Finite Group Databasebeta

← #26127422

Subgroups — #26127422

Order 12032.

440 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 99 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#20247705C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241414C214yesmaximal, complemented
4C47×#1800C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic
6C2×C22⋊(C47⋊C8)—21no—
7#15689062C2212yes—
8#15689059C412yes—
9#15689021—21no—
10#15689003—21no—
11#15688995—21no—
12#15688983—21no—
13#15688983C411yescharacteristic
14#15688849C2212yes—
15#15688847—21no—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#423—21no—
18C2×#100×C47—21no—
19C47×#343C2212yes—
20C23×Dic47—21no—
21C22×C47⋊C8—41no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×C22⋊Dic47—21no—
29C2×C22⋊Dic47—22no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—41no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95C2×C412yes—
35C22×D4×C47D411yescharacteristic
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C2×M16×C47—22no—
39C4×D4×C47—41no—
40C4×D4×C47—41no—
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C22⋊C4×C47C2×C411yescharacteristic
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C4⋊C8×C47—41no—
46C4⋊C8×C47—41no—
47#100×C47—42no—
48C24×C47C22⋊C411yescharacteristic
49C24×C47C22⋊C411yescharacteristic
50C2×D4×C47—41no—
51C2×D4×C47—41no—
52C2×D4×C47SD1612yes—
53C2×D4×C47Q1612yes—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—22no—
66C22×Dic47—22no—
67C22×Dic47—42no—
68C22×Dic47—82no—
69M16×C47—42no—
70M16×C47—42no—
71C2×C8×C47—42no—
72C4⋊C4×C47—41no—
73C22⋊C4×C47—41no—
74C22⋊C4×C47—41no—
75C42×C47—21no—
76C42×C47—21no—
77C22⋊Dic47—42no—
78C22⋊Dic47—82no—
79C4⋊Dic47—42no—
80C4⋊Dic47—41no—
81C4⋊Dic47—41no—
82C4⋊Dic47—42no—
83Dic47⋊C4—82no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—82no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47#10312yes—
95C23×C47#10312yes—
96C23×C47C2×C22⋊C411yescharacteristic
97D4×C47—42no—
98D4×C47—42no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—81no—
112C2×C4×C47#9911yescharacteristic
113C2×C4×C47#9911yescharacteristic
114C2×C4×C47C2×SD1611yescharacteristic
115C2×C4×C47C2×Q1611yescharacteristic
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—84no—
123C8×C47—82no—
124C47⋊C8—82no—
125#11903—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—22no—
129C22×C47—22no—
130C22×C47—42no—
131C22×C47—42no—
132C22×C47—42no—
133C22×C47—41no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—81no—
138C22×C47C2×#9911yescharacteristic
139C22×C47C2×#10311yescharacteristic
140C22×C47#42111yescharacteristic
141C4×C47—21no—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—41no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47#35212yes—
148Dic47—82no—
149Dic47—82no—
150#2487—471no—
151#1800D4711yescharacteristic
152#1459—471no—
153#1353—474no—
154C2×C47—21no—
155C2×C47—21no—
156C2×C47—42no—
157C2×C47—42no—
158C2×C47#152011yescharacteristic
159C2×C47C2×#35211yescharacteristic
160C2×C47#213411yescharacteristic
161C2×C4×D4C2×D4711yescharacteristic
162C22×C4⋊C4—471no—
163#423—21no—
164C2×#100—21no—
165C2×C22⋊C8—941no—
166#400—472no—
167#399—472no—
168#389—941no—
169#386—941no—
170#385—941no—
171#380—471no—
172#380—941no—
173#343C2×D4712yes—
174#324—472no—
175#323—941no—
176C47#1190311yescharacteristic, complemented, Sylow
177C22×D4C22⋊D4711yescharacteristic
178C23×C4—941no—
179C23×C4C4×D4711yescharacteristic
180C23×C4D18811yescharacteristic
181C2×M16—22no—
182C22×C8—1881no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—472no—
186C2×C4⋊C4—942no—
187C2×C4⋊C4—941no—
188C2×C4⋊C4—941no—
189C2×C4⋊C4—1881no—
190C2×C4⋊C4—1881no—
191C2×C4⋊C4C22×D4711yescharacteristic
192C2×C22⋊C4—942no—
193C2×C22⋊C4—941no—
194C2×C22⋊C4C4×D4711yescharacteristic
195C2×C22⋊C4D18811yescharacteristic
196C2×C42C22⋊D4711yescharacteristic
197C4⋊C8—41no—
198C4⋊C8—41no—
199#100—42no—
200C22⋊C8—942no—
201C22⋊C8—1881no—
202#95—472no—
203#95—942no—
204#95—942no—
205C24(C2×D47)⋊C411yescharacteristic
206C24(C2×D47)⋊C411yescharacteristic
207C2×D4—41no—
208C2×D4—41no—
209C2×D4#1062533512yes—
210C2×D4#1062533712yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—942no—
218C22×C4—942no—
219C22×C4—3762no—
220C22×C4—1882no—
221C22×C4(C2×D47)⋊C411yescharacteristic
222C22×C4(C2×D47)⋊C411yescharacteristic
223C22×C4C2×C4×D4711yescharacteristic
224C22×C4C2×D18811yescharacteristic
225C22×C4C2×C22⋊D4711yescharacteristic
226M16—42no—
227M16—42no—
228C2×C8—42no—
229C2×C8—3762no—
230C2×C8—1882no—
231C4⋊C4—41no—
232C4⋊C4—3762no—
233C4⋊C4—1882no—
234C4⋊C4—1882no—
235C4⋊C4—1881no—
236C4⋊C4—1881no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—3762no—
240C22⋊C4—1882no—
241C42—21no—
242C42—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1309394712yes—
252C23#1309394712yes—
253C23C2×(C2×D47)⋊C411yescharacteristic
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—81no—
269C2×C4—3764no—
270C2×C4—1882no—
271C2×C4—1882no—
272C2×C4—1882no—
273C2×C4—1882no—
274C2×C4—1882no—
275C2×C4—1882no—
276C2×C4#1309393811yescharacteristic
277C2×C4#1309393811yescharacteristic
278C2×C4C2×#1062533511yescharacteristic
279C2×C4C2×#1062533711yescharacteristic
280C8—82no—
281C8—3762no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22C2×#1309393811yescharacteristic
295C22C2×#1309394711yescharacteristic
296C22#1568914111yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—3762no—
304C4—3762no—
305C4#1568891412yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#2024195911yescharacteristic
311C2C2×#1568891411yescharacteristic
312C2#2024507911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1800C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C4⋊C4×C47C23
hypercenter#1800D47
hypocenterC47#11903
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20241414C2semidirect
176. C47#11903semidirect

Maximal subgroups

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

subgroupindex
#202477052
#202417062
#202414142
C47×#18002
#1190347