Finite Group Databasebeta

← #26103572

Subgroups — #26103572

Order 12032.

470 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 67 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#20243306C211yesmaximal, characteristic
2#20243306C211yesmaximal, characteristic
3C47×#1785C211yesmaximal, characteristic
4C2×#13093939—21no—
5C2×#13093939—21no—
6C2×#13093939—21no—
7C2×#13093939C412yescomplemented
8#15688986—21no—
9#15688986C412yescomplemented
10C22×#41×C47C2211yescharacteristic
11C2×#101×C47C412yescomplemented
12C2×#100×C47—21no—
13C47×#407—21no—
14C2×C22⋊Dic47—22no—
15C2×C22⋊Dic47—22no—
16#13093939—42no—
17#13093939—42no—
18#13093939—42no—
19#13093939—42no—
20C47⋊#95—42no—
21C47⋊#95—42no—
22C2×#41×C47—42no—
23C2×#41×C47—42no—
24C2×#41×C47—42no—
25C22×Q8×C47C2×C411yescharacteristic
26C22×D4×C47C2×C411yescharacteristic
27C22×D4×C47D411yescharacteristic
28C22×D4×C47D411yescharacteristic
29C23×C4×C47C2×C411yescharacteristic
30C23×C4×C47D411yescharacteristic
31C23×C4×C47Q811yescharacteristic
32C2×M16×C47—22no—
33#101×C47—42no—
34#100×C47—42no—
35C22⋊C8×C47—42no—
36C24×C47C22⋊C411yescharacteristic
37C24×C47C22⋊C411yescharacteristic
38C24×C47C4⋊C411yescharacteristic
39#41×C47—416no—
40C2×Q8×C47—22no—
41C2×Q8×C47—41no—
42C2×Q8×C47—41no—
43C2×D4×C47—22no—
44C2×D4×C47—22no—
45C2×D4×C47—22no—
46C2×D4×C47—22no—
47C2×D4×C47—22no—
48C2×D4×C47—41no—
49C2×D4×C47—41no—
50C2×D4×C47—41no—
51C2×D4×C47—41no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47C4211yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C4⋊C411yescharacteristic
64C22×C4×C47C4⋊C411yescharacteristic
65C22×Dic47—42no—
66C22×Dic47—42no—
67M16×C47—42no—
68M16×C47—42no—
69C2×C8×C47—42no—
70C22⋊Dic47—42no—
71C22⋊Dic47—42no—
72C22⋊Dic47—42no—
73C22⋊Dic47—42no—
74C23×C47—21no—
75C23×C47—21no—
76C23×C47—21no—
77C23×C47—21no—
78C23×C47—21no—
79C23×C47—21no—
80C23×C47—21no—
81C23×C47—81no—
82C23×C47—81no—
83C23×C47—81no—
84C23×C47#9511yescharacteristic
85C23×C47#9912yes—
86C23×C47#9912yes—
87Q8×C47—44no—
88D4×C47—44no—
89D4×C47—44no—
90D4×C47—44no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—44no—
103C2×C4×C47—44no—
104C2×C4×C47—44no—
105C2×C4×C47#9912yes—
106C2×Dic47—82no—
107C2×Dic47—82no—
108C2×Dic47—84no—
109C2×Dic47—84no—
110C8×C47—82no—
111#7313—471nomaximal, Sylow
112C22×C47—22no—
113C22×C47—22no—
114C22×C47—41no—
115C22×C47—42no—
116C22×C47—42no—
117C22×C47—42no—
118C22×C47—41no—
119C22×C47—81no—
120C22×C47—81no—
121C22×C47—81no—
122C22×C47—81no—
123C22×C47—81no—
124C22×C47—81no—
125C22×C47#34211yescharacteristic
126C22×C47#34211yescharacteristic
127C22×C47#34211yescharacteristic
128C4×C47—22no—
129C4×C47—41no—
130C4×C47—41no—
131C4×C47—41no—
132Dic47—162no—
133Dic47—162no—
134#1785D4711yescharacteristic
135#1782—471no—
136#1782—471no—
137C2×C47—21no—
138C2×C47—21no—
139C2×C47—21no—
140C2×C47—42no—
141C2×C47—42no—
142C2×C47—81no—
143C2×C47#130411yescharacteristic
144C22×#41C2×D4711yescharacteristic
145C2×#101Dic4712yescomplemented
146C2×#100—21no—
147C2×#99—472no—
148C2×#99—941no—
149C2×#99—941no—
150C2×#99—941no—
151#407—21no—
152#381—472no—
153#381—941no—
154C47#731311yescharacteristic, complemented, Sylow
155C2×#41—42no—
156C2×#41—42no—
157C2×#41—42no—
158C22×Q8C2×Dic4711yescharacteristic
159C22×D4C4×D4711yescharacteristic
160C22×D4D18811yescharacteristic
161C22×D4C22⋊D4711yescharacteristic
162C23×C4Dic9411yescharacteristic
163C23×C4C4×D4711yescharacteristic
164C23×C4C22⋊D4711yescharacteristic
165C2×M16—22no—
166C2×C22⋊C4—942no—
167C2×C22⋊C4—942no—
168#101—42no—
169#100—42no—
170#99—1882no—
171#99—1882no—
172#99—1882no—
173#99—1882no—
174C22⋊C8—42no—
175#95—1882no—
176#95—1882no—
177C24Dic47⋊C411yescharacteristic
178C24(C2×D47)⋊C411yescharacteristic
179C24(C2×D47)⋊C411yescharacteristic
180#41—416no—
181C2×Q8—22no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—1882no—
203C22×C4—1882no—
204C22×C4C4×Dic4711yescharacteristic
205C22×C4Dic47⋊C411yescharacteristic
206C22×C4C4⋊Dic4711yescharacteristic
207C22×C4C22⋊Dic4711yescharacteristic
208M16—42no—
209M16—42no—
210C2×C8—42no—
211C22⋊C4—1882no—
212C22⋊C4—1882no—
213C22⋊C4—1882no—
214C22⋊C4—1882no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—81no—
223C23—81no—
224C23—81no—
225C23C47⋊#9511yescharacteristic
226C23#1309393812yes—
227C23#1309393812yes—
228Q8—44no—
229D4—44no—
230D4—44no—
231D4—44no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—3762no—
247C2×C4—3762no—
248C2×C4—3764no—
249C2×C4—3764no—
250C2×C4#1309393912yes—
251C8—82no—
252C22—22no—
253C22—22no—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22#1568888911yescharacteristic
266C22#1568888911yescharacteristic
267C22#1568889011yescharacteristic
268C4—22no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—7522no—
273C4—7522no—
274C2—21no—
275C2—21no—
276C2—21no—
277C2—42no—
278C2—42no—
279C2—81no—
280C2C47⋊#130411yescharacteristic
subgroupG/H
centerC2C47⋊#1304
commutatorC22×C4×C47C42
FrattiniC22×#41C2×D47
FittingC47×#1785C2
socleC2×C47#1304
prime radical, cosocleC22×#41×C47C22
hypercenter#1785D47
hypocenterC47#7313
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
7. C2×#13093939C4semidirect
9. #15688986C4semidirect
11. C2×#101×C47C4semidirect
145. C2×#101Dic47semidirect
154. C47#7313semidirect

Maximal subgroups

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

subgroupindex
#202433062
C47×#17852
#731347