Finite Group Databasebeta

← #26134590

Subgroups — #26134590

Order 12032.

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

normal subgroups only

This group has 552 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#20248380C211yesmaximal, characteristic
2C2×#15688917C214yesmaximal
3C4×#13093939C211yesmaximal, characteristic
4C47×#1799C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13094011C2212yes—
7C2×#13093939C2212yes—
8#15689048—21no—
9#15689048—21no—
10#15689028C412yescomplemented
11#15688983—21no—
12#15688983—21no—
13C4×C4⋊Dic47C411yescharacteristic, complemented
14C4×C22⋊Dic47—21no—
15#15688917—28no—
16#15688890—21no—
17C2×C4×D4×C47C2211yescharacteristic
18C2×#99×C47C2212yes—
19C47×#380—21no—
20C47×#342—21no—
21C22×Dic94C2×C412yes—
22#13094024—41no—
23#13094024—41no—
24#13094011—24no—
25#13094011—24no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C22⋊Dic47—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C4×Dic47—41no—
33C2×C4×Dic47C2×C412yes—
34#13093939—22no—
35#13093939D414yes—
36C47⋊#95—21no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—41no—
40C22×D4×C47C2311yescharacteristic
41C23×C4×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C4×D4×C47—41no—
44C4×D4×C47—41no—
45C2×C4⋊C4×C47—41no—
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C42×C47C2×C411yescharacteristic
51#99×C47—24no—
52#95×C47—41no—
53C24×C47C2×D411yescharacteristic
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×C47C22×C411yescharacteristic
71C22×Dic47—22no—
72C22×Dic47—22no—
73C22×Dic47—42no—
74C2×Dic94—24no—
75C2×Dic94—42no—
76C2×Dic94—42no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—82no—
79C22⋊C4×C47—41no—
80C22⋊C4×C47—41no—
81C22⋊C4×C47—42no—
82C22⋊C4×C47—42no—
83C42×C47—21no—
84C42×C47—21no—
85C22⋊Dic47—24no—
86C22⋊Dic47—42no—
87C22⋊Dic47—44no—
88C22⋊Dic47—42no—
89C22⋊Dic47—42no—
90C4⋊Dic47—42no—
91C4⋊Dic47—42no—
92C4⋊Dic47—41no—
93C4⋊Dic47—41no—
94C4×Dic47—24no—
95C4×Dic47—82no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47C2×C22⋊C411yescharacteristic
105C23×C47#12112yes—
106C23×C47#12411yescharacteristic
107C23×C47#12711yescharacteristic
108D4×C47—42no—
109D4×C47—42no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—22no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
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—81no—
123C2×C4×C47—82no—
124C2×C4×C47—82no—
125C2×C4×C47—82no—
126C2×C4×C47#9911yescharacteristic
127C2×C4×C47#9911yescharacteristic
128C2×C4×C47C4×D412yes—
129C2×Dic47—42no—
130C2×Dic47—44no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—82no—
137C2×Dic47—82no—
138Dic94—84no—
139#13224—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—22no—
143C22×C47—21no—
144C22×C47—21no—
145C22×C47—42no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—81no—
153C22×C47—81no—
154C22×C47#39011yescharacteristic
155C22×C47C2×#9911yescharacteristic
156C22×C47#41011yescharacteristic
157C4×C47—21no—
158C4×C47—22no—
159C4×C47—41no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—162no—
163C4×C47#35412yescomplemented
164Dic47—82no—
165Dic47—84no—
166Dic47—82no—
167#2547—471no—
168C2×#354—474no—
169#1799D4711yescharacteristic
170C4×#99—471no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—42no—
174C2×C47—41no—
175C2×C47—41no—
176C2×C47#196411yescharacteristic
177C2×C47C2×#35411yescharacteristic
178C2×C47#212911yescharacteristic
179C2×C4⋊Q8—471no—
180C2×#124—472no—
181C2×C4×D4C2×D4711yescharacteristic
182C2×#99—472no—
183C2×#99C2×D4712yes—
184#397—941no—
185#397—941no—
186#391—472no—
187#380—21no—
188#380—941no—
189#380—941no—
190C4×C4⋊C4—471no—
191C4×C22⋊C4—941no—
192#354—948no—
193#342—21no—
194#342—941no—
195C47#1322411yescharacteristic, complemented, Sylow
196C22×Q8—472no—
197C22×D4C22×D4711yescharacteristic
198C23×C4D18811yescharacteristic
199C23×C4C22⋊D4711yescharacteristic
200C4⋊Q8—1881no—
201C4⋊Q8—1881no—
202#124—944no—
203#124—944no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—941no—
208C2×C4⋊C4—941no—
209C2×C4⋊C4—941no—
210C2×C4⋊C4C4×D4711yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—942no—
213C2×C22⋊C4—942no—
214C2×C22⋊C4—942no—
215C2×C22⋊C4D18811yescharacteristic
216C2×C22⋊C4C22⋊D4711yescharacteristic
217C2×C42—472no—
218C2×C42—1881no—
219C2×C42C4×D4711yescharacteristic
220#99—24no—
221#99—474no—
222#99—942no—
223#95—41no—
224#95—941no—
225#95—942no—
226#95—942no—
227#95—1881no—
228C24C2×D18811yescharacteristic
229C24C2×C22⋊D4711yescharacteristic
230C2×Q8—944no—
231C2×Q8—1882no—
232C2×Q8—1882no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4D4×D4712yes—
236C2×D4#1062535412yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—942no—
245C22×C4—942no—
246C22×C4—1882no—
247C22×C4(C2×D47)⋊C411yescharacteristic
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4(C2×D47)⋊C411yescharacteristic
251C22×C4C2×C4×D4711yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—82no—
254C4⋊C4—1882no—
255C4⋊C4—1882no—
256C4⋊C4—1881no—
257C4⋊C4—1881no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—944no—
263C22⋊C4—1882no—
264C22⋊C4—1884no—
265C22⋊C4—1882no—
266C22⋊C4—1882no—
267C42—21no—
268C42—21no—
269C42—944no—
270C42—3762no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×(C2×D47)⋊C411yescharacteristic
280C23#1309399912yes—
281C23#1309401311yescharacteristic
282C23#1309402011yescharacteristic
283Q8—3764no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—3762no—
303C2×C4—3762no—
304C2×C4—1884no—
305C2×C4—1882no—
306C2×C4—1882no—
307C2×C4—1882no—
308C2×C4—1882no—
309C2×C4—1882no—
310C2×C4—1882no—
311C2×C4#1309393811yescharacteristic
312C2×C4#1309393811yescharacteristic
313C2×C4#1309398612yes—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—21no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22#1568902411yescharacteristic
329C22C2×#1309393811yescharacteristic
330C22#1568909611yescharacteristic
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—162no—
337C4—3762no—
338C4—3764no—
339C4—3762no—
340C4#1568891712yescomplemented
341C2—21no—
342C2—21no—
343C2—41no—
344C2—42no—
345C2—41no—
346C2#2024422311yescharacteristic
347C2C2×#1568891711yescharacteristic
348C2#2024506211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#1799C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenter#1799D47
hypocenterC47#13224
LCSG ▷ C22×C4×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
5. C2×#13094024C4semidirect
10. #15689028C4semidirect
13. C4×C4⋊Dic47C4semidirect
163. C4×C47#354semidirect
195. C47#13224semidirect
340. C4#15688917semidirect

Maximal subgroups

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

subgroupindex
#202483802
C2×#156889172
C4×#130939392
C47×#17992
#1322447