Finite Group Databasebeta

← #26134781

Subgroups — #26134781

Order 12032.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689507C211yesmaximal, characteristic
2C2×#15688944C211yesmaximal, characteristic, complemented
3C2×#15688938C211yesmaximal, characteristic, complemented
4C2×#15688935C211yesmaximal, characteristic
5#20245107C211yesmaximal, characteristic, complemented
6#20245103C211yesmaximal, characteristic
7C47×#1772C211yesmaximal, characteristic, complemented
8C22×Dic188C2211yescharacteristic
9C22×#10625325—21no—
10C22×D376C2211yescharacteristic
11C2×#13094012—21no—
12C2×#13093964C2211yescharacteristic
13C2×#13093964C2211yescharacteristic
14#15689507—22no—
15#15689507C414yescomplemented
16#15688944—22no—
17#15688938—22no—
18#15688935—22no—
19#15688926—22no—
20#15688923—22no—
21C2×#108×C47C2211yescharacteristic
22C47×#427C2211yescharacteristic
23C47×#405—21no—
24C2×C4×C8×C47C2211yescharacteristic
25C47×#339—21no—
26C22×Dic94—21no—
27C22×D188—21no—
28C2×Dic188—21no—
29C2×Dic188—21no—
30C2×Dic188—21no—
31C2×Dic188—21no—
32C2×Dic188—21no—
33C2×Dic188C2×C412yes—
34C2×#10625325—24no—
35C2×#10625325—41no—
36C2×#10625325—41no—
37C2×#10625325—42no—
38C2×D376—21no—
39C2×D376—21no—
40C2×D376—21no—
41C2×D376—21no—
42C2×D376—21no—
43C2×D376C2×C412yes—
44#13094012—24no—
45#13094012—42no—
46C2×(C2×D47)⋊C4—42no—
47#13093964—21no—
48#13093964—21no—
49#13093964—21no—
50#13093964—21no—
51C2×C47⋊C16—21no—
52C2×C47⋊C16—21no—
53C2×M16×C47—21no—
54C22×C8×C47D411yescharacteristic
55C22×C8×C47C2311yescharacteristic
56#117×C47—21no—
57C2×C42×C47D411yescharacteristic
58#108×C47—21no—
59#108×C47D412yes—
60#107×C47D412yes—
61#106×C47—21no—
62#97×C47—41no—
63C4×C8×C47—41no—
64C4×C8×C47C2×C412yes—
65C4×C8×C47D412yes—
66C23×D47—41no—
67C22×C4×C47—21no—
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—41no—
70C2×D188—22no—
71C2×D188—41no—
72C2×D188—41no—
73C2×D188—41no—
74C2×D188—41no—
75C2×D188—41no—
76C2×Dic94—22no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94—41no—
81C2×Dic94—41no—
82M16×C47—22no—
83M16×C47—41no—
84C2×C8×C47—41no—
85C2×C8×C47C22⋊C412yes—
86C2×C8×C47D811yescharacteristic
87C2×C8×C47D811yescharacteristic
88C2×C8×C47SD1612yes—
89C2×C8×C47C22×C411yescharacteristic
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C2×C8×C47#4111yescharacteristic
95C4⋊C4×C47—22no—
96C22⋊C4×C47—41no—
97C42×C47—21no—
98C42×C47—41no—
99C42×C47C22⋊C412yes—
100(C2×D47)⋊C4—48no—
101Dic188—22no—
102Dic188—22no—
103Dic188—41no—
104Dic188—41no—
105D376—22no—
106D376—22no—
107D376—41no—
108D376—41no—
109#10625325—44no—
110#10625325—44no—
111C47⋊C16—41no—
112C47⋊C16—41no—
113C23×C47AGL(1,ℤ/8)11yescharacteristic
114C22×D47—42no—
115C22×D47—84no—
116C22×D47—81no—
117C22×D47—81no—
118C2×C4×C47—22no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47C2×C22⋊C411yescharacteristic
122C2×C4×C47#12111yescharacteristic
123C2×C4×C47#12111yescharacteristic
124C2×C4×C47C2×D811yescharacteristic
125C2×C4×C47C2×SD1611yescharacteristic
126C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
127C2×Dic47—42no—
128C2×Dic47—81no—
129C2×Dic47—81no—
130D188—42no—
131D188—42no—
132D188—81no—
133D188—81no—
134Dic94—42no—
135Dic94—42no—
136Dic94—81no—
137Dic94—81no—
138C8×C47—42no—
139C8×C47#10212yes—
140C8×C47#10212yes—
141C8×C47C4×D411yescharacteristic
142C8×C47C4×D411yescharacteristic
143C8×C47#12411yescharacteristic
144C8×C47#12711yescharacteristic
145#13260—471nomaximal, Sylow
146C22×C47—22no—
147C22×C47#41811yescharacteristic
148C22×C47#45911yescharacteristic
149C22×C47#46011yescharacteristic
150C2×D47—84no—
151C2×D47—82no—
152C2×D47—162no—
153C2×D47—162no—
154C4×C47—42no—
155C4×C47—82no—
156C4×C47#39011yescharacteristic
157C4×C47C2×#10211yescharacteristic
158C4×C47#45911yescharacteristic
159C4×C47#46011yescharacteristic
160Dic47—82no—
161C2×#495—471no—
162C2×#364—471no—
163C2×#362—471no—
164C2×#361—471no—
165#2141—471no—
166#2140—471no—
167#1772D4711yescharacteristic, complemented
168C2×C47—21no—
169C2×C47—21no—
170C2×C47#198211yescharacteristic
171C2×C47#221212yes—
172D47—162no—
173C22×Q16—471no—
174C22×SD16—941no—
175C22×D8—471no—
176C2×#124—941no—
177C2×M32—471no—
178C2×M32—471no—
179#495—474no—
180#495—942no—
181C2×#108C2×D4711yescharacteristic
182#427C2×D4711yescharacteristic
183#405—21no—
184C2×C4×C8C2×D4711yescharacteristic
185#364—942no—
186#362—942no—
187#361—942no—
188#358—942no—
189#357—942no—
190#339—21no—
191C47#1326011yescharacteristic, complemented, Sylow
192C22×Q8—941no—
193C22×D4—941no—
194C2×Q16—472no—
195C2×Q16—941no—
196C2×Q16—941no—
197C2×Q16—941no—
198C2×Q16—941no—
199C2×Q16—941no—
200C2×SD16—944no—
201C2×SD16—1881no—
202C2×SD16—1881no—
203C2×SD16—1882no—
204C2×D8—472no—
205C2×D8—941no—
206C2×D8—941no—
207C2×D8—941no—
208C2×D8—941no—
209C2×D8—941no—
210C2×M16—21no—
211C22×C8C22⋊D4711yescharacteristic
212C22×C8C22×D4711yescharacteristic
213#124—944no—
214#124—1882no—
215#117—21no—
216C2×C22⋊C4—1882no—
217C2×C42C22⋊D4711yescharacteristic
218M32—941no—
219M32—941no—
220M32—941no—
221M32—941no—
222C2×C16—941no—
223C2×C16—941no—
224#108—21no—
225#108D18812yes—
226#107D18812yes—
227#106—21no—
228#97—41no—
229C4×C8—41no—
230C4×C8C4×D4712yes—
231C4×C8D18812yes—
232C24—1881no—
233C2×Q8—942no—
234C2×Q8—1881no—
235C2×Q8—1881no—
236C2×Q8—1881no—
237C2×Q8—1881no—
238C2×Q8—1881no—
239C2×D4—942no—
240C2×D4—1881no—
241C2×D4—1881no—
242C2×D4—1881no—
243C2×D4—1881no—
244C2×D4—1881no—
245C22×C4—21no—
246C22×C4—1881no—
247C22×C4C2×C22⋊D4711yescharacteristic
248Q16—942no—
249Q16—942no—
250Q16—1881no—
251Q16—1881no—
252SD16—1884no—
253SD16—1884no—
254D8—942no—
255D8—942no—
256D8—1881no—
257D8—1881no—
258M16—22no—
259M16—41no—
260C2×C8—41no—
261C2×C8(C2×D47)⋊C412yes—
262C2×C8#1062533411yescharacteristic
263C2×C8#1062533411yescharacteristic
264C2×C8#1062533612yes—
265C2×C8C2×C4×D4711yescharacteristic
266C2×C8C2×D18811yescharacteristic
267C2×C8C2×D18811yescharacteristic
268C2×C8C2×D18811yescharacteristic
269C2×C8#1062535011yescharacteristic
270C2×C8#1062535011yescharacteristic
271C4⋊C4—22no—
272C22⋊C4—41no—
273C22⋊C4—1888no—
274C42—21no—
275C42—41no—
276C42(C2×D47)⋊C412yes—
277C16—1881no—
278C16—1881no—
279C23—3764no—
280C23—3761no—
281C23—3761no—
282C23—1882no—
283C23#1309404911yescharacteristic
284Q8—3761no—
285Q8—3761no—
286Q8—1882no—
287Q8—1882no—
288D4—3761no—
289D4—3761no—
290D4—1882no—
291D4—1882no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—3761no—
296C2×C4—3761no—
297C2×C4—1882no—
298C2×C4C2×(C2×D47)⋊C411yescharacteristic
299C2×C4#1309399811yescharacteristic
300C2×C4#1309399811yescharacteristic
301C2×C4C2×#1062533411yescharacteristic
302C2×C4C2×#1062533611yescharacteristic
303C2×C4#1309404911yescharacteristic
304C8—42no—
305C8#1309394412yes—
306C8#1309394412yes—
307C8C4×D18811yescharacteristic
308C8C4×D18811yescharacteristic
309C8#1309401211yescharacteristic
310C8#1309402111yescharacteristic
311C22—22no—
312C22—3764no—
313C22—3762no—
314C22—7522no—
315C22—7522no—
316C22#1568912711yescharacteristic
317C22#1568931811yescharacteristic
318C22#1568932311yescharacteristic
319C4—42no—
320C4—82no—
321C4—3762no—
322C4#1568902511yescharacteristic
323C4C2×#1309394411yescharacteristic
324C4#1568931811yescharacteristic
325C4#1568932311yescharacteristic
326C2—21no—
327C2—21no—
328C2—7522no—
329C2#2024431511yescharacteristic
330C2#2024545712yes—
subgroupG/H
centerC22#15689318
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1772C2
socleC22×C47#459
prime radicalC22×D376C22
cosocleC22×C8×C47C23
hypercenter#1772D47
hypocenterC47#13260
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1772

Complements

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

NH ≅ G/Ntype
2. C2×#15688944C2semidirect
3. C2×#15688938C2semidirect
5. #20245107C2semidirect
7. C47×#1772C2semidirect
15. #15689507C4semidirect
167. #1772D47semidirect
191. C47#13260semidirect

Maximal subgroups

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

subgroupindex
C2×#156895072
C2×#156889442
C2×#156889382
C2×#156889352
#202451072
#202451032
C47×#17722
#1326047