Finite Group Databasebeta

← #26133685

Subgroups — #26133685

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689393C211yesmaximal, characteristic
2C2×#15688937C211yesmaximal, characteristic, complemented
3#20245105C211yesmaximal, characteristic, complemented
4#20243442C211yesmaximal, characteristic, complemented
5#20241503C211yesmaximal, characteristic, complemented
6#20241492C211yesmaximal, characteristic, complemented
7C47×#3407C211yesmaximal, characteristic, complemented
8C2×#13094049—21no—
9C22×D376C2211yescharacteristic
10#15689393—22no—
11#15689393C414yescomplemented
12C2×#13093960C2211yescharacteristic
13C2×#13093960C2211yescharacteristic
14#15689161C2211yescharacteristic
15#15689140—21no—
16C2×#13093940—21no—
17#15688937—22no—
18#15688925—22no—
19#15688895—21no—
20C2×#131×C47C2211yescharacteristic
21C47×#504—22no—
22C47×#504—22no—
23C2×M32×C47—22no—
24C2×M32×C47C2211yescharacteristic
25C2×M32×C47C2211yescharacteristic
26C22×D188—21no—
27#13094049—24no—
28#13094049—42no—
29C2×#10625336—41no—
30C2×D376—21no—
31C2×D376—21no—
32C2×D376—21no—
33C2×D376—21no—
34C2×D376—21no—
35C2×D376C2×C412yes—
36C2×#10625334—41no—
37C2×#10625329—21no—
38C2×#10625329—21no—
39#13093983—21no—
40#13093960—21no—
41#13093960—21no—
42#13093960—21no—
43#13093960C2×C412yes—
44#13093958—21no—
45#13093956—21no—
46#13093951—42no—
47#13093940—24no—
48C2×#41×C47D411yescharacteristic
49#131×C47—21no—
50#131×C47—21no—
51#131×C47C2×C412yes—
52#131×C47D412yes—
53C2×M16×C47—21no—
54C2×M16×C47D411yescharacteristic
55C22×C8×C47—21no—
56C22×C8×C47C2311yescharacteristic
57M32×C47—24no—
58M32×C47—21no—
59M32×C47—21no—
60M32×C47—21no—
61M32×C47D412yes—
62C2×C16×C47—24no—
63C2×C16×C47—21no—
64C2×C16×C47D412yes—
65C23×D47—41no—
66#41×C47—21no—
67#41×C47—21no—
68#41×C47C22⋊C412yes—
69#41×C47D812yes—
70C2×Q8×C47SD1611yescharacteristic
71C2×D4×C47—21no—
72C2×D4×C47SD1611yescharacteristic
73C22×C4×C47—21no—
74C22×C4×C47C2×D411yescharacteristic
75C2×D188—22no—
76C2×D188—41no—
77C2×D188—41no—
78C2×D188—41no—
79C2×D188—41no—
80C2×D188—41no—
81M16×C47—21no—
82M16×C47—22no—
83M16×C47—22no—
84M16×C47C22⋊C412yes—
85C2×C8×C47—21no—
86C2×C8×C47—22no—
87C2×C8×C47—22no—
88C2×C8×C47C22×C411yescharacteristic
89C2×C8×C47C2×D411yescharacteristic
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47#4111yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C22⋊Dic47—41no—
95#10625336—44no—
96#10625334—44no—
97C4⋊Dic47—41no—
98C4×Dic47—41no—
99#10625329—22no—
100#10625329—41no—
101#10625329—41no—
102#10625329—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105D376—22no—
106D376—22no—
107D376—41no—
108D376—41no—
109C16×C47—22no—
110C16×C47—22no—
111C23×C47—21no—
112C23×C47#13511yescharacteristic
113C22×D47—42no—
114C22×D47—81no—
115C22×D47—81no—
116C22×D47—162no—
117Q8×C47—21no—
118Q8×C47#10212yes—
119D4×C47—22no—
120D4×C47—22no—
121D4×C47—21no—
122D4×C47#10212yes—
123C2×C4×C47—21no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47C2×C22⋊C411yescharacteristic
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47#12111yescharacteristic
129C2×C4×C47C2×D811yescharacteristic
130C2×C4×C47C2×SD1611yescharacteristic
131C2×C4×C47#13511yescharacteristic
132C2×Dic47—81no—
133D188—42no—
134D188—42no—
135D188—81no—
136D188—81no—
137C8×C47—22no—
138C8×C47C4×D411yescharacteristic
139C8×C47C4×D411yescharacteristic
140C8×C47#12011yescharacteristic
141C8×C47#12311yescharacteristic
142C47⋊C8—42no—
143C47⋊C8—81no—
144#13068—471nomaximal, Sylow
145C22×C47—22no—
146C22×C47—41no—
147C22×C47—81no—
148C22×C47#41611yescharacteristic
149C22×C47#46511yescharacteristic
150C22×C47#46611yescharacteristic
151C2×D47—82no—
152C2×D47—162no—
153C2×D47—162no—
154C2×D47—162no—
155C4×C47—22no—
156C4×C47#38611yescharacteristic
157C4×C47C2×#10211yescharacteristic
158C4×C47#46511yescharacteristic
159C4×C47#46611yescharacteristic
160Dic47—161no—
161#3407D4711yescharacteristic, complemented
162C2×#472—471no—
163C2×#361—471no—
164#2140—471no—
165#1814—471no—
166#1389—471no—
167#1385—471no—
168C2×C47—21no—
169C2×C47—21no—
170C2×C47—21no—
171C2×C47—42no—
172C2×C47#189311yescharacteristic
173D47—162no—
174C2×AGL(1,ℤ/8)—941no—
175C22×D8—471no—
176C2×#131C2×D4711yescharacteristic
177#504—22no—
178#504—22no—
179C2×M32—22no—
180C2×M32C2×D4711yescharacteristic
181C2×M32C2×D4711yescharacteristic
182#472—474no—
183#472—942no—
184C2×#108—471no—
185C2×#108—471no—
186#427—471no—
187#421—941no—
188C2×#100—941no—
189#361—942no—
190#357—942no—
191#344—941no—
192C47#1306811yescharacteristic, complemented, Sylow
193C2×#41D18811yescharacteristic
194C22×D4—941no—
195AGL(1,ℤ/8)—944no—
196AGL(1,ℤ/8)—1882no—
197C2×SD16—1881no—
198C2×D8—472no—
199C2×D8—941no—
200C2×D8—941no—
201C2×D8—941no—
202C2×D8—941no—
203C2×D8—941no—
204C2×D8—1881no—
205#131—21no—
206#131—21no—
207#131C4×D4712yes—
208#131C22⋊D4712yes—
209C2×M16—21no—
210C2×M16—941no—
211C2×M16—941no—
212C2×M16D18811yescharacteristic
213C22×C8—21no—
214C22×C8C22×D4711yescharacteristic
215#117—941no—
216M32—24no—
217M32—21no—
218M32—21no—
219M32—21no—
220M32C22⋊D4712yes—
221C2×C16—24no—
222C2×C16—21no—
223C2×C16C22⋊D4712yes—
224#108—472no—
225#108—941no—
226#108—941no—
227#108—941no—
228#107—941no—
229#106—941no—
230#104—1882no—
231#100—944no—
232C24—1881no—
233#41—21no—
234#41—21no—
235#41D37612yes—
236#41(C2×D47)⋊C412yes—
237C2×Q8#1062532511yescharacteristic
238C2×D4—21no—
239C2×D4—942no—
240C2×D4—1881no—
241C2×D4—1881no—
242C2×D4—1881no—
243C2×D4—1881no—
244C2×D4—1881no—
245C2×D4#1062532511yescharacteristic
246C22×C4—21no—
247C22×C4C2×D18811yescharacteristic
248SD16—1884no—
249D8—942no—
250D8—942no—
251D8—1884no—
252D8—1881no—
253D8—1881no—
254M16—21no—
255M16—22no—
256M16—22no—
257M16—942no—
258M16—1881no—
259M16—1881no—
260M16—1881no—
261M16(C2×D47)⋊C412yes—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—1881no—
266C2×C8—1881no—
267C2×C8C2×C4×D4711yescharacteristic
268C2×C8#1062535011yescharacteristic
269C2×C8#1062535011yescharacteristic
270C2×C8C2×C22⋊D4711yescharacteristic
271C2×C8C2×C22⋊D4711yescharacteristic
272C2×C8C2×C22⋊D4711yescharacteristic
273C4⋊C4—1881no—
274C22⋊C4—1881no—
275C42—1881no—
276C16—22no—
277C16—22no—
278C23—21no—
279C23—3761no—
280C23—3761no—
281C23—1882no—
282C23—7522no—
283C23#1309404611yescharacteristic
284Q8—21no—
285Q8#1309394612yes—
286D4—22no—
287D4—22no—
288D4—21no—
289D4—3761no—
290D4—3761no—
291D4—1882no—
292D4—1882no—
293D4#1309394612yes—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—3761no—
298C2×C4C2×(C2×D47)⋊C411yescharacteristic
299C2×C4#1309399811yescharacteristic
300C2×C4#1309399811yescharacteristic
301C2×C4C2×D37611yescharacteristic
302C2×C4C2×#1062532511yescharacteristic
303C2×C4#1309404611yescharacteristic
304C8—22no—
305C8—3761no—
306C8—1882no—
307C8C4×C22⋊D4711yescharacteristic
308C8C4×C22⋊D4711yescharacteristic
309C8#1309399311yescharacteristic
310C8#1309400711yescharacteristic
311C22—22no—
312C22—41no—
313C22—81no—
314C22—3762no—
315C22—7522no—
316C22—7522no—
317C22—7522no—
318C22#1568912011yescharacteristic
319C22#1568935211yescharacteristic
320C22#1568935911yescharacteristic
321C4—22no—
322C4—7521no—
323C4#1568900211yescharacteristic
324C4C2×#1309394611yescharacteristic
325C4#1568935211yescharacteristic
326C4#1568935911yescharacteristic
327C2—21no—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—7522no—
332C2#2024381611yescharacteristic
subgroupG/H
centerC2#20243816
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#3407C2
socleC2×C47#1893
prime radicalC2×#15689393C2
cosocleC22×C8×C47C23
hypercenter#3407D47
hypocenterC47#13068
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3407

Complements

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

NH ≅ G/Ntype
2. C2×#15688937C2semidirect
3. #20245105C2semidirect
4. #20243442C2semidirect
5. #20241503C2semidirect
6. #20241492C2semidirect
7. C47×#3407C2semidirect
11. #15689393C4semidirect
161. #3407D47semidirect
192. C47#13068semidirect

Maximal subgroups

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

subgroupindex
C2×#156893932
C2×#156889372
#202451052
#202434422
#202415032
#202414922
C47×#34072
#1306847