Finite Group Databasebeta

← #26119018

Subgroups — #26119018

Order 12032.

516 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 105 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#20254333C211yesmaximal, characteristic
2#20251765C211yesmaximal, characteristic
3#20244366C211yesmaximal, characteristic
4#20242068C211yesmaximal, characteristic
5#20242060C211yesmaximal, characteristic
6#20241220C211yesmaximal, characteristic
7C47×#3229C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9#15689502—21no—
10#15689475—21no—
11#15689333—22no—
12#15689278—22no—
13#15689176C2211yescharacteristic
14#15689147C2211yescharacteristic
15#15689030—22no—
16#15689028—22no—
17#15688988—22no—
18C42×Dic47C2211yescharacteristic
19#15688882—22no—
20#15688871—22no—
21#15688869—22no—
22C47×#533C2211yescharacteristic
23C2×C4⋊Q8×C47C2211yescharacteristic
24C47×#498—21no—
25C47×#487—21no—
26C47×#483—22no—
27C47×#475—22no—
28C47×#431C2211yescharacteristic
29C22×Dic94D412yes—
30C2×#10625337—24no—
31C2×#10625329—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024—21no—
35#13094024—21no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—21no—
38C2×C4×Dic47—21no—
39C2×C4×Dic47—21no—
40C2×C4×Dic47D412yes—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)D412yes—
44#13093947—24no—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C47⋊#95—28no—
48C22×Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47D411yescharacteristic
54C4⋊Q8×C47D411yescharacteristic
55#125×C47—21no—
56C22⋊Q8×C47—21no—
57C22⋊Q8×C47—21no—
58C4×Q8×C47—21no—
59C4×Q8×C47—21no—
60#117×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47D412yes—
63C2×C42×C47C2311yescharacteristic
64#106×C47—24no—
65C4⋊C8×C47D412yes—
66#103×C47—24no—
67C22⋊C8×C47—21no—
68C4×C8×C47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47—21no—
71C2×Q8×C47—41no—
72C2×Q8×C47—41no—
73C2×Q8×C47—42no—
74C2×Q8×C47—42no—
75C22×C4×C47—22no—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—22no—
80C22×Dic47—22no—
81C2×Dic94—22no—
82C2×Dic94—22no—
83C2×Dic94—44no—
84C2×C8×C47—22no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—41no—
95C22⋊C4×C47—41no—
96C42×C47—41no—
97C42×C47C2×D411yescharacteristic
98C42×C47C2×D411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47#4111yescharacteristic
101#10625337—84no—
102C4⋊Dic47—41no—
103C4⋊Dic47—41no—
104C4⋊Dic47—41no—
105C4⋊Dic47—41no—
106C4×Dic47—24no—
107C4×Dic47—22no—
108C4×Dic47—22no—
109C4×Dic47—42no—
110#10625329—42no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—41no—
113C23×C47#12011yescharacteristic
114Q8×C47—42no—
115Q8×C47—42no—
116Q8×C47—82no—
117Q8×C47—81no—
118Q8×C47—81no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—44no—
125C2×C4×C47#12011yescharacteristic
126C2×C4×C47#12011yescharacteristic
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47#12111yescharacteristic
129C2×C4×C47#12111yescharacteristic
130C2×C4×C47#12711yescharacteristic
131C2×C4×C47#13512yes—
132C2×C4×C47#13712yes—
133C2×Dic47—24no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—44no—
137C2×Dic47—42no—
138Dic94—44no—
139Dic94—82no—
140C8×C47—42no—
141C47⋊C8—42no—
142C47⋊C8—81no—
143#10298—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—21no—
146C22×C47#39211yescharacteristic
147C22×C47#45212yes—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—81no—
151C4×C47—81no—
152C4×C47—82no—
153C4×C47#45512yes—
154C4×C47#45612yes—
155C4×C47#46412yes—
156C4×C47#49511yescharacteristic
157C4×C47#49711yescharacteristic
158Dic47—44no—
159Dic47—82no—
160#3229D4711yescharacteristic
161#3209—471no—
162#2862—471no—
163#1994—471no—
164#1548—471no—
165#1546—471no—
166#1276—471no—
167C2×C47—22no—
168C2×C47#200411yescharacteristic
169C2×C47#200511yescharacteristic
170C2×C47#201511yescharacteristic
171#533C2×D4711yescharacteristic
172C2×C4⋊Q8—471no—
173C2×C4⋊Q8C2×D4711yescharacteristic
174#498—21no—
175#494—941no—
176#487—21no—
177#487—941no—
178#483—22no—
179#475—22no—
180#462—942no—
181#451—942no—
182#431—471no—
183#431C2×D4711yescharacteristic
184#423—471no—
185#391—942no—
186#391—942no—
187#382—942no—
188C43—471no—
189#339—942no—
190#333—942no—
191#332—942no—
192C47#1029811yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8—472no—
195C2×Q16—944no—
196C2×M16—941no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—941no—
202C4⋊Q8—941no—
203C4⋊Q8—941no—
204C4⋊Q8—941no—
205C4⋊Q8C22⋊D4711yescharacteristic
206C4⋊Q8C22⋊D4711yescharacteristic
207#125—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212#117—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4C22⋊D4712yes—
217C2×C42—472no—
218C2×C42—941no—
219C2×C42—941no—
220C2×C42C22×D4711yescharacteristic
221#106—24no—
222C4⋊C8—472no—
223C4⋊C8—941no—
224C4⋊C8—941no—
225C4⋊C8C22⋊D4712yes—
226#103—24no—
227#103—944no—
228C22⋊C8—21no—
229C22⋊C8—941no—
230C4×C8—21no—
231C4×C8—941no—
232#95—948no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—942no—
240C2×Q8—942no—
241C2×Q8—1884no—
242C22×C4—22no—
243C22×C4—942no—
244C22×C4—942no—
245C22×C4D4×D4711yescharacteristic
246C22×C4D4×D4711yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248Q16—3764no—
249M16—1882no—
250C2×C8—22no—
251C2×C8—942no—
252C2×C8—1881no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—1881no—
264C4⋊C4—1881no—
265C4⋊C4—1881no—
266C4⋊C4—1881no—
267C22⋊C4—41no—
268C42—41no—
269C42—944no—
270C42—942no—
271C42—942no—
272C42—1882no—
273C42D4×D4711yescharacteristic
274C42#1062535211yescharacteristic
275C42C2×C22⋊D4711yescharacteristic
276C42C2×C22⋊D4711yescharacteristic
277C23#1309399211yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—81no—
281Q8—81no—
282Q8—82no—
283Q8—3762no—
284Q8—1884no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—944no—
292C2×C4—1884no—
293C2×C4—1882no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4#1309399211yescharacteristic
297C2×C4#1309399311yescharacteristic
298C2×C4#1309399511yescharacteristic
299C2×C4#1309399611yescharacteristic
300C2×C4#1309399611yescharacteristic
301C2×C4#1309402011yescharacteristic
302C2×C4#1309404512yes—
303C2×C4#1309405312yes—
304C8—42no—
305C8—3761no—
306C8—1882no—
307C22—21no—
308C22—21no—
309C22#1568903111yescharacteristic
310C22#1568928212yes—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—81no—
315C4—81no—
316C4—3762no—
317C4—1884no—
318C4#1568929912yes—
319C4#1568930612yes—
320C4#1568934312yes—
321C4#1568950411yescharacteristic
322C4#1568951411yescharacteristic
323C2—22no—
324C2#2024441411yescharacteristic
325C2#2024442211yescharacteristic
326C2#2024448011yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3229C2
socleC22×C47#392
prime radicalC23×C47#120
hypercenter#3229D47
hypocenterC47#10298
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
192. C47#10298semidirect

Maximal subgroups

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

subgroupindex
#202543332
#202517652
#202443662
#202420682
#202420602
#202412202
C47×#32292
#1029847