Finite Group Databasebeta

← #26108054

Subgroups — #26108054

Order 12032.

782 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15689833C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic, complemented
3#20242072C212yesmaximal, complemented
4#20241814C212yesmaximal
5C47×#1461C211yesmaximal, characteristic, complemented
6#15689833—24no—
7#15689833—24no—
8C2×#13094024C2211yescharacteristic
9C2×#13094016C412yes—
10C2×#13094002—22no—
11C2×#13094000—21no—
12C2×C4×D188C412yes—
13C2×C4×D188C2211yescharacteristic
14C2×C4⋊C4×D47—21no—
15#15689147C2212yes—
16C2×#13093943—21no—
17#15688892—21no—
18#15688870—24no—
19#15688852—24no—
20C2×C4×Q8×C47C2211yescharacteristic
21C47×#431C2212yes—
22C47×#324—22no—
23C22×Dic94—21no—
24C22×D188C2×C411yescharacteristic
25C22×C4×D47—21no—
26C22×C4×D47—22no—
27C2×#10625329—22no—
28#13094024—22no—
29#13094024D414yes—
30#13094016—24no—
31#13094016—24no—
32#13094002—216no—
33#13094000—28no—
34C4×D188—24no—
35C4×D188—22no—
36C4×D188—24no—
37C4×D188D414yes—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47C2×C412yes—
40C2×C4⋊Dic47C2×C411yescharacteristic
41C2×Dic47⋊C4—22no—
42C2×Dic47⋊C4—22no—
43C4⋊C4×D47—28no—
44C2×(C2×D47)⋊C4—22no—
45C2×(C2×D47)⋊C4—21no—
46C4⋊(C47⋊C8)—22no—
47C4⋊(C47⋊C8)—22no—
48#13093943—42no—
49C22×Q8×C47D411yescharacteristic
50C4×Q8×C47—22no—
51C4×Q8×C47—22no—
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C4⋊C4×C47D412yes—
54C2×C42×C47C2×C412yes—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—22no—
57C22⋊C8×C47—22no—
58C4×C8×C47—22no—
59C23×D47—21no—
60C2×Q8×C47—22no—
61C2×Q8×C47—22no—
62C2×Q8×C47—21no—
63C2×Q8×C47—21no—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—21no—
70C22×Dic47—22no—
71C2×D188—21no—
72C2×D188—24no—
73C2×D188C22⋊C412yes—
74C2×C4×D47—24no—
75C2×C4×D47—28no—
76C2×C4×D47—42no—
77C2×C4×D47—41no—
78C2×C4×D47—41no—
79C2×C4×D47—84no—
80C2×Dic94—24no—
81C2×Dic94—82no—
82C2×C8×C47—42no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—24no—
85C42×C47—22no—
86C42×C47—22no—
87C42×C47C2×D412yes—
88C42×C47C2×D412yes—
89(C2×D47)⋊C4—28no—
90(C2×D47)⋊C4—24no—
91C4⋊Dic47—21no—
92C4⋊Dic47—22no—
93C4⋊Dic47—24no—
94C4⋊Dic47C22⋊C414yes—
95C4⋊Dic47C22⋊C412yes—
96Dic47⋊C4—28no—
97Dic47⋊C4—28no—
98#10625329—42no—
99#10625329—42no—
100C2×C47⋊C8—42no—
101C23×C47C2×C22⋊C411yescharacteristic
102C22×D47—24no—
103C22×D47—22no—
104C22×D47—42no—
105C22×D47—41no—
106C22×D47—41no—
107Q8×C47—42no—
108Q8×C47—42no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—41no—
115C2×C4×C47—42no—
116C2×C4×C47C2×C22⋊C411yescharacteristic
117C2×C4×C47C2×C22⋊C411yescharacteristic
118C2×C4×C47#12012yes—
119C2×C4×C47#12012yes—
120C2×C4×C47#13512yes—
121C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C47#13711yescharacteristic
123C2×Dic47—22no—
124C2×Dic47—24no—
125C2×Dic47—44no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128D188—24no—
129D188—22no—
130D188#10114yescomplemented
131C4×D47—44no—
132C4×D47—44no—
133C4×D47—88no—
134Dic94—84no—
135C8×C47—82no—
136C47⋊C8—82no—
137#8398—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47#37911yescharacteristic
141C22×C47#41611yescharacteristic
142C22×C47#41711yescharacteristic
143C2×D47—24no—
144C2×D47—44no—
145C2×D47—44no—
146C2×D47—42no—
147C2×D47—42no—
148C4×C47—22no—
149C4×C47—21no—
150C4×C47—42no—
151C4×C47—42no—
152C4×C47C2×#10112yes—
153C4×C47#44912yes—
154C4×C47#45212yes—
155Dic47—44no—
156Dic47—42no—
157C2×#554—471no—
158#1801—471no—
159#1549—472no—
160#1486—472no—
161#1461D4711yescharacteristic, complemented
162C2×C47—21no—
163C2×C47—21no—
164C2×C47#178411yescharacteristic
165C2×C47#178811yescharacteristic
166C2×C47#179511yescharacteristic
167D47—44no—
168#554—944no—
169#554—944no—
170C2×C4⋊Q8—471no—
171C2×#125—472no—
172C2×C22⋊Q8—941no—
173C2×C22⋊Q8—942no—
174C2×C4×Q8C2×D4711yescharacteristic
175C2×C4×D4—472no—
176C2×C4×D4—471no—
177C22×C4⋊C4—941no—
178#431C2×D4712yes—
179#423—472no—
180C2×#101—941no—
181#343—941no—
182#332—944no—
183D4⋊C8—944no—
184#324—22no—
185C47#839811yescharacteristic, complemented, Sylow
186C22×Q8—941no—
187C22×Q8D18811yescharacteristic
188C22×D4—471no—
189C23×C4—942no—
190C23×C4—941no—
191C2×M16—942no—
192C4⋊Q8—474no—
193C4⋊Q8—942no—
194#125—944no—
195#125—944no—
196C22⋊Q8—9416no—
197C22⋊Q8—948no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—474no—
201C4×D4—944no—
202C4×D4—944no—
203C4×D4—942no—
204C2×C4⋊C4—471no—
205C2×C4⋊C4—472no—
206C2×C4⋊C4—948no—
207C2×C4⋊C4—942no—
208C2×C4⋊C4—942no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4D18811yescharacteristic
211C2×C4⋊C4C22⋊D4712yes—
212C2×C22⋊C4—942no—
213C2×C22⋊C4—941no—
214C2×C42C4×D4712yes—
215C2×C42C22×D4711yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—942no—
218C4⋊C8—942no—
219#101—1882no—
220C22⋊C8—22no—
221C4×C8—22no—
222C24—941no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—944no—
228C2×Q8—3762no—
229C2×D4—472no—
230C2×D4—944no—
231C2×D4—941no—
232C22×C4—948no—
233C22×C4—941no—
234C22×C4—942no—
235C22×C4—944no—
236C22×C4—3764no—
237C22×C4—1882no—
238C22×C4—1881no—
239C22×C4—1881no—
240C22×C4(C2×D47)⋊C412yes—
241C22×C4(C2×D47)⋊C412yes—
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4C2×D18811yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245M16—1882no—
246M16—1882no—
247C2×C8—42no—
248C2×C8—1882no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—472no—
252C4⋊C4—474no—
253C4⋊C4—948no—
254C4⋊C4—948no—
255C4⋊C4—941no—
256C4⋊C4—942no—
257C4⋊C4—944no—
258C22⋊C4—948no—
259C22⋊C4—944no—
260C42—22no—
261C42—22no—
262C42D4×D4712yes—
263C42D4×D4712yes—
264C23—944no—
265C23—942no—
266C23—1882no—
267C23—1881no—
268C23—1881no—
269C23C2×(C2×D47)⋊C411yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—3764no—
273D4—474no—
274D4—944no—
275D4—942no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—942no—
284C2×C4—944no—
285C2×C4—3768no—
286C2×C4—1884no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1884no—
290C2×C4—1884no—
291C2×C4C22⋊C4×D4711yescharacteristic
292C2×C4C22⋊C4×D4711yescharacteristic
293C2×C4#1309399212yes—
294C2×C4#1309399412yes—
295C2×C4#1309404612yes—
296C2×C4#1309405211yescharacteristic
297C2×C4#1309405711yescharacteristic
298C8—82no—
299C8—3762no—
300C22—21no—
301C22—21no—
302C22—944no—
303C22—1884no—
304C22—1884no—
305C22—1882no—
306C22—1882no—
307C22#1568897811yescharacteristic
308C22#1568912011yescharacteristic
309C22#1568912511yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—1882no—
315C4—1884no—
316C4#101×D4712yes—
317C4#1568927012yes—
318C4#1568928712yes—
319C2—21no—
320C2—21no—
321C2—1884no—
322C2#2024331011yescharacteristic
323C2#2024332911yescharacteristic
324C2#2024335611yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#379
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1461D47
hypocenterC47#8398
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #20243384C2semidirect
3. #20242072C2semidirect
5. C47×#1461C2semidirect
130. D188#101semidirect
161. #1461D47semidirect
185. C47#8398semidirect

Maximal subgroups

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

subgroupindex
C2×#156898332
#202433842
#202420722
#202418142
C47×#14612
#839847