Finite Group Databasebeta

← #26120058

Subgroups — #26120058

Order 12032.

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

normal subgroups only

This group has 366 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
1#20243403C211yesmaximal, characteristic
2#20242164C211yesmaximal, characteristic
3#20242045C211yesmaximal, characteristic
4#20241952C211yesmaximal, characteristic
5#20241712C211yesmaximal, characteristic
6#20241248C211yesmaximal, characteristic
7C47×#3078C211yesmaximal, characteristic
8#15689176C2211yescharacteristic
9#15689176C2211yescharacteristic
10C2×C4⋊(C47⋊C8)C2211yescharacteristic
11C2×#13093948—21no—
12#15688989C2211yescharacteristic
13#15688885—21no—
14#15688875—22no—
15#15688872—21no—
16#15688869—21no—
17#15688860—22no—
18#15688849—22no—
19C47×#533C2211yescharacteristic
20C2×C4×Q8×C47C2211yescharacteristic
21C47×#479—22no—
22C47×#477—21no—
23C47×#475—21no—
24C47×#428—21no—
25C47×#423C2211yescharacteristic
26C47×#419—21no—
27C22×C47⋊C8—21no—
28C2×C4⋊Dic47—21no—
29C4⋊(C47⋊C8)—21no—
30C4⋊(C47⋊C8)—21no—
31C4⋊(C47⋊C8)—21no—
32C4⋊(C47⋊C8)—21no—
33#13093948—42no—
34C22⋊(C47⋊C8)—21no—
35C22⋊(C47⋊C8)D412yes—
36C4×C47⋊C8—21no—
37C4×C47⋊C8—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C22×Q8×C47D411yescharacteristic
41C2×M16×C47—21no—
42C4⋊Q8×C47D411yescharacteristic
43C4⋊Q8×C47D411yescharacteristic
44#125×C47—21no—
45C22⋊Q8×C47—21no—
46C22⋊Q8×C47—21no—
47C4×Q8×C47—21no—
48C4×Q8×C47—21no—
49C4×Q8×C47—21no—
50C4×Q8×C47—21no—
51C4×Q8×C47—21no—
52C4×Q8×C47—21no—
53#117×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47D411yescharacteristic
56C2×C42×C47—21no—
57C2×C42×C47C2311yescharacteristic
58#107×C47—22no—
59#106×C47—22no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47D412yes—
62#103×C47—22no—
63#103×C47—22no—
64C2×Q8×C47—21no—
65C2×Q8×C47—21no—
66C2×Q8×C47—21no—
67C2×Q8×C47—21no—
68C2×Q8×C47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47—21no—
71C2×Q8×C47Q1612yes—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47#4111yescharacteristic
77C22×Dic47—41no—
78M16×C47—41no—
79M16×C47—41no—
80C2×C8×C47—22no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—21no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C42×C47—41no—
93C42×C47—41no—
94C42×C47—41no—
95C42×C47C2×D411yescharacteristic
96C42×C47C2×D411yescharacteristic
97C42×C47#4111yescharacteristic
98C42×C47#4111yescharacteristic
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C23×C47#12111yescharacteristic
107Q8×C47—22no—
108Q8×C47—22no—
109Q8×C47—41no—
110Q8×C47—41no—
111Q8×C47—81no—
112Q8×C47—81no—
113C2×C4×C47—22no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—81no—
122C2×C4×C47#12011yescharacteristic
123C2×C4×C47#12111yescharacteristic
124C2×C4×C47#12111yescharacteristic
125C2×C4×C47#12311yescharacteristic
126C2×C4×C47#12311yescharacteristic
127C2×C4×C47#12411yescharacteristic
128C2×C4×C47C2×Q1611yescharacteristic
129C2×C4×C47#13511yescharacteristic
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C47#13711yescharacteristic
132C2×Dic47—81no—
133C2×Dic47—81no—
134C2×Dic47—81no—
135C8×C47—42no—
136C47⋊C8—42no—
137C47⋊C8—81no—
138C47⋊C8—81no—
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—42no—
150C4×C47—81no—
151C4×C47—81no—
152C4×C47—81no—
153C4×C47#44911yescharacteristic
154C4×C47#45111yescharacteristic
155C4×C47#48411yescharacteristic
156C4×C47#48511yescharacteristic
157Dic47—161no—
158#3078D4711yescharacteristic
159#1805—471no—
160#1571—471no—
161#1542—471no—
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×Q8C2×D4711yescharacteristic
172#479—22no—
173#477—21no—
174#475—21no—
175#431—471no—
176#431—471no—
177#428—21no—
178#423C2×D4711yescharacteristic
179C2×C4⋊C8—471no—
180#419—21no—
181C2×#103—941no—
182#382—471no—
183#340—941no—
184#335—942no—
185#333—941no—
186#332—941no—
187#328—942no—
188#324—942no—
189C47#1047111yescharacteristic, complemented, Sylow
190C22×Q8C22⋊D4711yescharacteristic
191C2×M16—21no—
192C22×C8—941no—
193C4⋊Q8C22⋊D4711yescharacteristic
194C4⋊Q8C22⋊D4711yescharacteristic
195#125—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204#117—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—941no—
207C2×C4⋊C4C22⋊D4711yescharacteristic
208C2×C42—21no—
209C2×C42C22×D4711yescharacteristic
210#107—22no—
211#106—22no—
212C4⋊C8—21no—
213C4⋊C8—941no—
214C4⋊C8—941no—
215C4⋊C8—941no—
216C4⋊C8—941no—
217C4⋊C8C22⋊D4712yes—
218#103—22no—
219#103—22no—
220#103—1882no—
221C22⋊C8—472no—
222C22⋊C8—941no—
223C4×C8—941no—
224C4×C8—941no—
225#95—941no—
226#95—941no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8#1062533712yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—1881no—
238C22×C4D4×D4711yescharacteristic
239C22×C4#1062535211yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241M16—41no—
242M16—41no—
243C2×C8—22no—
244C2×C8—942no—
245C2×C8—1881no—
246C2×C8—1881no—
247C2×C8—1881no—
248C2×C8—1881no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—1881no—
260C4⋊C4—1881no—
261C22⋊C4—41no—
262C42—41no—
263C42—41no—
264C42—41no—
265C42#1062535211yescharacteristic
266C42#1062535211yescharacteristic
267C42C2×C22⋊D4711yescharacteristic
268C42C2×C22⋊D4711yescharacteristic
269C23#1309399611yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—41no—
273Q8—41no—
274Q8—81no—
275Q8—81no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—3761no—
286C2×C4—3761no—
287C2×C4—3761no—
288C2×C4#1309399311yescharacteristic
289C2×C4#1309399511yescharacteristic
290C2×C4#1309399611yescharacteristic
291C2×C4#1309400511yescharacteristic
292C2×C4#1309400511yescharacteristic
293C2×C4#1309401111yescharacteristic
294C2×C4C2×#1062533711yescharacteristic
295C2×C4#1309404411yescharacteristic
296C2×C4#1309404911yescharacteristic
297C2×C4#1309405511yescharacteristic
298C8—42no—
299C8—3761no—
300C8—3761no—
301C8—1882no—
302C22—21no—
303C22—21no—
304C22#1568903711yescharacteristic
305C22#1568936211yescharacteristic
306C22#1568937111yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—7521no—
316C4#1568926811yescharacteristic
317C4#1568927911yescharacteristic
318C4#1568945611yescharacteristic
319C4#1568946111yescharacteristic
320C2—21no—
321C2—21no—
322C2#2024458811yescharacteristic
323C2#2024466511yescharacteristic
324C2#2024469111yescharacteristic
subgroupG/H
centerC22#15689037
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3078C2
socleC22×C47#394
prime radicalC23×C47#121
hypercenter#3078D47
hypocenterC47#10471
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

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
#202434032
#202421642
#202420452
#202419522
#202417122
#202412482
C47×#30782
#1047147