Finite Group Databasebeta

← #26111337

Subgroups — #26111337

Order 12032.

660 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#15689053C211yesmaximal, characteristic
2#20243472C212yesmaximal
3C2×#15688849C211yesmaximal, characteristic
4#20241266C212yesmaximal
5C47×#2316C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7C22×Dic47⋊C4—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689053—22no—
10#15689053C414yescomplemented
11C2×C47⋊#95—21no—
12C2×C47⋊#95C2211yescharacteristic
13#15688885—24no—
14#15688849—22no—
15C2×#117×C47—21no—
16C22×C4⋊C4×C47C2211yescharacteristic
17C47×#389—24no—
18C47×#380—21no—
19C47×#380C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—24no—
28C2×Dic47⋊C4—42no—
29C2×Dic47⋊C4—42no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32C47⋊#95—24no—
33C47⋊#95—21no—
34C47⋊#95C2×C412yes—
35C23×C4×C47D411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47C2311yescharacteristic
38#117×C47—42no—
39#117×C47—42no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47C2×C412yes—
45C2×C4⋊C4×C47D412yes—
46C2×C4⋊C4×C47D414yes—
47C2×C22⋊C4×C47—22no—
48C2×C42×C47—22no—
49#95×C47—22no—
50#95×C47—22no—
51C24×C47C2×D411yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—24no—
59C22×C4×C47—42no—
60C22×C4×C47—42no—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22⋊C412yes—
63C22×C4×C47D812yes—
64C22×C4×C47SD1612yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D412yes—
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47#4112yes—
69C22×Dic47—24no—
70C22×Dic47—22no—
71C22×Dic47—22no—
72C22×Dic47—42no—
73C22×Dic47—41no—
74C22×Dic47—41no—
75C22×Dic47—41no—
76C22×Dic47—41no—
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C22×Dic47—41no—
81C4⋊C4×C47—24no—
82C4⋊C4×C47—24no—
83C4⋊C4×C47—44no—
84C4⋊C4×C47—44no—
85C4⋊C4×C47—82no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—42no—
88C42×C47—44no—
89C4⋊Dic47—24no—
90C4⋊Dic47—42no—
91Dic47⋊C4—48no—
92C2×C47⋊C8—44no—
93C2×C47⋊C8—42no—
94C23×C47—22no—
95C23×C47—22no—
96C23×C47C2×C22⋊C411yescharacteristic
97C23×C47#12011yescharacteristic
98C23×C47#12311yescharacteristic
99C23×C47C2×D811yescharacteristic
100C23×C47C2×SD1611yescharacteristic
101C23×C47AGL(1,ℤ/8)11yescharacteristic
102C23×C47#13711yescharacteristic
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—24no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—44no—
117C2×C4×C47—44no—
118C2×C4×C47—44no—
119C2×C4×C47—81no—
120C2×C4×C47#10214yes—
121C2×C4×C47C4×D412yes—
122C2×C4×C47#12112yes—
123C2×Dic47—24no—
124C2×Dic47—44no—
125C2×Dic47—44no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—81no—
132C2×Dic47—81no—
133C2×Dic47—81no—
134C2×Dic47—81no—
135C47⋊C8—82no—
136#9027—471nomaximal, Sylow
137C22×C47—22no—
138C22×C47—22no—
139C22×C47—22no—
140C22×C47—22no—
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—21no—
145C22×C47#38611yescharacteristic
146C22×C47C2×#10211yescharacteristic
147C22×C47#41711yescharacteristic
148C22×C47#44711yescharacteristic
149C22×C47#44811yescharacteristic
150C22×C47#45612yes—
151C22×C47#45812yes—
152C22×C47#48011yescharacteristic
153C22×C47#48111yescharacteristic
154C4×C47—24no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—42no—
158C4×C47—84no—
159Dic47—44no—
160Dic47—82no—
161C2×#398—471no—
162#2316D4711yescharacteristic
163#1823—472no—
164C2×#324—471no—
165#1294—472no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47—21no—
170C2×C47#161012yes—
171C2×C47#161112yes—
172C2×C47#187411yescharacteristic
173C2×C47#188611yescharacteristic
174C2×C47#189011yescharacteristic
175C2×#117—21no—
176C22×C4⋊C4—471no—
177C22×C4⋊C4—941no—
178C22×C4⋊C4C2×D4711yescharacteristic
179C2×C22⋊C8—472no—
180#398—474no—
181#398—942no—
182#389—24no—
183#380—21no—
184#380C2×D4712yes—
185C2×#95—471no—
186C2×#95—941no—
187#340—944no—
188#324—942no—
189C47#902711yescharacteristic, complemented, Sylow
190C23×C4—941no—
191C23×C4—941no—
192C23×C4C22⋊D4711yescharacteristic
193C23×C4C22⋊D4711yescharacteristic
194C23×C4C22×D4711yescharacteristic
195C22×C8—942no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—472no—
203C2×C4⋊C4—942no—
204C2×C4⋊C4—942no—
205C2×C4⋊C4—944no—
206C2×C4⋊C4—941no—
207C2×C4⋊C4—1882no—
208C2×C4⋊C4—1882no—
209C2×C4⋊C4C4×D4712yes—
210C2×C4⋊C4D18812yes—
211C2×C4⋊C4C22⋊D4714yes—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C22⋊C8—942no—
215C22⋊C8—942no—
216#95—22no—
217#95—22no—
218#95—472no—
219#95—944no—
220#95—941no—
221C24C2×C22⋊D4711yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—944no—
232C22×C4—942no—
233C22×C4—942no—
234C22×C4—1881no—
235C22×C4—1881no—
236C22×C4—1881no—
237C22×C4—1881no—
238C22×C4—1881no—
239C22×C4—1881no—
240C22×C4—1882no—
241C22×C4—1881no—
242C22×C4—1881no—
243C22×C4(C2×D47)⋊C412yes—
244C22×C4(C2×D47)⋊C412yes—
245C22×C4#1062533412yes—
246C22×C4#1062533612yes—
247C22×C4C2×C4×D4711yescharacteristic
248C22×C4C2×D18811yescharacteristic
249C22×C4#1062535012yes—
250C22×C4C2×C22⋊D4712yes—
251C2×C8—1884no—
252C2×C8—1882no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—82no—
258C4⋊C4—944no—
259C4⋊C4—1888no—
260C4⋊C4—1882no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—44no—
264C23—22no—
265C23—22no—
266C23C2×(C2×D47)⋊C411yescharacteristic
267C23#1309399311yescharacteristic
268C23#1309400711yescharacteristic
269C23C2×#1062533411yescharacteristic
270C23C2×#1062533611yescharacteristic
271C23#1309405011yescharacteristic
272C23#1309405511yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—81no—
290C2×C4—944no—
291C2×C4—3761no—
292C2×C4—3761no—
293C2×C4—3761no—
294C2×C4—3761no—
295C2×C4—1882no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4—1884no—
300C2×C4—1884no—
301C2×C4—1882no—
302C2×C4#1309394414yes—
303C2×C4C4×C22⋊D4712yes—
304C2×C4#1309399812yes—
305C8—3762no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22#1568900211yescharacteristic
315C22C2×#1309394411yescharacteristic
316C22#1568912211yescharacteristic
317C22#1568925711yescharacteristic
318C22#1568926211yescharacteristic
319C22#1568930612yes—
320C22#1568931312yes—
321C22#1568943611yescharacteristic
322C22#1568944111yescharacteristic
323C4—22no—
324C4—22no—
325C4—24no—
326C4—42no—
327C4—84no—
328C4—3762no—
329C4—1884no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#2024239112yes—
335C2#2024239612yes—
336C2#2024370511yescharacteristic
337C2#2024378111yescharacteristic
338C2#2024380111yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2316C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2316D47
hypocenterC47#9027
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
10. #15689053C4semidirect
189. C47#9027semidirect

Maximal subgroups

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

subgroupindex
C2×#156890532
#202434722
C2×#156888492
#202412662
C47×#23162
#902747