Finite Group Databasebeta

← #26111314

Subgroups — #26111314

Order 12032.

616 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689061C211yesmaximal, characteristic
2#20246005C211yesmaximal, characteristic
3#20243478C211yesmaximal, characteristic
4C2×#15688848C211yesmaximal, characteristic
5#20241268C211yesmaximal, characteristic
6#20241266C211yesmaximal, characteristic
7C47×#1824C211yesmaximal, characteristic
8C2×#13093980—21no—
9C22×Dic47⋊C4C2211yescharacteristic
10C2×C22⋊(C47⋊C8)C2211yescharacteristic
11#15689061—22no—
12#15689061C414yescomplemented
13#15689020—22no—
14#15689018—22no—
15#15688983C2211yescharacteristic
16#15688981—21no—
17C2×C47⋊#95—21no—
18C2×C47⋊#95—21no—
19C2×C47⋊#95C2211yescharacteristic
20#15688885—22no—
21#15688848—22no—
22C22×C4⋊C4×C47C2211yescharacteristic
23C2×C22⋊C8×C47C2211yescharacteristic
24C47×#380C2211yescharacteristic
25C47×#340—22no—
26C23×Dic47—21no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C22×C47⋊C8—21no—
30#13093980—42no—
31#13093980—42no—
32C2×C4⋊Dic47D412yes—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—21no—
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—22no—
38C2×Dic47⋊C4—22no—
39C2×Dic47⋊C4C2×C412yes—
40C2×Dic47⋊C4D412yes—
41C2×C22⋊Dic47—21no—
42C2×C4×Dic47—22no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C47⋊#95—22no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95—24no—
49C47⋊#95—24no—
50C47⋊#95C2×C412yes—
51C23×C4×C47—21no—
52C23×C4×C47C2311yescharacteristic
53C22×C8×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47C2×C412yes—
60C2×C4⋊C4×C47D412yes—
61C2×C22⋊C4×C47—21no—
62C22⋊C8×C47—21no—
63C22⋊C8×C47—21no—
64#95×C47—21no—
65C24×C47C2×D411yescharacteristic
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—42no—
69C22×C4×C47—44no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47#4111yescharacteristic
77C22×C4×C47#4111yescharacteristic
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—22no—
85C22×Dic47—22no—
86C22×Dic47—42no—
87C22×Dic47—42no—
88C22×Dic47—42no—
89C22×Dic47—41no—
90C22×Dic47—41no—
91C22×Dic47C22⋊C412yes—
92C22×Dic47C22⋊C412yes—
93C22×Dic47SD1612yes—
94C22×Dic47Q1612yes—
95C2×C8×C47—42no—
96C2×C8×C47—41no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—42no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—41no—
102C22⋊C4×C47—41no—
103C22⋊C4×C47—41no—
104C22⋊Dic47—41no—
105C22⋊Dic47—41no—
106C4⋊Dic47—42no—
107Dic47⋊C4—22no—
108Dic47⋊C4—22no—
109Dic47⋊C4—22no—
110Dic47⋊C4—22no—
111Dic47⋊C4—44no—
112Dic47⋊C4—82no—
113C4×Dic47—44no—
114C2×C47⋊C8—42no—
115C2×C47⋊C8—41no—
116C23×C47—22no—
117C23×C47—22no—
118C23×C47C2×C22⋊C411yescharacteristic
119C23×C47#12011yescharacteristic
120C23×C47#12311yescharacteristic
121C23×C47C2×SD1611yescharacteristic
122C23×C47C2×Q1611yescharacteristic
123C23×C47AGL(1,ℤ/8)11yescharacteristic
124C23×C47#13711yescharacteristic
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—44no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—82no—
137C2×C4×C47—82no—
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×Dic47—22no—
143C2×Dic47—22no—
144C2×Dic47—22no—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—81no—
158C2×Dic47—82no—
159C2×Dic47—82no—
160C2×Dic47#10314yes—
161C8×C47—81no—
162C47⋊C8—81no—
163#9023—471nomaximal, Sylow
164C22×C47—22no—
165C22×C47—22no—
166C22×C47—22no—
167C22×C47—22no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47#38611yescharacteristic
173C22×C47C2×#10311yescharacteristic
174C22×C47#41711yescharacteristic
175C22×C47#45011yescharacteristic
176C22×C47#45111yescharacteristic
177C22×C47#45412yes—
178C22×C47#45812yes—
179C22×C47#48311yescharacteristic
180C22×C47#48411yescharacteristic
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—21no—
185C4×C47—82no—
186C4×C47—82no—
187Dic47—24no—
188Dic47—42no—
189Dic47—82no—
190Dic47—82no—
191C2×#400—471no—
192#2316—471no—
193#1824—471no—
194#1824D4711yescharacteristic
195C2×#324—471no—
196#1294—471no—
197#1294—471no—
198C2×C47—21no—
199C2×C47—21no—
200C2×C47—21no—
201C2×C47—21no—
202C2×C47#161612yes—
203C2×C47#161712yes—
204C2×C47#187211yescharacteristic
205C2×C47#188811yescharacteristic
206C2×C47#189011yescharacteristic
207C2×#117—941no—
208C22×C4⋊C4—471no—
209C22×C4⋊C4C2×D4711yescharacteristic
210C2×C22⋊C8—471no—
211C2×C22⋊C8C2×D4711yescharacteristic
212#400—474no—
213#400—942no—
214#389—942no—
215#389—942no—
216#380—471no—
217#380—941no—
218#380C2×D4711yescharacteristic
219C2×#95—471no—
220C2×#95—941no—
221C2×#95—941no—
222#340—22no—
223#340—942no—
224#324—942no—
225C47#902311yescharacteristic, complemented, Sylow
226C23×C4—21no—
227C23×C4—471no—
228C23×C4—471no—
229C23×C4—941no—
230C23×C4C22×D4711yescharacteristic
231C22×C8—21no—
232C22×C8—941no—
233#117—1882no—
234#117—1882no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—472no—
241C2×C4⋊C4—472no—
242C2×C4⋊C4—472no—
243C2×C4⋊C4—941no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4—942no—
248C2×C4⋊C4—942no—
249C2×C4⋊C4C4×D4712yes—
250C2×C4⋊C4C22⋊D4712yes—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—941no—
253C2×C42—942no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—941no—
257C22⋊C8—941no—
258#95—21no—
259#95—472no—
260#95—944no—
261#95—944no—
262#95—942no—
263#95—941no—
264#95—941no—
265C24D4×D4711yescharacteristic
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—472no—
273C22×C4—472no—
274C22×C4—472no—
275C22×C4—472no—
276C22×C4—941no—
277C22×C4—941no—
278C22×C4—941no—
279C22×C4—941no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4—942no—
283C22×C4—942no—
284C22×C4—1882no—
285C22×C4—1882no—
286C22×C4—1881no—
287C22×C4—1881no—
288C22×C4—1882no—
289C22×C4C2×C4×D4711yescharacteristic
290C22×C4#1062535011yescharacteristic
291C22×C4D4×D4711yescharacteristic
292C22×C4D4×D4711yescharacteristic
293C22×C4#1062535211yescharacteristic
294C22×C4C2×C22⋊D4711yescharacteristic
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—1882no—
298C2×C8—1881no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—942no—
305C4⋊C4—942no—
306C4⋊C4—942no—
307C4⋊C4—942no—
308C4⋊C4—3762no—
309C4⋊C4—1882no—
310C4⋊C4—1884no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—1881no—
314C22⋊C4—1881no—
315C42—1884no—
316C23—22no—
317C23—22no—
318C23C22⋊C4×D4711yescharacteristic
319C23#1309399211yescharacteristic
320C23#1309400611yescharacteristic
321C23SD16×D4711yescharacteristic
322C23Q16×D4711yescharacteristic
323C23#1309405111yescharacteristic
324C23#1309405611yescharacteristic
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—474no—
339C2×C4—942no—
340C2×C4—942no—
341C2×C4—942no—
342C2×C4—942no—
343C2×C4—942no—
344C2×C4—942no—
345C2×C4—942no—
346C2×C4—3762no—
347C2×C4—3762no—
348C2×C4—3761no—
349C2×C4—1882no—
350C2×C4—1882no—
351C2×C4—1882no—
352C2×C4—1882no—
353C2×C4—1882no—
354C2×C4—1882no—
355C2×C4—1882no—
356C2×C4—1882no—
357C2×C4(C22⋊D47)⋊C411yescharacteristic
358C2×C4C4×C22⋊D4711yescharacteristic
359C2×C4#1309399511yescharacteristic
360C2×C4#1309399711yescharacteristic
361C8—81no—
362C8—3761no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22#1568899911yescharacteristic
372C22#103×D4711yescharacteristic
373C22#1568912311yescharacteristic
374C22#1568927211yescharacteristic
375C22#1568927711yescharacteristic
376C22#1568929412yes—
377C22#1568931212yes—
378C22#1568945211yescharacteristic
379C22#1568945711yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—944no—
387C4—3762no—
388C4—3762no—
389C4—1882no—
390C2—21no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2#2024242312yes—
395C2#2024242812yes—
396C2#2024369211yescharacteristic
397C2#2024378811yescharacteristic
398C2#2024379811yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1824C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1824D47
hypocenterC47#9023
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
12. #15689061C4semidirect
225. C47#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#156890612
#202460052
#202434782
C2×#156888482
#202412682
#202412662
C47×#18242
#902347