Finite Group Databasebeta

← #26133682

Subgroups — #26133682

Order 12032.

470 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 109 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#20256490C211yesmaximal, characteristic, complemented
2C2×#15689390C211yesmaximal, characteristic
3C2×#15688936C211yesmaximal, characteristic, complemented
4#20243439C211yesmaximal, characteristic, complemented
5#20241504C211yesmaximal, characteristic, complemented
6#20241492C211yesmaximal, characteristic, complemented
7C47×#2140C211yesmaximal, characteristic, complemented
8C2×#13094050—21no—
9C2×#13094030C2211yescharacteristic
10#15689541—22no—
11#15689539—22no—
12C2×#13093964C2211yescharacteristic
13M32×D47—22no—
14#15689390—22no—
15#15689390C414yescomplemented
16C2×#13093960C2211yescharacteristic
17C2×#13093960C2211yescharacteristic
18#15689141—21no—
19C2×#13093941—21no—
20#15688936—22no—
21#15688893—21no—
22C22×D8×C47C2211yescharacteristic
23C2×M32×C47C2211yescharacteristic
24C47×#427C2211yescharacteristic
25C47×#357—22no—
26C2×#10625350D411yescharacteristic
27#13094050—24no—
28#13094050—42no—
29C2×#10625335—41no—
30C2×#10625334—41no—
31#13094030—21no—
32#13094030—21no—
33#13094030C2×C412yes—
34#13094030D412yes—
35C2×#10625329—21no—
36C2×#10625329—21no—
37C2×#10625329D411yescharacteristic
38C2×#10625324—21no—
39C2×C8×D47—21no—
40#13093964—21no—
41#13093964D412yes—
42#13093963—24no—
43C2×C47⋊C16—21no—
44C16×D47—24no—
45#13093960—21no—
46#13093960—21no—
47#13093960—21no—
48#13093960C2×C412yes—
49#13093952—42no—
50#13093941—24no—
51C22×D4×C47—21no—
52C2×D8×C47—21no—
53C2×D8×C47—21no—
54C2×D8×C47—21no—
55C2×D8×C47—21no—
56C2×D8×C47—21no—
57C2×D8×C47C2×C412yes—
58C22×C8×C47C2311yescharacteristic
59#117×C47—21no—
60M32×C47—21no—
61M32×C47—21no—
62C2×C16×C47D412yes—
63#107×C47—21no—
64#106×C47—21no—
65C24×C47—41no—
66C2×D4×C47—22no—
67C2×D4×C47—41no—
68C2×D4×C47—41no—
69C2×D4×C47—41no—
70C2×D4×C47—41no—
71C2×D4×C47—41no—
72C22×C4×C47C2×D411yescharacteristic
73C2×C22⋊D47—21no—
74#10625350—21no—
75#10625350—21no—
76#10625350C22⋊C412yes—
77#10625350D812yes—
78C2×D188SD1611yescharacteristic
79C2×C4×D47—21no—
80C2×Dic94SD1611yescharacteristic
81D8×C47—22no—
82D8×C47—22no—
83D8×C47—41no—
84D8×C47—41no—
85C2×C8×C47C22×C411yescharacteristic
86C2×C8×C47C2×D411yescharacteristic
87C2×C8×C47C2×D411yescharacteristic
88C2×C8×C47C2×D411yescharacteristic
89C2×C8×C47#4111yescharacteristic
90C2×C8×C47#4111yescharacteristic
91C4⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C42×C47—41no—
94#10625335—44no—
95#10625334—44no—
96#10625329—21no—
97#10625329—22no—
98#10625329—41no—
99#10625329—41no—
100#10625329—41no—
101#10625329C22⋊C412yes—
102C2×C47⋊C8—21no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105#10625324—22no—
106#10625324—22no—
107C8×D47—22no—
108C8×D47—22no—
109C16×C47—22no—
110C47⋊C16—22no—
111C23×C47—42no—
112C23×C47—81no—
113C23×C47—81no—
114C23×C47—162no—
115C23×C47#13511yescharacteristic
116C22×D47—21no—
117D4×C47—42no—
118D4×C47—42no—
119D4×C47—81no—
120D4×C47—81no—
121C2×C4×C47—81no—
122C2×C4×C47C2×C22⋊C411yescharacteristic
123C2×C4×C47#12111yescharacteristic
124C2×C4×C47#12111yescharacteristic
125C2×C4×C47C2×D811yescharacteristic
126C2×C4×C47C2×SD1611yescharacteristic
127C2×C4×C47#13511yescharacteristic
128C22⋊D47—22no—
129C22⋊D47—22no—
130C2×Dic47—21no—
131D188—21no—
132D188#10212yes—
133C4×D47—22no—
134C4×D47—22no—
135Dic94—21no—
136Dic94#10212yes—
137C8×C47C4×D411yescharacteristic
138C8×C47C4×D411yescharacteristic
139C8×C47#12011yescharacteristic
140C8×C47#12311yescharacteristic
141C47⋊C8—22no—
142C47⋊C8—42no—
143C47⋊C8—81no—
144#13068—471nomaximal, Sylow
145C22×C47—41no—
146C22×C47—82no—
147C22×C47—162no—
148C22×C47—162no—
149C22×C47—162no—
150C22×C47#41611yescharacteristic
151C22×C47#46511yescharacteristic
152C22×C47#46611yescharacteristic
153C2×D47—22no—
154C2×D47—81no—
155C4×C47—161no—
156C4×C47#38611yescharacteristic
157C4×C47C2×#10211yescharacteristic
158C4×C47#46511yescharacteristic
159C4×C47#46611yescharacteristic
160Dic47—22no—
161#3407—471no—
162C2×#472—471no—
163C2×#361—471no—
164#2140D4711yescharacteristic, complemented
165#1814—471no—
166#1389—471no—
167#1385—471no—
168C2×C47—21no—
169C2×C47—21no—
170C2×C47—21no—
171C2×C47—162no—
172C2×C47#189311yescharacteristic
173D47—42no—
174C2×AGL(1,ℤ/8)—941no—
175C22×D8C2×D4711yescharacteristic
176C2×#131—471no—
177#504—942no—
178#504—942no—
179C2×M32—471no—
180C2×M32—942no—
181C2×M32C2×D4711yescharacteristic
182#472—474no—
183#472—942no—
184C2×#108—471no—
185C2×#108—471no—
186#427C2×D4711yescharacteristic
187#421—941no—
188C2×#100—941no—
189#361—942no—
190#357—22no—
191#344—941no—
192C47#1306811yescharacteristic, complemented, Sylow
193C2×#41—471no—
194C22×D4—21no—
195AGL(1,ℤ/8)—944no—
196AGL(1,ℤ/8)—1882no—
197C2×SD16—1881no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—21no—
201C2×D8—21no—
202C2×D8—21no—
203C2×D8—1881no—
204C2×D8C4×D4712yes—
205#131—472no—
206#131—472no—
207#131—941no—
208#131—941no—
209C2×M16—471no—
210C2×M16—941no—
211C2×M16—941no—
212C2×M16—941no—
213C22×C8—941no—
214C22×C8C22×D4711yescharacteristic
215#117—21no—
216M32—21no—
217M32—21no—
218M32—472no—
219M32—944no—
220M32—941no—
221C2×C16—944no—
222C2×C16—941no—
223C2×C16C22⋊D4712yes—
224#108—472no—
225#108—941no—
226#108—941no—
227#108—941no—
228#107—21no—
229#106—21no—
230#104—1882no—
231#100—944no—
232C24—41no—
233#41—472no—
234#41—472no—
235#41—941no—
236#41—941no—
237C2×Q8—471no—
238C2×D4—22no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—471no—
245C2×D4—941no—
246C22×C4—941no—
247C22×C4D4×D4711yescharacteristic
248SD16—1884no—
249D8—22no—
250D8—22no—
251D8—41no—
252D8—41no—
253D8—1884no—
254M16—472no—
255M16—941no—
256M16—942no—
257M16—942no—
258M16—942no—
259M16—1881no—
260M16—1881no—
261M16—1881no—
262C2×C8—941no—
263C2×C8—942no—
264C2×C8—942no—
265C2×C8—1881no—
266C2×C8—1881no—
267C2×C8C2×C4×D4711yescharacteristic
268C2×C8#1062535011yescharacteristic
269C2×C8D4×D4711yescharacteristic
270C2×C8D4×D4711yescharacteristic
271C2×C8#1062535211yescharacteristic
272C2×C8C2×C22⋊D4711yescharacteristic
273C4⋊C4—41no—
274C22⋊C4—41no—
275C42—41no—
276C16—22no—
277C16—942no—
278C23—42no—
279C23—81no—
280C23—81no—
281C23—162no—
282C23—941no—
283C23#1309404311yescharacteristic
284Q8—472no—
285Q8—941no—
286D4—42no—
287D4—42no—
288D4—81no—
289D4—81no—
290D4—472no—
291D4—942no—
292D4—942no—
293D4—941no—
294C2×C4—81no—
295C2×C4—941no—
296C2×C4—942no—
297C2×C4—942no—
298C2×C4C22⋊C4×D4711yescharacteristic
299C2×C4#1309399511yescharacteristic
300C2×C4#1309399711yescharacteristic
301C2×C4D8×D4711yescharacteristic
302C2×C4SD16×D4711yescharacteristic
303C2×C4#1309404511yescharacteristic
304C8—942no—
305C8—3761no—
306C8—1882no—
307C8(C22⋊D47)⋊C411yescharacteristic
308C8C4×C22⋊D4711yescharacteristic
309C8#1309399211yescharacteristic
310C8#1309400611yescharacteristic
311C22—41no—
312C22—82no—
313C22—162no—
314C22—162no—
315C22—162no—
316C22—942no—
317C22—3761no—
318C22#1568911611yescharacteristic
319C22#1568935111yescharacteristic
320C22#1568935611yescharacteristic
321C4—161no—
322C4—942no—
323C4#1568899911yescharacteristic
324C4#102×D4711yescharacteristic
325C4#1568934911yescharacteristic
326C4#1568935811yescharacteristic
327C2—21no—
328C2—21no—
329C2—21no—
330C2—162no—
331C2—1882no—
332C2#2024381311yescharacteristic
subgroupG/H
centerC2#20243813
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2140C2
socleC2×C47#1893
prime radicalC2×#15689390C2
cosocleC22×C8×C47C23
hypercenter#2140D47
hypocenterC47#13068
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2140

Complements

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

NH ≅ G/Ntype
1. #20256490C2semidirect
3. C2×#15688936C2semidirect
4. #20243439C2semidirect
5. #20241504C2semidirect
6. #20241492C2semidirect
7. C47×#2140C2semidirect
15. #15689390C4semidirect
164. #2140D47semidirect
192. C47#13068semidirect

Maximal subgroups

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

subgroupindex
#202564902
C2×#156893902
C2×#156889362
#202434392
#202415042
#202414922
C47×#21402
#1306847