Finite Group Databasebeta

← #26105595

Subgroups — #26105595

Order 12032.

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

normal subgroups only

This group has 438 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#20251334C211yesmaximal, characteristic
2#20244927C212yesmaximal, complemented
3#20241213C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#2094C211yesmaximal, characteristic
6C2×#13093957—22no—
7C2×#13093947—22no—
8#15689062—21no—
9#15689035—22no—
10#15689029C2212yes—
11#15689003—21no—
12#15688998—22no—
13#15688878C2212yes—
14C2×#106×C47—21no—
15C2×#102×C47—21no—
16C47×#390C2211yescharacteristic
17C47×#384C2211yescharacteristic
18C47×#336C2211yescharacteristic
19C22×Dic94D412yes—
20C23×Dic47—21no—
21C22×C47⋊C8—22no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47D412yes—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—21no—
26#13093957—44no—
27#13093947—44no—
28C47⋊#95—22no—
29C47⋊#95—21no—
30C47⋊#95—22no—
31C47⋊#95—22no—
32C22×D4×C47D411yescharacteristic
33C22×C8×C47—21no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47D411yescharacteristic
37C2×C22⋊C4×C47—22no—
38C2×C42×C47C2311yescharacteristic
39#106×C47—42no—
40#102×C47—42no—
41C24×C47—21no—
42C2×D4×C47—21no—
43C2×D4×C47—41no—
44C2×D4×C47—41no—
45C2×D4×C47Q1612yes—
46C22×C4×C47—21no—
47C22×C4×C47—41no—
48C22×C4×C47C2×D411yescharacteristic
49C22×C4×C47C2×D412yes—
50C22×Dic47—22no—
51C22×Dic47—22no—
52C22×Dic47—82no—
53C22×Dic47—81no—
54C2×Dic94—22no—
55C2×Dic94—22no—
56C2×Dic94—42no—
57C2×Dic94—42no—
58C2×C8×C47—22no—
59C2×C8×C47—41no—
60C2×C8×C47—41no—
61C4⋊C4×C47—41no—
62C4⋊C4×C47—41no—
63C4⋊C4×C47—42no—
64C4⋊C4×C47—42no—
65C4⋊C4×C47—41no—
66C22⋊C4×C47—42no—
67C22⋊C4×C47—42no—
68C42×C47—41no—
69C22⋊Dic47—82no—
70C22⋊Dic47—81no—
71C4⋊Dic47—42no—
72C4⋊Dic47—81no—
73C2×C47⋊C8—24no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—42no—
76C23×C47—22no—
77C23×C47—42no—
78C23×C47—42no—
79C23×C47—41no—
80C23×C47—41no—
81C23×C47#12011yescharacteristic
82D4×C47—42no—
83D4×C47—81no—
84C2×C4×C47—22no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—21no—
88C2×C4×C47—42no—
89C2×C4×C47—41no—
90C2×C4×C47—82no—
91C2×C4×C47—81no—
92C2×C4×C47C2×Q1611yescharacteristic
93C2×C4×C47#13512yes—
94C2×C4×C47#13712yes—
95C2×C4×C47#13711yescharacteristic
96C2×Dic47—42no—
97C2×Dic47—42no—
98C2×Dic47—42no—
99C2×Dic47—42no—
100C2×Dic47—42no—
101C2×Dic47—42no—
102C2×Dic47—84no—
103Dic94—44no—
104Dic94—82no—
105C8×C47—42no—
106C47⋊C8—44no—
107#7833—471nomaximal, Sylow
108C22×C47—42no—
109C22×C47—44no—
110C22×C47—82no—
111C22×C47—81no—
112C22×C47—81no—
113C22×C47#45111yescharacteristic
114C22×C47#45212yes—
115C22×C47#45312yes—
116C22×C47#45311yescharacteristic
117C22×C47#45811yescharacteristic
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—22no—
121C4×C47—22no—
122C4×C47—81no—
123C4×C47—161no—
124Dic47—82no—
125Dic47—82no—
126#2804—471no—
127#2095—472no—
128#2094D4711yescharacteristic
129#1272—472no—
130#1270—471no—
131C2×C47—82no—
132C2×C47#161411yescharacteristic
133C2×C47#161812yes—
134C2×C47#163011yescharacteristic
135C2×C47#163212yes—
136C2×C47#165511yescharacteristic
137C2×#107—942no—
138C2×#106—21no—
139C2×#103—942no—
140C2×#102—21no—
141#400—941no—
142#393—942no—
143#391—472no—
144#390C2×D4711yescharacteristic
145#386—941no—
146#385—942no—
147#384C2×D4711yescharacteristic
148#336—472no—
149#336C2×D4711yescharacteristic
150C47#783311yescharacteristic, complemented, Sylow
151C22×Q8—472no—
152C22×D4C22⋊D4711yescharacteristic
153C23×C4—941no—
154C22×C8—21no—
155C22×C8—942no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—472no—
159C2×C4⋊C4—941no—
160C2×C4⋊C4C22⋊D4711yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—942no—
163C2×C22⋊C4—941no—
164C2×C42C22×D4711yescharacteristic
165#107—1884no—
166#106—42no—
167#103—1884no—
168#102—42no—
169#95—942no—
170#95—941no—
171#95—942no—
172#95—942no—
173C24—21no—
174C2×Q8—942no—
175C2×Q8—942no—
176C2×Q8—1882no—
177C2×Q8—1882no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4#1062533712yes—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—942no—
185C22×C4—942no—
186C22×C4—3762no—
187C22×C4—3761no—
188C22×C4D4×D4712yes—
189C22×C4C2×C22⋊D4711yescharacteristic
190C2×C8—22no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—944no—
194C2×C8—1882no—
195C2×C8—1882no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—3761no—
202C4⋊C4—1882no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—3762no—
206C22⋊C4—3761no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#1309399211yescharacteristic
214Q8—3762no—
215Q8—1884no—
216D4—42no—
217D4—81no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—3764no—
227C2×C4—1882no—
228C2×C4—1882no—
229C2×C4—1882no—
230C2×C4—1882no—
231C2×C4—1882no—
232C2×C4—1882no—
233C2×C4C2×#1062533711yescharacteristic
234C2×C4#1309404512yes—
235C2×C4#1309405312yes—
236C2×C4#1309405411yescharacteristic
237C8—42no—
238C8—1884no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1568927811yescharacteristic
245C22#1568928212yes—
246C22#1568928912yes—
247C22#1568929011yescharacteristic
248C22#1568931211yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—161no—
255C4—3762no—
256C4—3762no—
257C2—82no—
258C2#2024241211yescharacteristic
259C2#2024243312yes—
260C2#2024250711yescharacteristic
261C2#2024252012yes—
262C2#2024267611yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2094C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2094D47
hypocenterC47#7833
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2094

Complements

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

NH ≅ G/Ntype
2. #20244927C2semidirect
3. #20241213C2semidirect
150. C47#7833semidirect

Maximal subgroups

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

subgroupindex
#202513342
#202449272
#202412132
#202412092
C47×#20942
#783347