Finite Group Databasebeta

← #26119012

Subgroups — #26119012

Order 12032.

528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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#20242069C211yesmaximal, characteristic
5#20242068C211yesmaximal, characteristic
6#20241220C211yesmaximal, characteristic
7C47×#3209C211yesmaximal, 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#15688872—22no—
21#15688871—22no—
22C47×#533C2211yescharacteristic
23C2×C4⋊Q8×C47C2211yescharacteristic
24C47×#494—21no—
25C47×#487—21no—
26C47×#462—22no—
27C47×#451—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×C47D412yes—
49C2×Q16×C47—24no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47D411yescharacteristic
55C4⋊Q8×C47D411yescharacteristic
56#125×C47—21no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C4×Q8×C47—21no—
60C4×Q8×C47—21no—
61#117×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C42×C47C2311yescharacteristic
65C4⋊C8×C47D412yes—
66#103×C47—24no—
67C22⋊C8×C47—21no—
68C4×C8×C47—21no—
69C2×Q8×C47—22no—
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47—22no—
73C2×Q8×C47—44no—
74C22×C4×C47—22no—
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×Dic47—22no—
79C22×Dic47—22no—
80C2×Dic94—22no—
81C2×Dic94—22no—
82C2×Dic94—44no—
83Q16×C47—84no—
84C2×C8×C47—22no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
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—22no—
107C4×Dic47—42no—
108C4×Dic47—42no—
109C4×Dic47Q1614yescomplemented
110#10625329—42no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—41no—
113C23×C47#12011yescharacteristic
114Q8×C47—42no—
115Q8×C47—42no—
116Q8×C47—44no—
117Q8×C47—82no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—44no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47#12011yescharacteristic
125C2×C4×C47#12011yescharacteristic
126C2×C4×C47#12111yescharacteristic
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47#12111yescharacteristic
129C2×C4×C47#12711yescharacteristic
130C2×C4×C47C2×Q1612yes—
131C2×C4×C47#13712yes—
132C2×Dic47—24no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—44no—
136C2×Dic47—42no—
137Dic94—44no—
138Dic94—82no—
139C8×C47—42no—
140C47⋊C8—42no—
141C47⋊C8—81no—
142#10297—471nomaximal, Sylow
143C22×C47—21no—
144C22×C47—21no—
145C22×C47#39211yescharacteristic
146C22×C47#45112yes—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—82no—
150C4×C47—81no—
151C4×C47—81no—
152C4×C47#45512yes—
153C4×C47#45512yes—
154C4×C47#46212yes—
155C4×C47#49411yescharacteristic
156C4×C47#49711yescharacteristic
157Dic47—44no—
158Dic47—82no—
159#3209—471no—
160#3209D4711yescharacteristic
161#2862—471no—
162#1994—471no—
163#1548—471no—
164#1548—471no—
165#1276—471no—
166C2×C47—22no—
167C2×C47#200411yescharacteristic
168C2×C47#200411yescharacteristic
169C2×C47#201611yescharacteristic
170#533C2×D4711yescharacteristic
171C2×C4⋊Q8—471no—
172C2×C4⋊Q8C2×D4711yescharacteristic
173#494—21no—
174#494—941no—
175#487—21no—
176#487—941no—
177#462—22no—
178#462—942no—
179#451—22no—
180#451—942no—
181#431—471no—
182#431C2×D4711yescharacteristic
183#423—471no—
184#391—942no—
185#391—942no—
186#382—942no—
187C43—471no—
188#339—942no—
189#333—942no—
190#333—942no—
191C47#1029711yescharacteristic, complemented, Sylow
192C22×Q8—472no—
193C22×Q8C22⋊D4712yes—
194C2×Q16—24no—
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—21no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4—941no—
217C2×C42—472no—
218C2×C42—941no—
219C2×C42—941no—
220C2×C42C22×D4711yescharacteristic
221C4⋊C8—472no—
222C4⋊C8—941no—
223C4⋊C8—941no—
224C4⋊C8C22⋊D4712yes—
225#103—24no—
226#103—944no—
227C22⋊C8—21no—
228C22⋊C8—941no—
229C4×C8—21no—
230C4×C8—941no—
231#95—948no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8—942no—
238C2×Q8—942no—
239C2×Q8—1884no—
240C22×C4—22no—
241C22×C4—942no—
242C22×C4—942no—
243C22×C4D4×D4711yescharacteristic
244C22×C4D4×D4711yescharacteristic
245C22×C4C2×C22⋊D4711yescharacteristic
246Q16—84no—
247Q16—3764no—
248M16—1882no—
249C2×C8—22no—
250C2×C8—942no—
251C2×C8—1881no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—1881no—
263C4⋊C4—1881no—
264C4⋊C4—1881no—
265C4⋊C4—1881no—
266C22⋊C4—41no—
267C42—41no—
268C42—474no—
269C42—942no—
270C42—1882no—
271C42—1882no—
272C42D4×D4711yescharacteristic
273C42#1062535211yescharacteristic
274C42C2×C22⋊D4711yescharacteristic
275C42C2×C22⋊D4711yescharacteristic
276C23#1309399211yescharacteristic
277Q8—42no—
278Q8—42no—
279Q8—44no—
280Q8—82no—
281Q8—3762no—
282Q8—1884no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—944no—
290C2×C4—1884no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4—1882no—
294C2×C4#1309399211yescharacteristic
295C2×C4#1309399311yescharacteristic
296C2×C4#1309399511yescharacteristic
297C2×C4#1309399611yescharacteristic
298C2×C4#1309399611yescharacteristic
299C2×C4#1309402011yescharacteristic
300C2×C4Q16×D4712yes—
301C2×C4#1309405612yes—
302C8—42no—
303C8—3761no—
304C8—1882no—
305C22—21no—
306C22—21no—
307C22#1568903111yescharacteristic
308C22#1568927712yes—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—82no—
313C4—81no—
314C4—3762no—
315C4—1884no—
316C4#1568929912yes—
317C4#1568930112yes—
318C4#1568933112yes—
319C4#1568950111yescharacteristic
320C4#1568951611yescharacteristic
321C2—22no—
322C2#2024441411yescharacteristic
323C2#2024441511yescharacteristic
324C2#2024448611yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3209C2
socleC22×C47#392
prime radicalC23×C47#120
hypercenter#3209D47
hypocenterC47#10297
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
109. C4×Dic47Q16semidirect
191. C47#10297semidirect

Maximal subgroups

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

subgroupindex
#202543332
#202517652
#202443662
#202420692
#202420682
#202412202
C47×#32092
#1029747