Finite Group Databasebeta

← #26105068

Subgroups — #26105068

Order 12032.

492 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 83 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#20250318C211yesmaximal, characteristic
2#20244713C212yesmaximal, complemented
3#20241212C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#2013C211yesmaximal, characteristic
6C2×#13094005—21no—
7C2×#13093957—22no—
8#15689053—21no—
9#15689048—22no—
10#15689021C2212yes—
11#15688998—22no—
12#15688885—22no—
13#15688878C2212yes—
14C22×SD16×C47—21no—
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#103×C47—21no—
17C47×#390C2211yescharacteristic
18C47×#336C2211yescharacteristic
19C23×Dic47—21no—
20C22×C47⋊C8—22no—
21#13094005—42no—
22#13094005—42no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47D412yes—
25C2×C4⋊Dic47D412yes—
26C2×Dic47⋊C4—22no—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—21no—
29#13093957—44no—
30C47⋊#95—22no—
31C47⋊#95—21no—
32C47⋊#95—22no—
33C22×Q8×C47—21no—
34C22×D4×C47D411yescharacteristic
35C2×SD16×C47—42no—
36C2×SD16×C47—42no—
37C2×SD16×C47—42no—
38C22×C8×C47—21no—
39C4⋊Q8×C47—42no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47—22no—
43C2×C42×C47C2311yescharacteristic
44#103×C47—42no—
45C24×C47—21no—
46C2×Q8×C47—22no—
47C2×Q8×C47—41no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C2×Q8×C47—41no—
52C2×D4×C47—21no—
53C2×D4×C47—41no—
54C2×D4×C47—41no—
55C2×D4×C47D812yes—
56C22×C4×C47—21no—
57C22×C4×C47—41no—
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47C2×D412yes—
60C22×Dic47—22no—
61C22×Dic47—22no—
62C22×Dic47—42no—
63C22×Dic47—81no—
64C22×Dic47—81no—
65SD16×C47—48no—
66C2×C8×C47—22no—
67C2×C8×C47—41no—
68C2×C8×C47—41no—
69C4⋊C4×C47—24no—
70C4⋊C4×C47—42no—
71C4⋊C4×C47—41no—
72C22⋊C4×C47—42no—
73C22⋊C4×C47—42no—
74C42×C47—41no—
75C22⋊Dic47—42no—
76C22⋊Dic47—44no—
77C4⋊Dic47—22no—
78C4⋊Dic47—22no—
79C4⋊Dic47—42no—
80C4⋊Dic47—81no—
81Dic47⋊C4—42no—
82Dic47⋊C4—42no—
83C2×C47⋊C8—24no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—42no—
86C23×C47—22no—
87C23×C47—42no—
88C23×C47—41no—
89C23×C47—42no—
90C23×C47—41no—
91C23×C47#12011yescharacteristic
92Q8×C47—42no—
93Q8×C47—42no—
94Q8×C47—81no—
95Q8×C47—81no—
96D4×C47—42no—
97D4×C47—81no—
98C2×C4×C47—21no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—42no—
103C2×C4×C47—41no—
104C2×C4×C47—42no—
105C2×C4×C47—81no—
106C2×C4×C47—81no—
107C2×C4×C47C2×D811yescharacteristic
108C2×C4×C47#13512yes—
109C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
110C2×C4×C47#13712yes—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—84no—
118C8×C47—42no—
119C47⋊C8—44no—
120#7751—471nomaximal, Sylow
121C22×C47—42no—
122C22×C47—44no—
123C22×C47—82no—
124C22×C47—81no—
125C22×C47—81no—
126C22×C47#44711yescharacteristic
127C22×C47#45212yes—
128C22×C47#45311yescharacteristic
129C22×C47#45612yes—
130C22×C47B(4,𝔽2)11yescharacteristic
131C4×C47—21no—
132C4×C47—21no—
133C4×C47—22no—
134C4×C47—22no—
135C4×C47—82no—
136C4×C47—81no—
137Dic47—82no—
138Dic47—82no—
139#2724—471no—
140#2057—472no—
141#2013D4711yescharacteristic
142#1271—472no—
143#1270—471no—
144C2×C47—82no—
145C2×C47#159511yescharacteristic
146C2×C47#160312yes—
147C2×C47#163011yescharacteristic
148C2×C47#164712yes—
149C2×C47#166011yescharacteristic
150C22×SD16—21no—
151C2×C4⋊Q8C2×D4711yescharacteristic
152C2×#123—941no—
153C2×#107—942no—
154C2×#103—21no—
155#398—941no—
156#397—942no—
157#390C2×D4711yescharacteristic
158#389—472no—
159#385—942no—
160#340—942no—
161#336—472no—
162#336C2×D4711yescharacteristic
163C47#775111yescharacteristic, complemented, Sylow
164C22×Q8—21no—
165C22×D4C22⋊D4711yescharacteristic
166C23×C4—941no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×SD16—42no—
170C22×C8—21no—
171C22×C8—942no—
172C4⋊Q8—42no—
173#123—1882no—
174#123—1882no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—472no—
177C2×C4⋊C4—472no—
178C2×C4⋊C4—941no—
179C2×C4⋊C4—942no—
180C2×C4⋊C4C22⋊D4711yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—942no—
183C2×C22⋊C4—941no—
184C2×C42C22×D4711yescharacteristic
185#107—1884no—
186#103—42no—
187#95—942no—
188#95—941no—
189#95—942no—
190C24—21no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4#1062533412yes—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4—942no—
204C22×C4—942no—
205C22×C4—3761no—
206C22×C4—3761no—
207C22×C4—1882no—
208C22×C4D4×D4712yes—
209C22×C4C2×C22⋊D4711yescharacteristic
210SD16—48no—
211C2×C8—22no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—944no—
215C2×C8—1882no—
216C2×C8—1882no—
217C4⋊C4—24no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—942no—
221C4⋊C4—942no—
222C4⋊C4—3761no—
223C4⋊C4—1882no—
224C4⋊C4—1882no—
225C4⋊C4—1882no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—1882no—
229C22⋊C4—1884no—
230C42—41no—
231C23—22no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23—41no—
236C23#1309399211yescharacteristic
237Q8—42no—
238Q8—42no—
239Q8—81no—
240Q8—81no—
241D4—42no—
242D4—81no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—3764no—
253C2×C4—1882no—
254C2×C4—1882no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4C2×#1062533411yescharacteristic
260C2×C4#1309404512yes—
261C2×C4#1309404911yescharacteristic
262C2×C4#1309405312yes—
263C8—42no—
264C8—1884no—
265C22—42no—
266C22—44no—
267C22—81no—
268C22—81no—
269C22—82no—
270C22#1568925611yescharacteristic
271C22#1568928212yes—
272C22#1568929011yescharacteristic
273C22#1568930412yes—
274C22#1568930911yescharacteristic
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—81no—
280C4—82no—
281C4—3762no—
282C4—3762no—
283C2—82no—
284C2#2024230511yescharacteristic
285C2#2024234812yes—
286C2#2024250711yescharacteristic
287C2#2024262112yes—
288C2#2024269711yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2013C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2013D47
hypocenterC47#7751
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #20244713C2semidirect
3. #20241212C2semidirect
163. C47#7751semidirect

Maximal subgroups

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

subgroupindex
#202503182
#202447132
#202412122
#202412092
C47×#20132
#775147