Finite Group Databasebeta

← #26110402

Subgroups — #26110402

Order 12032.

360 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252524C211yesmaximal, characteristic
2#20242171C211yesmaximal, characteristic, complemented
3#20241872C211yesmaximal, characteristic
4#20241799C211yesmaximal, characteristic
5#20241281C211yesmaximal, characteristic, complemented
6#20241238C211yesmaximal, characteristic, complemented
7C47×#1832C211yesmaximal, characteristic, complemented
8#15689657C2211yescharacteristic
9#15689252—21no—
10#15689194—21no—
11#15689193—21no—
12C8×C22⋊D47—21no—
13#15689183—21no—
14#15689180C2211yescharacteristic
15#15689176C2211yescharacteristic
16#15689148—21no—
17C4×#10625329C2211yescharacteristic
18#15688875—21no—
19#15688873—21no—
20#15688859C411yescharacteristic
21#15688859C411yescharacteristic
22#15688856C411yescharacteristic
23#15688856C411yescharacteristic
24#15688853—21no—
25#15688850—21no—
26C2×C4⋊C8×C47C2211yescharacteristic
27C2×C4⋊C8×C47C2211yescharacteristic
28C47×#376C2211yescharacteristic
29C47×#336—21no—
30C47×#336—21no—
31C2×#10625329—21no—
32C2×#10625324—21no—
33C2×C8×D47—21no—
34#13094018—21no—
35#13094016Q811yescharacteristic
36#13094012Q811yescharacteristic
37#13094007—21no—
38#13094003C2×C411yescharacteristic
39#13093998C2×C411yescharacteristic
40C4×Dic94D411yescharacteristic
41C4×D188D411yescharacteristic
42C4×C22⋊D47—21no—
43#13093981—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)D411yescharacteristic
46C4⋊(C47⋊C8)D411yescharacteristic
47Dic47⋊C8—21no—
48Dic47⋊C8—21no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)C2×C411yescharacteristic
51C22⋊(C47⋊C8)C2×C411yescharacteristic
52(C2×D47)⋊C8—21no—
53(C2×D47)⋊C8—21no—
54#13093935—21no—
55#13093935—21no—
56#13093934—21no—
57C4×C47⋊C8—21no—
58C4×C47⋊C8—21no—
59C8×Dic47—21no—
60C22×C8×C47—21no—
61C22×C8×C47D411yescharacteristic
62C22×C8×C47D411yescharacteristic
63C2×C42×C47—21no—
64C2×C42×C47C2311yescharacteristic
65C4⋊C8×C47—21no—
66C4⋊C8×C47—21no—
67C4⋊C8×C47—21no—
68C4⋊C8×C47—21no—
69#95×C47—21no—
70#95×C47C2×C411yescharacteristic
71#95×C47C2×C411yescharacteristic
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47C22×C411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47#4111yescharacteristic
77C2×C22⋊D47—21no—
78C2×D188C4⋊C411yescharacteristic
79C2×C4×D47—21no—
80C2×Dic94C4⋊C411yescharacteristic
81C2×C8×C47—21no—
82C2×C8×C47—21no—
83C2×C8×C47—41no—
84C2×C8×C47—41no—
85C2×C8×C47—42no—
86C2×C8×C47—41no—
87C42×C47—41no—
88C42×C47—41no—
89C42×C47C2×D411yescharacteristic
90C42×C47C2×D411yescharacteristic
91C42×C47C2×Q811yescharacteristic
92C42×C47#4111yescharacteristic
93C22⋊Dic47—21no—
94(C2×D47)⋊C4—21no—
95(C2×D47)⋊C4—21no—
96(C2×D47)⋊C4—21no—
97(C2×D47)⋊C4—21no—
98C4⋊Dic47C4⋊C411yescharacteristic
99C4⋊Dic47C4⋊C411yescharacteristic
100Dic47⋊C4—21no—
101Dic47⋊C4—21no—
102Dic47⋊C4—21no—
103Dic47⋊C4—21no—
104C4×Dic47—21no—
105#10625329—22no—
106#10625329—41no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—21no—
111C2×C47⋊C8—41no—
112#10625324—81no—
113C8×D47—81no—
114C23×C47C4×D411yescharacteristic
115C22×D47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47C2×C4⋊C411yescharacteristic
128C2×C4×C47C4×D411yescharacteristic
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47C22⋊Q811yescharacteristic
131C2×C4×C47C22⋊Q811yescharacteristic
132C2×C4×C47#12311yescharacteristic
133C22⋊D47—41no—
134C22⋊D47—41no—
135C2×Dic47—21no—
136C2×Dic47—21no—
137C2×Dic47—21no—
138D188—41no—
139C4×D47—41no—
140C4×D47—41no—
141Dic94—41no—
142C8×C47—41no—
143C8×C47—41no—
144C8×C47—81no—
145C47⋊C8—41no—
146C47⋊C8—41no—
147C47⋊C8—42no—
148C47⋊C8—41no—
149C47⋊C8—41no—
150#8877—471nomaximal, Sylow
151C22×C47—21no—
152C22×C47—21no—
153C22×C47#38511yescharacteristic
154C22×C47#44311yescharacteristic
155C22×C47#44411yescharacteristic
156C2×D47—41no—
157C2×D47—41no—
158C2×D47—41no—
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—81no—
165C4×C47—81no—
166C4×C47#45411yescharacteristic
167C4×C47#45511yescharacteristic
168Dic47—41no—
169Dic47—41no—
170Dic47—41no—
171#2992—471no—
172#1832D4711yescharacteristic, complemented
173#1573—471no—
174#1500—471no—
175#1483—471no—
176#1300—471no—
177#1283—471no—
178C2×C47—21no—
179C2×C47—21no—
180C2×C47#186011yescharacteristic
181C2×C47#186611yescharacteristic
182C2×C47#186811yescharacteristic
183D47—81no—
184#529—471no—
185#446—941no—
186#435—941no—
187#435—941no—
188C8×D4—941no—
189#433—941no—
190#432—471no—
191#431—471no—
192#424—941no—
193C2×C4⋊C8C2×D4711yescharacteristic
194C2×C4⋊C8C2×D4711yescharacteristic
195C4×M16—471no—
196#376C2×D4711yescharacteristic
197#336—21no—
198#336—21no—
199#335—941no—
200#334—941no—
201#328—471no—
202#328—471no—
203#327—471no—
204#327—471no—
205Q8⋊C8—941no—
206D4⋊C8—941no—
207C47#887711yescharacteristic, complemented, Sylow
208C2×M16—941no—
209C2×M16—941no—
210C22×C8—21no—
211C22×C8—941no—
212C22×C8C22⋊D4711yescharacteristic
213C22×C8C22⋊D4711yescharacteristic
214#126—941no—
215#125—471no—
216#124—471no—
217#123—941no—
218C22⋊Q8—471no—
219#121—471no—
220C4×Q8—471no—
221C4×D4—471no—
222C4×D4—941no—
223#117—941no—
224C2×C42—21no—
225C2×C42C22×D4711yescharacteristic
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—471no—
231C4⋊C8—471no—
232C4⋊C8—941no—
233C4⋊C8—941no—
234C4⋊C8—941no—
235C22⋊C8—471no—
236C22⋊C8—471no—
237C22⋊C8—941no—
238C22⋊C8—941no—
239C22⋊C8—941no—
240#97—941no—
241#97—941no—
242#97—941no—
243C4×C8—941no—
244C4×C8—941no—
245C4×C8—941no—
246#95—21no—
247#95C4×D4711yescharacteristic
248#95C4×D4711yescharacteristic
249C2×Q8—471no—
250C2×D4—471no—
251C2×D4—941no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—941no—
255C22×C4C2×C4×D4711yescharacteristic
256C22×C4#1062535011yescharacteristic
257C22×C4C2×C22⋊D4711yescharacteristic
258M16—942no—
259M16—3761no—
260M16—1881no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—941no—
268C2×C8—941no—
269C2×C8—941no—
270C2×C8—941no—
271C2×C8—3761no—
272C2×C8—1881no—
273C4⋊C4—471no—
274C4⋊C4—471no—
275C4⋊C4—941no—
276C4⋊C4—941no—
277C4⋊C4—941no—
278C4⋊C4—941no—
279C22⋊C4—941no—
280C22⋊C4—941no—
281C22⋊C4—941no—
282C22⋊C4—941no—
283C22⋊C4—941no—
284C42—41no—
285C42—41no—
286C42—941no—
287C42D4×D4711yescharacteristic
288C42D4×D4711yescharacteristic
289C42Q8×D4711yescharacteristic
290C42#1062535411yescharacteristic
291C23—941no—
292C23C4×C22⋊D4711yescharacteristic
293Q8—1881no—
294D4—1881no—
295D4—1881no—
296D4—1881no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—941no—
309C2×C4—941no—
310C2×C4—941no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4C4⋊C4×D4711yescharacteristic
314C2×C4#1309398611yescharacteristic
315C2×C4#1309399211yescharacteristic
316C2×C4#1309400011yescharacteristic
317C2×C4#1309400211yescharacteristic
318C2×C4#1309400911yescharacteristic
319C8—41no—
320C8—41no—
321C8—81no—
322C8—1881no—
323C8—1881no—
324C8—1882no—
325C8—1881no—
326C8—1881no—
327C22—21no—
328C22—21no—
329C22—1881no—
330C22—1881no—
331C22—1881no—
332C22#1568899411yescharacteristic
333C22#1568923811yescharacteristic
334C22#1568924311yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—81no—
341C4—81no—
342C4—1881no—
343C4—1881no—
344C4—1881no—
345C4#1568929511yescharacteristic
346C4#1568930011yescharacteristic
347C2—21no—
348C2—21no—
349C2—3761no—
350C2#2024362011yescharacteristic
351C2#2024366211yescharacteristic
352C2#2024367211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1832C2
socleC22×C47#385
prime radical#13093998C2×C4
cosocleC2×C42×C47C23
hypercenter#1832D47
hypocenterC47#8877
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1832

Complements

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

NH ≅ G/Ntype
2. #20242171C2semidirect
5. #20241281C2semidirect
6. #20241238C2semidirect
7. C47×#1832C2semidirect
172. #1832D47semidirect
207. C47#8877semidirect

Maximal subgroups

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

subgroupindex
#202525242
#202421712
#202418722
#202417992
#202412812
#202412382
C47×#18322
#887747