Finite Group Databasebeta

← #26107838

Subgroups — #26107838

Order 12032.

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

normal subgroups only

This group has 774 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
1#20257191C211yesmaximal, characteristic
2#20244183C211yesmaximal, characteristic
3#20242079C212yesmaximal, complemented
4#20241842C212yesmaximal, complemented
5C47×#1800C211yesmaximal, characteristic
6#15689832—24no—
7#15689830—22no—
8D4×Dic94—22no—
9#15689754—21no—
10#15689749C414yes—
11#15689680C2212yes—
12C2×#13094003—22no—
13#15689598—21no—
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×C22⋊(C47⋊C8)—22no—
16C2×C4×C47⋊C8C2211yescharacteristic
17#15688871—22no—
18#15688868—22no—
19#15688853—24no—
20C2×C4×D4×C47C2211yescharacteristic
21C47×#423C2212yes—
22C2×#100×C47—21no—
23C47×#343—21no—
24C22×Dic94—22no—
25C22×C47⋊C8—21no—
26C22×C47⋊C8—21no—
27#13094024D412yes—
28#13094024D412yes—
29#13094023—24no—
30#13094016—22no—
31#13094015—24no—
32#13094003—24no—
33#13094003—24no—
34#13094003—24no—
35#13094003—22no—
36#13094003C2×C414yes—
37#13094001—28no—
38#13094001—28no—
39C4×Dic94—22no—
40C4×Dic94D414yes—
41D4×Dic47—24no—
42D4×Dic47—24no—
43#13093983—22no—
44C2×C4⋊Dic47—22no—
45C2×Dic47⋊C4—24no—
46C2×C22⋊Dic47—24no—
47C4⋊(C47⋊C8)—22no—
48C22⋊(C47⋊C8)—44no—
49C4×C47⋊C8—22no—
50C22×D4×C47D411yescharacteristic
51C23×C4×C47D412yes—
52C2×M16×C47—22no—
53C4×D4×C47—22no—
54C4×D4×C47—22no—
55C2×C4⋊C4×C47D411yescharacteristic
56C2×C22⋊C4×C47C2×C412yes—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—22no—
59C4⋊C8×C47—22no—
60#100×C47—42no—
61C24×C47C22⋊C412yes—
62C2×D4×C47—22no—
63C2×D4×C47—22no—
64C2×D4×C47SD1612yes—
65C2×D4×C47Q1612yes—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—42no—
69C22×C4×C47—42no—
70C22×C4×C47C22⋊C412yes—
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—24no—
75C22×Dic47—24no—
76C2×Dic94—22no—
77C2×Dic94—24no—
78C2×Dic94—82no—
79C2×Dic94—82no—
80C2×Dic94C22⋊C414yes—
81M16×C47—42no—
82M16×C47—42no—
83C2×C8×C47—42no—
84C4⋊C4×C47—22no—
85C22⋊C4×C47—24no—
86C42×C47C2×D412yes—
87C42×C47C2×D412yes—
88C22⋊Dic47—28no—
89C22⋊Dic47—24no—
90C4⋊Dic47—24no—
91C4⋊Dic47—22no—
92C4⋊Dic47—22no—
93C4⋊Dic47—22no—
94C4⋊Dic47C22⋊C414yes—
95Dic47⋊C4—28no—
96Dic47⋊C4—24no—
97Dic47⋊C4—24no—
98C4×Dic47—24no—
99C2×C47⋊C8—22no—
100C2×C47⋊C8—41no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8—42no—
103C2×C47⋊C8—41no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47#10314yes—
109C23×C47C2×C22⋊C411yescharacteristic
110D4×C47—24no—
111D4×C47—24no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—22no—
115C2×C4×C47—41no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—82no—
120C2×C4×C47C2×C22⋊C412yes—
121C2×C4×C47#12012yes—
122C2×C4×C47#12011yescharacteristic
123C2×C4×C47#12011yescharacteristic
124C2×C4×C47C2×SD1611yescharacteristic
125C2×C4×C47C2×Q1611yescharacteristic
126C2×C4×C47#13712yes—
127C2×Dic47—24no—
128C2×Dic47—24no—
129C2×Dic47—84no—
130C2×Dic47—84no—
131Dic94—42no—
132Dic94—42no—
133Dic94—44no—
134Dic94—84no—
135C8×C47—82no—
136C47⋊C8—42no—
137C47⋊C8—81no—
138#8355—471nomaximal, Sylow
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—24no—
142C22×C47—44no—
143C22×C47—42no—
144C22×C47—42no—
145C22×C47—82no—
146C22×C47#37911yescharacteristic
147C22×C47C2×#10311yescharacteristic
148C22×C47#41911yescharacteristic
149C4×C47—21no—
150C4×C47—22no—
151C4×C47—42no—
152C4×C47#44812yes—
153C4×C47#45112yes—
154C4×C47#45311yescharacteristic
155C4×C47#45511yescharacteristic
156Dic47—44no—
157Dic47—44no—
158#3501—471no—
159#1954—471no—
160#1800D4711yescharacteristic
161#1550—472no—
162#1493—472no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—44no—
166C2×C47#178711yescharacteristic
167C2×C47#179111yescharacteristic
168C2×C47#179211yescharacteristic
169#554—944no—
170#554—942no—
171D4×Q8—942no—
172#544—941no—
173#543—474no—
174#533—472no—
175C2×C22⋊Q8—942no—
176#518—941no—
177C2×C4×D4C2×D4711yescharacteristic
178#423C2×D4712yes—
179C2×C4⋊C8—471no—
180C2×#100—21no—
181C2×C22⋊C8—942no—
182C2×C4×C8—471no—
183#343—21no—
184#333—942no—
185#332—942no—
186Q8⋊C8—944no—
187C47#835511yescharacteristic, complemented, Sylow
188C22×Q8—942no—
189C22×D4C22⋊D4711yescharacteristic
190C23×C4D18812yes—
191C2×M16—22no—
192C22×C8—941no—
193C22×C8—941no—
194C4⋊Q8—472no—
195C4⋊Q8—472no—
196C4⋊Q8—944no—
197#125—944no—
198#125—942no—
199C22⋊Q8—474no—
200C22⋊Q8—948no—
201C22⋊Q8—948no—
202C22⋊Q8—944no—
203C22⋊Q8—944no—
204C22⋊Q8—944no—
205C22⋊Q8—942no—
206C4×Q8—474no—
207C4×Q8—942no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—944no—
211C4×D4—944no—
212#117—942no—
213C2×C4⋊C4—944no—
214C2×C4⋊C4—942no—
215C2×C4⋊C4C22⋊D4711yescharacteristic
216C2×C22⋊C4—944no—
217C2×C22⋊C4C4×D4712yes—
218C2×C42C22×D4711yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—942no—
222#100—42no—
223C22⋊C8—1884no—
224C4×C8—942no—
225C24(C2×D47)⋊C412yes—
226C2×Q8—474no—
227C2×Q8—944no—
228C2×Q8—942no—
229C2×Q8—3762no—
230C2×Q8—3762no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4#1062533512yes—
234C2×D4#1062533712yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—944no—
240C22×C4—944no—
241C22×C4(C2×D47)⋊C412yes—
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4C2×D18811yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C2×C8—942no—
249C2×C8—1882no—
250C2×C8—1881no—
251C2×C8—1881no—
252C2×C8—1881no—
253C4⋊C4—22no—
254C4⋊C4—474no—
255C4⋊C4—948no—
256C4⋊C4—944no—
257C4⋊C4—944no—
258C4⋊C4—942no—
259C4⋊C4—942no—
260C4⋊C4—944no—
261C4⋊C4—942no—
262C22⋊C4—24no—
263C22⋊C4—948no—
264C22⋊C4—944no—
265C42—944no—
266C42D4×D4712yes—
267C42D4×D4712yes—
268C23—22no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23#1309394714yes—
273C23C2×(C2×D47)⋊C411yescharacteristic
274Q8—3764no—
275Q8—1884no—
276Q8—1882no—
277Q8—1882no—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—944no—
289C2×C4—944no—
290C2×C4—3764no—
291C2×C4—3764no—
292C2×C4C22⋊C4×D4712yes—
293C2×C4#1309399212yes—
294C2×C4#1309399411yescharacteristic
295C2×C4#1309399411yescharacteristic
296C2×C4C2×#1062533511yescharacteristic
297C2×C4C2×#1062533711yescharacteristic
298C2×C4#1309405412yes—
299C8—82no—
300C8—3761no—
301C8—1882no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—44no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22#1568897811yescharacteristic
310C22C2×#1309394711yescharacteristic
311C22#1568913111yescharacteristic
312C4—22no—
313C4—21no—
314C4—42no—
315C4—1884no—
316C4—1884no—
317C4#1568926112yes—
318C4#1568927812yes—
319C4#1568929311yescharacteristic
320C4#1568930311yescharacteristic
321C2—21no—
322C2—21no—
323C2—44no—
324C2#2024332211yescharacteristic
325C2#2024333711yescharacteristic
326C2#2024334411yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1800C2
socleC22×C47#379
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#1800D47
hypocenterC47#8355
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20242079C2semidirect
4. #20241842C2semidirect
187. C47#8355semidirect

Maximal subgroups

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

subgroupindex
#202571912
#202441832
#202420792
#202418422
C47×#18002
#835547