Finite Group Databasebeta

← #26133091

Subgroups — #26133091

Order 12032.

424 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 91 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#20252455C211yesmaximal, characteristic
2#20245053C211yesmaximal, characteristic
3#20243674C212yesmaximal, complemented
4#20241508C212yesmaximal, complemented
5C47×#1884C211yesmaximal, characteristic
6#15689201—22no—
7D4×C47⋊C8—22no—
8#15689176—21no—
9#15689151—22no—
10#15689097C2212yes—
11#15689090—21no—
12#15689090C2212yes—
13#15688918C412yescomplemented
14#15688916C412yescomplemented
15#15688894—22no—
16C47×#568C2211yescharacteristic
17C47×#532C2211yescharacteristic
18C47×#421—21no—
19C47×#410C2211yescharacteristic
20C47×#339—21no—
21C2×#10625329—21no—
22C2×#10625329D412yes—
23C22×C47⋊C8—21no—
24C22×C47⋊C8D412yes—
25#13093983—22no—
26C4⋊(C47⋊C8)—21no—
27C4⋊(C47⋊C8)—21no—
28Dic47⋊C8—44no—
29#13093939—22no—
30#13093939C2×C414yes—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C4×C47⋊C8—22no—
35C2×#41×C47D411yescharacteristic
36C2×#41×C47C2311yescharacteristic
37#131×C47—42no—
38C2×M16×C47—22no—
39C2×M16×C47—21no—
40C4⋊Q8×C47Q811yescharacteristic
41#127×C47C2×C411yescharacteristic
42#124×C47C2×C411yescharacteristic
43#124×C47Q811yescharacteristic
44#121×C47—22no—
45C4×D4×C47—22no—
46#117×C47—21no—
47C2×C42×C47D411yescharacteristic
48#104×C47—42no—
49#99×C47—22no—
50#41×C47—41no—
51#41×C47—41no—
52#41×C47—41no—
53#41×C47—41no—
54C2×Q8×C47C4⋊C411yescharacteristic
55C2×Q8×C47C2×Q811yescharacteristic
56C2×D4×C47—22no—
57C2×D4×C47—21no—
58C2×D4×C47C4⋊C411yescharacteristic
59C2×D4×C47C22×C411yescharacteristic
60C2×D4×C47#4112yes—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D412yes—
65M16×C47—22no—
66M16×C47—42no—
67M16×C47—41no—
68C2×C8×C47—24no—
69C2×C8×C47—41no—
70C4⋊C4×C47—22no—
71C4⋊C4×C47—41no—
72C22⋊C4×C47—22no—
73C22⋊C4×C47—22no—
74C22⋊C4×C47—41no—
75C42×C47—21no—
76C42×C47—41no—
77C42×C47C4⋊C411yescharacteristic
78C42×C47C4⋊C411yescharacteristic
79C22⋊Dic47—24no—
80C4⋊Dic47—42no—
81C4×Dic47—42no—
82#10625329—42no—
83#10625329—81no—
84C2×C47⋊C8—22no—
85C2×C47⋊C8—22no—
86C2×C47⋊C8—42no—
87C2×C47⋊C8—81no—
88C23×C47—21no—
89C23×C47C4×D412yes—
90C23×C47#12311yescharacteristic
91Q8×C47—41no—
92Q8×C47—41no—
93D4×C47—41no—
94D4×C47—42no—
95D4×C47—41no—
96D4×C47—44no—
97D4×C47—81no—
98C2×C4×C47—21no—
99C2×C4×C47—22no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—42no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—81no—
109C2×C4×C47C2×C4⋊C411yescharacteristic
110C2×C4×C47#12011yescharacteristic
111C2×C4×C47C22⋊Q812yes—
112C2×Dic47—44no—
113C8×C47—42no—
114C47⋊C8—42no—
115C47⋊C8—42no—
116#12968—471nomaximal, Sylow
117C22×C47—22no—
118C22×C47—21no—
119C22×C47—41no—
120C22×C47—42no—
121C22×C47—42no—
122C22×C47—41no—
123C22×C47#38511yescharacteristic
124C4×C47—22no—
125C4×C47—21no—
126C4×C47—22no—
127C4×C47—41no—
128C4×C47—41no—
129C4×C47—161no—
130C4×C47B(4,𝔽2)11yescharacteristic
131C4×C47#45811yescharacteristic
132Dic47—84no—
133#2979—471no—
134#2126—471no—
135#1884D4711yescharacteristic
136#1868—472no—
137#1391—472no—
138C2×C47—21no—
139C2×C47—42no—
140C2×C47—41no—
141C2×C47—81no—
142C2×C47#186711yescharacteristic
143#568C2×D4711yescharacteristic
144#532C2×D4711yescharacteristic
145#436—942no—
146C8×D4—942no—
147#431—941no—
148#424—942no—
149#421—21no—
150#410—472no—
151#410C2×D4711yescharacteristic
152#408—472no—
153#408—941no—
154#354—472no—
155#353—472no—
156#344—942no—
157#339—21no—
158C47#1296811yescharacteristic, complemented, Sylow
159C2×#41C22⋊D4711yescharacteristic
160C2×#41C22×D4711yescharacteristic
161#131—42no—
162C2×M16—22no—
163C2×M16—21no—
164C2×M16—472no—
165C2×M16—941no—
166C22×C8—472no—
167C22×C8—941no—
168C4⋊Q8Dic9411yescharacteristic
169#127C4×D4711yescharacteristic
170#124Dic9411yescharacteristic
171#124C4×D4711yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—21no—
175#117—942no—
176C2×C42C22⋊D4711yescharacteristic
177C4⋊C8—941no—
178C4⋊C8—941no—
179C4⋊C8—1884no—
180#104—42no—
181#99—22no—
182#99—474no—
183#99—942no—
184C22⋊C8—942no—
185C22⋊C8—942no—
186C22⋊C8—942no—
187C4×C8—942no—
188#41—41no—
189#41—41no—
190#41—41no—
191#41—41no—
192C2×Q8Dic47⋊C411yescharacteristic
193C2×Q8C2×Dic9411yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4Dic47⋊C411yescharacteristic
197C2×D4C2×C4×D4711yescharacteristic
198C2×D4#1062535212yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4D4×D4712yes—
202C22×C4C2×C22⋊D4711yescharacteristic
203M16—22no—
204M16—42no—
205M16—41no—
206M16—3761no—
207M16—1882no—
208C2×C8—24no—
209C2×C8—41no—
210C2×C8—942no—
211C2×C8—942no—
212C2×C8—3761no—
213C2×C8—1882no—
214C4⋊C4—22no—
215C4⋊C4—41no—
216C4⋊C4—1882no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—41no—
220C22⋊C4—944no—
221C42—21no—
222C42—41no—
223C42—1882no—
224C42Dic47⋊C411yescharacteristic
225C42Dic47⋊C411yescharacteristic
226C23—21no—
227C23(C22⋊D47)⋊C412yes—
228C23#1309400511yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—41no—
232D4—42no—
233D4—41no—
234D4—44no—
235D4—81no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—1884no—
248C2×C4C2×Dic47⋊C411yescharacteristic
249C2×C4#1309399211yescharacteristic
250C2×C4#1309400112yes—
251C8—42no—
252C8—1882no—
253C8—1882no—
254C22—21no—
255C22—22no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22#1568899511yescharacteristic
261C4—22no—
262C4—21no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—161no—
267C4—3764no—
268C4#1568930911yescharacteristic
269C4#1568931211yescharacteristic
270C2—21no—
271C2—41no—
272C2—42no—
273C2—81no—
274C2#2024366811yescharacteristic
subgroupG/H
centerC4#15689309
commutatorC2×D4×C47C22×C4
FrattiniC2×#41C22×D47
FittingC47×#1884C2
socleC2×C47#1867
prime radicalC47×#532C22
cosocleC2×#41×C47C23
hypercenter#1884D47
hypocenterC47#12968
LCSG ▷ C2×D4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1884

Complements

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

NH ≅ G/Ntype
3. #20243674C2semidirect
4. #20241508C2semidirect
13. #15688918C4semidirect
14. #15688916C4semidirect
158. C47#12968semidirect

Maximal subgroups

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

subgroupindex
#202524552
#202450532
#202436742
#202415082
C47×#18842
#1296847