Finite Group Databasebeta

← #26133101

Subgroups — #26133101

Order 12032.

472 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 95 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#20246562C211yesmaximal, characteristic
2C2×#15688920C211yesmaximal, characteristic
3#20243603C212yesmaximal, complemented
4#20241514C212yesmaximal, complemented
5C47×#1888C211yesmaximal, characteristic
6#15689166—22no—
7C2×#13093943C2212yes—
8#15689030C2212yes—
9#15689016—22no—
10#15689003—21no—
11#15688998—22no—
12#15688989—21no—
13#15688920C414yescomplemented
14#15688882—22no—
15C2×#124×C47C2211yescharacteristic
16C47×#519C2211yescharacteristic
17C47×#418—21no—
18C2×#101×C47C2211yescharacteristic
19C47×#339—21no—
20C23×Dic47—21no—
21C2×#10625329—22no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47D412yes—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—21no—
27C2×C4×Dic47D412yes—
28#13093957—42no—
29#13093955—42no—
30#13093943—22no—
31#13093943C2×C414yescomplemented
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C22×Q8×C47C2311yescharacteristic
36C22×D4×C47D411yescharacteristic
37C2×M16×C47—21no—
38#124×C47—22no—
39#124×C47C2×C412yes—
40#124×C47Q812yes—
41C4×Q8×C47—41no—
42C4×Q8×C47—41no—
43#117×C47—21no—
44#117×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C42×C47D411yescharacteristic
47#102×C47—41no—
48#102×C47—41no—
49#101×C47—22no—
50C24×C47—21no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47C22×C411yescharacteristic
54C2×Q8×C47C2×Q811yescharacteristic
55C2×Q8×C47#4112yes—
56C2×D4×C47—21no—
57C2×D4×C47—41no—
58C2×D4×C47—41no—
59C2×D4×C47C4⋊C412yes—
60C22×C4×C47—21no—
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×Dic47—22no—
64C22×Dic47—22no—
65C22×Dic47—82no—
66C22×Dic47—81no—
67M16×C47—41no—
68M16×C47—41no—
69C2×C8×C47—41no—
70C4⋊C4×C47—22no—
71C4⋊C4×C47—22no—
72C4⋊C4×C47—21no—
73C4⋊C4×C47—21no—
74C22⋊C4×C47—24no—
75C22⋊C4×C47—24no—
76C22⋊C4×C47—41no—
77C42×C47—21no—
78C42×C47—42no—
79C42×C47—41no—
80C42×C47C4⋊C412yes—
81C22⋊Dic47—82no—
82C22⋊Dic47—81no—
83C4⋊Dic47—42no—
84C4⋊Dic47—81no—
85C4⋊Dic47—81no—
86C4×Dic47—42no—
87#10625329—24no—
88#10625329—42no—
89C2×C47⋊C8—42no—
90C23×C47—22no—
91C23×C47—42no—
92C23×C47—42no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47#12011yescharacteristic
96Q8×C47—41no—
97Q8×C47—41no—
98Q8×C47—42no—
99D4×C47—42no—
100D4×C47—81no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—22no—
104C2×C4×C47—21no—
105C2×C4×C47—22no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47C2×C4⋊C411yescharacteristic
111C2×C4×C47C4×D412yes—
112C2×C4×C47C22⋊Q812yes—
113C2×C4×C47#12311yescharacteristic
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—84no—
121C8×C47—81no—
122C47⋊C8—44no—
123#12970—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—44no—
127C22×C47—42no—
128C22×C47—82no—
129C22×C47—81no—
130C22×C47—81no—
131C22×C47#38511yescharacteristic
132C22×C47B(4,𝔽2)11yescharacteristic
133C22×C47#45811yescharacteristic
134C4×C47—22no—
135C4×C47—21no—
136C4×C47—21no—
137C4×C47—22no—
138C4×C47—42no—
139C4×C47—81no—
140C4×C47—81no—
141Dic47—82no—
142Dic47—82no—
143#2379—471no—
144C2×#355—471no—
145#1888D4711yescharacteristic
146#1857—472no—
147#1394—472no—
148C2×C47—21no—
149C2×C47—21no—
150C2×C47—82no—
151C2×C47#186711yescharacteristic
152C2×C47#220211yescharacteristic
153C2×C47#220311yescharacteristic
154C2×#124C2×D4711yescharacteristic
155#519C2×D4711yescharacteristic
156#428—942no—
157#418—21no—
158C2×#101—472no—
159C2×#101C2×D4711yescharacteristic
160#391—472no—
161#388—942no—
162#386—941no—
163#385—942no—
164#382—941no—
165#355—474no—
166#339—21no—
167#339—942no—
168C47#1297011yescharacteristic, complemented, Sylow
169C22×Q8C22×D4711yescharacteristic
170C22×D4C22⋊D4711yescharacteristic
171C23×C4—941no—
172C2×M16—21no—
173C2×M16—942no—
174#124—22no—
175#124Dic9412yes—
176#124C4×D4712yes—
177C4×Q8—41no—
178C4×Q8—41no—
179#117—22no—
180#117—21no—
181C2×C4⋊C4—472no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4—941no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—942no—
186C2×C22⋊C4—941no—
187C2×C42—472no—
188C2×C42C22⋊D4711yescharacteristic
189#107—1882no—
190#106—1882no—
191#102—41no—
192#102—41no—
193#101—22no—
194#101—474no—
195#101—942no—
196#95—942no—
197#95—942no—
198#95—942no—
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×Dic9411yescharacteristic
203C2×Q8C2×C4×D4711yescharacteristic
204C2×Q8#1062535212yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4Dic47⋊C412yes—
209C22×C4—21no—
210C22×C4—942no—
211C22×C4—942no—
212C22×C4—3762no—
213C22×C4—3761no—
214C22×C4D4×D4712yes—
215C22×C4C2×C22⋊D4711yescharacteristic
216M16—41no—
217M16—41no—
218M16—944no—
219M16—1882no—
220C2×C8—41no—
221C2×C8—1882no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—3761no—
227C4⋊C4—3761no—
228C4⋊C4—1882no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—3762no—
233C22⋊C4—3761no—
234C42—21no—
235C42—42no—
236C42—41no—
237C42—1882no—
238C42Dic47⋊C412yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23#1309399211yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—42no—
249D4—81no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—3764no—
260C2×C4—1882no—
261C2×C4—1882no—
262C2×C4—1882no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4C2×Dic47⋊C411yescharacteristic
267C2×C4(C22⋊D47)⋊C412yes—
268C2×C4#1309400112yes—
269C2×C4#1309400511yescharacteristic
270C8—81no—
271C8—1884no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#1568899511yescharacteristic
280C22#1568930911yescharacteristic
281C22#1568931211yescharacteristic
282C4—21no—
283C4—22no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—81no—
288C4—81no—
289C4—3762no—
290C4—3762no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2#2024366811yescharacteristic
295C2#2024540411yescharacteristic
296C2#2024540811yescharacteristic
subgroupG/H
centerC22#15689309
commutatorC2×Q8×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#1888C2
socleC22×C47B(4,𝔽2)
prime radicalC22×D4×C47D4
cosocleC22×Q8×C47C23
hypercenter#1888D47
hypocenterC47#12970
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1888

Complements

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

NH ≅ G/Ntype
3. #20243603C2semidirect
4. #20241514C2semidirect
13. #15688920C4semidirect
31. #13093943C2×C4semidirect
168. C47#12970semidirect

Maximal subgroups

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

subgroupindex
#202465622
C2×#156889202
#202436032
#202415142
C47×#18882
#1297047