Finite Group Databasebeta

← #26125570

Subgroups — #26125570

Order 12032.

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

normal subgroups only

This group has 460 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#20246100C211yesmaximal, characteristic
2#20241706C211yesmaximal, characteristic
3#20241333C212yesmaximal, complemented
4#20241321C212yesmaximal, complemented
5C47×#1459C211yesmaximal, characteristic
6C2×#13094011C411yescharacteristic
7C2×C4×Dic94C411yescharacteristic
8C2×C22⋊(C47⋊C8)—21no—
9#15689028C2211yescharacteristic
10#15689026C411yescharacteristic
11#15689011—21no—
12#15689011—21no—
13#15688996—21no—
14#15688996—21no—
15#15688992C2211yescharacteristic
16#15688969C411yescharacteristic
17#15688849C2212yes—
18#15688847—21no—
19C2×C4×D4×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#324C2212yes—
22C47×#323—21no—
23C22×Dic94C2×C411yescharacteristic
24C22×C47⋊C8—41no—
25#13094011—22no—
26#13094011C814yescomplemented
27C4×Dic94—41no—
28C4×Dic94—41no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C4×Dic47—21no—
38C2×C4×Dic47C2×C411yescharacteristic
39C2×C4×Dic47C2×C411yescharacteristic
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)—41no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C22×D4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C22×C8×C47—41no—
50C4×D4×C47—41no—
51C4×D4×C47—41no—
52C2×C4⋊C4×C47C2311yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47D411yescharacteristic
55C2×C42×C47C2×C411yescharacteristic
56C22⋊C8×C47—22no—
57C22⋊C8×C47—41no—
58C24×C47C22⋊C411yescharacteristic
59C24×C47C22⋊C411yescharacteristic
60C2×D4×C47—41no—
61C2×D4×C47—41no—
62C2×D4×C47C2×C812yes—
63C2×D4×C47M1612yes—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—81no—
69C22×C4×C47—81no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×Dic47—21no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C2×Dic94—21no—
80C2×Dic94—41no—
81C2×Dic94—41no—
82C2×Dic94C2×C812yescomplemented
83C2×C8×C47—42no—
84C2×C8×C47—82no—
85C4⋊C4×C47—41no—
86C22⋊C4×C47—41no—
87C22⋊C4×C47—41no—
88C42×C47—21no—
89C42×C47—21no—
90C22⋊Dic47—24no—
91C22⋊Dic47—24no—
92C22⋊Dic47—81no—
93C22⋊Dic47—81no—
94C4⋊Dic47—41no—
95Dic47⋊C4—41no—
96Dic47⋊C4—41no—
97Dic47⋊C4—81no—
98Dic47⋊C4—81no—
99C4×Dic47—21no—
100C4×Dic47—42no—
101C4×Dic47—41no—
102C4×Dic47C2×C812yescomplemented
103C2×C47⋊C8—42no—
104C2×C47⋊C8—82no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47C22⋊C812yes—
114C23×C47C22⋊C812yes—
115C23×C47C2×C22⋊C411yescharacteristic
116D4×C47—42no—
117D4×C47—42no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—41no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—81no—
130C2×C4×C47—81no—
131C2×C4×C47#9911yescharacteristic
132C2×C4×C47#9911yescharacteristic
133C2×C4×C47C22×C811yescharacteristic
134C2×C4×C47C2×M1611yescharacteristic
135C2×Dic47—22no—
136C2×Dic47—22no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147Dic94—42no—
148Dic94—42no—
149C8×C47—82no—
150C47⋊C8—82no—
151#11481—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—42no—
157C22×C47—42no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47—41no—
162C22×C47—81no—
163C22×C47—81no—
164C22×C47C2×C22⋊C811yescharacteristic
165C22×C47C2×#9911yescharacteristic
166C22×C47#41311yescharacteristic
167C4×C47—21no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—41no—
172C4×C47—41no—
173C4×C47—41no—
174Dic47—42no—
175Dic47—42no—
176Dic47—81no—
177Dic47—81no—
178#2326—471no—
179#1459—471no—
180#1459D4711yescharacteristic
181#1325—472no—
182#1319—472no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—42no—
186C2×C47—42no—
187C2×C47#145911yescharacteristic
188C2×C47#212711yescharacteristic
189C2×C47#213511yescharacteristic
190C2×#124—471no—
191C2×C4×Q8—471no—
192C2×C4×D4C2×D4711yescharacteristic
193C2×C22⋊C8—21no—
194C2×C22⋊C8—941no—
195#391—471no—
196#390—471no—
197#388—941no—
198#388—941no—
199#385—941no—
200#385—941no—
201#384—471no—
202#376—471no—
203#324—472no—
204#324C2×D4712yes—
205#323—21no—
206#323—941no—
207C47#1148111yescharacteristic, complemented, Sylow
208C22×Q8—471no—
209C22×D4C4×D4711yescharacteristic
210C23×C4D18811yescharacteristic
211C23×C4C22⋊D4711yescharacteristic
212C22×C8—41no—
213C22×C8—1881no—
214#124—474no—
215#124—942no—
216C4×Q8—1881no—
217C4×Q8—1881no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4—941no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4C22×D4711yescharacteristic
225C2×C22⋊C4—941no—
226C2×C22⋊C4—941no—
227C2×C22⋊C4—941no—
228C2×C22⋊C4—941no—
229C2×C22⋊C4D18811yescharacteristic
230C2×C22⋊C4C22⋊D4711yescharacteristic
231C2×C42—471no—
232C2×C42—471no—
233C2×C42—941no—
234C2×C42C4×D4711yescharacteristic
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—942no—
238C22⋊C8—1881no—
239#95—941no—
240#95—941no—
241#95—941no—
242#95—941no—
243C24(C2×D47)⋊C411yescharacteristic
244C24(C2×D47)⋊C411yescharacteristic
245C2×Q8—472no—
246C2×Q8—941no—
247C2×Q8—1881no—
248C2×Q8—1881no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4C8×D4712yes—
252C2×D4#1062532412yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—941no—
260C22×C4—941no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4(C2×D47)⋊C411yescharacteristic
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4C2×C4×D4711yescharacteristic
266C22×C4C2×D18811yescharacteristic
267C22×C4C2×C22⋊D4711yescharacteristic
268C2×C8—42no—
269C2×C8—82no—
270C2×C8—3762no—
271C2×C8—1882no—
272C4⋊C4—41no—
273C4⋊C4—3761no—
274C4⋊C4—3761no—
275C4⋊C4—1881no—
276C4⋊C4—1881no—
277C4⋊C4—1881no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—944no—
281C22⋊C4—944no—
282C22⋊C4—3761no—
283C22⋊C4—3761no—
284C42—21no—
285C42—21no—
286C42—472no—
287C42—941no—
288C42—1881no—
289C42—1882no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23(C2×D47)⋊C812yes—
299C23(C2×D47)⋊C812yes—
300C23C2×(C2×D47)⋊C411yescharacteristic
301Q8—1882no—
302Q8—1882no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—942no—
319C2×C4—942no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4—1881no—
329C2×C4—1881no—
330C2×C4#1309393811yescharacteristic
331C2×C4#1309393811yescharacteristic
332C2×C4C2×C8×D4711yescharacteristic
333C2×C4C2×#1062532411yescharacteristic
334C8—82no—
335C8—3762no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22C2×(C2×D47)⋊C811yescharacteristic
349C22C2×#1309393811yescharacteristic
350C22#1568910511yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3761no—
359C4—3761no—
360C4—1882no—
361C4—1882no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—42no—
366C2#2024170411yescharacteristic
367C2#2024505511yescharacteristic
368C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC2×C4×C47C22×C8
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1459C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1459D47
hypocenterC47#11481
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20241333C2semidirect
4. #20241321C2semidirect
26. #13094011C8semidirect
82. C2×Dic94C2×C8semidirect
102. C4×Dic47C2×C8semidirect
207. C47#11481semidirect

Maximal subgroups

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

subgroupindex
#202461002
#202417062
#202413332
#202413212
C47×#14592
#1148147