Finite Group Databasebeta

← #26125899

Subgroups — #26125899

Order 12032.

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

normal subgroups only

This group has 616 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#20241726C211yesmaximal, characteristic, complemented
2#20241706C211yesmaximal, characteristic, complemented
3#20241320C214yesmaximal, complemented
4C47×#2541C211yesmaximal, characteristic
5#15689087—21no—
6C2×C22⋊(C47⋊C8)—21no—
7#15688849C2212yes—
8#15688849C2212yes—
9#15688847—21no—
10#15688847—21no—
11C2×#127×C47C411yescharacteristic
12C2×#127×C47C412yes—
13C2×#121×C47—21no—
14C2×#121×C47—21no—
15C2×C4×D4×C47C2211yescharacteristic
16C47×#392—21no—
17C47×#392—21no—
18C47×#390C2212yes—
19C4×C4⋊C4×C47C411yescharacteristic
20C2×#10625329—41no—
21C22×C47⋊C8—41no—
22C22⋊(C47⋊C8)—22no—
23C22⋊(C47⋊C8)—22no—
24C22⋊(C47⋊C8)—41no—
25C22⋊(C47⋊C8)—41no—
26C22×D4×C47—22no—
27C22×D4×C47—22no—
28C22×D4×C47—21no—
29C22×D4×C47—21no—
30C22×D4×C47C2×C411yescharacteristic
31C22×D4×C47C2×C412yes—
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34#127×C47—41no—
35#127×C47—41no—
36#127×C47—44no—
37#121×C47—44no—
38#121×C47—44no—
39C4×D4×C47—41no—
40C4×D4×C47—41no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47C2311yescharacteristic
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47C2×C412yes—
49#95×C47—21no—
50C24×C47—22no—
51C24×C47C22⋊C411yescharacteristic
52C24×C47C22⋊C411yescharacteristic
53C2×D4×C47—22no—
54C2×D4×C47—22no—
55C2×D4×C47—24no—
56C2×D4×C47—41no—
57C2×D4×C47—42no—
58C2×D4×C47—42no—
59C2×D4×C47—44no—
60C2×D4×C47—44no—
61C2×D4×C47—42no—
62C2×D4×C47—42no—
63C2×D4×C47—42no—
64C2×D4×C47—44no—
65C2×D4×C47—44no—
66C2×D4×C47—41no—
67C2×D4×C47—41no—
68C2×D4×C47—82no—
69C2×D4×C47—82no—
70C2×D4×C47—82no—
71C22×C4×C47—22no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—42no—
77C22×C4×C47—42no—
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C4⋊C4×C47—24no—
84C4⋊C4×C47—41no—
85C22⋊C4×C47—41no—
86C22⋊C4×C47—41no—
87C22⋊C4×C47—44no—
88C22⋊C4×C47—44no—
89C42×C47—21no—
90C42×C47—21no—
91C42×C47—42no—
92#10625329—82no—
93C2×C47⋊C8—42no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—82no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—42no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47—42no—
109C23×C47—42no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47#10012yes—
113C23×C47C2×C22⋊C411yescharacteristic
114D4×C47—44no—
115D4×C47—84no—
116D4×C47—82no—
117D4×C47—88no—
118D4×C47—88no—
119D4×C47—84no—
120D4×C47—82no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—22no—
124C2×C4×C47—24no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—81no—
136C2×C4×C47—81no—
137C2×C4×C47#9911yescharacteristic
138C2×C4×C47#9911yescharacteristic
139C2×C4×C47#13112yes—
140C47⋊C8—82no—
141C47⋊C8—82no—
142#11563—471nomaximal, Sylow
143C22×C47—21no—
144C22×C47—21no—
145C22×C47—22no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—81no—
154C22×C47—81no—
155C22×C47—82no—
156C22×C47—82no—
157C22×C47—82no—
158C22×C47—82no—
159C22×C47—82no—
160C22×C47—82no—
161C22×C47—82no—
162C22×C47#40811yescharacteristic
163C22×C47C2×#9911yescharacteristic
164C22×C47C2×#10011yescharacteristic
165C4×C47—21no—
166C4×C47—22no—
167C4×C47—44no—
168C4×C47—41no—
169C4×C47—41no—
170C4×C47—41no—
171C4×C47#35312yes—
172#2541D4711yescharacteristic
173#1465—471no—
174#1459—471no—
175#1318—474no—
176C2×C47—21no—
177C2×C47—21no—
178C2×C47—42no—
179C2×C47—81no—
180C2×C47—162no—
181C2×C47#147111yescharacteristic
182C2×C47C2×#35311yescharacteristic
183C2×C47#213411yescharacteristic
184C2×#127Dic4711yescharacteristic
185C2×#127Dic4712yes—
186C2×#121—21no—
187C2×#121—21no—
188C2×C4×D4C2×D4711yescharacteristic
189#407—941no—
190C2×C22⋊C8—941no—
191#392—21no—
192#392—21no—
193#390C2×D4712yes—
194C4×C4⋊C4Dic4711yescharacteristic
195#324—472no—
196#324—472no—
197#323—941no—
198#323—941no—
199C47#1156311yescharacteristic, complemented, Sylow
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C2×Dic4711yescharacteristic
205C22×D4C2×Dic4712yes—
206C23×C4C22⋊D4711yescharacteristic
207C23×C4C22⋊D4711yescharacteristic
208C2×M16—1881no—
209C22×C8—1881no—
210#127—41no—
211#127—41no—
212#127—44no—
213#121—44no—
214#121—44no—
215C4×D4—41no—
216C4×D4—41no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22×D4711yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C22⋊D4711yescharacteristic
222C2×C22⋊C4C22⋊D4711yescharacteristic
223C2×C42C2×Dic4711yescharacteristic
224C2×C42C2×Dic4712yes—
225C22⋊C8—942no—
226C22⋊C8—942no—
227C22⋊C8—1881no—
228C22⋊C8—1881no—
229#95—21no—
230C24—22no—
231C24C22⋊Dic4711yescharacteristic
232C24C22⋊Dic4711yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C22⋊Dic4711yescharacteristic
259C22×C4C22⋊Dic4711yescharacteristic
260C22×C4C22×Dic4711yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262C22×C4C2×C22⋊D4711yescharacteristic
263M16—3762no—
264C2×C8—3762no—
265C2×C8—1882no—
266C2×C8—1882no—
267C4⋊C4—24no—
268C4⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C42—21no—
274C42—21no—
275C42—42no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23#1309394112yes—
293C23C2×C22⋊Dic4711yescharacteristic
294D4—44no—
295D4—84no—
296D4—82no—
297D4—88no—
298D4—88no—
299D4—84no—
300D4—82no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#1309393911yescharacteristic
318C2×C4#1309393911yescharacteristic
319C2×C4#1309403112yes—
320C8—3762no—
321C8—3762no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22#1568909011yescharacteristic
342C22C2×#1309393911yescharacteristic
343C22C2×#1309394111yescharacteristic
344C4—21no—
345C4—22no—
346C4—44no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4#1568891612yes—
351C2—21no—
352C2—21no—
353C2—42no—
354C2—81no—
355C2—162no—
356C2#2024174811yescharacteristic
357C2C2×#1568891611yescharacteristic
358C2#2024508011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
Fitting, prime radicalC47×#2541C2
socleC22×C47C2×#99
cosocleC2×C4⋊C4×C47C23
hypercenter#2541D47
hypocenterC47#11563
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. #20241726C2semidirect
2. #20241706C2semidirect
3. #20241320C2semidirect
199. C47#11563semidirect

Maximal subgroups

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

subgroupindex
#202417262
#202417062
#202413202
C47×#25412
#1156347