Finite Group Databasebeta

← #26224239

Subgroups — #26224239

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689247C211yesmaximal, characteristic, complemented
2C2×#15689201C211yesmaximal, characteristic
3#20246672C211yesmaximal, characteristic
4#20244328C211yesmaximal, characteristic, complemented
5#20243991C214yesmaximal, complemented
6#20243507C211yesmaximal, characteristic, complemented
7#20243174C214yesmaximal, complemented
8#20243076C211yesmaximal, characteristic, complemented
9C47×#1955C211yesmaximal, characteristic
10C22×#10625329C2212yes—
11C2×C4×Dic94C2211yescharacteristic
12#15689247—24no—
13#15689201—24no—
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×Dic47⋊C8C2212yes—
16C2×C22⋊(C47⋊C8)C2212yes—
17C2×#13093934C2212yes—
18C2×C4×C47⋊C8C2211yescharacteristic
19C2×C8×Dic47C2212yes—
20C2×C8×Dic47C2212yes—
21#15689061C412yes—
22#15689048C412yes—
23#15689037C412yes—
24#15689035C412yes—
25C4×C22⋊Dic47C2214yes—
26#15688969C2211yescharacteristic
27#15688877C2214yes—
28#15688877C2214yes—
29C2×C4×D4×C47C2211yescharacteristic
30C2×C4⋊C8×C47C2211yescharacteristic
31C2×C22⋊C8×C47C2214yes—
32C2×#97×C47C2211yescharacteristic
33C22×Dic94C2×C411yescharacteristic
34C2×#10625329—24no—
35C2×#10625329—28no—
36C22×C47⋊C8C2312yes—
37C22×C47⋊C8C2311yescharacteristic
38C22×C47⋊C8C2311yescharacteristic
39C4×Dic94—22no—
40C4×Dic94—22no—
41C2×C4⋊Dic47C2×C411yescharacteristic
42C2×Dic47⋊C4C2×C412yes—
43C2×C22⋊Dic47C2×C414yes—
44C2×C22⋊Dic47C2×C414yes—
45C2×C4×Dic47C2312yes—
46C2×C4×Dic47C2312yes—
47C4⋊(C47⋊C8)—22no—
48Dic47⋊C8—24no—
49C22⋊(C47⋊C8)—24no—
50#13093934—24no—
51C4×C47⋊C8D414yes—
52C8×Dic47—24no—
53C8×Dic47—24no—
54C47⋊#95C2×C412yes—
55C47⋊#95C2×C412yes—
56C47⋊#95C2×C414yes—
57C47⋊#95C2×C414yes—
58C22×D4×C47C2×C411yescharacteristic
59C23×C4×C47C2312yes—
60C22×C8×C47C2312yes—
61C22×C8×C47C2312yes—
62C4×D4×C47—22no—
63C4×D4×C47—22no—
64C2×C4⋊C4×C47C2×C411yescharacteristic
65C2×C22⋊C4×C47C2×C412yes—
66C2×C42×C47C2311yescharacteristic
67C4⋊C8×C47—22no—
68C22⋊C8×C47—28no—
69#97×C47—22no—
70C24×C47C22×C412yes—
71C2×D4×C47—22no—
72C2×D4×C47—42no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—24no—
76C22×C4×C47—24no—
77C22×C4×C47C22×C412yes—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C22×C411yescharacteristic
80C22×C4×C47C2411yescharacteristic
81C22×Dic47C22×C414yes—
82C22×Dic47C22×C412yes—
83C22×Dic47C22×C412yes—
84C2×Dic94—22no—
85C2×Dic94—24no—
86C2×C8×C47—22no—
87C2×C8×C47—22no—
88C2×C8×C47—22no—
89C2×C8×C47—24no—
90C2×C8×C47#4114yes—
91C4⋊C4×C47—22no—
92C22⋊C4×C47—24no—
93C42×C47C2×D412yes—
94C42×C47#4112yes—
95C22⋊Dic47—28no—
96C22⋊Dic47—44no—
97C4⋊Dic47—22no—
98Dic47⋊C4—24no—
99C4×Dic47—22no—
100C4×Dic47—22no—
101C4×Dic47—22no—
102C4×Dic47—22no—
103#10625329—48no—
104C2×C47⋊C8—22no—
105C2×C47⋊C8—22no—
106C2×C47⋊C8—24no—
107C2×C47⋊C8C2×D412yes—
108C2×C47⋊C8C2×D412yes—
109C2×C47⋊C8#4114yes—
110C23×C47—22no—
111C23×C47—24no—
112C23×C47—24no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47C23×C411yescharacteristic
116D4×C47—44no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—24no—
121C2×C4×C47—42no—
122C2×C4×C47—44no—
123C2×C4×C47—42no—
124C2×C4×C47C4×D412yes—
125C2×C4×C47C4×D412yes—
126C2×C4×C47C22×D411yescharacteristic
127C2×C4×C47C2×#4112yes—
128C2×C4×C47C2×#4111yescharacteristic
129C2×C4×C47C2×#4112yes—
130C2×Dic47—22no—
131C2×Dic47—22no—
132C2×Dic47—22no—
133C2×Dic47—24no—
134C2×Dic47—24no—
135C2×Dic47—24no—
136C2×Dic47—24no—
137C2×Dic47#13114yes—
138Dic94—28no—
139C8×C47—24no—
140C8×C47—42no—
141C47⋊C8—22no—
142C47⋊C8—42no—
143C47⋊C8#12414yes—
144#21639—471nomaximal, Sylow
145C22×C47—42no—
146C22×C47—42no—
147C22×C47—44no—
148C22×C47—44no—
149C22×C47—42no—
150C22×C47C2×C4×D411yescharacteristic
151C22×C47C4×#4112yes—
152C22×C47C2×#13112yes—
153C22×C47#56812yes—
154C4×C47—22no—
155C4×C47—22no—
156C4×C47—42no—
157C4×C47C2×#12411yescharacteristic
158C4×C47#52912yes—
159C4×C47#53211yescharacteristic
160Dic47—24no—
161Dic47—42no—
162Dic47—44no—
163C2×C8×Q8—471no—
164C2×#436—471no—
165#2398—471no—
166#1985—471no—
167#1955D4711yescharacteristic
168#1923—474no—
169#1832—471no—
170#1753—474no—
171#1726—471no—
172C2×C47—82no—
173C2×C47C4×#12411yescharacteristic
174C2×C47#292911yescharacteristic
175C2×C47#293011yescharacteristic
176C2×C47#296914yes—
177C22×M16—472no—
178C2×C4×Q8—471no—
179C2×C4×D4C2×D4711yescharacteristic
180C8×Q8—944no—
181#436—944no—
182C2×C4⋊C8—472no—
183C2×C4⋊C8—471no—
184C2×C4⋊C8C2×D4711yescharacteristic
185C2×C22⋊C8—472no—
186C2×C22⋊C8C2×D4714yes—
187C2×#97—472no—
188C2×#97C2×D4711yescharacteristic
189C2×C4×C8—471no—
190C2×C4×C8—472no—
191C2×C4×C8—472no—
192#400—472no—
193#397—472no—
194#394—472no—
195#393—472no—
196C4×C22⋊C4—474no—
197#376—471no—
198#336—474no—
199#336—474no—
200C47#2163911yescharacteristic, complemented, Sylow
201C22×Q8—471no—
202C22×D4C4×D4711yescharacteristic
203C23×C4C22×D4712yes—
204C2×M16—948no—
205C2×M16—944no—
206C22×C8—471no—
207C22×C8—471no—
208C22×C8—472no—
209C22×C8C22×D4712yes—
210C22×C8C22×D4712yes—
211C4×Q8—942no—
212C4×Q8—942no—
213C4×D4—22no—
214C4×D4—22no—
215C2×C4⋊C4—472no—
216C2×C4⋊C4—471no—
217C2×C4⋊C4C4×D4711yescharacteristic
218C2×C22⋊C4—474no—
219C2×C22⋊C4—474no—
220C2×C22⋊C4C4×D4712yes—
221C2×C42—472no—
222C2×C42—472no—
223C2×C42C22×D4711yescharacteristic
224C4⋊C8—22no—
225C4⋊C8—944no—
226C4⋊C8—942no—
227C22⋊C8—28no—
228C22⋊C8—944no—
229#97—22no—
230#97—944no—
231C4×C8—474no—
232C4×C8—944no—
233C4×C8—944no—
234#95—472no—
235#95—472no—
236#95—474no—
237#95—474no—
238C24C2×C4×D4712yes—
239C2×Q8—944no—
240C2×Q8—942no—
241C2×D4—22no—
242C2×D4—42no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—472no—
248C22×C4—474no—
249C22×C4—472no—
250C22×C4C2×C4×D4712yes—
251C22×C4C2×C4×D4711yescharacteristic
252C22×C4C2×C4×D4711yescharacteristic
253C22×C4C23×D4711yescharacteristic
254M16—1888no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—472no—
260C2×C8—474no—
261C2×C8—472no—
262C2×C8—942no—
263C2×C8—942no—
264C2×C8—944no—
265C2×C8#1062535214yes—
266C4⋊C4—22no—
267C4⋊C4—944no—
268C4⋊C4—942no—
269C22⋊C4—24no—
270C22⋊C4—948no—
271C22⋊C4—1884no—
272C42—942no—
273C42—942no—
274C42—942no—
275C42—942no—
276C42D4×D4712yes—
277C42#1062535212yes—
278C23—22no—
279C23—24no—
280C23—24no—
281C23—22no—
282C23—22no—
283C23C22×C4×D4711yescharacteristic
284Q8—948no—
285D4—44no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—474no—
294C2×C4—942no—
295C2×C4—942no—
296C2×C4—942no—
297C2×C4—944no—
298C2×C4—944no—
299C2×C4—944no—
300C2×C4—944no—
301C2×C4(C22⋊D47)⋊C412yes—
302C2×C4(C22⋊D47)⋊C412yes—
303C2×C4C2×D4×D4711yescharacteristic
304C2×C4#41×D4712yes—
305C2×C4C2×#1062535211yescharacteristic
306C2×C4C2×#1062535212yes—
307C8—24no—
308C8—42no—
309C8—474no—
310C8—942no—
311C8—1882no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—44no—
316C22—44no—
317C22C2×(C22⋊D47)⋊C411yescharacteristic
318C22C4×#1062535212yes—
319C22#131×D4712yes—
320C22#1568988912yes—
321C4—22no—
322C4—22no—
323C4—42no—
324C4—944no—
325C4—1882no—
326C4—1884no—
327C4#124×D4711yescharacteristic
328C4#1568965412yes—
329C4#1568967111yescharacteristic
330C2—82no—
331C2#2024593511yescharacteristic
332C2#2025208511yescharacteristic
333C2#2025208711yescharacteristic
334C2#2025238514yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1955C2
prime radicalC22×D4×C47C2×C4
cosocleC22×C4×C47C24
hypercenter#1955D47
hypocenterC47#21639
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. C2×#15689247C2semidirect
4. #20244328C2semidirect
5. #20243991C2semidirect
6. #20243507C2semidirect
7. #20243174C2semidirect
8. #20243076C2semidirect
200. C47#21639semidirect

Maximal subgroups

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

subgroupindex
C2×#156892472
C2×#156892012
#202466722
#202443282
#202439912
#202435072
#202431742
#202430762
C47×#19552
#2163947