Finite Group Databasebeta

← #26110680

Subgroups — #26110680

Order 12032.

392 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 95 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#20252459C211yesmaximal, characteristic
2#20242153C212yesmaximal, complemented
3#20241980C211yesmaximal, characteristic
4#20241281C212yesmaximal, complemented
5C47×#1963C211yesmaximal, characteristic
6#15689195—22no—
7#15689195—22no—
8#15689180C2212yes—
9#15689090—21no—
10#15689090—21no—
11C4×#10625329C2212yes—
12#15688875—22no—
13#15688873—22no—
14#15688863C411yescharacteristic, complemented
15#15688863C412yescomplemented
16#15688863C411yescharacteristic, complemented
17C47×#532C2211yescharacteristic
18C47×#423C2211yescharacteristic
19C2×#104×C47—21no—
20C2×#104×C47—21no—
21C47×#376C2211yescharacteristic
22C2×#10625329—22no—
23C2×#10625329—21no—
24C2×#10625329—21no—
25C22×C47⋊C8—21no—
26C22×C47⋊C8—21no—
27C4⋊(C47⋊C8)D412yes—
28C4⋊(C47⋊C8)D412yes—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32#13093935—22no—
33#13093935C2×C412yes—
34#13093935C2×C412yes—
35C4×C47⋊C8—22no—
36C2×#41×C47D411yescharacteristic
37C2×#41×C47D411yescharacteristic
38C2×M16×C47—21no—
39C4⋊Q8×C47Q811yescharacteristic
40#127×C47Q811yescharacteristic
41#124×C47C2×C411yescharacteristic
42#124×C47C2×C411yescharacteristic
43#121×C47—22no—
44C4×D4×C47—22no—
45C2×C42×C47—21no—
46C2×C42×C47C2311yescharacteristic
47C4⋊C8×C47—22no—
48#104×C47—42no—
49#104×C47—42no—
50#95×C47—22no—
51#41×C47—41no—
52#41×C47—41no—
53#41×C47—41no—
54#41×C47—41no—
55C2×Q8×C47C4⋊C411yescharacteristic
56C2×Q8×C47C4⋊C411yescharacteristic
57C2×D4×C47—21no—
58C2×D4×C47—21no—
59C2×D4×C47—22no—
60C2×D4×C47C4⋊C411yescharacteristic
61C2×D4×C47C4⋊C411yescharacteristic
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—41no—
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47#4112yes—
67M16×C47—41no—
68M16×C47—41no—
69C2×C8×C47—41no—
70C4⋊C4×C47—22no—
71C22⋊C4×C47—22no—
72C22⋊C4×C47—22no—
73C42×C47—22no—
74C42×C47—41no—
75C42×C47C22×C411yescharacteristic
76C42×C47C2×D412yes—
77C42×C47C2×Q811yescharacteristic
78#10625329—44no—
79#10625329—81no—
80#10625329—81no—
81C2×C47⋊C8—22no—
82C2×C47⋊C8—22no—
83C2×C47⋊C8—22no—
84C2×C47⋊C8—22no—
85C2×C47⋊C8—81no—
86C2×C47⋊C8—81no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47#12311yescharacteristic
90Q8×C47—41no—
91Q8×C47—41no—
92D4×C47—41no—
93D4×C47—41no—
94D4×C47—82no—
95D4×C47—81no—
96D4×C47—81no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—42no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—42no—
107C2×C4×C47—82no—
108C2×C4×C47C2×C4⋊C411yescharacteristic
109C2×C4×C47C4×D412yes—
110C2×C4×C47#12011yescharacteristic
111C2×C4×C47C22⋊Q812yes—
112C8×C47—81no—
113C47⋊C8—24no—
114C47⋊C8—42no—
115C47⋊C8—42no—
116C47⋊C8—42no—
117#8918—471nomaximal, Sylow
118C22×C47—21no—
119C22×C47—22no—
120C22×C47—41no—
121C22×C47—41no—
122C22×C47—42no—
123C22×C47—42no—
124C22×C47#38511yescharacteristic
125C4×C47—21no—
126C4×C47—22no—
127C4×C47—22no—
128C4×C47—41no—
129C4×C47—41no—
130C4×C47—82no—
131C4×C47#45311yescharacteristic
132C4×C47#45511yescharacteristic
133#2980—471no—
134#1963D4711yescharacteristic
135#1569—472no—
136#1525—471no—
137#1300—472no—
138C2×C47—41no—
139C2×C47—81no—
140C2×C47—81no—
141C2×C47#185611yescharacteristic
142C2×C47#186112yes—
143#532C2×D4711yescharacteristic
144#435—942no—
145#435—942no—
146#432—472no—
147#423C2×D4711yescharacteristic
148C2×#104—21no—
149C2×#104—21no—
150#408—941no—
151#408—941no—
152C4×M16—472no—
153#376C2×D4711yescharacteristic
154#335—942no—
155#334—942no—
156#329—471no—
157#329—472no—
158#329—471no—
159C47#891811yescharacteristic, complemented, Sylow
160C2×#41C22⋊D4711yescharacteristic
161C2×#41C22⋊D4711yescharacteristic
162C2×M16—21no—
163C2×M16—941no—
164C2×M16—941no—
165C2×M16—942no—
166C22×C8—941no—
167C22×C8—941no—
168C4⋊Q8Dic9411yescharacteristic
169#127Dic9411yescharacteristic
170#124C4×D4711yescharacteristic
171#124C4×D4711yescharacteristic
172#121—22no—
173C4×D4—22no—
174C2×C42—21no—
175C2×C42C22×D4711yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—472no—
178C4⋊C8—472no—
179#104—42no—
180#104—42no—
181C22⋊C8—942no—
182C22⋊C8—942no—
183C22⋊C8—942no—
184#97—472no—
185#97—472no—
186#97—942no—
187C4×C8—942no—
188#95—22no—
189#41—41no—
190#41—41no—
191#41—41no—
192#41—41no—
193C2×Q8Dic47⋊C411yescharacteristic
194C2×Q8Dic47⋊C411yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4Dic47⋊C411yescharacteristic
199C2×D4Dic47⋊C411yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4#1062535212yes—
204C22×C4C2×C22⋊D4711yescharacteristic
205M16—41no—
206M16—41no—
207M16—3761no—
208M16—3761no—
209M16—1884no—
210C2×C8—41no—
211C2×C8—942no—
212C2×C8—942no—
213C2×C8—942no—
214C2×C8—942no—
215C2×C8—3761no—
216C2×C8—3761no—
217C4⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C42—22no—
221C42—41no—
222C42C2×Dic9411yescharacteristic
223C42C2×C4×D4711yescharacteristic
224C42D4×D4712yes—
225C23—21no—
226C23—21no—
227C23#1309400511yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—41no—
232D4—82no—
233D4—81no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4C2×Dic47⋊C411yescharacteristic
247C2×C4(C22⋊D47)⋊C412yes—
248C2×C4#1309399211yescharacteristic
249C2×C4#1309400112yes—
250C8—81no—
251C8—944no—
252C8—1882no—
253C8—1882no—
254C8—1882no—
255C22—21no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22#1568899511yescharacteristic
262C4—21no—
263C4—22no—
264C4—22no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4#1568929011yescharacteristic
269C4#1568930011yescharacteristic
270C2—41no—
271C2—81no—
272C2—81no—
273C2#2024359711yescharacteristic
274C2#2024362612yes—
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1963C2
socleC22×C47#385
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#1963D47
hypocenterC47#8918
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #20242153C2semidirect
4. #20241281C2semidirect
14. #15688863C4semidirect
15. #15688863C4semidirect
16. #15688863C4semidirect
159. C47#8918semidirect

Maximal subgroups

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

subgroupindex
#202524592
#202421532
#202419802
#202412812
C47×#19632
#891847