Finite Group Databasebeta

← #26127409

Subgroups — #26127409

Order 12032.

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

normal subgroups only

This group has 452 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#20248561C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241408C214yesmaximal, complemented
4C47×#1799C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×C22⋊(C47⋊C8)—21no—
7#15689035C412yescomplemented
8#15689035C2212yes—
9#15689026—21no—
10#15688988—21no—
11#15688983—21no—
12#15688983—21no—
13C4×C4⋊Dic47C411yescharacteristic, complemented
14#15688849—21no—
15#15688847C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C2×#99×C47—21no—
18C47×#380—21no—
19C47×#342C2212yes—
20C22×Dic94C2×C412yes—
21C22×C47⋊C8—41no—
22#13094024—41no—
23#13094024—41no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C2×C4×Dic47—21no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—41no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C47⋊#95C2×C412yes—
35C22×D4×C47D411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C4×D4×C47—41no—
39C4×D4×C47—41no—
40C2×C4⋊C4×C47—41no—
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47C2311yescharacteristic
45C2×C42×C47C2×C411yescharacteristic
46#99×C47—42no—
47#95×C47—41no—
48C24×C47C22⋊C411yescharacteristic
49C24×C47C2×D411yescharacteristic
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×C47—42no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×Dic47—22no—
67C22×Dic47—22no—
68C2×Dic94—42no—
69C2×Dic94—82no—
70C4⋊C4×C47—41no—
71C4⋊C4×C47—82no—
72C22⋊C4×C47—41no—
73C22⋊C4×C47—41no—
74C22⋊C4×C47—42no—
75C22⋊C4×C47—42no—
76C42×C47—21no—
77C42×C47—21no—
78C22⋊Dic47—82no—
79C22⋊Dic47—82no—
80C4⋊Dic47—42no—
81C4⋊Dic47—42no—
82C4⋊Dic47—41no—
83C4⋊Dic47—41no—
84C4×Dic47—42no—
85C2×C47⋊C8—42no—
86C2×C47⋊C8—82no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47#10312yes—
96C23×C47C2×C22⋊C411yescharacteristic
97C23×C47C2×SD1611yescharacteristic
98C23×C47C2×Q1611yescharacteristic
99D4×C47—42no—
100D4×C47—42no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—22no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—81no—
113C2×C4×C47—81no—
114C2×C4×C47—82no—
115C2×C4×C47—82no—
116C2×C4×C47—82no—
117C2×C4×C47#9911yescharacteristic
118C2×C4×C47#9911yescharacteristic
119C2×C4×C47#10312yes—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126Dic94—84no—
127C47⋊C8—82no—
128#11900—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—22no—
133C22×C47—21no—
134C22×C47—42no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47—81no—
143C22×C47C2×#9911yescharacteristic
144C22×C47C2×#10311yescharacteristic
145C22×C47#42111yescharacteristic
146C4×C47—21no—
147C4×C47—22no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—162no—
152C4×C47#35412yes—
153Dic47—82no—
154Dic47—82no—
155#2563—471no—
156#1799D4711yescharacteristic
157#1459—471no—
158#1351—474no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47—41no—
162C2×C47—42no—
163C2×C47—41no—
164C2×C47#152011yescharacteristic
165C2×C47#212211yescharacteristic
166C2×C47C2×#35411yescharacteristic
167C2×C4⋊Q8—471no—
168C2×C4×D4C2×D4711yescharacteristic
169C2×#99—21no—
170C2×C22⋊C8—941no—
171#393—472no—
172#393—472no—
173#390—941no—
174#382—941no—
175#380—21no—
176#380—941no—
177#380—941no—
178C4×C4⋊C4—471no—
179#342C2×D4712yes—
180#324—941no—
181#323—472no—
182C47#1190011yescharacteristic, complemented, Sylow
183C22×Q8—472no—
184C22×D4C22⋊D4711yescharacteristic
185C23×C4D18811yescharacteristic
186C23×C4C22⋊D4711yescharacteristic
187C22×C8—1881no—
188C4⋊Q8—1881no—
189C4⋊Q8—1881no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4D18811yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—942no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4C4×D4711yescharacteristic
201C2×C22⋊C4C22×D4711yescharacteristic
202C2×C42—941no—
203C2×C42C4×D4711yescharacteristic
204#99—42no—
205C22⋊C8—942no—
206C22⋊C8—1881no—
207#95—41no—
208#95—472no—
209#95—942no—
210#95—942no—
211C24(C2×D47)⋊C411yescharacteristic
212C24C2×C22⋊D4711yescharacteristic
213C2×Q8—3762no—
214C2×Q8—1882no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4#1062533512yes—
218C2×D4#1062533712yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—942no—
227C22×C4—942no—
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4(C2×D47)⋊C411yescharacteristic
231C22×C4C2×C4×D4711yescharacteristic
232C22×C4C2×D18811yescharacteristic
233C2×C8—3762no—
234C2×C8—1882no—
235C4⋊C4—41no—
236C4⋊C4—82no—
237C4⋊C4—1882no—
238C4⋊C4—1882no—
239C4⋊C4—1881no—
240C4⋊C4—1881no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—3762no—
246C22⋊C4—3762no—
247C42—21no—
248C42—21no—
249C42—1882no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1309394712yes—
259C23C2×(C2×D47)⋊C411yescharacteristic
260C23C2×#1062533511yescharacteristic
261C23C2×#1062533711yescharacteristic
262Q8—3764no—
263D4—42no—
264D4—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—1882no—
282C2×C4—1882no—
283C2×C4—1882no—
284C2×C4—1882no—
285C2×C4—1882no—
286C2×C4—1882no—
287C2×C4#1309393811yescharacteristic
288C2×C4#1309393811yescharacteristic
289C2×C4#1309394712yes—
290C8—3762no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#1309393811yescharacteristic
306C22C2×#1309394711yescharacteristic
307C22#1568914111yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—162no—
314C4—3762no—
315C4—3762no—
316C4#1568891712yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2#2024195911yescharacteristic
323C2#2024503711yescharacteristic
324C2C2×#1568891711yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1799C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1799D47
hypocenterC47#11900
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20241408C2semidirect
5. C2×#13094024C4semidirect
7. #15689035C4semidirect
13. C4×C4⋊Dic47C4semidirect
182. C47#11900semidirect

Maximal subgroups

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

subgroupindex
#202485612
#202417062
#202414082
C47×#17992
#1190047