Finite Group Databasebeta

← #26111305

Subgroups — #26111305

Order 12032.

812 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#15689059C211yesmaximal, characteristic
2#20246002C211yesmaximal, characteristic
3#20243479C212yesmaximal, complemented
4#20241268C212yesmaximal, complemented
5C2×C47×#323C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7C22×Dic47⋊C4—21no—
8C22×C22⋊Dic47—21no—
9C22×C22⋊Dic47—21no—
10#15689059—22no—
11#15689059C414yescomplemented
12#15688998—24no—
13#15688983—21no—
14#15688983C2212yes—
15C2×C47⋊#95C2211yescharacteristic
16#15688886—24no—
17C22×C22⋊C4×C47C2211yescharacteristic
18C2×C22⋊C8×C47C2212yes—
19C47×#323—22no—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C23×Dic47—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47C2×C412yescomplemented
28C2×C4⋊Dic47D414yes—
29C2×Dic47⋊C4—24no—
30C2×Dic47⋊C4—44no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—24no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—44no—
35C2×C22⋊Dic47—42no—
36C2×C22⋊Dic47—42no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95—22no—
40C47⋊#95C2×C412yescomplemented
41C25×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C22×C8×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47C2×C412yes—
48C2×C22⋊C4×C47D412yes—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—21no—
54C24×C47—21no—
55C24×C47—42no—
56C24×C47—42no—
57C24×C47C22⋊C412yes—
58C24×C47SD1612yes—
59C24×C47Q1612yes—
60C24×C47C2×D411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D412yes—
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4112yes—
70C22×Dic47—24no—
71C22×Dic47—22no—
72C22×Dic47—22no—
73C22×Dic47—24no—
74C22×Dic47—42no—
75C22×Dic47—41no—
76C22×Dic47—41no—
77C22×Dic47—42no—
78C22×Dic47—44no—
79C22×Dic47—42no—
80C22×Dic47—41no—
81C22×Dic47—41no—
82C22×Dic47—82no—
83C22×Dic47—82no—
84C2×C8×C47—44no—
85C2×C8×C47—42no—
86C22⋊C4×C47—24no—
87C22⋊C4×C47—24no—
88C22⋊Dic47—48no—
89C22⋊Dic47—44no—
90C22⋊Dic47—42no—
91C22⋊Dic47—42no—
92C4⋊Dic47—24no—
93C4⋊Dic47—42no—
94C4⋊Dic47—44no—
95C4⋊Dic47—82no—
96Dic47⋊C4—48no—
97C23×C47—22no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—42no—
106C23×C47—42no—
107C23×C47—44no—
108C23×C47—44no—
109C23×C47—42no—
110C23×C47—42no—
111C23×C47—82no—
112C23×C47—82no—
113C23×C47—81no—
114C23×C47—81no—
115C23×C47#10314yes—
116C23×C47C2×C22⋊C411yescharacteristic
117C23×C47#12011yescharacteristic
118C23×C47#12311yescharacteristic
119C23×C47C2×SD1611yescharacteristic
120C23×C47C2×Q1611yescharacteristic
121C23×C47AGL(1,ℤ/8)11yescharacteristic
122C23×C47#13711yescharacteristic
123C2×C4×C47—24no—
124C2×C4×C47—22no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—81no—
131C2×C4×C47C4×D412yes—
132C2×C4×C47#12112yes—
133C2×Dic47—24no—
134C2×Dic47—44no—
135C2×Dic47—44no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—44no—
142C2×Dic47—44no—
143C2×Dic47—44no—
144C2×Dic47—82no—
145C2×Dic47—88no—
146C2×Dic47—82no—
147C8×C47—82no—
148#9021—471nomaximal, Sylow
149C22×C47—22no—
150C22×C47—22no—
151C22×C47—22no—
152C22×C47—22no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—24no—
158C22×C47—44no—
159C22×C47—44no—
160C22×C47—44no—
161C22×C47—42no—
162C22×C47—42no—
163C22×C47—82no—
164C22×C47—82no—
165C22×C47—82no—
166C22×C47#38611yescharacteristic
167C22×C47C2×#10311yescharacteristic
168C22×C47#41711yescharacteristic
169C22×C47#45011yescharacteristic
170C22×C47#45111yescharacteristic
171C22×C47#45312yes—
172C22×C47#45812yes—
173C22×C47#48311yescharacteristic
174C22×C47#48411yescharacteristic
175C4×C47—22no—
176C4×C47—22no—
177C4×C47—42no—
178Dic47—44no—
179Dic47—84no—
180Dic47—82no—
181C2×#399—471no—
182#2315—471no—
183#1824—472no—
184C2×#323D4711yescharacteristic
185#1294—472no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47—44no—
191C2×C47#161312yes—
192C2×C47#161412yes—
193C2×C47#187211yescharacteristic
194C2×C47#188711yescharacteristic
195C2×C47#189011yescharacteristic
196C22×C4⋊C4—471no—
197C22×C4⋊C4—941no—
198C22×C22⋊C4—941no—
199C22×C22⋊C4—941no—
200C22×C22⋊C4C2×D4711yescharacteristic
201C2×C22⋊C8C2×D4712yes—
202#399—474no—
203#399—942no—
204#385—944no—
205#380—472no—
206#380—941no—
207C2×#95—471no—
208#340—944no—
209#323—22no—
210C47#902111yescharacteristic, complemented, Sylow
211C25D18811yescharacteristic
212C23×C4—941no—
213C23×C4—942no—
214C23×C4—941no—
215C23×C4D18811yescharacteristic
216C23×C4C22×D4711yescharacteristic
217C22×C8—22no—
218C2×C4⋊C4—472no—
219C2×C4⋊C4—474no—
220C2×C4⋊C4—942no—
221C2×C4⋊C4—942no—
222C2×C4⋊C4—942no—
223C2×C4⋊C4—944no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4—1884no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—942no—
229C2×C22⋊C4—944no—
230C2×C22⋊C4—942no—
231C2×C22⋊C4—1884no—
232C2×C22⋊C4—1882no—
233C2×C22⋊C4—1882no—
234C2×C22⋊C4C4×D4712yes—
235C2×C22⋊C4C22⋊D4712yes—
236C22⋊C8—22no—
237C22⋊C8—22no—
238#95—472no—
239#95—942no—
240#95—942no—
241#95—941no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—42no—
247C24—42no—
248C24#1062532512yes—
249C24Dic18812yes—
250C24(C2×D47)⋊C412yes—
251C24C2×D18811yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—944no—
257C22×C4—942no—
258C22×C4—942no—
259C22×C4—944no—
260C22×C4—3762no—
261C22×C4—3762no—
262C22×C4—1882no—
263C22×C4—1884no—
264C22×C4—1882no—
265C22×C4—1881no—
266C22×C4—1881no—
267C22×C4—1882no—
268C22×C4—1881no—
269C22×C4—1881no—
270C22×C4(C2×D47)⋊C412yes—
271C22×C4C2×C4×D4711yescharacteristic
272C22×C4D4×D4712yes—
273C22×C4#1062535212yes—
274C22×C4C2×C22⋊D4711yescharacteristic
275C2×C8—44no—
276C2×C8—42no—
277C4⋊C4—944no—
278C4⋊C4—3762no—
279C4⋊C4—1884no—
280C4⋊C4—1888no—
281C4⋊C4—1882no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—1888no—
285C22⋊C4—1884no—
286C22⋊C4—1882no—
287C22⋊C4—1882no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—44no—
298C23—44no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—81no—
303C23—81no—
304C23—82no—
305C23—82no—
306C23#1309394914yes—
307C23C2×(C2×D47)⋊C411yescharacteristic
308C23#1309399411yescharacteristic
309C23#1309400811yescharacteristic
310C23C2×#1062532511yescharacteristic
311C23C2×Dic18811yescharacteristic
312C23#1309405211yescharacteristic
313C23#1309405711yescharacteristic
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—944no—
323C2×C4—3762no—
324C2×C4—3768no—
325C2×C4—3762no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1884no—
330C2×C4—1884no—
331C2×C4—1884no—
332C2×C4—1882no—
333C2×C4—1884no—
334C2×C4—1884no—
335C2×C4—1882no—
336C2×C4(C22⋊D47)⋊C412yes—
337C2×C4#1309399612yes—
338C8—82no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—24no—
348C22—44no—
349C22—42no—
350C22—42no—
351C22—44no—
352C22—44no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22#1568900011yescharacteristic
357C22C2×#1309394911yescharacteristic
358C22#1568912511yescharacteristic
359C22#1568927611yescharacteristic
360C22#1568928111yescharacteristic
361C22#1568929312yes—
362C22#1568931412yes—
363C22#1568945411yescharacteristic
364C22#1568945911yescharacteristic
365C4—22no—
366C4—22no—
367C4—42no—
368C4—3762no—
369C4—3764no—
370C4—1884no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—44no—
376C2#2024241012yes—
377C2#2024241512yes—
378C2#2024369311yescharacteristic
379C2#2024378411yescharacteristic
380C2#2024379911yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#323C2
socleC23×C47#120
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC2×#323D47
hypocenterC47#9021
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20243479C2semidirect
4. #20241268C2semidirect
11. #15689059C4semidirect
27. C2×C4⋊Dic47C2×C4semidirect
40. C47⋊#95C2×C4semidirect
210. C47#9021semidirect

Maximal subgroups

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

subgroupindex
C2×#156890592
#202460022
#202434792
#202412682
C2×C47×#3232
#902147