Finite Group Databasebeta

← #26112021

Subgroups — #26112021

Order 12032.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 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#20251719C211yesmaximal, characteristic
2C2×#15688969C211yesmaximal, characteristic, complemented
3C2×#15688858C212yesmaximal
4#20241268C212yesmaximal, complemented
5C47×#1817C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8C22×C4×Dic47—21no—
9#15689025C412yescomplemented
10#15689021C412yescomplemented
11#15689014—22no—
12#15689002—21no—
13#15688983C2211yescharacteristic
14#15688969—24no—
15C2×C47⋊#95—21no—
16C2×C47⋊#95C2212yes—
17#15688858—24no—
18C22×C42×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#336—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23C23×Dic47—22no—
24#13094007—44no—
25#13093998—22no—
26#13093998—22no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—21no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—21no—
34C2×C4×Dic47—24no—
35C2×C4×Dic47—28no—
36C47⋊#95—24no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—21no—
40C47⋊#95D414yes—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C22×C8×C47—22no—
45C2×C42×C47—22no—
46C2×C42×C47—22no—
47C2×C42×C47C2×C412yes—
48C2×C42×C47D412yes—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C24×C47C2×D411yescharacteristic
52C23×D47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—24no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—42no—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22⋊C412yes—
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D412yes—
66C22×C4×C47#4112yes—
67C2×C22⋊D47—41no—
68C2×C22⋊D47—41no—
69C2×C22⋊D47—42no—
70C2×C22⋊D47—42no—
71C22×Dic47—28no—
72C22×Dic47—24no—
73C22×Dic47—22no—
74C22×Dic47—21no—
75C22×Dic47—44no—
76C22×Dic47—42no—
77C22×Dic47—42no—
78C2×D188—22no—
79C2×D188—42no—
80C2×C8×C47—44no—
81C2×C8×C47—42no—
82C42×C47—24no—
83C42×C47—24no—
84C22⋊Dic47—42no—
85(C2×D47)⋊C4—44no—
86(C2×D47)⋊C4—44no—
87(C2×D47)⋊C4—42no—
88C4⋊Dic47—22no—
89C4⋊Dic47—42no—
90Dic47⋊C4—44no—
91C4×Dic47—216no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47C2×C22⋊C411yescharacteristic
95C23×C47#12011yescharacteristic
96C23×C47#12311yescharacteristic
97C23×C47#13512yes—
98C23×C47AGL(1,ℤ/8)12yes—
99C22×D47—41no—
100C22×D47—42no—
101C22×D47—41no—
102C22×D47—41no—
103C22×D47—42no—
104C2×C4×C47—24no—
105C2×C4×C47—24no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47#10414yes—
118C2×C4×C47C4×D412yes—
119C2×C4×C47#12112yes—
120C22⋊D47—82no—
121C22⋊D47—82no—
122C2×Dic47—28no—
123C2×Dic47—44no—
124C2×Dic47—48no—
125C2×Dic47—48no—
126C2×Dic47—44no—
127C2×Dic47—41no—
128C2×Dic47—42no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133D188—82no—
134C8×C47—82no—
135#9149—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—24no—
141C22×C47—24no—
142C22×C47#38611yescharacteristic
143C22×C47#41012yes—
144C22×C47#41611yescharacteristic
145C22×C47#41811yescharacteristic
146C22×C47C2×#10412yes—
147C22×C47#44912yes—
148C22×C47#48212yes—
149C2×D47—81no—
150C2×D47—81no—
151C2×D47—81no—
152C2×D47—82no—
153C2×D47—82no—
154C4×C47—24no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—42no—
158Dic47—48no—
159Dic47—81no—
160Dic47—82no—
161#2854—471no—
162C2×#376—471no—
163#1817D4711yescharacteristic, complemented
164C2×#327—472no—
165#1294—472no—
166C2×C47—22no—
167C2×C47—22no—
168C2×C47#150012yes—
169C2×C47#150512yes—
170C2×C47#187511yescharacteristic
171C2×C47#189611yescharacteristic
172C2×C47#189911yescharacteristic
173D47—161no—
174C2×#123—941no—
175C2×#121—471no—
176C22×C42—941no—
177C22×C42C2×D4711yescharacteristic
178C2×C22⋊C8C2×D4712yes—
179#390—472no—
180#389—472no—
181#388—942no—
182#386—941no—
183#380—471no—
184#376—944no—
185C2×#95—472no—
186C2×#95—941no—
187#336—22no—
188#327—944no—
189C47#914911yescharacteristic, complemented, Sylow
190C22×D4—471no—
191C22×D4—941no—
192C23×C4—942no—
193C23×C4D18811yescharacteristic
194C23×C4D18811yescharacteristic
195C23×C4C22×D4711yescharacteristic
196C22×C8—22no—
197#123—1884no—
198#121—942no—
199#121—942no—
200C2×C4⋊C4—471no—
201C2×C4⋊C4—472no—
202C2×C4⋊C4—942no—
203C2×C22⋊C4—941no—
204C2×C22⋊C4—942no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4—941no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—944no—
210C2×C42—948no—
211C2×C42C4×D4712yes—
212C2×C42C22⋊D4712yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—474no—
216#95—944no—
217#95—942no—
218#95—942no—
219#95—941no—
220C24—941no—
221C24C2×D18811yescharacteristic
222C2×D4—942no—
223C2×D4—1882no—
224C2×D4—1881no—
225C2×D4—1881no—
226C2×D4—1882no—
227C2×D4—1882no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—948no—
237C22×C4—944no—
238C22×C4—942no—
239C22×C4—941no—
240C22×C4—1884no—
241C22×C4—1882no—
242C22×C4—1882no—
243C22×C4(C2×D47)⋊C412yes—
244C22×C4(C2×D47)⋊C412yes—
245C22×C4C2×C4×D4711yescharacteristic
246C22×C4D4×D4712yes—
247C22×C4#1062535212yes—
248C22×C4C2×C22⋊D4711yescharacteristic
249C2×C8—44no—
250C2×C8—42no—
251C4⋊C4—942no—
252C4⋊C4—1882no—
253C4⋊C4—1884no—
254C22⋊C4—1884no—
255C22⋊C4—1882no—
256C22⋊C4—1884no—
257C22⋊C4—1882no—
258C42—24no—
259C42—24no—
260C42—9416no—
261C23—22no—
262C23—22no—
263C23—1881no—
264C23—1882no—
265C23—1881no—
266C23—1881no—
267C23—1882no—
268C23C2×(C2×D47)⋊C411yescharacteristic
269C23#1309399411yescharacteristic
270C23#1309400811yescharacteristic
271C23#1309404612yes—
272C23#1309405212yes—
273D4—3762no—
274D4—3762no—
275D4—3762no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—948no—
290C2×C4—1881no—
291C2×C4—1882no—
292C2×C4—1881no—
293C2×C4—1882no—
294C2×C4—1881no—
295C2×C4—1882no—
296C2×C4—1884no—
297C2×C4—1888no—
298C2×C4—1888no—
299C2×C4—1884no—
300C2×C4#1309395014yes—
301C2×C4(C22⋊D47)⋊C412yes—
302C2×C4#1309399612yes—
303C8—82no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—24no—
310C22—3761no—
311C22—3761no—
312C22—3762no—
313C22—3762no—
314C22—3761no—
315C22#1568900011yescharacteristic
316C22#1568909612yes—
317C22#1568912011yescharacteristic
318C22#1568912911yescharacteristic
319C22C2×#1309395012yes—
320C22#1568927012yes—
321C22#1568944812yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—3762no—
327C4—3761no—
328C4—1888no—
329C2—22no—
330C2—22no—
331C2—7521no—
332C2#2024187512yes—
333C2#2024189712yes—
334C2#2024370811yescharacteristic
335C2#2024382911yescharacteristic
336C2#2024384411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1817D47
hypocenterC47#9149
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#15688969C2semidirect
4. #20241268C2semidirect
5. C47×#1817C2semidirect
9. #15689025C4semidirect
10. #15689021C4semidirect
163. #1817D47semidirect
189. C47#9149semidirect

Maximal subgroups

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

subgroupindex
#202517192
C2×#156889692
C2×#156888582
#202412682
C47×#18172
#914947