Finite Group Databasebeta

← #26133691

Subgroups — #26133691

Order 12032.

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

normal subgroups only

This group has 450 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
2C2×#15689399C211yesmaximal, characteristic
3C2×#15688940C211yesmaximal, characteristic, complemented
4#20245107C211yesmaximal, characteristic, complemented
5#20243439C211yesmaximal, characteristic
6#20241491C211yesmaximal, characteristic, complemented
7C47×#1389C211yesmaximal, characteristic, complemented
8C2×#13094057—21no—
9C22×Dic188C2211yescharacteristic
10C2×#13094030C2211yescharacteristic
11#15689541—22no—
12#15689539—22no—
13C2×#13093964C2211yescharacteristic
14M32×D47—22no—
15#15689399—22no—
16#15689399C414yescomplemented
17C2×#13093960C2211yescharacteristic
18#15689141—21no—
19C2×#13093942—21no—
20#15688940—22no—
21#15688926—22no—
22#15688893—21no—
23C2×M32×C47C2211yescharacteristic
24C2×#108×C47C2211yescharacteristic
25C47×#427C2211yescharacteristic
26C2×#10625350D411yescharacteristic
27C22×Dic94—21no—
28#13094057—24no—
29#13094057—42no—
30C2×Dic188—21no—
31C2×Dic188—21no—
32C2×Dic188—21no—
33C2×Dic188—21no—
34C2×Dic188—21no—
35C2×Dic188—41no—
36C2×Dic188C2×C412yes—
37C2×#10625325—41no—
38#13094030—21no—
39#13094030—21no—
40#13094030C2×C412yes—
41#13094030D412yes—
42C2×#10625329—21no—
43C2×#10625329D411yescharacteristic
44C2×#10625324—21no—
45C2×C8×D47—21no—
46#13093964—21no—
47#13093964—21no—
48#13093963—24no—
49C2×C47⋊C16D412yes—
50C16×D47—24no—
51#13093960—21no—
52#13093960—21no—
53#13093952—42no—
54#13093942—24no—
55C2×M16×C47—21no—
56C22×C8×C47C2311yescharacteristic
57#117×C47—21no—
58M32×C47—21no—
59M32×C47D412yes—
60C2×C16×C47—21no—
61#108×C47—21no—
62#108×C47C2×C412yes—
63#107×C47—21no—
64#106×C47—21no—
65C22×C4×C47C2×D411yescharacteristic
66C2×C22⋊D47—21no—
67C22×Dic47—41no—
68#10625350—21no—
69#10625350—21no—
70#10625350C22⋊C412yes—
71#10625350D812yes—
72C2×D188SD1611yescharacteristic
73C2×C4×D47—21no—
74C2×Dic94—22no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94SD1611yescharacteristic
81M16×C47—22no—
82M16×C47—41no—
83C2×C8×C47—41no—
84C2×C8×C47C22×C411yescharacteristic
85C2×C8×C47C2×D411yescharacteristic
86C2×C8×C47C2×D411yescharacteristic
87C2×C8×C47C2×D411yescharacteristic
88C2×C8×C47#4111yescharacteristic
89C2×C8×C47#4111yescharacteristic
90C4⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C42×C47—41no—
93#10625329—21no—
94#10625329—41no—
95#10625329—41no—
96#10625329C22⋊C412yes—
97C2×C47⋊C8—21no—
98C2×C47⋊C8—41no—
99Dic188—22no—
100Dic188—22no—
101Dic188—44no—
102Dic188—41no—
103Dic188—41no—
104#10625325—44no—
105#10625324—22no—
106#10625324—22no—
107C8×D47—22no—
108C8×D47—22no—
109C16×C47—22no—
110C47⋊C16—22no—
111C23×C47#13511yescharacteristic
112C22×D47—21no—
113C2×C4×C47—81no—
114C2×C4×C47C2×C22⋊C411yescharacteristic
115C2×C4×C47#12111yescharacteristic
116C2×C4×C47#12111yescharacteristic
117C2×C4×C47C2×D811yescharacteristic
118C2×C4×C47C2×SD1611yescharacteristic
119C2×C4×C47#13511yescharacteristic
120C22⋊D47—22no—
121C22⋊D47—22no—
122C2×Dic47—21no—
123C2×Dic47—42no—
124C2×Dic47—81no—
125C2×Dic47—81no—
126D188—21no—
127D188#10212yes—
128C4×D47—22no—
129C4×D47—22no—
130Dic94—21no—
131Dic94—42no—
132Dic94—42no—
133Dic94—81no—
134Dic94—81no—
135Dic94#10212yes—
136C8×C47—42no—
137C8×C47C4×D411yescharacteristic
138C8×C47C4×D411yescharacteristic
139C8×C47#12011yescharacteristic
140C8×C47#12311yescharacteristic
141C47⋊C8—22no—
142C47⋊C8—81no—
143#13069—471nomaximal, Sylow
144C22×C47—41no—
145C22×C47#41611yescharacteristic
146C22×C47#46511yescharacteristic
147C22×C47#46611yescharacteristic
148C2×D47—22no—
149C2×D47—81no—
150C4×C47—161no—
151C4×C47#38611yescharacteristic
152C4×C47C2×#10211yescharacteristic
153C4×C47#46511yescharacteristic
154C4×C47#46611yescharacteristic
155Dic47—22no—
156Dic47—82no—
157#3407—471no—
158C2×#473—471no—
159C2×#362—471no—
160#2141—471no—
161#1814—471no—
162#1389D4711yescharacteristic, complemented
163#1385—471no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47#189311yescharacteristic
168D47—42no—
169C2×#137—941no—
170C22×Q16—471no—
171C2×#131—471no—
172#504—942no—
173#504—942no—
174C2×M32—471no—
175C2×M32—942no—
176C2×M32C2×D4711yescharacteristic
177#473—474no—
178#473—942no—
179C2×#108—471no—
180C2×#108C2×D4711yescharacteristic
181#427C2×D4711yescharacteristic
182#421—941no—
183C2×#101—941no—
184#362—942no—
185#358—942no—
186#344—941no—
187C47#1306911yescharacteristic, complemented, Sylow
188C2×#41—471no—
189C22×Q8—941no—
190#137—944no—
191#137—1882no—
192C2×Q16—472no—
193C2×Q16—941no—
194C2×Q16—941no—
195C2×Q16—941no—
196C2×Q16—941no—
197C2×Q16—941no—
198C2×Q16—1881no—
199C2×SD16—1881no—
200#131—472no—
201#131—472no—
202#131—941no—
203#131—941no—
204C2×M16—21no—
205C2×M16—471no—
206C2×M16—941no—
207C2×M16—941no—
208C22×C8—941no—
209C22×C8C22×D4711yescharacteristic
210#117—21no—
211M32—21no—
212M32—944no—
213M32—941no—
214M32—941no—
215M32D18812yes—
216C2×C16—21no—
217C2×C16—472no—
218C2×C16—944no—
219#108—21no—
220#108—941no—
221#108—941no—
222#108C4×D4712yes—
223#107—21no—
224#106—21no—
225#104—1882no—
226#101—944no—
227#41—472no—
228#41—472no—
229#41—941no—
230#41—941no—
231C2×Q8—471no—
232C2×Q8—942no—
233C2×Q8—1881no—
234C2×Q8—1881no—
235C2×Q8—1881no—
236C2×Q8—1881no—
237C2×Q8—1881no—
238C2×D4—471no—
239C2×D4—941no—
240C22×C4—941no—
241C22×C4—1881no—
242C22×C4D4×D4711yescharacteristic
243Q16—942no—
244Q16—942no—
245Q16—1884no—
246Q16—1881no—
247Q16—1881no—
248SD16—1884no—
249M16—22no—
250M16—41no—
251M16—472no—
252M16—941no—
253M16—942no—
254M16—942no—
255M16—1881no—
256M16—1881no—
257C2×C8—41no—
258C2×C8—941no—
259C2×C8—942no—
260C2×C8—942no—
261C2×C8—1881no—
262C2×C8C2×C4×D4711yescharacteristic
263C2×C8C2×D18811yescharacteristic
264C2×C8#1062535011yescharacteristic
265C2×C8D4×D4711yescharacteristic
266C2×C8D4×D4711yescharacteristic
267C2×C8#1062535411yescharacteristic
268C4⋊C4—41no—
269C22⋊C4—41no—
270C42—41no—
271C16—22no—
272C16—942no—
273C23—941no—
274C23#1309404511yescharacteristic
275Q8—472no—
276Q8—941no—
277Q8—3761no—
278Q8—3761no—
279Q8—1882no—
280Q8—1882no—
281D4—472no—
282D4—942no—
283D4—942no—
284D4—941no—
285C2×C4—81no—
286C2×C4—941no—
287C2×C4—942no—
288C2×C4—942no—
289C2×C4—3761no—
290C2×C4—3761no—
291C2×C4—1882no—
292C2×C4C22⋊C4×D4711yescharacteristic
293C2×C4#1309399711yescharacteristic
294C2×C4#1309399911yescharacteristic
295C2×C4D8×D4711yescharacteristic
296C2×C4SD16×D4711yescharacteristic
297C2×C4#1309404711yescharacteristic
298C8—42no—
299C8—942no—
300C8—3761no—
301C8#1309398611yescharacteristic
302C8C4×D18811yescharacteristic
303C8#1309399411yescharacteristic
304C8#1309400911yescharacteristic
305C22—41no—
306C22—942no—
307C22—3761no—
308C22#1568911811yescharacteristic
309C22#1568935311yescharacteristic
310C22#1568935811yescharacteristic
311C4—161no—
312C4—942no—
313C4—3762no—
314C4#1568900111yescharacteristic
315C4#102×D4711yescharacteristic
316C4#1568935511yescharacteristic
317C4#1568936011yescharacteristic
318C2—21no—
319C2—21no—
320C2—21no—
321C2—1882no—
322C2#2024381511yescharacteristic
subgroupG/H
centerC2#20243815
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1389C2
socleC2×C47#1893
prime radicalC2×#10625350D4
cosocleC22×C8×C47C23
hypercenter#1389D47
hypocenterC47#13069
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #1389

Complements

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

NH ≅ G/Ntype
3. C2×#15688940C2semidirect
4. #20245107C2semidirect
6. #20241491C2semidirect
7. C47×#1389C2semidirect
16. #15689399C4semidirect
162. #1389D47semidirect
187. C47#13069semidirect

Maximal subgroups

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

subgroupindex
#202564902
C2×#156893992
C2×#156889402
#202451072
#202434392
#202414912
C47×#13892
#1306947