Finite Group Databasebeta

← #26134952

Subgroups — #26134952

Order 12032.

500 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 107 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#20248561C211yesmaximal, characteristic
2#20243376C211yesmaximal, characteristic
3#20241512C214yesmaximal
4C4×#99×C47C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13093939C2212yes—
7#15689035—21no—
8#15689035—21no—
9#15689026C2212yes—
10#15688988C412yescomplemented
11#15688983—21no—
12#15688983—21no—
13#15688983—21no—
14C4×C4⋊Dic47C411yescharacteristic, complemented
15#15688890—21no—
16C2×C4×D4×C47C2211yescharacteristic
17C2×#99×C47C2212yes—
18C4×C22⋊C4×C47—21no—
19C47×#342—21no—
20C22×Dic94—21no—
21#13094024—41no—
22#13094024—41no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yes—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30C2×C4×Dic47C2×C412yes—
31#13093939—24no—
32C47⋊#95—21no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95—41no—
36C22×D4×C47C2311yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C4×D4×C47—41no—
40C4×D4×C47—41no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C42×C47—41no—
46C2×C42×C47C2×C411yescharacteristic
47#99×C47—22no—
48#99×C47Q814yescomplemented
49#95×C47—41no—
50C24×C47C2×D411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47C2×Q812yes—
55C2×D4×C47#4112yes—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×Dic47—22no—
69C22×Dic47—22no—
70C22×Dic47—42no—
71C2×Dic94—42no—
72C2×Dic94—82no—
73C4⋊C4×C47—41no—
74C22⋊C4×C47—24no—
75C22⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—42no—
78C42×C47—21no—
79C42×C47—21no—
80C42×C47—82no—
81C22⋊Dic47—42no—
82C22⋊Dic47—42no—
83C22⋊Dic47—82no—
84C22⋊Dic47—82no—
85C4⋊Dic47—42no—
86C4⋊Dic47—41no—
87C4⋊Dic47—41no—
88C4⋊Dic47—42no—
89C4⋊Dic47—82no—
90C4×Dic47—42no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47C2×C22⋊C411yescharacteristic
100C23×C47C22⋊Q812yes—
101C23×C47#12411yescharacteristic
102C23×C47C4⋊Q811yescharacteristic
103D4×C47—42no—
104D4×C47—42no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—44no—
117C2×C4×C47—81no—
118C2×C4×C47—81no—
119C2×C4×C47—82no—
120C2×C4×C47—82no—
121C2×C4×C47#9911yescharacteristic
122C2×C4×C47#9911yescharacteristic
123C2×C4×C47C4×Q812yes—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—82no—
131C2×Dic47—82no—
132C2×Dic47—82no—
133Dic94—84no—
134#13299—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—22no—
139C22×C47—21no—
140C22×C47—42no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47—81no—
148C22×C47—81no—
149C22×C47#39111yescharacteristic
150C22×C47C2×#9911yescharacteristic
151C22×C47#41011yescharacteristic
152C4×C47—21no—
153C4×C47—22no—
154C4×C47—41no—
155C4×C47—41no—
156C4×C47—41no—
157C4×C47—84no—
158C4×C47#35412yes—
159Dic47—82no—
160Dic47—82no—
161Dic47—162no—
162#2563—471no—
163#1799—471no—
164C4×#99D4711yescharacteristic
165#1393—474no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—41no—
169C2×C47—42no—
170C2×C47—41no—
171C2×C47#199611yescharacteristic
172C2×C47C2×#35411yescharacteristic
173C2×C47#212811yescharacteristic
174C2×C4⋊Q8—471no—
175C2×C4×D4C2×D4711yescharacteristic
176C2×#99—472no—
177C2×#99C2×D4712yes—
178#393—941no—
179#393—941no—
180#390—472no—
181#382—472no—
182#380—941no—
183#380—941no—
184#380—941no—
185C4×C4⋊C4—471no—
186C4×C22⋊C4—21no—
187#342—21no—
188#342—941no—
189C47#1329911yescharacteristic, complemented, Sylow
190C22×Q8—941no—
191C22×D4C22×D4711yescharacteristic
192C23×C4D18811yescharacteristic
193C23×C4C22⋊D4711yescharacteristic
194C4⋊Q8—1881no—
195C4⋊Q8—1881no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—472no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4—941no—
201C2×C4⋊C4—1881no—
202C2×C4⋊C4C4×D4711yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—942no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4—942no—
207C2×C22⋊C4D18811yescharacteristic
208C2×C22⋊C4C22⋊D4711yescharacteristic
209C2×C42—41no—
210C2×C42—472no—
211C2×C42C4×D4711yescharacteristic
212#99—22no—
213#99—944no—
214#99Dic9414yescomplemented
215#95—41no—
216#95—941no—
217#95—942no—
218#95—942no—
219#95—1881no—
220C24C2×D18811yescharacteristic
221C24C2×C22⋊D4711yescharacteristic
222C2×Q8—3762no—
223C2×Q8—1882no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4C2×Dic9412yes—
227C2×D4#1062535012yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—942no—
236C22×C4—942no—
237C22×C4—1882no—
238C22×C4(C2×D47)⋊C411yescharacteristic
239C22×C4(C2×D47)⋊C411yescharacteristic
240C22×C4(C2×D47)⋊C411yescharacteristic
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4C2×C4×D4711yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—3762no—
245C4⋊C4—1882no—
246C4⋊C4—1882no—
247C4⋊C4—1881no—
248C4⋊C4—1881no—
249C22⋊C4—24no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—42no—
253C22⋊C4—3762no—
254C22⋊C4—3762no—
255C22⋊C4—1882no—
256C22⋊C4—1882no—
257C42—21no—
258C42—21no—
259C42—82no—
260C42—1882no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23C2×(C2×D47)⋊C411yescharacteristic
270C23#1309400312yes—
271C23#1309401211yescharacteristic
272C23#1309402411yescharacteristic
273Q8—3764no—
274D4—42no—
275D4—42no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—3762no—
293C2×C4—3762no—
294C2×C4—3762no—
295C2×C4—1882no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4#1309393811yescharacteristic
302C2×C4#1309393811yescharacteristic
303C2×C4C4×Dic9412yes—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—81no—
318C22#1568902911yescharacteristic
319C22C2×#1309393811yescharacteristic
320C22#1568909611yescharacteristic
321C4—21no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—84no—
327C4—3762no—
328C4—3762no—
329C4—7522no—
330C4#1568891712yes—
331C2—21no—
332C2—21no—
333C2—42no—
334C2—41no—
335C2—41no—
336C2#2024437311yescharacteristic
337C2C2×#1568891711yescharacteristic
338C2#2024505911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC22×D4C22×D47
FittingC4×#99×C47C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenterC4×#99D47
hypocenterC47#13299
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
5. C2×#13094024C4semidirect
10. #15688988C4semidirect
14. C4×C4⋊Dic47C4semidirect
48. #99×C47Q8semidirect
189. C47#13299semidirect
214. #99Dic94semidirect

Maximal subgroups

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

subgroupindex
#202485612
#202433762
#202415122
C4×#99×C472
#1329947