Finite Group Databasebeta

← #26134560

Subgroups — #26134560

Order 12032.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#20247696C211yesmaximal, characteristic
2C2×#15688912C214yesmaximal
3C4×#13093941C211yesmaximal, characteristic
4C47×#1799C211yesmaximal, characteristic
5C2×#13094005C2212yes—
6C22×C4⋊Dic47C411yescharacteristic, complemented
7C2×#13093941C2212yes—
8C4×#10625329—21no—
9#15689053—21no—
10#15689048—21no—
11#15689021—21no—
12#15688984C412yescomplemented
13#15688983—21no—
14#15688983C411yescharacteristic, complemented
15#15688912—28no—
16#15688892—21no—
17C2×C4×D4×C47C2211yescharacteristic
18C2×#99×C47C2212yes—
19C47×#380—21no—
20C47×#342—21no—
21C23×Dic47C2×C412yes—
22C2×#10625329—22no—
23#13094005—24no—
24#13094005—24no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—41no—
29C2×C4⋊Dic47—41no—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47C2×C412yes—
33#13093941—22no—
34#13093941D414yes—
35#13093935—41no—
36C4×C47⋊C8—41no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95—22no—
40C22×D4×C47C2311yescharacteristic
41C23×C4×C47C2×C411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C4×D4×C47—41no—
44C4×D4×C47—41no—
45C2×C4⋊C4×C47—41no—
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51#99×C47—24no—
52#95×C47—41no—
53C24×C47C22×C411yescharacteristic
54C24×C47C2×D411yescharacteristic
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47C2×D412yes—
58C2×D4×C47#4112yes—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—22no—
72C22×Dic47—22no—
73C22×Dic47—24no—
74C22×Dic47—42no—
75C22×Dic47—42no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—82no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—42no—
81C22⋊C4×C47—42no—
82C42×C47—21no—
83C42×C47—21no—
84C22⋊Dic47—24no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C4⋊Dic47—42no—
88C4⋊Dic47—42no—
89C4⋊Dic47—41no—
90C4⋊Dic47—41no—
91Dic47⋊C4—44no—
92#10625329—24no—
93#10625329—42no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47C2×C22⋊C411yescharacteristic
104C23×C47C4×D412yes—
105C23×C47#12411yescharacteristic
106C23×C47#12711yescharacteristic
107D4×C47—42no—
108D4×C47—42no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—22no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—81no—
122C2×C4×C47—82no—
123C2×C4×C47—82no—
124C2×C4×C47—82no—
125C2×C4×C47#9911yescharacteristic
126C2×C4×C47#9911yescharacteristic
127C2×C4×C47#12112yes—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—84no—
135C47⋊C8—44no—
136#13217—471nomaximal, Sylow
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—22no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—42no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—42no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—81no—
151C22×C47#39011yescharacteristic
152C22×C47C2×#9911yescharacteristic
153C22×C47#41011yescharacteristic
154C4×C47—21no—
155C4×C47—22no—
156C4×C47—41no—
157C4×C47—41no—
158C4×C47—41no—
159C4×C47—162no—
160C4×C47#35212yescomplemented
161Dic47—82no—
162Dic47—82no—
163#2486—471no—
164C2×#352—474no—
165#1799D4711yescharacteristic
166C4×#100—471no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47—41no—
170C2×C47—42no—
171C2×C47—41no—
172C2×C47#196411yescharacteristic
173C2×C47C2×#35211yescharacteristic
174C2×C47#212311yescharacteristic
175C2×#123—472no—
176C2×C4×D4C2×D4711yescharacteristic
177C22×C4⋊C4—471no—
178C2×#100—472no—
179C2×#99C2×D4712yes—
180C4×M16—941no—
181#398—941no—
182#397—941no—
183#389—941no—
184#381—472no—
185#380—21no—
186#380—471no—
187#380—941no—
188#352—948no—
189#343—941no—
190#342—21no—
191C47#1321711yescharacteristic, complemented, Sylow
192C22×D4C22×D4711yescharacteristic
193C23×C4—472no—
194C23×C4C4×D4711yescharacteristic
195C23×C4C22⋊D4711yescharacteristic
196C2×M16—942no—
197#123—944no—
198#123—944no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—941no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—942no—
206C2×C4⋊C4—1881no—
207C2×C4⋊C4—1881no—
208C2×C4⋊C4D18811yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—472no—
211C2×C22⋊C4—942no—
212C2×C22⋊C4C4×D4711yescharacteristic
213C2×C22⋊C4C22⋊D4711yescharacteristic
214C2×C42D18811yescharacteristic
215#100—474no—
216#100—942no—
217#99—24no—
218#97—1881no—
219C4×C8—1881no—
220#95—41no—
221#95—941no—
222#95—942no—
223#95—942no—
224C24C2×C4×D4711yescharacteristic
225C24C2×C22⋊D4711yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4D4×D4712yes—
229C2×D4#1062535412yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—944no—
238C22×C4—942no—
239C22×C4—942no—
240C22×C4—1882no—
241C22×C4—1882no—
242C22×C4(C2×D47)⋊C411yescharacteristic
243C22×C4(C2×D47)⋊C411yescharacteristic
244C22×C4(C2×D47)⋊C411yescharacteristic
245C22×C4(C2×D47)⋊C411yescharacteristic
246C22×C4C2×D18811yescharacteristic
247M16—944no—
248M16—1882no—
249C2×C8—1882no—
250C4⋊C4—41no—
251C4⋊C4—82no—
252C4⋊C4—1882no—
253C4⋊C4—1882no—
254C4⋊C4—1881no—
255C4⋊C4—1881no—
256C4⋊C4—1884no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—944no—
262C22⋊C4—1882no—
263C22⋊C4—1882no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×(C2×D47)⋊C411yescharacteristic
275C23#1309398612yes—
276C23#1309401311yescharacteristic
277C23#1309402011yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—3764no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4—1882no—
302C2×C4—1882no—
303C2×C4#1309393811yescharacteristic
304C2×C4#1309393811yescharacteristic
305C2×C4#1309399912yes—
306C8—1884no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1568902411yescharacteristic
322C22C2×#1309393811yescharacteristic
323C22#1568909611yescharacteristic
324C4—21no—
325C4—22no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—162no—
330C4—3762no—
331C4—3762no—
332C4#1568891212yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#2024422311yescharacteristic
339C2C2×#1568891211yescharacteristic
340C2#2024504111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC24×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#1799C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenter#1799D47
hypocenterC47#13217
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Dic47C4semidirect
12. #15688984C4semidirect
14. #15688983C4semidirect
160. C4×C47#352semidirect
191. C47#13217semidirect
332. C4#15688912semidirect

Maximal subgroups

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

subgroupindex
#202476962
C2×#156889122
C4×#130939412
C47×#17992
#1321747