Finite Group Databasebeta

← #26134100

Subgroups — #26134100

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×#15688952C211yesmaximal, characteristic, complemented
2C2×#15688930C211yesmaximal, characteristic
3#20245095C211yesmaximal, characteristic
4#20243267C211yesmaximal, characteristic, complemented
5#20241485C211yesmaximal, characteristic, complemented
6#20241466C211yesmaximal, characteristic, complemented
7C47×#1974C211yesmaximal, characteristic
8C22×C47⋊C16C2211yescharacteristic
9C22×C47⋊C16C2211yescharacteristic
10C2×#13093960C2211yescharacteristic
11#15689161C2211yescharacteristic
12#15689078—21no—
13#15688952—22no—
14#15688930—22no—
15#15688927C411yescharacteristic, complemented
16#15688927C411yescharacteristic, complemented
17#15688924C411yescharacteristic, complemented
18#15688924C411yescharacteristic, complemented
19#15688883—21no—
20C2×#135×C47C2211yescharacteristic
21C2×#108×C47C2211yescharacteristic
22C2×#104×C47—21no—
23C47×#408—21no—
24C2×C4×C8×C47C2211yescharacteristic
25C2×#10625329—21no—
26#13093983—21no—
27C2×C47⋊C16—21no—
28C2×C47⋊C16—21no—
29C2×C47⋊C16—21no—
30C2×C47⋊C16—21no—
31C2×C47⋊C16—21no—
32C2×C47⋊C16C2×C411yescharacteristic
33C2×C47⋊C16C2×C411yescharacteristic
34#13093960—21no—
35#13093960D412yes—
36#13093958C2×C411yescharacteristic
37#13093958C2×C411yescharacteristic
38#13093956—21no—
39#13093934—41no—
40C8×Dic47—41no—
41C2×#41×C47—21no—
42#135×C47—21no—
43#135×C47—21no—
44#135×C47—21no—
45#135×C47—21no—
46C2×Q16×C47C2×C411yescharacteristic
47C2×SD16×C47—21no—
48C2×D8×C47C2×C411yescharacteristic
49C2×M16×C47—21no—
50C22×C8×C47D411yescharacteristic
51C22×C8×C47C2311yescharacteristic
52C2×C42×C47D411yescharacteristic
53#108×C47—21no—
54#108×C47—21no—
55#104×C47—42no—
56C22⋊C8×C47—41no—
57C22⋊C8×C47—41no—
58C4×C8×C47—21no—
59C4×C8×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—21no—
68C22×C4×C47—41no—
69C22×C4×C47C2×D411yescharacteristic
70Q16×C47—21no—
71Q16×C47—21no—
72SD16×C47—41no—
73SD16×C47—41no—
74D8×C47—21no—
75D8×C47—21no—
76M16×C47—41no—
77M16×C47—41no—
78C2×C8×C47—21no—
79C2×C8×C47—21no—
80C2×C8×C47—41no—
81C2×C8×C47D811yescharacteristic
82C2×C8×C47D811yescharacteristic
83C2×C8×C47C22×C411yescharacteristic
84C2×C8×C47C2×D411yescharacteristic
85C2×C8×C47#4111yescharacteristic
86C2×C8×C47#4111yescharacteristic
87C2×C8×C47#4111yescharacteristic
88C2×C8×C47#4111yescharacteristic
89C42×C47—21no—
90C42×C47—21no—
91C22⋊Dic47—41no—
92C4⋊Dic47—21no—
93C4⋊Dic47—21no—
94C4×Dic47—41no—
95#10625329—22no—
96#10625329—41no—
97C2×C47⋊C8—41no—
98C47⋊C16—21no—
99C47⋊C16—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—22no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—81no—
114C2×C4×C47—81no—
115C2×C4×C47C4×D411yescharacteristic
116C2×C4×C47#12111yescharacteristic
117C2×C4×C47#12311yescharacteristic
118C2×C4×C47C2×D811yescharacteristic
119C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C47#13711yescharacteristic
121C2×Dic47—41no—
122C2×Dic47—41no—
123C8×C47—21no—
124C8×C47—21no—
125C8×C47—81no—
126C8×C47#11711yescharacteristic
127C8×C47C4×D411yescharacteristic
128C8×C47#12411yescharacteristic
129C8×C47#12611yescharacteristic
130C47⋊C8—42no—
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—42no—
141C4×C47—81no—
142C4×C47#38811yescharacteristic
143C4×C47#44011yescharacteristic
144C4×C47#45911yescharacteristic
145C4×C47#48011yescharacteristic
146Dic47—81no—
147Dic47—81no—
148C2×#368—471no—
149C2×#359—471no—
150#2138—471no—
151#1974D4711yescharacteristic
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×C16—471no—
164C2×#108—471no—
165C2×#108C2×D4711yescharacteristic
166#427—471no—
167C2×#104—21no—
168#408—21no—
169#405—941no—
170C2×C4×C8C2×D4711yescharacteristic
171#368—942no—
172#359—942no—
173#358—471no—
174#358—471no—
175#357—471no—
176#357—471no—
177#339—941no—
178C47#1313411yescharacteristic, complemented, Sylow
179C2×#41—21no—
180#135—21no—
181#135—21no—
182#135—21no—
183#135—21no—
184C2×Q16C4×D4711yescharacteristic
185C2×SD16—21no—
186C2×D8C4×D4711yescharacteristic
187C2×M16—21no—
188C2×M16—941no—
189C22×C8C22⋊D4711yescharacteristic
190C22×C8C22×D4711yescharacteristic
191#117—941no—
192C2×C42C22⋊D4711yescharacteristic
193C2×C16—471no—
194C2×C16—471no—
195C2×C16—941no—
196C2×C16—941no—
197C2×C16—941no—
198C2×C16—941no—
199C2×C16—941no—
200#108—21no—
201#108—21no—
202#108—472no—
203#108—941no—
204#107—471no—
205#107—471no—
206#106—941no—
207#104—42no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210#97—1881no—
211C4×C8—21no—
212C4×C8—21no—
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—21no—
222C22×C4—41no—
223C22×C4C2×C22⋊D4711yescharacteristic
224Q16—21no—
225Q16—21no—
226SD16—41no—
227SD16—41no—
228D8—21no—
229D8—21no—
230M16—41no—
231M16—41no—
232M16—942no—
233M16—1881no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—1881no—
238C2×C8#1062533411yescharacteristic
239C2×C8#1062533411yescharacteristic
240C2×C8C2×C4×D4711yescharacteristic
241C2×C8#1062535011yescharacteristic
242C2×C8D4×D4711yescharacteristic
243C2×C8#1062535211yescharacteristic
244C2×C8#1062535211yescharacteristic
245C2×C8#1062535211yescharacteristic
246C4⋊C4—941no—
247C4⋊C4—941no—
248C22⋊C4—1881no—
249C42—21no—
250C42—21no—
251C42—1881no—
252C16—941no—
253C16—941no—
254C16—1881no—
255C23—41no—
256C23#1309404411yescharacteristic
257Q8—41no—
258Q8—41no—
259D4—41no—
260D4—41no—
261D4—81no—
262D4—81no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—1881no—
270C2×C4—1881no—
271C2×C4C4×C22⋊D4711yescharacteristic
272C2×C4#1309399511yescharacteristic
273C2×C4#1309400511yescharacteristic
274C2×C4C2×#1062533411yescharacteristic
275C2×C4#1309405011yescharacteristic
276C2×C4#1309405411yescharacteristic
277C8—21no—
278C8—21no—
279C8—81no—
280C8—1882no—
281C8#1309398011yescharacteristic
282C8(C22⋊D47)⋊C411yescharacteristic
283C8#1309401011yescharacteristic
284C8#1309401711yescharacteristic
285C22—21no—
286C22—21no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22C4×#1062533411yescharacteristic
291C22#1568934211yescharacteristic
292C22#1568944411yescharacteristic
293C4—42no—
294C4—81no—
295C4—3761no—
296C4—3761no—
297C4#1568901111yescharacteristic
298C4#1568922311yescharacteristic
299C4#1568931511yescharacteristic
300C4#1568943411yescharacteristic
301C2—21no—
302C2—21no—
303C2—161no—
304C2#2024400511yescharacteristic
305C2#2024526711yescharacteristic
306C2#2024527211yescharacteristic
subgroupG/H
centerC2×C4C2×#10625334
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1974C2
socleC22×C47C4×D8
prime radicalC2×#135×C47C22
cosocleC22×C8×C47C23
hypercenter#1974D47
hypocenterC47#13134
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ #1974

Complements

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

NH ≅ G/Ntype
1. C2×#15688952C2semidirect
4. #20243267C2semidirect
5. #20241485C2semidirect
6. #20241466C2semidirect
15. #15688927C4semidirect
16. #15688927C4semidirect
17. #15688924C4semidirect
18. #15688924C4semidirect
178. C47#13134semidirect

Maximal subgroups

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

subgroupindex
C2×#156889522
C2×#156889302
#202450952
#202432672
#202414852
#202414662
C47×#19742
#1313447