Finite Group Databasebeta

← #26371801

Subgroups — #26371801

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253901C211yesmaximal, characteristic
2#500×D47C211yesmaximal, characteristic
3#20245675C211yesmaximal, characteristic, complemented
4#20245674C211yesmaximal, characteristic, complemented
5#20245671C211yesmaximal, characteristic, complemented
6#20245669C212yesmaximal, complemented
7#20245662C212yesmaximal, complemented
8#20245175C212yesmaximal
9#20245173C212yesmaximal
10#20244981C211yesmaximal, characteristic
11C47×#2255C211yesmaximal, characteristic, complemented
12#15689679—21no—
13C4⋊Q8×D47—21no—
14C2×#13093963C2212yes—
15#15689528C2211yescharacteristic
16#15689527—21no—
17#15689527C2211yescharacteristic
18#15689527C2212yes—
19#15689527C2211yescharacteristic
20#15689526—21no—
21#15689525C2211yescharacteristic
22#15689522—21no—
23#15689520—21no—
24#15689160C2211yescharacteristic
25#15689160C2211yescharacteristic
26#15689158—21no—
27#107×D47—21no—
28#107×D47C2212yes—
29C4×C8×D47C2211yescharacteristic
30#15688951C2212yes—
31#15688950C2212yes—
32#15688950C2212yes—
33#15688949C2212yes—
34#15688948C2212yes—
35#15688948C2212yes—
36#15688899C2211yescharacteristic
37C16×Dic47C2212yes—
38C47×#500C2211yescharacteristic
39C47×#499C2211yescharacteristic
40C47×#367C2212yes—
41C47×#366C2212yes—
42C47×#346C2211yescharacteristic
43C2×Q8×D47—41no—
44C2×C8×D47D411yescharacteristic
45C2×C8×D47C2311yescharacteristic
46#13094024—21no—
47#13094023—21no—
48#13094023—21no—
49#13094023—22no—
50#13094023—41no—
51#13094022—41no—
52#13094016—21no—
53#13094002—41no—
54#13094002—41no—
55#13093989—41no—
56#13093989—41no—
57#13093982—21no—
58#13093982—21no—
59C4⋊C4×D47—22no—
60C4⋊C4×D47—41no—
61C42×D47D411yescharacteristic
62#13093963Q818yescomplemented
63C2×C47⋊C16C2312yes—
64#13093958—21no—
65#13093958C2311yescharacteristic
66#13093958C2311yescharacteristic
67#13093958C2312yes—
68#13093957—21no—
69#13093957—21no—
70#13093957—22no—
71#13093957—22no—
72#13093956—21no—
73#13093955—21no—
74#13093955—21no—
75C4×C47⋊C8D411yescharacteristic
76C8×Dic47D411yescharacteristic
77C8×Dic47C2311yescharacteristic
78C4⋊Q8×C47—21no—
79#125×C47—21no—
80C2×C16×C47C2312yes—
81#107×C47—21no—
82#107×C47C2311yescharacteristic
83#107×C47C2311yescharacteristic
84#107×C47C2312yes—
85#106×C47—21no—
86C4×C8×C47C2311yescharacteristic
87C2×Q8×C47—41no—
88Q8×D47—84no—
89C2×C4×D47—42no—
90C2×C4×D47D811yescharacteristic
91C2×C4×D47D811yescharacteristic
92C2×C4×D47C2×D411yescharacteristic
93C2×Dic94—41no—
94C2×Dic94—41no—
95C2×Dic94—41no—
96C2×Dic94—42no—
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2411yescharacteristic
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47—41no—
105C42×C47C2×D411yescharacteristic
106(C2×D47)⋊C4—41no—
107(C2×D47)⋊C4—41no—
108C4⋊Dic47—21no—
109C4⋊Dic47—21no—
110C4⋊Dic47—22no—
111C4⋊Dic47—41no—
112Dic47⋊C4—41no—
113Dic47⋊C4—41no—
114Dic47⋊C4—42no—
115Dic47⋊C4—42no—
116Dic47⋊C4—41no—
117Dic47⋊C4—41no—
118C4×Dic47—41no—
119C4×Dic47—41no—
120C4×Dic47D811yescharacteristic
121C4×Dic47D811yescharacteristic
122C4×Dic47C2×D411yescharacteristic
123C2×C47⋊C8C2×D411yescharacteristic
124C2×C47⋊C8C2×D411yescharacteristic
125C8×D47Q1614yes—
126C8×D47C2×Q814yes—
127C16×C47C2×Q814yescomplemented
128C47⋊C16C2×Q814yescomplemented
129C22×D47C2×D811yescharacteristic
130Q8×C47—82no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47C2×D811yescharacteristic
135C2×C4×C47C2×D811yescharacteristic
136C2×C4×C47C22×D411yescharacteristic
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—42no—
140C2×Dic47C2×D811yescharacteristic
141C2×Dic47C2×D811yescharacteristic
142C2×Dic47C2×D811yescharacteristic
143C4×D47—22no—
144C4×D47—22no—
145C4×D47—84no—
146C4×D47C4⋊Q812yes—
147C4×D47C2×Q1612yes—
148Dic94—82no—
149Dic94—82no—
150Dic94—84no—
151Dic94—82no—
152C8×C47C2×Q1612yes—
153C8×C47C22×Q812yes—
154C47⋊C8C4⋊Q812yes—
155C47⋊C8C2×Q1612yes—
156#33731—471nomaximal, Sylow
157C22×C47C22×D811yescharacteristic
158C2×D47—21no—
159C2×D47—21no—
160C2×D47#50012yes—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—82no—
164C4×C47—81no—
165C4×C47—81no—
166C4×C47C2×C4⋊Q811yescharacteristic
167C4×C47C22×Q1611yescharacteristic
168Dic47—82no—
169Dic47—81no—
170Dic47—81no—
171Dic47#50012yes—
172Dic47#50912yes—
173Dic47#51012yes—
174#3160—471no—
175C2×#500—471no—
176#2255—471no—
177#2255—471no—
178#2255—471no—
179#2255D4711yescharacteristic, complemented
180#2254—472no—
181#2252—472no—
182#2159—472no—
183#2159—472no—
184C4×M32—471no—
185C2×C47C2×#50011yescharacteristic
186C2×C47C2×#50911yescharacteristic
187C2×C47C2×#51011yescharacteristic
188D47—22no—
189#533—941no—
190C2×C4⋊Q8—941no—
191C2×M32—472no—
192#500—472no—
193#500—471no—
194#500—471no—
195#500—471no—
196#500—941no—
197#500—941no—
198#500C2×D4711yescharacteristic
199#499—471no—
200#499—941no—
201#499C2×D4711yescharacteristic
202#498—941no—
203#427—471no—
204#427—471no—
205#427—941no—
206C2×#107—472no—
207C2×#107—941no—
208C2×C4×C8—471no—
209#367—472no—
210#367—472no—
211#367—472no—
212#367C2×D4712yes—
213#366—472no—
214#366—472no—
215#366—472no—
216#366C2×D4712yes—
217#346—471no—
218#346C2×D4711yescharacteristic
219C4×C16—472no—
220C47#3373111yescharacteristic, complemented, Sylow
221C22×Q8—1881no—
222C22×C8—471no—
223C22×C8—471no—
224C4⋊Q8—21no—
225C4⋊Q8—942no—
226C4⋊Q8—941no—
227C4⋊Q8—941no—
228C4⋊Q8—941no—
229C4⋊Q8—1881no—
230C4⋊Q8—1881no—
231#125—21no—
232#125—941no—
233C22⋊Q8—1881no—
234C22⋊Q8—1881no—
235C4×Q8—1881no—
236C4×Q8—1881no—
237#117—941no—
238#117—941no—
239C2×C4⋊C4—942no—
240C2×C4⋊C4—1881no—
241C2×C42—471no—
242M32—478no—
243C2×C16—472no—
244C2×C16C22×D4712yes—
245#107—21no—
246#107—471no—
247#107—472no—
248#107—471no—
249#107—942no—
250#107—942no—
251#107—941no—
252#107—941no—
253#107—941no—
254#107C22×D4711yescharacteristic
255#107C22×D4712yes—
256#107C22×D4711yescharacteristic
257#106—21no—
258#106—941no—
259#106—941no—
260#106—941no—
261C4×C8—471no—
262C4×C8—471no—
263C4×C8—471no—
264C4×C8C22×D4711yescharacteristic
265C2×Q8—41no—
266C2×Q8—3764no—
267C2×Q8—1881no—
268C2×Q8—1881no—
269C2×Q8—1882no—
270C2×Q8—1881no—
271C22×C4—471no—
272C22×C4—471no—
273C22×C4—471no—
274C22×C4—1882no—
275C2×C8—471no—
276C2×C8—471no—
277C2×C8—474no—
278C2×C8—474no—
279C2×C8D4×D4711yescharacteristic
280C2×C8C23×D4711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—941no—
288C4⋊C4—941no—
289C4⋊C4—942no—
290C4⋊C4—1882no—
291C4⋊C4—1882no—
292C4⋊C4—1881no—
293C4⋊C4—1881no—
294C4⋊C4—1881no—
295C4⋊C4—1881no—
296C4⋊C4—1881no—
297C22⋊C4—1881no—
298C22⋊C4—1881no—
299C42—471no—
300C42—471no—
301C42—471no—
302C42—1881no—
303C42—1881no—
304C42D4×D4711yescharacteristic
305C16—474no—
306C16Q8×D4714yescomplemented
307C23—471no—
308Q8—82no—
309Q8—3762no—
310Q8—3764no—
311Q8—3762no—
312Q8—3762no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—471no—
317C2×C4—471no—
318C2×C4—471no—
319C2×C4—472no—
320C2×C4—472no—
321C2×C4—942no—
322C2×C4—942no—
323C2×C4—3764no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1882no—
327C2×C4D8×D4711yescharacteristic
328C2×C4D8×D4711yescharacteristic
329C2×C4C2×D4×D4711yescharacteristic
330C8—472no—
331C8—472no—
332C8Q16×D4712yes—
333C8C2×Q8×D4712yes—
334C22—472no—
335C22—941no—
336C22—941no—
337C22C2×D8×D4711yescharacteristic
338C4—21no—
339C4—21no—
340C4—81no—
341C4—82no—
342C4—81no—
343C4—472no—
344C4—472no—
345C4—472no—
346C4—3761no—
347C4—3762no—
348C4—3761no—
349C4C4⋊Q8×D4711yescharacteristic
350C4C2×Q16×D4711yescharacteristic
351C2—942no—
352C2#500×D4711yescharacteristic
353C2#509×D4711yescharacteristic
354C2#510×D4711yescharacteristic
subgroupG/H
centerC22C2×D8×D47
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2255C2
socleC22×C47C22×D8
prime radicalC22×D47C2×D8
hypercenter#2255D47
hypocenterC47#33731
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #20245675C2semidirect
4. #20245674C2semidirect
5. #20245671C2semidirect
6. #20245669C2semidirect
7. #20245662C2semidirect
11. C47×#2255C2semidirect
62. #13093963Q8semidirect
127. C16×C47C2×Q8semidirect
128. C47⋊C16C2×Q8semidirect
179. #2255D47semidirect
220. C47#33731semidirect
306. C16Q8×D47semidirect

Maximal subgroups

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

subgroupindex
#202539012
#500×D472
#202456752
#202456742
#202456712
#202456692
#202456622
#202451752
#202451732
#202449812
C47×#22552
#3373147