Finite Group Databasebeta

← #26114128

Subgroups — #26114128

Order 12032.

626 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 135 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#20252412C211yesmaximal, characteristic, complemented
2#20243380C211yesmaximal, characteristic, complemented
3#20241813C212yesmaximal, complemented
4#20241246C212yesmaximal, complemented
5C47×#2535C211yesmaximal, characteristic
6D4×C47⋊C8—24no—
7#15689176C2212yes—
8#15689147C2212yes—
9C2×#13093941—21no—
10C2×C22⋊(C47⋊C8)—22no—
11#15688892—21no—
12D4⋊(C47⋊C8)—24no—
13C2×#121×C47—21no—
14C2×C4×D4×C47C2211yescharacteristic
15C2×C4×D4×C47C2211yescharacteristic
16C47×#397C412yes—
17C47×#392C412yes—
18C47×#386—22no—
19C47×#382C2211yescharacteristic
20C47×#380—21no—
21C2×#10625329—22no—
22C22×C47⋊C8—24no—
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26#13093941—42no—
27C22⋊(C47⋊C8)—24no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8D414yescomplemented
30C22×D4×C47—22no—
31C22×D4×C47C2×C411yescharacteristic
32C22×D4×C47D411yescharacteristic
33C23×C4×C47—21no—
34C23×C4×C47D412yes—
35#121×C47—42no—
36#121×C47—42no—
37C4×D4×C47—22no—
38C4×D4×C47—22no—
39C4×D4×C47—22no—
40C4×D4×C47—22no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47C2×C412yes—
48C2×C42×C47C2311yescharacteristic
49#95×C47—22no—
50#95×C47—21no—
51#95×C47C2×C412yes—
52C24×C47—21no—
53C24×C47C22⋊C412yes—
54C2×D4×C47—22no—
55C2×D4×C47—22no—
56C2×D4×C47—22no—
57C2×D4×C47—42no—
58C2×D4×C47—44no—
59C2×D4×C47—44no—
60C2×D4×C47—82no—
61C2×D4×C47D812yes—
62C2×D4×C47SD1612yes—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—22no—
67C22×C4×C47—42no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47—42no—
71C22×C4×C47—82no—
72C22×C4×C47C22⋊C412yes—
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C4⋊C4×C47—22no—
77C4⋊C4×C47—22no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C22⋊C4×C47—24no—
81C22⋊C4×C47—44no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—41no—
84C22⋊C4×C47—82no—
85C42×C47C2×D412yes—
86C42×C47#4112yes—
87#10625329—42no—
88#10625329—42no—
89C2×C47⋊C8—24no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—84no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47—42no—
96C23×C47—42no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47#10214yes—
104C23×C47C2×C22⋊C411yescharacteristic
105D4×C47—24no—
106D4×C47—24no—
107D4×C47—24no—
108D4×C47—42no—
109D4×C47—88no—
110C2×C4×C47—22no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—44no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—41no—
121C2×C4×C47—82no—
122C2×C4×C47—82no—
123C2×C4×C47—82no—
124C2×C4×C47C4×D412yes—
125C2×C4×C47#12011yescharacteristic
126C2×C4×C47#12112yes—
127C2×C4×C47#12311yescharacteristic
128C2×C4×C47C2×D811yescharacteristic
129C2×C4×C47C2×SD1611yescharacteristic
130C2×C4×C47#13512yes—
131C47⋊C8—44no—
132C47⋊C8—82no—
133#9485—471nomaximal, Sylow
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—24no—
137C22×C47—44no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47—42no—
141C22×C47—42no—
142C22×C47—42no—
143C22×C47—82no—
144C22×C47—81no—
145C22×C47—81no—
146C22×C47—81no—
147C22×C47—81no—
148C22×C47#38611yescharacteristic
149C22×C47C2×#10211yescharacteristic
150C22×C47#41611yescharacteristic
151C4×C47—22no—
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—42no—
155C4×C47—82no—
156C4×C47—81no—
157C4×C47#46512yes—
158C4×C47#46612yes—
159#2972—471no—
160#2535D4711yescharacteristic
161#1800—471no—
162#1486—472no—
163#1286—472no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—44no—
167C2×C47—82no—
168C2×C47#189111yescharacteristic
169C2×C47#189311yescharacteristic
170C2×C47#191111yescharacteristic
171C2×#121—21no—
172C2×C4×D4C2×D4711yescharacteristic
173C2×C4×D4C2×D4711yescharacteristic
174C8×D4—944no—
175#431—472no—
176#423—472no—
177C2×#100—941no—
178C2×C22⋊C8—942no—
179#397Dic4712yes—
180#392Dic4712yes—
181#386—22no—
182#382C2×D4711yescharacteristic
183#380—21no—
184#343—941no—
185D4⋊C8—944no—
186C47#948511yescharacteristic, complemented, Sylow
187C22×D4—22no—
188C22×D4C2×Dic4711yescharacteristic
189C22×D4C22⋊D4711yescharacteristic
190C23×C4—21no—
191C23×C4C22⋊D4712yes—
192C2×M16—942no—
193C22×C8—944no—
194#121—42no—
195#121—42no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic4711yescharacteristic
202C2×C4⋊C4C22⋊D4711yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C2×Dic4712yes—
207C2×C42C22×D4711yescharacteristic
208C4⋊C8—942no—
209C4⋊C8—942no—
210C4⋊C8—942no—
211#100—1882no—
212C22⋊C8—944no—
213C22⋊C8—942no—
214C4×C8—474no—
215#95—22no—
216#95—21no—
217#95C2×Dic4712yes—
218C24—21no—
219C24C22⋊Dic4712yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—82no—
227C2×D4#1062533412yes—
228C2×D4#1062533512yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—82no—
238C22×C4C22⋊Dic4712yes—
239C22×C4C22×Dic4711yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241C22×C4C2×C22⋊D4711yescharacteristic
242M16—1882no—
243M16—1882no—
244C2×C8—944no—
245C2×C8—3764no—
246C2×C8—1882no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C22⋊C4—24no—
252C22⋊C4—44no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—82no—
256C42D4×D4712yes—
257C42#1062535212yes—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1309394514yes—
270C23C2×C22⋊Dic4711yescharacteristic
271D4—24no—
272D4—24no—
273D4—24no—
274D4—42no—
275D4—88no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4D4×Dic4712yes—
291C2×C4#1309399211yescharacteristic
292C2×C4#1309399612yes—
293C2×C4#1309400511yescharacteristic
294C2×C4C2×#1062533411yescharacteristic
295C2×C4C2×#1062533511yescharacteristic
296C2×C4#1309404412yes—
297C8—3762no—
298C8—1884no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#1568900311yescharacteristic
314C22C2×#1309394511yescharacteristic
315C22#1568911911yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4#1568935012yes—
323C4#1568935712yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2#2024380711yescharacteristic
329C2#2024381711yescharacteristic
330C2#2024392111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#2535C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#2535D47
hypocenterC47#9485
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #20252412C2semidirect
2. #20243380C2semidirect
3. #20241813C2semidirect
4. #20241246C2semidirect
29. C4×C47⋊C8D4semidirect
186. C47#9485semidirect

Maximal subgroups

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

subgroupindex
#202524122
#202433802
#202418132
#202412462
C47×#25352
#948547