Finite Group Databasebeta

← #26133104

Subgroups — #26133104

Order 12032.

472 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 103 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
2#20243792C211yesmaximal, characteristic
3#20243605C212yesmaximal, complemented
4#20241514C212yesmaximal, complemented
5C2×C47×#355C211yesmaximal, characteristic
6#15689601C2211yescharacteristic
7#15689165—22no—
8#15689128—21no—
9C2×#13093943C2211yescharacteristic
10#15689030C2212yes—
11#15689016—22no—
12#15689003—21no—
13#15688998—22no—
14#15688989—21no—
15#15688884—22no—
16#15688883—21no—
17C2×#124×C47C2211yescharacteristic
18C2×#101×C47C2212yes—
19C47×#355C414yescomplemented
20C23×Dic47—21no—
21C2×#10625329—21no—
22Q8×Dic47—41no—
23Q8×Dic47—41no—
24#13093983—21no—
25#13093983—22no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47D412yes—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—21no—
31C2×C4×Dic47D412yes—
32#13093958—42no—
33#13093956—42no—
34#13093945—41no—
35#13093945—41no—
36#13093943—22no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C22×Q8×C47C2311yescharacteristic
41C22×D4×C47D411yescharacteristic
42C2×M16×C47—22no—
43#124×C47—22no—
44#124×C47C2×C412yes—
45#124×C47Q812yes—
46C2×C22⋊C4×C47—22no—
47C2×C42×C47D411yescharacteristic
48#101×C47—22no—
49#101×C47C2×C414yescomplemented
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—24no—
68M16×C47—42no—
69C2×C8×C47—42no—
70C22⋊C4×C47—24no—
71C22⋊C4×C47—24no—
72C42×C47—21no—
73C42×C47C4⋊C412yes—
74C22⋊Dic47—41no—
75C22⋊Dic47—82no—
76C22⋊Dic47—81no—
77C4⋊Dic47—21no—
78C4⋊Dic47—21no—
79C4⋊Dic47—22no—
80C4⋊Dic47—22no—
81C4⋊Dic47—42no—
82C4⋊Dic47—81no—
83C4⋊Dic47—81no—
84C4×Dic47—41no—
85C4×Dic47—42no—
86C4×Dic47—42no—
87#10625329—41no—
88#10625329—41no—
89C2×C47⋊C8—41no—
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—21no—
102C2×C4×C47—22no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47C2×C4⋊C411yescharacteristic
109C2×C4×C47C4×D412yes—
110C2×C4×C47C22⋊Q812yes—
111C2×C4×C47#12311yescharacteristic
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—84no—
121C8×C47—44no—
122C47⋊C8—81no—
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—21no—
135C4×C47—22no—
136C4×C47—21no—
137C4×C47—22no—
138C4×C47—42no—
139Dic47—82no—
140Dic47—81no—
141Dic47—82no—
142Dic47—81no—
143#2379—471no—
144C2×#355D4711yescharacteristic
145#1888—471no—
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#519—471no—
156#428—942no—
157#418—941no—
158C2×#101—471no—
159C2×#101C2×D4712yes—
160#391—472no—
161#388—942no—
162#386—941no—
163#385—942no—
164#382—941no—
165#355Dic4714yescomplemented
166#339—942no—
167#339—941no—
168C47#1297011yescharacteristic, complemented, Sylow
169C22×Q8C22×D4711yescharacteristic
170C22×D4D18811yescharacteristic
171C23×C4—941no—
172C2×M16—22no—
173C2×M16—941no—
174#124—22no—
175#124Dic9412yes—
176#124C2×Dic4712yes—
177C4×Q8—1881no—
178C4×Q8—1881no—
179#117—941no—
180#117—942no—
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×C42D18811yescharacteristic
189#107—1882no—
190#106—1882no—
191#102—1881no—
192#102—1881no—
193#101—22no—
194#101—942no—
195#101C2×Dic4714yescomplemented
196#95—942no—
197#95—942no—
198#95—942no—
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×Dic9411yescharacteristic
203C2×Q8#1062535212yes—
204C2×Q8C22×Dic4711yescharacteristic
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊Dic4712yes—
209C22×C4—21no—
210C22×C4—942no—
211C22×C4—942no—
212C22×C4—3762no—
213C22×C4—3761no—
214C22×C4C2×D18811yescharacteristic
215C22×C4D4×D4712yes—
216M16—24no—
217M16—42no—
218M16—1881no—
219M16—1881no—
220C2×C8—42no—
221C2×C8—1881no—
222C4⋊C4—941no—
223C4⋊C4—941no—
224C4⋊C4—942no—
225C4⋊C4—942no—
226C4⋊C4—3761no—
227C4⋊C4—3761no—
228C4⋊C4—1882no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—3762no—
232C22⋊C4—3761no—
233C22⋊C4—1881no—
234C42—21no—
235C42—1882no—
236C42—1882no—
237C42—1881no—
238C42C4⋊Dic4712yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23#1309399411yescharacteristic
245Q8—41no—
246Q8—42no—
247Q8—41no—
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—3764no—
258C2×C4—1882no—
259C2×C4—1882no—
260C2×C4—1882no—
261C2×C4—1882no—
262C2×C4—1881no—
263C2×C4—1881no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4C2×C4⋊Dic4711yescharacteristic
267C2×C4D4×Dic4712yes—
268C2×C4#1309400112yes—
269C2×C4#1309400811yescharacteristic
270C8—44no—
271C8—3761no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#1568899811yescharacteristic
280C22#1568931111yescharacteristic
281C22#1568931411yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—3762no—
288C4—3762no—
289C4—3761no—
290C4—3761no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2#2024367111yescharacteristic
295C2#2024540611yescharacteristic
296C2#2024541111yescharacteristic
subgroupG/H
centerC22#15689311
commutatorC2×Q8×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC2×C47×#355C2
socleC22×C47B(4,𝔽2)
prime radicalC22×D4×C47D4
cosocleC22×Q8×C47C23
hypercenterC2×#355D47
hypocenterC47#12970
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #20243605C2semidirect
4. #20241514C2semidirect
19. C47×#355C4semidirect
49. #101×C47C2×C4semidirect
165. #355Dic47semidirect
168. C47#12970semidirect
195. #101C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202465622
#202437922
#202436052
#202415142
C2×C47×#3552
#1297047