
21#15688848—22no—
22C47×#533C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C47×#333—21no—
26C47×#332—21no—
27C22×Dic94D411yescharacteristic
28C2×#10625329—21no—
29C4×Dic94—21no—
30C4×Dic94—21no—
31C4×Dic94—21no—
32C4×Dic94—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47D411yescharacteristic
35C2×Dic47⋊C4—21no—
36C2×C4×Dic47—21no—
37#13093957—22no—
38#13093955—22no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)D412yes—
42#13093949—42no—
43#13093947—22no—
44#13093947—22no—
45C22⋊(C47⋊C8)D412yes—
46C4×C47⋊C8—21no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C22×C8×C47—21no—
50C4⋊Q8×C47D411yescharacteristic
51C4⋊Q8×C47D411yescharacteristic
52#125×C47—21no—
53C22⋊Q8×C47—21no—
54C22⋊Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57#117×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47—21no—
62C22⋊C8×C47—21no—
63C4×C8×C47—21no—
64C2×Q8×C47—21no—
65C2×Q8×C47—21no—
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47#4111yescharacteristic
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—41no—
72C2×Dic94—21no—
73C2×Dic94—21no—
74C2×Dic94—21no—
75C2×Dic94—21no—
76C2×Dic94—21no—
77C2×Dic94Q1612yes—
78C2×C8×C47—41no—
79C2×C8×C47—41no—
80C2×C8×C47—41no—
81C2×C8×C47—41no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—21no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C42×C47—41no—
90C42×C47C2×D411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47#4111yescharacteristic
93C42×C47#4111yescharacteristic
94C4⋊Dic47—21no—
95C4⋊Dic47—21no—
96C4⋊Dic47—41no—
97C4⋊Dic47—41no—
98Dic47⋊C4—41no—
99Dic47⋊C4—41no—
100C4×Dic47—41no—
101C4×Dic47—41no—
102#10625329—41no—
103#10625329—41no—
104C2×C47⋊C8—22no—
105C2×C47⋊C8—22no—
106C23×C47#12111yescharacteristic
107Q8×C47—81no—
108Q8×C47—81no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47#12011yescharacteristic
114C2×C4×C47#12111yescharacteristic
115C2×C4×C47#12111yescharacteristic
116C2×C4×C47#12311yescharacteristic
117C2×C4×C47#12311yescharacteristic
118C2×C4×C47#12411yescharacteristic
119C2×C4×C47C2×Q1611yescharacteristic
120C2×C4×C47#13511yescharacteristic
121C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C47#13711yescharacteristic
123C2×Dic47—22no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—81no—
128C2×Dic47—81no—
129C2×Dic47—81no—
130C2×Dic47—81no—
131Dic94—22no—
132Dic94—22no—
133Dic94—41no—
134Dic94—41no—
135C8×C47—81no—
136C8×C47—81no—
137C47⋊C8—42no—
138C47⋊C8—42no—
139#10471—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47#39411yescharacteristic
143C22×C47#46711yescharacteristic
144C22×C47#46811yescharacteristic
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—81no—
150C4×C47—81no—
151C4×C47#44911yescharacteristic
152C4×C47#45111yescharacteristic
153C4×C47#48411yescharacteristic
154C4×C47#48511yescharacteristic
155Dic47—42no—
156Dic47—81no—
157Dic47—161no—
158#3078—471no—
159#1805—471no—
160#1571—471no—
161#1542D4711yescharacteristic
162#1518—471no—
163#1461—471no—
164#1286—471no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47#203611yescharacteristic
168C2×C47#204911yescharacteristic
169C2×C47#205311yescharacteristic
170#533C2×D4711yescharacteristic
171C2×C4×Q8—471no—
172#479—942no—
173#477—941no—
174#475—941no—
175#431—471no—
176#431C2×D4711yescharacteristic
177#428—941no—
178#423—471no—
179C2×C4⋊C8C2×D4711yescharacteristic
180#419—941no—
181C2×#103—941no—
182#382—471no—
183#340—941no—
184#335—942no—
185#333—21no—
186#332—21no—
187#328—942no—
188#324—942no—
189C47#1047111yescharacteristic, complemented, Sylow
190C22×Q8—471no—
191C2×M16—941no—
192C22×C8—21no—
193C4⋊Q8C22⋊D4711yescharacteristic
194C4⋊Q8C22⋊D4711yescharacteristic
195#125—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—941no—
201C4×Q8—941no—
202C4×Q8—941no—
203C4×Q8—941no—
204#117—21no—
205C2×C4⋊C4—471no—
206C2×C4⋊C4—941no—
207C2×C4⋊C4—941no—
208C2×C42—941no—
209C2×C42C22×D4711yescharacteristic
210#107—942no—
211#106—942no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—472no—
216C4⋊C8—941no—
217C4⋊C8—941no—
218#103—942no—
219#103—942no—
220#103—1882no—
221C22⋊C8—21no—
222C22⋊C8—472no—
223C4×C8—21no—
224C4×C8—941no—
225#95—941no—
226#95—941no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—472no—
230C2×Q8—941no—
231C2×Q8—941no—
232C2×Q8—941no—
233C2×Q8—941no—
234C2×Q8—941no—
235C22×C4—941no—
236C22×C4—941no—
237C22×C4—1881no—
238C22×C4#1062535011yescharacteristic
239C22×C4D4×D4711yescharacteristic
240C22×C4D4×D4711yescharacteristic
241M16—1881no—
242M16—1881no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—942no—
248C2×C8—942no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—941no—
256C4⋊C4—941no—
257C4⋊C4—1881no—
258C4⋊C4—1881no—
259C4⋊C4—1881no—
260C4⋊C4—1881no—
261C22⋊C4—41no—
262C42—41no—
263C42—1881no—
264C42—1881no—
265C42#1062535011yescharacteristic
266C42D4×D4711yescharacteristic
267C42#1062535211yescharacteristic
268C42C2×C22⋊D4711yescharacteristic
269C23#1309399711yescharacteristic
270Q8—81no—
271Q8—81no—
272Q8—942no—
273Q8—942no—
274Q8—1881no—
275Q8—1881no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—942no—
281C2×C4—3761no—
282C2×C4—3761no—
283C2×C4—3761no—
284C2×C4—3761no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4#1309399211yescharacteristic
289C2×C4#1309399611yescharacteristic
290C2×C4#1309399711yescharacteristic
291C2×C4#1309400611yescharacteristic
292C2×C4#1309400711yescharacteristic
293C2×C4#1309401011yescharacteristic
294C2×C4Q16×D4711yescharacteristic
295C2×C4#1309404511yescharacteristic
296C2×C4#1309405111yescharacteristic
297C2×C4#1309405611yescharacteristic
298C8—81no—
299C8—81no—
300C8—1882no—
301C8—1882no—
302C22—21no—
303C22—21no—
304C22#1568903811yescharacteristic
305C22#1568936311yescharacteristic
306C22#1568937211yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—81no—
313C4—3761no—
314C4—1882no—
315C4—7521no—
316C4#1568926711yescharacteristic
317C4#1568927711yescharacteristic
318C4#1568945711yescharacteristic
319C4#1568946211yescharacteristic
320C2—21no—
321C2—21no—
322C2#2024458911yescharacteristic
323C2#2024466811yescharacteristic
324C2#2024469211yescharacteristic
subgroupG/H
centerC22#15689038
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1542C2
socleC22×C47#394
prime radicalC23×C47#121
hypercenter#1542D47
hypocenterC47#10471
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
189. C47#10471semidirect

Maximal subgroups

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

subgroupindex
#202531892
#202434042
#202421622
#202419522
#202417112
#202412472
C47×#15422
#1047147