Finite Group Databasebeta

← #26111340

Subgroups — #26111340

Order 12032.

696 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 127 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×#15689048C211yesmaximal, characteristic
2#20245986C211yesmaximal, characteristic
3#20243475C212yesmaximal, complemented
4#20241268C212yesmaximal, complemented
5C2×C47×#323C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7C22×C22⋊Dic47—21no—
8#15689048—22no—
9#15689048C414yescomplemented
10#15689016—24no—
11#15688983C2212yes—
12#15688979—21no—
13C4×C22⋊Dic47—21no—
14C2×C47⋊#95—21no—
15C2×C47⋊#95C2211yescharacteristic
16#15688886—24no—
17C22×C22⋊C4×C47C2211yescharacteristic
18C2×C22⋊C8×C47C2212yes—
19C47×#323—22no—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C2×C4⋊Dic47—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47C2×C412yes—
26C2×C4⋊Dic47D414yes—
27C2×C22⋊Dic47—24no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—44no—
32C2×C4×Dic47—22no—
33C47⋊#95—22no—
34C47⋊#95—24no—
35C47⋊#95—21no—
36C47⋊#95—22no—
37C47⋊#95C2×C412yes—
38C25×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C22×C8×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47C2×C412yes—
45C2×C22⋊C4×C47D412yes—
46C22⋊C8×C47—22no—
47C22⋊C8×C47—22no—
48C24×C47—21no—
49C24×C47—22no—
50C24×C47—21no—
51C24×C47—22no—
52C24×C47—42no—
53C24×C47—42no—
54C24×C47C22⋊C412yes—
55C24×C47C2×D411yescharacteristic
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47#4112yes—
65C22×Dic47—22no—
66C22×Dic47—22no—
67C22×Dic47—24no—
68C22×Dic47—42no—
69C22×Dic47—42no—
70C22×Dic47—41no—
71C22×Dic47—41no—
72C22×Dic47—42no—
73C22×Dic47—42no—
74C22×Dic47—42no—
75C2×C8×C47—44no—
76C2×C8×C47—42no—
77C22⋊C4×C47—24no—
78C22⋊C4×C47—24no—
79C22⋊Dic47—48no—
80C22⋊Dic47—42no—
81C22⋊Dic47—42no—
82C22⋊Dic47—82no—
83C22⋊Dic47—82no—
84C4⋊Dic47—24no—
85C4⋊Dic47—42no—
86C4⋊Dic47—44no—
87C4×Dic47—82no—
88C23×C47—22no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—22no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47—48no—
96C23×C47—42no—
97C23×C47—42no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47—81no—
101C23×C47—81no—
102C23×C47—84no—
103C23×C47C2×C22⋊C411yescharacteristic
104C23×C47#12011yescharacteristic
105C23×C47#12311yescharacteristic
106C23×C47#13512yes—
107C23×C47AGL(1,ℤ/8)12yes—
108C2×C4×C47—24no—
109C2×C4×C47—22no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—81no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12112yes—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—44no—
125C2×Dic47—44no—
126C2×Dic47—44no—
127C2×Dic47—84no—
128C2×Dic47—82no—
129C2×Dic47—82no—
130C8×C47—82no—
131#9028—471nomaximal, Sylow
132C22×C47—24no—
133C22×C47—24no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—44no—
139C22×C47—42no—
140C22×C47—42no—
141C22×C47—42no—
142C22×C47—82no—
143C22×C47—84no—
144C22×C47—84no—
145C22×C47#38611yescharacteristic
146C22×C47#41611yescharacteristic
147C22×C47#41811yescharacteristic
148C22×C47#44912yes—
149C22×C47#45512yes—
150C22×C47#45812yes—
151C22×C47#48212yes—
152C4×C47—22no—
153C4×C47—22no—
154C4×C47—42no—
155Dic47—82no—
156Dic47—84no—
157Dic47—82no—
158C2×#397—471no—
159#2312—471no—
160#1823—472no—
161C2×#323D4711yescharacteristic
162#1294—472no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—22no—
166C2×C47—82no—
167C2×C47#161214yes—
168C2×C47#187511yescharacteristic
169C2×C47#188511yescharacteristic
170C2×C47#189011yescharacteristic
171C22×C4⋊C4—471no—
172C22×C22⋊C4—941no—
173C22×C22⋊C4C2×D4711yescharacteristic
174C2×C22⋊C8C2×D4712yes—
175#397—474no—
176#397—942no—
177#388—944no—
178#380—472no—
179#379—941no—
180C4×C22⋊C4—941no—
181C2×#95—471no—
182C2×#95—941no—
183#340—944no—
184#323—22no—
185C47#902811yescharacteristic, complemented, Sylow
186C25D18811yescharacteristic
187C23×C4—941no—
188C23×C4—941no—
189C23×C4D18811yescharacteristic
190C23×C4C22×D4711yescharacteristic
191C22×C8—22no—
192C2×C4⋊C4—472no—
193C2×C4⋊C4—474no—
194C2×C4⋊C4—942no—
195C2×C4⋊C4—942no—
196C2×C4⋊C4—941no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4—942no—
201C2×C22⋊C4—944no—
202C2×C22⋊C4—942no—
203C2×C22⋊C4—1884no—
204C2×C22⋊C4C4×D4712yes—
205C2×C22⋊C4C22⋊D4712yes—
206C2×C42—942no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#95—472no—
210#95—942no—
211#95—944no—
212#95—942no—
213#95—941no—
214C24—21no—
215C24—22no—
216C24—21no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24(C2×D47)⋊C412yes—
221C24C2×D18811yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—942no—
227C22×C4—942no—
228C22×C4—944no—
229C22×C4—1882no—
230C22×C4—1882no—
231C22×C4—1881no—
232C22×C4—1881no—
233C22×C4—1882no—
234C22×C4—1882no—
235C22×C4—1882no—
236C22×C4(C2×D47)⋊C412yes—
237C22×C4C2×C4×D4711yescharacteristic
238C22×C4D4×D4712yes—
239C22×C4#1062535212yes—
240C22×C4C2×C22⋊D4711yescharacteristic
241C2×C8—44no—
242C2×C8—42no—
243C4⋊C4—944no—
244C4⋊C4—1884no—
245C4⋊C4—1882no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—3762no—
249C22⋊C4—3762no—
250C22⋊C4—1888no—
251C22⋊C4—1882no—
252C22⋊C4—1882no—
253C42—3762no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—48no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—81no—
267C23—81no—
268C23—84no—
269C23C2×(C2×D47)⋊C411yescharacteristic
270C23#1309399411yescharacteristic
271C23#1309400811yescharacteristic
272C23#1309404612yes—
273C23#1309405212yes—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—3764no—
283C2×C4—3762no—
284C2×C4—3762no—
285C2×C4—1882no—
286C2×C4—1882no—
287C2×C4—1882no—
288C2×C4—1884no—
289C2×C4—1884no—
290C2×C4—1884no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4—1882no—
294C2×C4(C22⋊D47)⋊C412yes—
295C2×C4#1309399612yes—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—24no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—84no—
309C22—84no—
310C22#1568900011yescharacteristic
311C22#1568912011yescharacteristic
312C22#1568912911yescharacteristic
313C22#1568927012yes—
314C22#1568930312yes—
315C22#1568931412yes—
316C22#1568944812yes—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—3762no—
321C4—3764no—
322C4—3762no—
323C2—21no—
324C2—21no—
325C2—22no—
326C2—82no—
327C2#2024240414yes—
328C2#2024370811yescharacteristic
329C2#2024377411yescharacteristic
330C2#2024379911yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#323C2
socleC23×C47#120
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC2×#323D47
hypocenterC47#9028
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20243475C2semidirect
4. #20241268C2semidirect
9. #15689048C4semidirect
185. C47#9028semidirect

Maximal subgroups

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

subgroupindex
C2×#156890482
#202459862
#202434752
#202412682
C2×C47×#3232
#902847