Finite Group Databasebeta

← #26112056

Subgroups — #26112056

Order 12032.

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

normal subgroups only

This group has 692 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
1#20251719C211yesmaximal, characteristic
2C2×#15688989C211yesmaximal, characteristic, complemented
3C2×#15688858C212yesmaximal
4#20241268C212yesmaximal, complemented
5C47×#1817C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8C22×C4⋊Dic47—21no—
9#15689025C412yescomplemented
10#15689021C412yescomplemented
11#15689014—22no—
12#15689002—21no—
13#15688989—24no—
14#15688983C2211yescharacteristic
15C2×C47⋊#95—21no—
16C2×C47⋊#95C2212yes—
17#15688858—24no—
18C22×C42×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#336—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23C23×Dic47—22no—
24#13094007—42no—
25#13094007—42no—
26#13093998—22no—
27#13093998—22no—
28C2×C4⋊Dic47—24no—
29C2×C4⋊Dic47—42no—
30C2×C4⋊Dic47—42no—
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×C4⋊Dic47C2×C412yes—
33C2×Dic47⋊C4—22no—
34C2×C22⋊Dic47—21no—
35C2×(C2×D47)⋊C4—22no—
36C2×(C2×D47)⋊C4—22no—
37C2×(C2×D47)⋊C4—21no—
38C47⋊#95—24no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C47⋊#95—21no—
42C47⋊#95D414yes—
43C23×C4×C47D411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47C2311yescharacteristic
46C22×C8×C47—22no—
47C2×C42×C47—22no—
48C2×C42×C47—22no—
49C2×C42×C47C2×C412yes—
50C2×C42×C47D412yes—
51C22⋊C8×C47—22no—
52C22⋊C8×C47—22no—
53C24×C47C2×D411yescharacteristic
54C23×D47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47C22⋊C412yes—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47D812yes—
66C22×C4×C47SD1612yes—
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D412yes—
70C22×C4×C47#4112yes—
71C2×C22⋊D47—41no—
72C2×C22⋊D47—41no—
73C2×C22⋊D47—41no—
74C2×C22⋊D47—41no—
75C2×C22⋊D47—41no—
76C2×C22⋊D47—41no—
77C22×Dic47—22no—
78C22×Dic47—21no—
79C22×Dic47—24no—
80C22×Dic47—44no—
81C22×Dic47—44no—
82C22×Dic47—42no—
83C22×Dic47—42no—
84C2×D188—22no—
85C2×D188—42no—
86C2×C8×C47—44no—
87C2×C8×C47—42no—
88C42×C47—24no—
89C42×C47—24no—
90C22⋊Dic47—41no—
91C22⋊Dic47—41no—
92(C2×D47)⋊C4—44no—
93(C2×D47)⋊C4—44no—
94(C2×D47)⋊C4—42no—
95C4⋊Dic47—22no—
96C4⋊Dic47—48no—
97C4⋊Dic47—42no—
98Dic47⋊C4—44no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47C2×C22⋊C411yescharacteristic
102C23×C47#12011yescharacteristic
103C23×C47#12311yescharacteristic
104C23×C47C2×D811yescharacteristic
105C23×C47C2×SD1611yescharacteristic
106C23×C47AGL(1,ℤ/8)11yescharacteristic
107C23×C47#13711yescharacteristic
108C22×D47—41no—
109C22×D47—42no—
110C22×D47—41no—
111C22×D47—41no—
112C22×D47—42no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47#10214yes—
129C2×C4×C47C4×D412yes—
130C2×C4×C47#12112yes—
131C22⋊D47—81no—
132C22⋊D47—81no—
133C22⋊D47—81no—
134C22⋊D47—81no—
135C2×Dic47—44no—
136C2×Dic47—44no—
137C2×Dic47—41no—
138C2×Dic47—42no—
139C2×Dic47—41no—
140C2×Dic47—42no—
141C2×Dic47—41no—
142C2×Dic47—42no—
143C2×Dic47—44no—
144C2×Dic47—84no—
145C2×Dic47—84no—
146D188—82no—
147C8×C47—82no—
148#9156—471nomaximal, Sylow
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—22no—
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47#38611yescharacteristic
158C22×C47#41012yes—
159C22×C47C2×#10211yescharacteristic
160C22×C47#41711yescharacteristic
161C22×C47#42112yes—
162C22×C47#44711yescharacteristic
163C22×C47#44811yescharacteristic
164C22×C47#48011yescharacteristic
165C22×C47#48111yescharacteristic
166C2×D47—81no—
167C2×D47—81no—
168C2×D47—81no—
169C2×D47—82no—
170C2×D47—82no—
171C4×C47—24no—
172C4×C47—22no—
173C4×C47—22no—
174C4×C47—42no—
175Dic47—84no—
176Dic47—81no—
177Dic47—82no—
178#2854—471no—
179C2×#382—471no—
180#1817D4711yescharacteristic, complemented
181C2×#327—472no—
182#1294—472no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47#151212yes—
188C2×C47#151712yes—
189C2×C47#187411yescharacteristic
190C2×C47#189711yescharacteristic
191C2×C47#189911yescharacteristic
192D47—161no—
193C2×#123—941no—
194C2×#121—471no—
195C22×C4⋊C4—941no—
196C22×C42C2×D4711yescharacteristic
197C2×C22⋊C8C2×D4712yes—
198#390—472no—
199#389—472no—
200#388—942no—
201#386—941no—
202#382—944no—
203#380—471no—
204C2×#95—472no—
205C2×#95—941no—
206#336—22no—
207#327—944no—
208C47#915611yescharacteristic, complemented, Sylow
209C22×D4—471no—
210C22×D4—941no—
211C23×C4—942no—
212C23×C4D18811yescharacteristic
213C23×C4D18811yescharacteristic
214C23×C4C22×D4711yescharacteristic
215C22×C8—22no—
216#123—1882no—
217#123—1882no—
218#121—942no—
219#121—942no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4—472no—
222C2×C4⋊C4—944no—
223C2×C4⋊C4—942no—
224C2×C4⋊C4—1882no—
225C2×C4⋊C4—1882no—
226C2×C22⋊C4—941no—
227C2×C22⋊C4—942no—
228C2×C22⋊C4—942no—
229C2×C22⋊C4—941no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C4×D4712yes—
233C2×C42C22⋊D4712yes—
234C22⋊C8—22no—
235C22⋊C8—22no—
236#95—474no—
237#95—944no—
238#95—942no—
239#95—942no—
240#95—941no—
241C24—941no—
242C24C2×D18811yescharacteristic
243C2×D4—942no—
244C2×D4—1882no—
245C2×D4—1881no—
246C2×D4—1881no—
247C2×D4—1881no—
248C2×D4—1881no—
249C2×D4—1881no—
250C2×D4—1881no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—942no—
260C22×C4—941no—
261C22×C4—944no—
262C22×C4—1884no—
263C22×C4—1884no—
264C22×C4—1882no—
265C22×C4—1882no—
266C22×C4#1062532512yes—
267C22×C4D37612yes—
268C22×C4(C2×D47)⋊C412yes—
269C22×C4(C2×D47)⋊C412yes—
270C22×C4C2×C4×D4711yescharacteristic
271C22×C4D4×D4712yes—
272C22×C4#1062535212yes—
273C22×C4C2×C22⋊D4711yescharacteristic
274C2×C8—44no—
275C2×C8—42no—
276C4⋊C4—942no—
277C4⋊C4—1882no—
278C4⋊C4—1888no—
279C4⋊C4—1884no—
280C22⋊C4—1884no—
281C22⋊C4—1881no—
282C22⋊C4—1881no—
283C22⋊C4—1884no—
284C22⋊C4—1882no—
285C42—24no—
286C42—24no—
287C23—22no—
288C23—22no—
289C23—1881no—
290C23—1882no—
291C23—1881no—
292C23—1881no—
293C23—1882no—
294C23C2×(C2×D47)⋊C411yescharacteristic
295C23#1309399411yescharacteristic
296C23#1309400811yescharacteristic
297C23C2×D37611yescharacteristic
298C23C2×#1062532511yescharacteristic
299C23#1309405211yescharacteristic
300C23#1309405711yescharacteristic
301D4—3762no—
302D4—3761no—
303D4—3761no—
304D4—3761no—
305D4—3761no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—3764no—
322C2×C4—3764no—
323C2×C4—1884no—
324C2×C4—1884no—
325C2×C4—1881no—
326C2×C4—1884no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4—1882no—
332C2×C4#1309394614yes—
333C2×C4(C22⋊D47)⋊C412yes—
334C2×C4#1309399612yes—
335C8—82no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—3762no—
345C22—3762no—
346C22—3761no—
347C22—3761no—
348C22—3761no—
349C22#1568900011yescharacteristic
350C22#1568909612yes—
351C22C2×#1309394611yescharacteristic
352C22#1568912511yescharacteristic
353C22#1568913912yes—
354C22#1568925911yescharacteristic
355C22#1568926411yescharacteristic
356C22#1568943711yescharacteristic
357C22#1568944211yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—3764no—
363C4—3762no—
364C4—3761no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—7521no—
370C2#2024192712yes—
371C2#2024194912yes—
372C2#2024370311yescharacteristic
373C2#2024383411yescharacteristic
374C2#2024384411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1817D47
hypocenterC47#9156
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#15688989C2semidirect
4. #20241268C2semidirect
5. C47×#1817C2semidirect
9. #15689025C4semidirect
10. #15689021C4semidirect
180. #1817D47semidirect
208. C47#9156semidirect

Maximal subgroups

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

subgroupindex
#202517192
C2×#156889892
C2×#156888582
#202412682
C47×#18172
#915647