Finite Group Databasebeta

← #26131351

Subgroups — #26131351

Order 12032.

444 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251815C211yesmaximal, characteristic
2#20245017C211yesmaximal, characteristic
3#20245015C211yesmaximal, characteristic
4C2×#15688900C211yesmaximal, characteristic, complemented
5C2×#15688900C211yesmaximal, characteristic, complemented
6#20241474C211yesmaximal, characteristic, complemented
7C47×#1378C211yesmaximal, characteristic, complemented
8C2×#13094032C411yescharacteristic
9C2×#13094030C2211yescharacteristic
10#15689591C411yescharacteristic
11C2×#13093964C2211yescharacteristic
12C2×#13093964C2211yescharacteristic
13#15689080—21no—
14#15689079C411yescharacteristic
15#15689077—21no—
16M16×Dic47C411yescharacteristic
17C4×#10625324—21no—
18C2×#13093934C2211yescharacteristic
19#97×D47—21no—
20#15688908C414yescomplemented
21#15688906C414yescomplemented
22#15688900—21no—
23#15688900—21no—
24#15688900—21no—
25#15688900—21no—
26C2×M32×C47C2211yescharacteristic
27C2×M32×C47C2211yescharacteristic
28C47×#405C2211yescharacteristic
29C2×#10625350C2×C411yescharacteristic
30#13094032—22no—
31#13094032—22no—
32#13094030—21no—
33#13094030—21no—
34#13094030C2×C412yes—
35#13094030C2×C412yes—
36C2×#10625329C2×C411yescharacteristic
37C2×#10625324—21no—
38C2×#10625324—21no—
39C22×C47⋊C8C2×C411yescharacteristic
40C2×C8×D47—21no—
41C2×C8×D47—21no—
42#13093989—21no—
43#13093986—21no—
44(C22⋊D47)⋊C4—22no—
45#13093981—21no—
46C2×C4×Dic47C2×C411yescharacteristic
47C42×D47—21no—
48#13093964—21no—
49#13093964—21no—
50#13093964—21no—
51#13093964C2×C412yescomplemented
52C2×C47⋊C16—21no—
53C2×C47⋊C16C2×C412yescomplemented
54#13093935—21no—
55#13093934—21no—
56#13093934D411yescharacteristic
57#13093934D411yescharacteristic
58#13093934D411yescharacteristic
59#13093934D411yescharacteristic
60C4×C47⋊C8—21no—
61C8×Dic47—21no—
62C2×M16×C47C2×C411yescharacteristic
63C22×C8×C47C2311yescharacteristic
64#117×C47C2×C411yescharacteristic
65M32×C47—21no—
66M32×C47—21no—
67M32×C47—21no—
68M32×C47C2×C412yescomplemented
69C2×C16×C47—21no—
70C2×C16×C47C2×C412yescomplemented
71#97×C47—21no—
72C4×C8×C47—21no—
73C22×C4×C47C22×C411yescharacteristic
74C2×C22⋊D47—21no—
75C22×Dic47—21no—
76#10625350—21no—
77#10625350—21no—
78#10625350C4212yes—
79#10625350M1612yes—
80C2×D188M1611yescharacteristic
81C2×C4×D47—21no—
82C2×C4×D47—22no—
83C2×Dic94M1611yescharacteristic
84M16×C47—21no—
85M16×C47M1612yes—
86C2×C8×C47—21no—
87C2×C8×C47C22×C411yescharacteristic
88C2×C8×C47C22×C411yescharacteristic
89C2×C8×C47C2×D411yescharacteristic
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47#4111yescharacteristic
92C2×C8×C47#4111yescharacteristic
93C4⋊C4×C47—21no—
94C22⋊C4×C47M1612yes—
95C42×C47—21no—
96(C2×D47)⋊C4—22no—
97Dic47⋊C4—22no—
98C4×Dic47—22no—
99C4×Dic47—21no—
100C4×Dic47C22⋊C411yescharacteristic
101C4×Dic47C22⋊C411yescharacteristic
102#10625329—21no—
103#10625329C4212yes—
104C2×C47⋊C8—21no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8C22⋊C411yescharacteristic
108C2×C47⋊C8C22⋊C411yescharacteristic
109#10625324—24no—
110#10625324—22no—
111#10625324—22no—
112C8×D47—24no—
113C8×D47—22no—
114C8×D47—22no—
115C16×C47—22no—
116C16×C47—22no—
117C47⋊C16—22no—
118C47⋊C16—22no—
119C23×C47C2×M1611yescharacteristic
120C22×D47—21no—
121C2×C4×C47—22no—
122C2×C4×C47C2×C4211yescharacteristic
123C2×C4×C47C2×C22⋊C411yescharacteristic
124C2×C4×C47#11711yescharacteristic
125C2×C4×C47C2×M1611yescharacteristic
126C2×C4×C47C2×M1611yescharacteristic
127C2×C4×C47C2×M1611yescharacteristic
128C22⋊D47—22no—
129C22⋊D47—22no—
130C2×Dic47—21no—
131C2×Dic47—21no—
132C2×Dic47—21no—
133C2×Dic47—81no—
134D188—21no—
135D188#9712yes—
136C4×D47—22no—
137C4×D47—22no—
138C4×D47—82no—
139Dic94—21no—
140Dic94#9712yes—
141C8×C47—22no—
142C8×C47C4×D411yescharacteristic
143C8×C47C4×D411yescharacteristic
144C8×C47C4×D411yescharacteristic
145C8×C47C4×D411yescharacteristic
146C47⋊C8—22no—
147C47⋊C8—21no—
148C47⋊C8—21no—
149#12609—471nomaximal, Sylow
150C22×C47—41no—
151C22×C47C4×M1611yescharacteristic
152C22×C47#40711yescharacteristic
153C22×C47#43211yescharacteristic
154C2×D47—22no—
155C2×D47—42no—
156C4×C47—42no—
157C4×C47C4×C22⋊C411yescharacteristic
158C4×C47C2×#9711yescharacteristic
159C4×C47#40711yescharacteristic
160C4×C47#43211yescharacteristic
161Dic47—22no—
162Dic47—41no—
163Dic47—41no—
164#2875—471no—
165#2117—471no—
166#2117—471no—
167C2×#347—471no—
168C2×#347—471no—
169#1378—471no—
170#1378D4711yescharacteristic, complemented
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47#174911yescharacteristic
175D47—42no—
176C2×#131—471no—
177C2×#131—471no—
178C4×#41—471no—
179C2×M32—471no—
180C2×M32—471no—
181C2×M32C2×D4711yescharacteristic
182C2×M32C2×D4711yescharacteristic
183#405—471no—
184#405—941no—
185#405—941no—
186#405C2×D4711yescharacteristic
187C4×M16—471no—
188C4×M16—941no—
189C2×#97—471no—
190C2×#97—941no—
191#350—474no—
192#350—474no—
193#347—941no—
194#347—941no—
195#347—941no—
196#347—941no—
197C47#1260911yescharacteristic, complemented, Sylow
198C2×#41—471no—
199#131—472no—
200#131—472no—
201#131—942no—
202#131—942no—
203#131—941no—
204#131—941no—
205C2×M16—471no—
206C2×M16—941no—
207C2×M16—941no—
208C2×M16C4×D4711yescharacteristic
209C22×C8—471no—
210C22×C8—941no—
211C22×C8—941no—
212C22×C8C22×D4711yescharacteristic
213C4×Q8—941no—
214C4×D4—941no—
215C4×D4—942no—
216#117—941no—
217#117C4×D4711yescharacteristic
218C2×C42—471no—
219C2×C42—941no—
220M32—21no—
221M32—21no—
222M32—21no—
223M32—472no—
224M32—941no—
225M32—941no—
226M32—941no—
227M32C4×D4712yescomplemented
228C2×C16—21no—
229C2×C16—472no—
230C2×C16—941no—
231C2×C16C4×D4712yescomplemented
232#97—21no—
233#97—471no—
234#97—471no—
235#97—471no—
236#97—471no—
237#97—941no—
238#97—941no—
239C4×C8—21no—
240C4×C8—941no—
241C4×C8—941no—
242#41—472no—
243#41—472no—
244#41—941no—
245#41—941no—
246C2×Q8—471no—
247C2×D4—471no—
248C2×D4—941no—
249C22×C4—942no—
250C22×C4—941no—
251C22×C4—941no—
252C22×C4C2×C4×D4711yescharacteristic
253M16—21no—
254M16—472no—
255M16—944no—
256M16—941no—
257M16—942no—
258M16—942no—
259M16#1062532412yes—
260C2×C8—21no—
261C2×C8—471no—
262C2×C8—471no—
263C2×C8—944no—
264C2×C8—941no—
265C2×C8—941no—
266C2×C8—941no—
267C2×C8—942no—
268C2×C8—942no—
269C2×C8C2×C4×D4711yescharacteristic
270C2×C8C2×C4×D4711yescharacteristic
271C2×C8D4×D4711yescharacteristic
272C2×C8D4×D4711yescharacteristic
273C2×C8#1062535211yescharacteristic
274C2×C8#1062535411yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—942no—
277C22⋊C4—942no—
278C22⋊C4#1062532412yes—
279C42—21no—
280C42—471no—
281C42—471no—
282C42—942no—
283C42—941no—
284C16—22no—
285C16—22no—
286C16—942no—
287C16—942no—
288C23—941no—
289C23C2×#1062532411yescharacteristic
290Q8—472no—
291Q8—941no—
292D4—472no—
293D4—942no—
294D4—942no—
295D4—941no—
296C2×C4—22no—
297C2×C4—941no—
298C2×C4—942no—
299C2×C4—942no—
300C2×C4—941no—
301C2×C4—941no—
302C2×C4—3762no—
303C2×C4—3761no—
304C2×C4C42×D4711yescharacteristic
305C2×C4C22⋊C4×D4711yescharacteristic
306C2×C4#1309398211yescharacteristic
307C2×C4M16×D4711yescharacteristic
308C2×C4M16×D4711yescharacteristic
309C2×C4C2×#1062532411yescharacteristic
310C8—22no—
311C8—942no—
312C8—941no—
313C8—941no—
314C8(C22⋊D47)⋊C411yescharacteristic
315C8(C22⋊D47)⋊C411yescharacteristic
316C8#1309398611yescharacteristic
317C8#1309398611yescharacteristic
318C22—41no—
319C22—942no—
320C22—1882no—
321C22C4×#1062532411yescharacteristic
322C22#1568908511yescharacteristic
323C22#1568917911yescharacteristic
324C4—42no—
325C4—942no—
326C4—1881no—
327C4—1881no—
328C4#1568897011yescharacteristic
329C4#97×D4711yescharacteristic
330C4#1568908511yescharacteristic
331C4#1568917911yescharacteristic
332C2—21no—
333C2—21no—
334C2—21no—
335C2—1882no—
336C2#2024314911yescharacteristic
subgroupG/H
centerC4#15688970
commutatorC2×C4×C47C2×C42
FrattiniC22×C8C22×D47
FittingC47×#1378C2
socleC2×C47#1749
prime radicalC2×#10625350C2×C4
cosocleC22×C8×C47C23
hypercenter#1378D47
hypocenterC47#12609
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1378

Complements

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

NH ≅ G/Ntype
4. C2×#15688900C2semidirect
5. C2×#15688900C2semidirect
6. #20241474C2semidirect
7. C47×#1378C2semidirect
20. #15688908C4semidirect
21. #15688906C4semidirect
51. #13093964C2×C4semidirect
53. C2×C47⋊C16C2×C4semidirect
68. M32×C47C2×C4semidirect
70. C2×C16×C47C2×C4semidirect
170. #1378D47semidirect
197. C47#12609semidirect
227. M32C4×D47semidirect
231. C2×C16C4×D47semidirect

Maximal subgroups

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

subgroupindex
#202518152
#202450172
#202450152
C2×#156889002
#202414742
C47×#13782
#1260947