Finite Group Databasebeta

← #26134105

Subgroups — #26134105

Order 12032.

324 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688953C211yesmaximal, characteristic, complemented
2C2×#15688930C211yesmaximal, characteristic
3#20244277C211yesmaximal, characteristic
4#20243269C211yesmaximal, characteristic, complemented
5#20241483C211yesmaximal, characteristic, complemented
6#20241465C211yesmaximal, characteristic, complemented
7C47×#2138C211yesmaximal, characteristic
8C22×C47⋊C16C2211yescharacteristic
9C2×#13093960C2211yescharacteristic
10C2×#13093960C2211yescharacteristic
11C2×#13093951—21no—
12#15689090—21no—
13#15689079—21no—
14C2×C8×Dic47C2211yescharacteristic
15#15688953—22no—
16#15688930—22no—
17#15688882—21no—
18C2×#135×C47C2211yescharacteristic
19C22×C16×C47C2211yescharacteristic
20C47×#427C2211yescharacteristic
21C47×#358C411yescharacteristic, complemented
22C47×#358C411yescharacteristic, complemented
23C47×#357C411yescharacteristic, complemented
24C47×#357C411yescharacteristic, complemented
25C2×#10625329—21no—
26C2×#10625329—21no—
27C22×C47⋊C8D411yescharacteristic
28C2×C4×Dic47D411yescharacteristic
29C2×C47⋊C16—21no—
30C2×C47⋊C16—21no—
31C2×C47⋊C16—21no—
32#13093960—21no—
33#13093960—21no—
34#13093960—21no—
35#13093960D412yes—
36#13093951—42no—
37C22⋊(C47⋊C8)—41no—
38C22⋊(C47⋊C8)—41no—
39#13093935—41no—
40C4×C47⋊C8—41no—
41C8×Dic47—21no—
42C8×Dic47—21no—
43C2×#41×C47—21no—
44#135×C47—21no—
45#135×C47—21no—
46#135×C47—21no—
47#135×C47—21no—
48C2×Q16×C47C2×C411yescharacteristic
49C2×SD16×C47—21no—
50C2×D8×C47C2×C411yescharacteristic
51C22×C8×C47C2311yescharacteristic
52#117×C47—21no—
53C2×C16×C47—21no—
54C2×C16×C47—21no—
55C2×C16×C47C2×C411yescharacteristic
56C2×C16×C47C2×C411yescharacteristic
57#107×C47C2×C411yescharacteristic
58#107×C47C2×C411yescharacteristic
59#106×C47—21no—
60#41×C47—41no—
61#41×C47—41no—
62#41×C47—41no—
63#41×C47—41no—
64C2×Q8×C47—21no—
65C2×D4×C47—21no—
66C2×D4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—21no—
70Q16×C47—21no—
71Q16×C47—21no—
72SD16×C47—41no—
73SD16×C47—41no—
74D8×C47—21no—
75D8×C47—21no—
76C2×C8×C47C22×C411yescharacteristic
77C2×C8×C47C2×D411yescharacteristic
78C2×C8×C47#4111yescharacteristic
79C2×C8×C47#4111yescharacteristic
80C2×C8×C47#4111yescharacteristic
81C2×C8×C47#4111yescharacteristic
82C4⋊C4×C47—21no—
83C4⋊C4×C47—21no—
84C22⋊C4×C47—41no—
85C42×C47—41no—
86C4×Dic47—21no—
87C4×Dic47—21no—
88#10625329—22no—
89#10625329—41no—
90#10625329—41no—
91#10625329—41no—
92C2×C47⋊C8—21no—
93C2×C47⋊C8—21no—
94C2×C47⋊C8—41no—
95C2×C47⋊C8—41no—
96C2×C47⋊C8D811yescharacteristic
97C2×C47⋊C8D811yescharacteristic
98C16×C47—21no—
99C16×C47—21no—
100C47⋊C16—41no—
101C23×C47—41no—
102C23×C47#13511yescharacteristic
103Q8×C47—41no—
104Q8×C47—41no—
105D4×C47—41no—
106D4×C47—41no—
107D4×C47—81no—
108D4×C47—81no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—81no—
113C2×C4×C47—81no—
114C2×C4×C47C4×D411yescharacteristic
115C2×C4×C47#12111yescharacteristic
116C2×C4×C47#12311yescharacteristic
117C2×C4×C47C2×D811yescharacteristic
118C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C47#13711yescharacteristic
120C2×Dic47—22no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C8×C47#11711yescharacteristic
124C8×C47C4×D411yescharacteristic
125C8×C47#12411yescharacteristic
126C8×C47#12611yescharacteristic
127C47⋊C8—21no—
128C47⋊C8—21no—
129C47⋊C8—42no—
130C47⋊C8—81no—
131#13134—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—81no—
135C22×C47—81no—
136C22×C47—81no—
137C22×C47C4×D811yescharacteristic
138C22×C47#46411yescharacteristic
139C22×C47#48211yescharacteristic
140C4×C47—81no—
141C4×C47—81no—
142C4×C47—81no—
143C4×C47#38811yescharacteristic
144C4×C47#44011yescharacteristic
145C4×C47#45911yescharacteristic
146C4×C47#48011yescharacteristic
147Dic47—42no—
148C2×#368—471no—
149C2×#359—471no—
150#2138D4711yescharacteristic
151#1974—471no—
152#1772—471no—
153#1382—471no—
154#1374—471no—
155C2×C47—21no—
156C2×C47—21no—
157C2×C47—161no—
158C2×C47#192511yescharacteristic
159C2×C47#217811yescharacteristic
160C2×C47#217911yescharacteristic
161C2×#135C2×D4711yescharacteristic
162C22×C16—471no—
163C22×C16C2×D4711yescharacteristic
164C2×#108—471no—
165C2×#108—471no—
166#427C2×D4711yescharacteristic
167C2×#104—941no—
168#408—941no—
169#405—941no—
170C2×C4×C8—471no—
171#368—942no—
172#359—942no—
173#358Dic4711yescharacteristic, complemented
174#358Dic4711yescharacteristic, complemented
175#357Dic4711yescharacteristic, complemented
176#357Dic4711yescharacteristic, complemented
177#339—941no—
178C47#1313411yescharacteristic, complemented, Sylow
179C2×#41—21no—
180#135—21no—
181#135—21no—
182#135—21no—
183#135—21no—
184C2×Q16C2×Dic4711yescharacteristic
185C2×SD16—21no—
186C2×D8C2×Dic4711yescharacteristic
187C2×M16—941no—
188C2×M16—941no—
189C22×C8—471no—
190C22×C8C22×D4711yescharacteristic
191#117—21no—
192C2×C42—471no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—941no—
196C2×C16—941no—
197C2×C16—941no—
198C2×C16C2×Dic4711yescharacteristic
199C2×C16C2×Dic4711yescharacteristic
200#108—472no—
201#108—941no—
202#108—941no—
203#108—941no—
204#107C2×Dic4711yescharacteristic
205#107C2×Dic4711yescharacteristic
206#106—21no—
207#104—1882no—
208C22⋊C8—1881no—
209C22⋊C8—1881no—
210#97—1881no—
211C4×C8—941no—
212C4×C8—941no—
213C4×C8—1881no—
214#41—41no—
215#41—41no—
216#41—41no—
217#41—41no—
218C2×Q8—21no—
219C2×D4—21no—
220C2×D4—41no—
221C22×C4—41no—
222C22×C4—941no—
223C22×C4D4×D4711yescharacteristic
224Q16—21no—
225Q16—21no—
226SD16—41no—
227SD16—41no—
228D8—21no—
229D8—21no—
230M16—942no—
231M16—1881no—
232M16—1881no—
233M16—1881no—
234C2×C8—471no—
235C2×C8—471no—
236C2×C8—941no—
237C2×C8—941no—
238C2×C8—1881no—
239C2×C8—1881no—
240C2×C8#1062535011yescharacteristic
241C2×C8#1062535011yescharacteristic
242C2×C8D4×D4711yescharacteristic
243C2×C8#1062535211yescharacteristic
244C2×C8#1062535211yescharacteristic
245C2×C8C22×Dic4711yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—21no—
248C22⋊C4—41no—
249C42—41no—
250C42—941no—
251C42—941no—
252C16—21no—
253C16—21no—
254C16—1881no—
255C23—41no—
256C23#1309404311yescharacteristic
257Q8—41no—
258Q8—41no—
259D4—41no—
260D4—41no—
261D4—81no—
262D4—81no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—942no—
269C2×C4—1881no—
270C2×C4—1881no—
271C2×C4D4×Dic4711yescharacteristic
272C2×C4#1309399711yescharacteristic
273C2×C4#1309400611yescharacteristic
274C2×C4D8×D4711yescharacteristic
275C2×C4#1309405111yescharacteristic
276C2×C4#1309405311yescharacteristic
277C8—941no—
278C8—941no—
279C8—3761no—
280C8—1882no—
281C8#1309398311yescharacteristic
282C8D4×Dic4711yescharacteristic
283C8#1309401011yescharacteristic
284C8#1309401711yescharacteristic
285C22—21no—
286C22—21no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22D8×Dic4711yescharacteristic
291C22#1568934411yescharacteristic
292C22#1568944511yescharacteristic
293C4—81no—
294C4—81no—
295C4—81no—
296C4—1882no—
297C4#1568901611yescharacteristic
298C4#1568922411yescharacteristic
299C4#1568931711yescharacteristic
300C4#1568943511yescharacteristic
301C2—21no—
302C2—21no—
303C2—161no—
304C2#2024401011yescharacteristic
305C2#2024526911yescharacteristic
306C2#2024527411yescharacteristic
subgroupG/H
centerC2×C4D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2138C2
socleC22×C47C4×D8
prime radicalC2×#135×C47C22
cosocleC22×C8×C47C23
hypercenter#2138D47
hypocenterC47#13134
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. C2×#15688953C2semidirect
4. #20243269C2semidirect
5. #20241483C2semidirect
6. #20241465C2semidirect
21. C47×#358C4semidirect
22. C47×#358C4semidirect
23. C47×#357C4semidirect
24. C47×#357C4semidirect
173. #358Dic47semidirect
174. #358Dic47semidirect
175. #357Dic47semidirect
176. #357Dic47semidirect
178. C47#13134semidirect

Maximal subgroups

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

subgroupindex
C2×#156889532
C2×#156889302
#202442772
#202432692
#202414832
#202414652
C47×#21382
#1313447