Finite Group Databasebeta

← #26133304

Subgroups — #26133304

Order 12032.

492 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689393C211yesmaximal, characteristic
2#20251999C211yesmaximal, characteristic, complemented
3C2×#15688937C211yesmaximal, characteristic, complemented
4C2×#15688937C211yesmaximal, characteristic, complemented
5#20241500C211yesmaximal, characteristic, complemented
6#20241492C211yesmaximal, characteristic, complemented
7C47×#2117C211yesmaximal, characteristic, complemented
8C2×#13094049—21no—
9C22×D376C2211yescharacteristic
10#15689890—21no—
11#15689393—22no—
12#15689393C414yes—
13#15689172—22no—
14#15689172—21no—
15C2×#13093960—22no—
16C2×#13093960C2211yescharacteristic
17C2×#13093960C2211yescharacteristic
18C2×#13093960C2211yescharacteristic
19C2×#13093940—21no—
20#15688937—22no—
21#15688937—22no—
22C2×#131×C47C2211yescharacteristic
23C2×M32×C47C2211yescharacteristic
24C2×M32×C47C2211yescharacteristic
25C47×#350—22no—
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—
37#13094031—42no—
38#13094031—42no—
39C2×#10625329—24no—
40C2×#10625329—21no—
41C2×#10625329—21no—
42C2×#10625329—21no—
43#13093960—24no—
44#13093960—21no—
45#13093960—21no—
46#13093960—21no—
47#13093960C2×C412yes—
48#13093960D412yes—
49#13093960D412yes—
50#13093940—24no—
51C2×#41×C47D411yescharacteristic
52#131×C47—21no—
53#131×C47—21no—
54#131×C47C2×C412yes—
55#131×C47D412yes—
56C2×M16×C47—21no—
57C2×M16×C47D411yescharacteristic
58C22×C8×C47—21no—
59C22×C8×C47C2311yescharacteristic
60M32×C47—21no—
61M32×C47—21no—
62M32×C47—21no—
63M32×C47—21no—
64C2×C16×C47—21no—
65C2×C16×C47—21no—
66C23×D47—41no—
67#41×C47—21no—
68#41×C47—21no—
69#41×C47C22⋊C412yes—
70#41×C47D812yes—
71C2×Q8×C47SD1611yescharacteristic
72C2×D4×C47—21no—
73C2×D4×C47SD1611yescharacteristic
74C22×C4×C47—21no—
75C22×C4×C47C2×D411yescharacteristic
76C2×D188—22no—
77C2×D188—41no—
78C2×D188—41no—
79C2×D188—41no—
80C2×D188—41no—
81C2×D188—41no—
82M16×C47—21no—
83M16×C47—22no—
84M16×C47—22no—
85M16×C47C22⋊C412yes—
86C2×C8×C47—21no—
87C2×C8×C47—22no—
88C2×C8×C47—22no—
89C2×C8×C47C22×C411yescharacteristic
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C2×C8×C47#4111yescharacteristic
95#10625336—44no—
96#10625334—44no—
97#10625329—22no—
98#10625329—22no—
99#10625329—22no—
100#10625329—44no—
101#10625329—41no—
102#10625329—41no—
103#10625329—41no—
104C2×C47⋊C8—44no—
105C2×C47⋊C8—41no—
106C2×C47⋊C8—41no—
107C2×C47⋊C8—41no—
108D376—22no—
109D376—22no—
110D376—41no—
111D376—41no—
112C16×C47—41no—
113C16×C47—41no—
114C23×C47—21no—
115C23×C47#13711yescharacteristic
116C22×D47—42no—
117C22×D47—81no—
118C22×D47—81no—
119C22×D47—162no—
120Q8×C47—21no—
121Q8×C47#10212yes—
122D4×C47—22no—
123D4×C47—22no—
124D4×C47—21no—
125D4×C47#10212yes—
126C2×C4×C47—21no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47C2×C22⋊C411yescharacteristic
130C2×C4×C47#12011yescharacteristic
131C2×C4×C47#12311yescharacteristic
132C2×C4×C47C2×D811yescharacteristic
133C2×C4×C47C2×SD1611yescharacteristic
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135D188—42no—
136D188—42no—
137D188—81no—
138D188—81no—
139C8×C47—22no—
140C8×C47C4×D411yescharacteristic
141C8×C47C4×D411yescharacteristic
142C8×C47#12111yescharacteristic
143C8×C47#12111yescharacteristic
144C47⋊C8—42no—
145C47⋊C8—42no—
146C47⋊C8—42no—
147#13007—471nomaximal, Sylow
148C22×C47—22no—
149C22×C47—41no—
150C22×C47—81no—
151C22×C47#41711yescharacteristic
152C22×C47#44811yescharacteristic
153C22×C47#48011yescharacteristic
154C2×D47—82no—
155C2×D47—162no—
156C2×D47—162no—
157C2×D47—162no—
158C4×C47—22no—
159C4×C47#38611yescharacteristic
160C4×C47C2×#10211yescharacteristic
161C4×C47#44711yescharacteristic
162C4×C47#48111yescharacteristic
163C2×#472—471no—
164#2916—471no—
165C2×#361—471no—
166C2×#361—471no—
167#2117D4711yescharacteristic, complemented
168#1388—471no—
169#1385—471no—
170C2×C47—21no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—42no—
174C2×C47#187411yescharacteristic
175D47—162no—
176C2×AGL(1,ℤ/8)—941no—
177C22×D8—471no—
178#568—941no—
179C2×#131C2×D4711yescharacteristic
180C2×M32C2×D4711yescharacteristic
181C2×M32C2×D4711yescharacteristic
182#472—474no—
183#472—942no—
184#430—942no—
185#430—941no—
186C2×#108—471no—
187C2×#108—471no—
188C2×#108—471no—
189C2×#108—942no—
190C2×#100—941no—
191#361—942no—
192#361—942no—
193#350—22no—
194C47#1300711yescharacteristic, complemented, Sylow
195C2×#41D18811yescharacteristic
196C22×D4—941no—
197AGL(1,ℤ/8)—944no—
198AGL(1,ℤ/8)—1882no—
199C2×SD16—1881no—
200C2×D8—472no—
201C2×D8—941no—
202C2×D8—941no—
203C2×D8—941no—
204C2×D8—941no—
205C2×D8—941no—
206C2×D8—1881no—
207#131—21no—
208#131—21no—
209#131—1882no—
210#131—1882no—
211#131C4×D4712yes—
212#131C22⋊D4712yes—
213C2×M16—21no—
214C2×M16—944no—
215C2×M16—941no—
216C2×M16—941no—
217C2×M16—941no—
218C2×M16D18811yescharacteristic
219C22×C8—21no—
220C22×C8C22×D4711yescharacteristic
221M32—21no—
222M32—21no—
223M32—21no—
224M32—21no—
225C2×C16—21no—
226C2×C16—21no—
227#108—472no—
228#108—472no—
229#108—472no—
230#108—944no—
231#108—941no—
232#108—941no—
233#108—941no—
234#100—944no—
235C24—1881no—
236#41—21no—
237#41—21no—
238#41D37612yes—
239#41(C2×D47)⋊C412yes—
240C2×Q8#1062532511yescharacteristic
241C2×D4—21no—
242C2×D4—942no—
243C2×D4—1881no—
244C2×D4—1881no—
245C2×D4—1881no—
246C2×D4—1881no—
247C2×D4—1881no—
248C2×D4#1062532511yescharacteristic
249C22×C4—21no—
250C22×C4C2×D18811yescharacteristic
251SD16—1884no—
252D8—942no—
253D8—942no—
254D8—1884no—
255D8—1881no—
256D8—1881no—
257M16—21no—
258M16—22no—
259M16—22no—
260M16—942no—
261M16—942no—
262M16—942no—
263M16—1884no—
264M16—1881no—
265M16—1881no—
266M16—1881no—
267M16(C2×D47)⋊C412yes—
268C2×C8—21no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—1884no—
272C2×C8—1881no—
273C2×C8—1881no—
274C2×C8—1881no—
275C2×C8C2×C4×D4711yescharacteristic
276C2×C8D4×D4711yescharacteristic
277C2×C8D4×D4711yescharacteristic
278C2×C8#1062535211yescharacteristic
279C2×C8#1062535211yescharacteristic
280C2×C8C2×C22⋊D4711yescharacteristic
281C16—41no—
282C16—41no—
283C23—21no—
284C23—3761no—
285C23—3761no—
286C23—1882no—
287C23—7522no—
288C23#1309405711yescharacteristic
289Q8—21no—
290Q8#1309394612yes—
291D4—22no—
292D4—22no—
293D4—21no—
294D4—3761no—
295D4—3761no—
296D4—1882no—
297D4—1882no—
298D4#1309394612yes—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4C2×(C2×D47)⋊C411yescharacteristic
303C2×C4#1309399411yescharacteristic
304C2×C4#1309400811yescharacteristic
305C2×C4C2×D37611yescharacteristic
306C2×C4C2×#1062532511yescharacteristic
307C2×C4#1309405211yescharacteristic
308C8—22no—
309C8—1882no—
310C8—1882no—
311C8—1882no—
312C8(C22⋊D47)⋊C411yescharacteristic
313C8(C22⋊D47)⋊C411yescharacteristic
314C8#1309399611yescharacteristic
315C8#1309399611yescharacteristic
316C22—22no—
317C22—41no—
318C22—81no—
319C22—3762no—
320C22—7522no—
321C22—7522no—
322C22—7522no—
323C22#1568912511yescharacteristic
324C22#1568926411yescharacteristic
325C22#1568943711yescharacteristic
326C4—22no—
327C4#1568900011yescharacteristic
328C4C2×#1309394611yescharacteristic
329C4#1568925911yescharacteristic
330C4#1568944211yescharacteristic
331C2—21no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—7522no—
336C2#2024370311yescharacteristic
subgroupG/H
centerC2#20243703
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2117C2
socleC2×C47#1874
prime radicalC2×#15689393C2
cosocleC22×C8×C47C23
hypercenter#2117D47
hypocenterC47#13007
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #20251999C2semidirect
3. C2×#15688937C2semidirect
4. C2×#15688937C2semidirect
5. #20241500C2semidirect
6. #20241492C2semidirect
7. C47×#2117C2semidirect
167. #2117D47semidirect
194. C47#13007semidirect

Maximal subgroups

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

subgroupindex
C2×#156893932
#202519992
C2×#156889372
#202415002
#202414922
C47×#21172
#1300747