Finite Group Databasebeta

← #26136594

Subgroups — #26136594

Order 12032.

474 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#15688916C212yesmaximal, complemented
2#20241553C212yesmaximal, complemented
3#20241551C212yesmaximal, complemented
4C47×#3151C211yesmaximal, characteristic
5C2×#13093939—22no—
6#15688916—22no—
7#15688916C2214yes—
8C2×AGL(1,ℤ/8)×C47—22no—
9C47×#532C411yescharacteristic, complemented
10C2×#127×C47C2211yescharacteristic
11C47×#496—22no—
12C47×#493C411yescharacteristic, complemented
13C47×#493C2211yescharacteristic
14C47×#492C411yescharacteristic, complemented
15C47×#492C2211yescharacteristic
16C4×M16×C47C411yescharacteristic, complemented
17C2×C22⋊Dic47—42no—
18#13093939—24no—
19#13093939—42no—
20C2×#41×C47—21no—
21C22×D4×C47—41no—
22C22×D4×C47D412yes—
23AGL(1,ℤ/8)×C47—44no—
24AGL(1,ℤ/8)×C47—44no—
25C2×SD16×C47—22no—
26C2×SD16×C47—22no—
27C2×D8×C47—22no—
28C2×D8×C47—22no—
29C2×M16×C47—21no—
30C4⋊Q8×C47C2×C411yescharacteristic, complemented
31#127×C47—41no—
32#127×C47C2×C411yescharacteristic
33#127×C47C2×C411yescharacteristic, complemented
34#127×C47D412yes—
35#127×C47C2311yescharacteristic
36#124×C47—21no—
37#121×C47—41no—
38C4×D4×C47—41no—
39C2×C42×C47C2×C411yescharacteristic
40#97×C47—21no—
41C4×C8×C47C2×C411yescharacteristic, complemented
42C4×C8×C47C2×C411yescharacteristic, complemented
43C24×C47—22no—
44#41×C47—42no—
45#41×C47—42no—
46C2×Q8×C47—21no—
47C2×D4×C47—22no—
48C2×D4×C47—21no—
49C2×D4×C47—42no—
50C2×D4×C47—42no—
51C2×D4×C47—41no—
52C2×D4×C47—42no—
53C2×D4×C47—41no—
54C2×D4×C47—41no—
55C2×D4×C47—41no—
56C2×D4×C47—84no—
57C2×D4×C47C22⋊C412yes—
58C2×D4×C47C2×D412yes—
59C22×C4×C47—21no—
60C22×C4×C47—41no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×Dic47—82no—
63SD16×C47—44no—
64SD16×C47—44no—
65D8×C47—44no—
66D8×C47—44no—
67M16×C47—24no—
68C2×C8×C47—21no—
69C2×C8×C47—21no—
70C4⋊C4×C47—41no—
71C22⋊C4×C47—42no—
72C42×C47—21no—
73C42×C47C22⋊C411yescharacteristic
74C42×C47C22×C411yescharacteristic
75C22⋊Dic47—44no—
76C22⋊Dic47—82no—
77C23×C47—22no—
78C23×C47—22no—
79C23×C47—42no—
80C23×C47—42no—
81C23×C47—41no—
82C23×C47—84no—
83C23×C47#9911yescharacteristic
84Q8×C47—42no—
85D4×C47—42no—
86D4×C47—42no—
87D4×C47—42no—
88D4×C47—82no—
89D4×C47—82no—
90D4×C47—82no—
91D4×C47—82no—
92D4×C47—84no—
93D4×C47—82no—
94C2×C4×C47—22no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—81no—
100C2×C4×C47—81no—
101C2×C4×C47#9911yescharacteristic
102C2×C4×C47C2×C22⋊C411yescharacteristic
103C2×Dic47—84no—
104C2×Dic47—164no—
105C8×C47—22no—
106C8×C47—22no—
107#13584—471nomaximal, Sylow
108C22×C47—41no—
109C22×C47—42no—
110C22×C47—42no—
111C22×C47—82no—
112C22×C47—82no—
113C22×C47—84no—
114C22×C47—82no—
115C22×C47—81no—
116C22×C47—162no—
117C22×C47#35312yes—
118C22×C47C2×#9911yescharacteristic
119C4×C47—22no—
120C4×C47—21no—
121C4×C47—21no—
122C4×C47—41no—
123C4×C47—81no—
124Dic47—164no—
125#3151D4711yescharacteristic
126C2×#353—472no—
127#1409—472no—
128#1408—472no—
129C2×C47—21no—
130C2×C47—22no—
131C2×C47—82no—
132C2×C47—82no—
133C2×C47—161no—
134C2×C47C2×#35311yescharacteristic
135C2×AGL(1,ℤ/8)—22no—
136#532Dic4711yescharacteristic, complemented
137C2×#127C2×D4711yescharacteristic
138#496—22no—
139#493Dic4711yescharacteristic, complemented
140#493C2×D4711yescharacteristic
141#492Dic4711yescharacteristic, complemented
142#492C2×D4711yescharacteristic
143C2×#99—942no—
144C4×M16Dic4711yescharacteristic, complemented
145#353—474no—
146#353—942no—
147C47#1358411yescharacteristic, complemented, Sylow
148C2×#41—21no—
149C22×D4—41no—
150C22×D4C22⋊D4712yes—
151AGL(1,ℤ/8)—44no—
152AGL(1,ℤ/8)—44no—
153C2×SD16—22no—
154C2×SD16—22no—
155C2×D8—22no—
156C2×D8—22no—
157C2×M16—21no—
158C4⋊Q8C2×Dic4711yescharacteristic, complemented
159#127—41no—
160#127C2×Dic4711yescharacteristic
161#127C2×Dic4711yescharacteristic, complemented
162#127C22⋊D4712yes—
163#127C22×D4711yescharacteristic
164#124—21no—
165#121—41no—
166C4×D4—41no—
167C2×C22⋊C4—1882no—
168C2×C42C2×Dic4711yescharacteristic
169#99—944no—
170#99—1882no—
171#97—21no—
172C4×C8C2×Dic4711yescharacteristic, complemented
173C4×C8C2×Dic4711yescharacteristic, complemented
174C24—22no—
175#41—42no—
176#41—42no—
177C2×Q8—21no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—84no—
188C2×D4C22⋊Dic4712yes—
189C2×D4C2×C22⋊D4712yes—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—3762no—
193C22×C4C22⋊Dic4711yescharacteristic
194SD16—44no—
195SD16—44no—
196D8—44no—
197D8—44no—
198M16—24no—
199C2×C8—21no—
200C2×C8—21no—
201C4⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—3762no—
204C22⋊C4—1884no—
205C42—21no—
206C42C22⋊Dic4711yescharacteristic
207C42C22×Dic4711yescharacteristic
208C23—22no—
209C23—22no—
210C23—42no—
211C23—42no—
212C23—41no—
213C23—84no—
214C23#1309393911yescharacteristic
215Q8—42no—
216D4—42no—
217D4—42no—
218D4—42no—
219D4—82no—
220D4—82no—
221D4—82no—
222D4—82no—
223D4—84no—
224D4—82no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—3764no—
233C2×C4—7524no—
234C2×C4#1309393911yescharacteristic
235C2×C4C2×C22⋊Dic4711yescharacteristic
236C8—22no—
237C8—22no—
238C22—41no—
239C22—42no—
240C22—42no—
241C22—82no—
242C22—82no—
243C22—84no—
244C22—82no—
245C22—81no—
246C22—162no—
247C22#1568891612yes—
248C22C2×#1309393911yescharacteristic
249C4—22no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—7524no—
255C2—21no—
256C2—22no—
257C2—82no—
258C2—82no—
259C2—161no—
260C2C2×#1568891611yescharacteristic
subgroupG/H
centerC2C2×#15688916
commutatorC42×C47C22×C4
Frattini#127C22×D47
Fitting, prime radicalC47×#3151C2
socleC2×C47C2×#353
cosocle#127×C47C23
hypercenter#3151D47
hypocenterC47#13584
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. C2×#15688916C2semidirect
2. #20241553C2semidirect
3. #20241551C2semidirect
9. C47×#532C4semidirect
12. C47×#493C4semidirect
14. C47×#492C4semidirect
16. C4×M16×C47C4semidirect
30. C4⋊Q8×C47C2×C4semidirect
33. #127×C47C2×C4semidirect
41. C4×C8×C47C2×C4semidirect
42. C4×C8×C47C2×C4semidirect
136. #532Dic47semidirect
139. #493Dic47semidirect
141. #492Dic47semidirect
144. C4×M16Dic47semidirect
147. C47#13584semidirect
158. C4⋊Q8C2×Dic47semidirect
161. #127C2×Dic47semidirect
172. C4×C8C2×Dic47semidirect
173. C4×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889162
#202415532
#202415512
C47×#31512
#1358447