Finite Group Databasebeta

← #26111315

Subgroups — #26111315

Order 12032.

616 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 143 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×#15689062C211yesmaximal, characteristic
2#20246006C211yesmaximal, characteristic
3#20243479C212yesmaximal
4#20241268C212yesmaximal
5C2×C47×#324C211yesmaximal, characteristic
6C2×#13093983—21no—
7C22×C4⋊Dic47C2211yescharacteristic
8#15689062—22no—
9#15689062C414yescomplemented
10#15689022—24no—
11#15688983—21no—
12#15688983C2212yes—
13C2×C47⋊#95—21no—
14C2×C47⋊#95—21no—
15C2×C47⋊#95C2211yescharacteristic
16#15688886—24no—
17C22×C4⋊C4×C47C2211yescharacteristic
18C2×C22⋊C8×C47C2212yes—
19C47×#324—22no—
20C23×Dic47—21no—
21C23×Dic47—21no—
22#13093983—42no—
23#13093983—42no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—22no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—22no—
29C2×C4⋊Dic47C2×C412yes—
30C2×C4⋊Dic47D414yes—
31C2×C22⋊Dic47—22no—
32C2×C4×Dic47—22no—
33C47⋊#95—22no—
34C47⋊#95—24no—
35C47⋊#95—24no—
36C47⋊#95—21no—
37C47⋊#95—22no—
38C47⋊#95C2×C412yes—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C23×C4×C47C2311yescharacteristic
42C22×C8×C47—22no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47C2×C412yes—
46C2×C4⋊C4×C47D412yes—
47C22⋊C8×C47—22no—
48C22⋊C8×C47—22no—
49C24×C47C2×D411yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—42no—
57C22×C4×C47—42no—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22⋊C412yes—
60C22×C4×C47SD1612yes—
61C22×C4×C47Q1612yes—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D412yes—
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47#4112yes—
66C22×Dic47—22no—
67C22×Dic47—22no—
68C22×Dic47—24no—
69C22×Dic47—42no—
70C22×Dic47—42no—
71C22×Dic47—44no—
72C22×Dic47—41no—
73C22×Dic47—41no—
74C22×Dic47—41no—
75C22×Dic47—41no—
76C2×C8×C47—44no—
77C2×C8×C47—42no—
78C4⋊C4×C47—24no—
79C4⋊C4×C47—24no—
80C22⋊Dic47—42no—
81C22⋊Dic47—42no—
82C4⋊Dic47—24no—
83C4⋊Dic47—44no—
84C4⋊Dic47—42no—
85C4⋊Dic47—44no—
86C4⋊Dic47—82no—
87C4×Dic47—44no—
88C23×C47—22no—
89C23×C47—22no—
90C23×C47C2×C22⋊C411yescharacteristic
91C23×C47#12011yescharacteristic
92C23×C47#12311yescharacteristic
93C23×C47C2×SD1611yescharacteristic
94C23×C47C2×Q1611yescharacteristic
95C23×C47AGL(1,ℤ/8)11yescharacteristic
96C23×C47#13711yescharacteristic
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—24no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—81no—
111C2×C4×C47#10314yes—
112C2×C4×C47C4×D412yes—
113C2×C4×C47#12112yes—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—44no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—44no—
120C2×Dic47—44no—
121C2×Dic47—44no—
122C2×Dic47—82no—
123C2×Dic47—82no—
124C2×Dic47—82no—
125C2×Dic47—82no—
126C8×C47—82no—
127#9023—471nomaximal, Sylow
128C22×C47—22no—
129C22×C47—22no—
130C22×C47—22no—
131C22×C47—22no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47#38611yescharacteristic
137C22×C47C2×#10311yescharacteristic
138C22×C47#41711yescharacteristic
139C22×C47#45011yescharacteristic
140C22×C47#45111yescharacteristic
141C22×C47#45412yes—
142C22×C47#45812yes—
143C22×C47#48311yescharacteristic
144C22×C47#48411yescharacteristic
145C4×C47—24no—
146C4×C47—22no—
147C4×C47—22no—
148C4×C47—42no—
149Dic47—82no—
150Dic47—84no—
151Dic47—82no—
152C2×#400—471no—
153#2316—471no—
154#1824—472no—
155C2×#324D4711yescharacteristic
156#1294—472no—
157C2×C47—21no—
158C2×C47—21no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47#161612yes—
162C2×C47#161712yes—
163C2×C47#187211yescharacteristic
164C2×C47#188811yescharacteristic
165C2×C47#189011yescharacteristic
166C2×#117—941no—
167C22×C4⋊C4—471no—
168C22×C4⋊C4C2×D4711yescharacteristic
169C2×C22⋊C8C2×D4712yes—
170#400—474no—
171#400—942no—
172#389—944no—
173#380—472no—
174#380—941no—
175C2×#95—471no—
176C2×#95—941no—
177C2×#95—941no—
178#340—944no—
179#324—22no—
180C47#902311yescharacteristic, complemented, Sylow
181C23×C4—941no—
182C23×C4—941no—
183C23×C4D18811yescharacteristic
184C23×C4D18811yescharacteristic
185C23×C4C22×D4711yescharacteristic
186C22×C8—22no—
187#117—1882no—
188#117—1882no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—472no—
192C2×C4⋊C4—474no—
193C2×C4⋊C4—942no—
194C2×C4⋊C4—942no—
195C2×C4⋊C4—942no—
196C2×C4⋊C4—942no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4C4×D4712yes—
199C2×C4⋊C4C22⋊D4712yes—
200C2×C22⋊C4—942no—
201C2×C42—942no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#95—472no—
205#95—942no—
206#95—944no—
207#95—944no—
208#95—942no—
209#95—941no—
210C24C2×D18811yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—942no—
220C22×C4—942no—
221C22×C4—944no—
222C22×C4—1882no—
223C22×C4—1882no—
224C22×C4—1884no—
225C22×C4—1881no—
226C22×C4—1881no—
227C22×C4—1881no—
228C22×C4—1881no—
229C22×C4#1062532512yes—
230C22×C4Dic18812yes—
231C22×C4(C2×D47)⋊C412yes—
232C22×C4(C2×D47)⋊C412yes—
233C22×C4C2×C4×D4711yescharacteristic
234C22×C4D4×D4712yes—
235C22×C4#1062535212yes—
236C22×C4C2×C22⋊D4711yescharacteristic
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—944no—
242C4⋊C4—3762no—
243C4⋊C4—1884no—
244C4⋊C4—1884no—
245C4⋊C4—1882no—
246C22⋊C4—1882no—
247C22⋊C4—1882no—
248C42—1884no—
249C23—22no—
250C23—22no—
251C23C2×(C2×D47)⋊C411yescharacteristic
252C23#1309399411yescharacteristic
253C23#1309400811yescharacteristic
254C23C2×#1062532511yescharacteristic
255C23C2×Dic18811yescharacteristic
256C23#1309405211yescharacteristic
257C23#1309405711yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—3762no—
273C2×C4—3762no—
274C2×C4—3762no—
275C2×C4—3762no—
276C2×C4—1882no—
277C2×C4—1882no—
278C2×C4—1884no—
279C2×C4—1882no—
280C2×C4—1884no—
281C2×C4—1884no—
282C2×C4—1884no—
283C2×C4—1882no—
284C2×C4#1309394914yes—
285C2×C4(C22⋊D47)⋊C412yes—
286C2×C4#1309399612yes—
287C8—82no—
288C22—22no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22#1568900011yescharacteristic
297C22C2×#1309394911yescharacteristic
298C22#1568912511yescharacteristic
299C22#1568927611yescharacteristic
300C22#1568928111yescharacteristic
301C22#1568929812yes—
302C22#1568931412yes—
303C22#1568945411yescharacteristic
304C22#1568945911yescharacteristic
305C4—22no—
306C4—22no—
307C4—24no—
308C4—42no—
309C4—3762no—
310C4—3762no—
311C4—3764no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#2024242712yes—
317C2#2024243212yes—
318C2#2024369311yescharacteristic
319C2#2024378911yescharacteristic
320C2#2024379911yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#9023
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
9. #15689062C4semidirect
180. C47#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#156890622
#202460062
#202434792
#202412682
C2×C47×#3242
#902347