Finite Group Databasebeta

← #26133205

Subgroups — #26133205

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256490C211yesmaximal, characteristic, complemented
2#20252093C211yesmaximal, characteristic
3#20245008C211yesmaximal, characteristic, complemented
4C2×#15688904C211yesmaximal, characteristic, complemented
5C2×#15688904C211yesmaximal, characteristic, complemented
6#20241474C211yesmaximal, characteristic, complemented
7C47×#1378C211yesmaximal, characteristic, complemented
8C2×#13094030C2211yescharacteristic
9C22×#10625324C2211yescharacteristic
10#15689635C411yescharacteristic
11#15689541—22no—
12#15689539—22no—
13C2×#13093964C2211yescharacteristic
14C2×#13093964C2211yescharacteristic
15M32×D47—22no—
16#15689202—22no—
17#15689196—22no—
18#15689150C411yescharacteristic
19#15689091C411yescharacteristic
20#15689086C411yescharacteristic
21#15688904—22no—
22#15688904—22no—
23#15688904—22no—
24C2×M32×C47C2211yescharacteristic
25C2×M32×C47C2211yescharacteristic
26C47×#405C2211yescharacteristic
27C2×#10625350C2×C411yescharacteristic
28C22×C4×D47C2×C411yescharacteristic
29#13094030—21no—
30#13094030—21no—
31#13094030—21no—
32#13094030—21no—
33C2×#10625329C2×C411yescharacteristic
34C2×#10625324—21no—
35C2×#10625324—21no—
36C2×#10625324—21no—
37C2×#10625324—21no—
38C2×#10625324—21no—
39C2×#10625324—21no—
40C2×#10625324D412yes—
41C22×C47⋊C8C2×C411yescharacteristic
42C2×C8×D47D412yes—
43#13094008C812yes—
44#13094004—21no—
45#13093999—21no—
46#13093994C812yes—
47C4×D188—22no—
48#13093964—21no—
49#13093964—21no—
50#13093964—21no—
51#13093964D412yescomplemented
52#13093963—24no—
53C2×C47⋊C16—21no—
54C2×C47⋊C16—21no—
55C16×D47—24no—
56C4⋊(C47⋊C8)—22no—
57(C2×D47)⋊C8—22no—
58(C2×D47)⋊C8—22no—
59C2×M16×C47C2×C411yescharacteristic
60C22×C8×C47C2311yescharacteristic
61#117×C47C2×C411yescharacteristic
62M32×C47—21no—
63M32×C47—21no—
64M32×C47—21no—
65M32×C47D412yescomplemented
66C2×C16×C47—21no—
67C2×C16×C47—21no—
68#97×C47—21no—
69C4×C8×C47—21no—
70C23×D47C2×C811yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C2×C22⋊D47C2×C812yes—
73C22×Dic47C2×C811yescharacteristic
74#10625350—21no—
75#10625350—21no—
76#10625350—21no—
77#10625350—21no—
78C2×D188—22no—
79C2×D188M1611yescharacteristic
80C2×C4×D47—21no—
81C2×C4×D47—21no—
82C2×C4×D47—21no—
83C2×C4×D47—21no—
84C2×C4×D47—21no—
85C2×C4×D47—21no—
86C2×C4×D47C22⋊C412yes—
87C2×Dic94M1611yescharacteristic
88M16×C47—21no—
89M16×C47—21no—
90C2×C8×C47—21no—
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47C2×D411yescharacteristic
94C2×C8×C47C2×D411yescharacteristic
95C2×C8×C47#4111yescharacteristic
96C2×C8×C47#4111yescharacteristic
97C4⋊C4×C47—21no—
98C22⋊C4×C47C2×C812yes—
99C42×C47—21no—
100(C2×D47)⋊C4—22no—
101(C2×D47)⋊C4—22no—
102C4⋊Dic47—22no—
103#10625329—21no—
104#10625329—21no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8C22⋊C412yes—
109#10625324—22no—
110#10625324—22no—
111#10625324—41no—
112#10625324—41no—
113#10625324—41no—
114#10625324—41no—
115C8×D47—22no—
116C8×D47—22no—
117C16×C47—22no—
118C16×C47—41no—
119C47⋊C16—22no—
120C47⋊C16—41no—
121C23×C47C22×C811yescharacteristic
122C22×D47—21no—
123C22×D47—21no—
124C22×D47—21no—
125C22×D47—81no—
126C22×D47C22⋊C812yes—
127C2×C4×C47—22no—
128C2×C4×C47C2×C22⋊C411yescharacteristic
129C2×C4×C47C4×D411yescharacteristic
130C2×C4×C47C4×D411yescharacteristic
131C2×C4×C47C2×M1611yescharacteristic
132C2×C4×C47#13111yescharacteristic
133C2×C4×C47#13111yescharacteristic
134C22⋊D47—22no—
135C22⋊D47—22no—
136C2×Dic47—21no—
137C2×Dic47—21no—
138C2×Dic47—21no—
139C2×Dic47C22⋊C812yes—
140D188—21no—
141D188—21no—
142D188—82no—
143C4×D47—22no—
144C4×D47—22no—
145C4×D47—41no—
146C4×D47—41no—
147C4×D47—41no—
148C4×D47—41no—
149Dic94—21no—
150Dic94—21no—
151C8×C47—41no—
152C8×C47#12011yescharacteristic
153C8×C47#12111yescharacteristic
154C8×C47#12111yescharacteristic
155C8×C47#12311yescharacteristic
156C47⋊C8—22no—
157C47⋊C8—41no—
158#12989—471nomaximal, Sylow
159C22×C47—41no—
160C22×C47C2×C22⋊C811yescharacteristic
161C22×C47C8×D411yescharacteristic
162C22×C47C8×D411yescharacteristic
163C2×D47—22no—
164C2×D47—42no—
165C2×D47—41no—
166C2×D47—41no—
167C2×D47—41no—
168C2×D47—81no—
169C2×D47—81no—
170C4×C47—42no—
171C4×C47#38611yescharacteristic
172C4×C47#40811yescharacteristic
173C4×C47#43511yescharacteristic
174C4×C47#43511yescharacteristic
175Dic47—22no—
176Dic47—41no—
177#3407—471no—
178#2930—471no—
179#2115—471no—
180C2×#349—471no—
181C2×#349—471no—
182#1378—471no—
183#1378D4711yescharacteristic, complemented
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47#186911yescharacteristic
188D47—42no—
189D47—81no—
190C2×#131—471no—
191C22×M16—471no—
192#525—471no—
193#504—942no—
194#504—942no—
195C2×M32—471no—
196C2×M32—471no—
197C2×M32—942no—
198C2×M32C2×D4711yescharacteristic
199C2×M32C2×D4711yescharacteristic
200#436—942no—
201#435—942no—
202#424—471no—
203#408—471no—
204#407—471no—
205#405C2×D4711yescharacteristic
206#349—942no—
207#349—942no—
208#349—942no—
209C47#1298911yescharacteristic, complemented, Sylow
210C2×#41—471no—
211C23×C4—471no—
212#131—941no—
213#131—941no—
214#131—941no—
215#131—941no—
216C2×M16—471no—
217C2×M16—472no—
218C2×M16—941no—
219C2×M16—941no—
220C2×M16—941no—
221C2×M16—941no—
222C2×M16—941no—
223C2×M16—941no—
224C2×M16C4×D4711yescharacteristic
225C22×C8—472no—
226C22×C8—471no—
227C22×C8C22×D4711yescharacteristic
228#123—472no—
229C22⋊Q8—941no—
230#121—941no—
231#120—472no—
232C4×D4—942no—
233#117C4×D4711yescharacteristic
234M32—21no—
235M32—21no—
236M32—21no—
237M32—472no—
238M32—944no—
239M32—941no—
240M32—941no—
241M32—941no—
242M32D18812yescomplemented
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—944no—
246C2×C16—941no—
247C2×C16—941no—
248C4⋊C8—942no—
249C22⋊C8—942no—
250C22⋊C8—942no—
251#97—21no—
252C4×C8—21no—
253C24—471no—
254#41—941no—
255#41—941no—
256#41—941no—
257#41—941no—
258C2×Q8—471no—
259C2×D4—472no—
260C2×D4—471no—
261C2×D4—942no—
262C22×C4—472no—
263C22×C4—471no—
264C22×C4—941no—
265C22×C4—941no—
266C22×C4—941no—
267C22×C4—941no—
268C22×C4—941no—
269C22×C4—941no—
270C22×C4C2×C4×D4711yescharacteristic
271M16—21no—
272M16—21no—
273M16—941no—
274M16—941no—
275M16—942no—
276M16—942no—
277M16—1881no—
278M16—1881no—
279M16—1881no—
280M16—1881no—
281C2×C8—21no—
282C2×C8—472no—
283C2×C8—942no—
284C2×C8—942no—
285C2×C8—941no—
286C2×C8—941no—
287C2×C8—941no—
288C2×C8C2×D18811yescharacteristic
289C2×C8#1062535011yescharacteristic
290C2×C8D4×D4711yescharacteristic
291C2×C8D4×D4711yescharacteristic
292C2×C8D4×D4711yescharacteristic
293C2×C8#1062535411yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—942no—
296C22⋊C4—942no—
297C22⋊C4—942no—
298C22⋊C4C8×D4712yes—
299C42—21no—
300C16—22no—
301C16—41no—
302C16—942no—
303C16—1881no—
304C23—472no—
305C23—941no—
306C23—941no—
307C23—941no—
308C23—3761no—
309C23C2×C8×D4711yescharacteristic
310Q8—941no—
311Q8—941no—
312D4—942no—
313D4—942no—
314D4—941no—
315D4—941no—
316D4—3762no—
317C2×C4—22no—
318C2×C4—472no—
319C2×C4—942no—
320C2×C4—942no—
321C2×C4—941no—
322C2×C4—941no—
323C2×C4—941no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4C22⋊C4×D4711yescharacteristic
329C2×C4#1309398611yescharacteristic
330C2×C4C4×D18811yescharacteristic
331C2×C4M16×D4711yescharacteristic
332C2×C4#1309403011yescharacteristic
333C2×C4#1309403211yescharacteristic
334C8—41no—
335C8—942no—
336C8—1881no—
337C8#1309399411yescharacteristic
338C8#1309399711yescharacteristic
339C8#1309399911yescharacteristic
340C8#1309400911yescharacteristic
341C22—41no—
342C22—942no—
343C22—3761no—
344C22—3761no—
345C22—1881no—
346C22—1881no—
347C22—1882no—
348C22—1881no—
349C22C22⋊C8×D4711yescharacteristic
350C22#1568918811yescharacteristic
351C22C8×D18811yescharacteristic
352C4—42no—
353C4—942no—
354C4—1881no—
355C4#1568900111yescharacteristic
356C4#1568908811yescharacteristic
357C4#1568919311yescharacteristic
358C4#1568919611yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—3761no—
363C2—1882no—
364C2#2024367911yescharacteristic
subgroupG/H
centerC4#15689001
commutatorC23×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#1378C2
socleC2×C47#1869
prime radical#15689635C4
cosocleC22×C8×C47C23
hypercenter#1378D47
hypocenterC47#12989
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1378

Complements

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

NH ≅ G/Ntype
1. #20256490C2semidirect
3. #20245008C2semidirect
4. C2×#15688904C2semidirect
5. C2×#15688904C2semidirect
6. #20241474C2semidirect
7. C47×#1378C2semidirect
51. #13093964D4semidirect
65. M32×C47D4semidirect
183. #1378D47semidirect
209. C47#12989semidirect
242. M32D188semidirect

Maximal subgroups

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

subgroupindex
#202564902
#202520932
#202450082
C2×#156889042
#202414742
C47×#13782
#1298947