Finite Group Databasebeta

← #26111886

Subgroups — #26111886

Order 12032.

614 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625329C211yesmaximal, characteristic
2#20251698C211yesmaximal, characteristic
3C2×#15688864C212yesmaximal, complemented
4#20241270C212yesmaximal, complemented
5C47×#1749C211yesmaximal, characteristic
6C22×#10625329—21no—
7C2×#13094016C2211yescharacteristic
8C4×#10625329—24no—
9C2×#13093935C2212yes—
10C2×C4×C47⋊C8—21no—
11#15689021C411yescharacteristic
12#15689021C411yescharacteristic
13#15689007—21no—
14#15689007—21no—
15#15688996—21no—
16#15688996—21no—
17#15688989C2211yescharacteristic
18#15688983C411yescharacteristic
19#15688983C411yescharacteristic
20#15688864—24no—
21C22×C42×C47C2211yescharacteristic
22C2×C22⋊C8×C47—21no—
23C2×#97×C47C2212yes—
24C47×#336—21no—
25C2×#10625329—24no—
26C2×#10625329—24no—
27C2×#10625329—24no—
28C22×C47⋊C8—22no—
29#13094016—22no—
30#13094016—22no—
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×C4⋊Dic47C2×C411yescharacteristic
33C2×C4⋊Dic47C2×C411yescharacteristic
34C2×C4⋊Dic47C2×C411yescharacteristic
35C2×Dic47⋊C4—21no—
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—21no—
38C2×Dic47⋊C4—21no—
39C2×C22⋊Dic47—21no—
40C2×C22⋊Dic47—21no—
41#13093935—22no—
42#13093935D414yes—
43C4×C47⋊C8—24no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C47⋊#95—21no—
50C23×C4×C47C2×C411yescharacteristic
51C23×C4×C47D411yescharacteristic
52C23×C4×C47D411yescharacteristic
53C22×C8×C47—22no—
54C2×C42×C47—22no—
55C2×C42×C47—22no—
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47D411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C2×C42×C47C2311yescharacteristic
60C22⋊C8×C47—24no—
61#97×C47—22no—
62#97×C47—22no—
63C24×C47C22⋊C411yescharacteristic
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—22no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C2×C8×C47—24no—
85C2×C8×C47—42no—
86C2×C8×C47—42no—
87C42×C47—22no—
88C42×C47—21no—
89C42×C47—22no—
90C42×C47—21no—
91C42×C47C2×D412yes—
92C42×C47#4112yes—
93C22⋊Dic47—42no—
94C22⋊Dic47—42no—
95C4⋊Dic47—22no—
96C4⋊Dic47—22no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99Dic47⋊C4—42no—
100Dic47⋊C4—42no—
101Dic47⋊C4—41no—
102Dic47⋊C4—42no—
103Dic47⋊C4—41no—
104#10625329—216no—
105C2×C47⋊C8—24no—
106C2×C47⋊C8—24no—
107C2×C47⋊C8—24no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47#10412yes—
115C23×C47C2×C22⋊C411yescharacteristic
116C2×C4×C47—24no—
117C2×C4×C47—24no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—44no—
131C2×C4×C47—44no—
132C2×C4×C47#10112yes—
133C2×C4×C47#10412yes—
134C2×C4×C47#10412yes—
135C2×C4×C47C4×D412yes—
136C2×C4×C47#12011yescharacteristic
137C2×C4×C47#12112yes—
138C2×C4×C47#12311yescharacteristic
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C8×C47—44no—
152C47⋊C8—28no—
153#9122—471nomaximal, Sylow
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47#38611yescharacteristic
162C22×C47C2×#10111yescharacteristic
163C22×C47#41311yescharacteristic
164C22×C47C2×#10411yescharacteristic
165C22×C47C2×#10411yescharacteristic
166C22×C47C2×#10411yescharacteristic
167C22×C47#42111yescharacteristic
168C4×C47—22no—
169C4×C47—22no—
170C4×C47—21no—
171C4×C47—22no—
172C4×C47—21no—
173C4×C47—22no—
174C4×C47—22no—
175Dic47—81no—
176Dic47—81no—
177Dic47—81no—
178Dic47—81no—
179C2×C4×M16—471no—
180#2851—471no—
181#1749D4711yescharacteristic
182C2×#330—472no—
183#1295—472no—
184C2×C47—42no—
185C2×C47#152411yescharacteristic
186C2×C47#152611yescharacteristic
187C2×C47#152811yescharacteristic
188C2×C47#152911yescharacteristic
189C2×C47#187111yescharacteristic
190C2×C47#189611yescharacteristic
191C2×C47#190111yescharacteristic
192C22×M16—941no—
193C2×#125—471no—
194C22×C42C2×D4711yescharacteristic
195C2×C22⋊C8—21no—
196C4×M16—944no—
197C2×#97—472no—
198C2×#97C2×D4712yes—
199C2×C4×C8—941no—
200#389—471no—
201#389—471no—
202#387—941no—
203#387—941no—
204#385—941no—
205#385—941no—
206#382—471no—
207#380—471no—
208#380—471no—
209#336—21no—
210#330—944no—
211C47#912211yescharacteristic, complemented, Sylow
212C23×C4C4×D4711yescharacteristic
213C23×C4D18811yescharacteristic
214C23×C4C22⋊D4711yescharacteristic
215C2×M16—944no—
216C2×M16—944no—
217C2×M16—944no—
218C22×C8—22no—
219C22×C8—942no—
220#125—942no—
221#125—942no—
222C2×C4⋊C4—471no—
223C2×C4⋊C4—471no—
224C2×C4⋊C4—471no—
225C2×C4⋊C4—471no—
226C2×C4⋊C4—941no—
227C2×C4⋊C4—941no—
228C2×C4⋊C4—941no—
229C2×C4⋊C4—941no—
230C2×C22⋊C4—941no—
231C2×C22⋊C4—941no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C4×D4711yescharacteristic
235C2×C42D18811yescharacteristic
236C2×C42C22⋊D4711yescharacteristic
237C2×C42C22×D4711yescharacteristic
238C22⋊C8—24no—
239#97—22no—
240#97—22no—
241#97—474no—
242#97—942no—
243C4×C8—944no—
244#95—941no—
245#95—941no—
246#95—941no—
247#95—941no—
248#95—941no—
249#95—941no—
250C24(C2×D47)⋊C411yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4—941no—
264C22×C4—941no—
265C22×C4(C2×D47)⋊C411yescharacteristic
266C22×C4(C2×D47)⋊C411yescharacteristic
267C22×C4(C2×D47)⋊C411yescharacteristic
268C22×C4C2×C4×D4711yescharacteristic
269C22×C4C2×D18811yescharacteristic
270C22×C4C2×C22⋊D4711yescharacteristic
271M16—9416no—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—944no—
276C2×C8—944no—
277C2×C8—944no—
278C4⋊C4—942no—
279C4⋊C4—942no—
280C4⋊C4—1881no—
281C4⋊C4—1881no—
282C4⋊C4—1882no—
283C4⋊C4—1882no—
284C4⋊C4—1881no—
285C4⋊C4—1882no—
286C4⋊C4—1881no—
287C22⋊C4—1882no—
288C22⋊C4—1882no—
289C42—22no—
290C42—21no—
291C42—22no—
292C42—21no—
293C42D4×D4712yes—
294C42#1062535212yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23#1309395012yes—
302C23C2×(C2×D47)⋊C411yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—1881no—
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—1881no—
331C2×C4#1309394212yes—
332C2×C4#1309395012yes—
333C2×C4#1309395212yes—
334C2×C4(C22⋊D47)⋊C412yes—
335C2×C4#1309399411yescharacteristic
336C2×C4#1309399612yes—
337C2×C4#1309400811yescharacteristic
338C8—44no—
339C8—948no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22#1568900011yescharacteristic
348C22C2×#1309394211yescharacteristic
349C22#1568910511yescharacteristic
350C22C2×#1309395011yescharacteristic
351C22C2×#1309395011yescharacteristic
352C22C2×#1309395211yescharacteristic
353C22#1568914111yescharacteristic
354C4—21no—
355C4—22no—
356C4—21no—
357C4—22no—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—3761no—
362C4—3761no—
363C4—3761no—
364C4—3761no—
365C2—42no—
366C2#2024197411yescharacteristic
367C2#2024198211yescharacteristic
368C2#2024199111yescharacteristic
369C2#2024199611yescharacteristic
370C2#2024368811yescharacteristic
371C2#2024382911yescharacteristic
372C2#2024385411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1749D47
hypocenterC47#9122
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. C2×#15688864C2semidirect
4. #20241270C2semidirect
211. C47#9122semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253292
#202516982
C2×#156888642
#202412702
C47×#17492
#912247